Introduction to Financial Mathematics

Transcription

1 Introduction to Financial Mathematics MTH 210 Fall 2016 Jie Zhong November 30, 2016 Mathematics Department, UR

2 Table of Contents Arbitrage Interest Rates, Discounting, and Basic Assets Forward Contracts Forward Rates and Libor Interest Rate Swaps Futures Contracts Options Probability Theory: Conditioning Fundamental Theorem of Asset Pricing and Binomial Tree Normal Distribution and Central Limit Theorem Continuous-Time Limit and Black-Scholes Formula 1

3 Arbitrage

4 Assets and Porfolios Asset A valuable thing that can be owned. Exmaple shares (stocks), bonds, cash (domestic or foreign currency), real estate, and resources rights Remark We assume that we can hold any amount of an asset, including negative amounts. For example, if you have no apples and you owe Johnny 3 apples, you have 3 apples. 2

5 Assets and Portfolios Portfolio A collection of assets. The value (price) of a portfolio is the sum of the values (prices) of the assets in the portfolio. Notation V A (T) - the value of a portfolio A at time T. If t is the current time and T > t, then V A (t) is a constant and V A (T) is a random variable. 3

6 Arbitrage Definition A portfolio A is called an arbitrage portfolio if the following conditions hold: its current value V A (t) 0, and for some future time T > t, P(V A (T) 0) = 1, and P(V A (T) > 0) > 0. In other words, arbitrage means free lunch. 4

7 No Arbitrage Assumption There exist no arbitrage portfolios! 5

8 Monotonicity Theorem Monotonicity Theorem (see Chapter 6) Assume no arbitrage. Let A and B be portfolios and let T > t, where t is the current time. (a) If P(V A (T) V B (T)) = 1, then V A (t) V B (t). (b) If in addition, P(V A (T) > V B (T)) > 0, then V A (t) > V B (t). 6

9 Proof of Monotonicity Theorem Proof of (a) We suppose V A (T) < V B (B), and show this leads to a contradiction. Define ε = V B (t) V A (t) > 0. Consider the portfolio C consisting of A minus B plus ε of cash. Then and V C (t) = V A (t) V B (t) + ε = 0, P(V C (T) > 0) P(V C (T) ε) = P(V A (T) V B (T) + ε ε) = P(V A (T) V B (T)) = 1. Thus, the portfolio C is an arbitrage portfolio! Contradiction. 7

10 Proof of Monotonicity Theorem Proof of (b) By the way of contradiction, we assume V A (t) V B (t). Consider the portfolio C consisting of A minus B. Then V C (t) = V A (t) V B (t) 0, P(V C (T) 0) = P(V A (T) V B (T) 0) = 1, and P(V C (T) > 0) = P(V A (T) V B (T) > 0) > 0. Thus, the portfolio C is an arbitrage portfolio! Contradiction. 8

11 Replication Theorem Let A and B be portfolios and let T > t, where t is the current time. If V A (T) = V B (T) with probability one, then V A (t) = V B (t). Proof V A (T) = V B (T) means V A (T) V B (T) and V A (T) V B (T). Then the Monotonicity Theorem implies V A (t) = V B (t). 9

12 Interest Rates, Discounting, and Basic Assets

13 Interest Rate and Compounding We will always deal with per-year interest rates. If we invest (lend or deposit) N dollars at interest rate r compounded annually, then After one year the value of the investment is N(1 + r) dollars After two years: N(1 + r) 2 dollars After T years: N(1 + r) T dollars N is called the notional or principle. Unless we are dealing with multiple currencies, we will usually omit writing dollar or currency symbols like $. If we borrow N at interest rate r compounded annually, then after T years we have N(1 + r) T. In other words, after T years we owe N(1 + r) T. Borrowing N means investing N. 10

14 Interest Rate and Compounding If we deposit N at rate r compounded semi-annually (twice per year), then: After 6 months we have N(1 + r/2). After T years we have N(1 + r/2) 2T. In general, if we invest N at interest rate r compounded m times per year, then: After 1/m years the value of the investment is N(1 + r/m). After 2/m years: N(1 + r/m) 2. After T years: N(1 + r/m) mt. m is called the compounding frequency. The time T is measured in years and can be any non-negative real number. 11

15 Interest Rate and Compounding If we invest N at interest rate r compounded continuously, then: After T years we have Ne rt. This is because lim (1 + m r/m)mt = e rt. Remark The function (1 + r/m) mt is increasing in m! 12

16 Interest Rate and Compounding Example If we invest 500 at rate 3% = 0.03 with compounding frequency 4, the value after 3 years is 500( /4) 4 3 Example If we invest 500 at rate 3% = 0.03 with continuous compounding, the value after 3 years is 500e

17 Interest Rate and Compounding Summary Suppose that some amount of cash has been deposited. At time t, the amount is A(t). At time T > t, the amount is A(T). If we have interest rate r with compounding frequency m from time t to time T, then A(T) = A(t)(1+r/m) m(t t), A(t) = A(T)(1+r/m) m(t t) If we have interest rate r with continuous compounding from time t to time T, then A(T) = A(t)e r(t t), A(t) = A(T)e r(t t) 14

18 Interest Rate and Compounding Relation between discrete and continuous compounding Under the No Arbitrage Assumption, we can claim that if the interest rate with compounding frequency m is r m and the interest rate with continuous compounding is r, then (a) ( ) 1 + rm mt m = e rt for all T > 0 (b) r = m log ( ) 1 + rm m (c) r m = m ( e r/m 1 ) 15

19 Interest Rate and Compounding Proof If one of the statements (a),(b),(c) holds, then we get the other two statements by algebra. We only prove (a). Suppose ( ) 1 + rm mt m < e rt for some T > 0. Fix t < T and consider Portfolio A: At time t, invest 1 at rate r m with compounding frequency m. Portfolio B: At time t, invest 1 with at rate r with continuous compounding. 16

20 Interest Rate and Compouding Continue the proof of (a) Then V A (t) = V B (t) = 1 and V A (T) = ( ) 1 + rm mt m < e rt = V B (T) with probability one. This contradicts the Monotonicity Theorem. If ( ) 1 + rm mt m > e rt for some T > 0, then a similar arguments also leads to a contradiction. The only remaining possibility is that ( ) 1 + rm mt m = e rt for all times T > 0. Question Can you prove (a) by the No Arbitrage Assumption directly? Hint: construct one portfolio is suffice. 17

21 Zero Coupon Bonds and Discounting Definition A zero coupon bond (ZCB) with maturity T is an asset that pays 1 at time T ( and nothing else). Question What is the value of a ZCB at time t T? In other words, what is the value today of the promise of a dollar tomorrow? Let Z(t, T) be the value at time t of a ZCB with maturity T, where t T. By definition, Z(T, T) = 1. 18

22 Zero Coupon Bonds and Discounting Let Z(t, T) be the value at time t of a ZCB with maturity T, where t T. By definition, Z(T, T) = 1. Result If the continuously compounded interest rate from time t to time T has constant value r, then Z(t, T) = e r(t t). If the interest rate with compounding frequency m from time t to time T has constant value r, then Z(t, T) = (1 + r/m) m(t t). 19

23 Zero Coupon Bonds and Discounting Proof (only the continuous compouding case) Let s consider Portfolio A: At time t, a ZCB with maturity T Portfolio B: At time t, investment of N = e r(t t) with continuously compounded interest rate r. Then V A (T) = 1 and V B (T) = e r(t t) e r(t t) = 1 with probability one. By the Replication Theorem, Z(t, T) = V A (t) = V ( B)(t) = e r(t t). Z(t, T) is also called a discount factor. It depends on the interest rate and compounding frequency over the period from t to T. 20

24 Zero Coupon Bonds and Discounting Determining the value of an asset at time t based on its value at some future time T > t is called discounting. The value at time t is called the present value. Example Consider an asset that pays 500 and matures 3 years from now. If the continuously compounded interest rate is 2.1%, what is its present value? If the present time is t, the asset is equivalent to 500 ZCBs with maturity T = 3 + t, and the present value is 500Z(t, T) = 500e r(t t) = 500e 0.021(3). 21

25 Zero Coupon Bonds and Discounting We can define interest rates in terms of Z(t, T). The interest rate r A such that Z(t, T) = (1 + r A ) (T t) is called the annually compounded zero rate or zero rate for the period t to T. The number r such that Z(t, T) = e r(t t) is called the continuous zero rate for the period t to T. The result above says that if the continuous interest rate is constant over the period t to T, then it equals the continuous zero rate for the period t to T. 22

26 Annuities Definition An annuity is a series of fixed payments C at times T 1,, T n. It is equivalent to the following collection of ZCBs: C ZCBs with maturity T 1 C ZCBs with maturity T 2. C ZCBs with maturity T n Its value at time t T 1 is n C Z(t, T i ). i=1 23

27 Annuities Example Consider an annuity starting at time 0 that pays 1 each year for M years. Assume the annually compounded zero rate is r A for all maturities T = 1,, M. This means that Z(t, T) = (1 + r A ) (T t) whenever t 1 and T {1,, M} The value of the annuity at time t = 0 is V 0 = M Z(0, T) = T=1 M T=1 1 (1 + r A ) T. 24

28 Annuities The value of the annuity at time t = 0 is V 0 = M Z(0, T) = T=1 M T=1 1 (1 + r A ) T. We can simplify this geometric sum by a standard trick. Observe that r A V 0 V 0 = and solve for V 0 to obtain 1 (1 + r A ) M+1 1, 1 + r A V 0 = 1 (1 + r A) M r A. 25

29 Annuities By arguing as in previous example we get: Result Consider an annuity starting at time 0 that pays C each year for M years. Assume the annually compounded zero rate is r A for all maturities T = 1,..., M. The value at time t = 0 is V 0 = C 1 (1 + r A) M r A. 26

30 Annuities Example In the US, a $100 million Powerball lottery jackpot is typically structured as an annuity paying $4 million per year for 25 years. With an annually compounded interest rate of 3%, the value of the jackpot at time t = 0 is T=1 25 Z(0, T) = T=1 1 ( ) T 6 1 ( ) 25 = million 27

31 Annuities Example A loan of 1000 is to be paid back in 5 equal installments due yearly. Interest of 15% of the balance is applied each year, before the installment is paid. This type of loan is called an amoritized loan. Note installments include both interest and a portion of the balance. Find the amount C of each installment. 28

32 Annuities Key: for the lender, the loan is equivalent to an annuity. Assume it starts at t = 0. So it pays C at times T = 1, 2, 3, 4, 5. The 15% yearly interest on the balance is equivalent to a 15% annually compounded interest rate. By the previous result, the value at time 0 is 1 (1 + (0.15)) 5 V 0 = C 0.15 On the other hand, we know V 0 = Therefore 0.15 C = ( )

33 Annuities Question 1. What is the outstanding balance of the loan after each installment is paid? 2. What is the amount of interest included in each installment? 3. How much of the loan is repaid as part of each installment? 30

34 Annuities 1st installment interest paid: = 150 capital repaid (how you paid back in this installment): = outstanding balance: = nd installment interest paid: = capital repaid: = outstanding balance: =

35 Annuities Question Can we generalize it to a formula? In other words, can we express the result in terms of V 0 (amount borrowed) and r (interest rate)? 32

36 Annuities Let n = 1, 2, 3, 4 or 5. The present value of the outstanding balance after n 1 installments are paid is equal to the amount borrowed minus the present value of the first n 1 installments: V 0 C 1 + r C (1 + r) n 1 = V (1 + r) 6 n 1 0 (1 + r) 5 1, by recalling that C = V 0 r 1 (1 + r) 5. 33

37 Annuities Then the actual outstanding balance remaining after n 1 installments are paid is: (1 + r) 6 n 1 V 0 (1 + r) 5 1 (1 + (1 + r) 5 (1 + r) n 1 r)n 1 = V 0 (1 + r) 5, 1 the interest included in the nth installment is, V 0 (1 + r) 5 (1 + r) n 1 (1 + r) 5 1 r, and the capital repaid as part of the nth installment is: C V 0 (1 + r) 5 (1 + r) n 1 (1 + r) 5 1 r = V 0 r(1 + r) n 1 (1 + r)

38 Stocks Definition A stock or share is an asset giving ownership of a fraction of a company. The price of a stock at time T is denoted by S T. If t is the current time, then the known price S t is called the spot price. S T is a random variable for T > t. A stock may sometimes pay a cash payment called a dividend. A dividend is often expressed as a percentage q of the stock price. q is called the dividend yield. 35

39 Bonds Definition A fixed rate bond with notional N, coupon c, start date T 0, maturity T n, and term length α is an asset that pays coupons αnc at times T 1,..., T n and pays the notional N at maturity date T n. Usually, the coupon times T 1,..., T n are regularly spaced (for example, annually, semi-annually, or quarterly) and the maturity date coincides with the final coupon date T = T n. 36

40 Bonds Remark A fixed rate bond with coupon c and notional 1 is equivalent to an annuity paying c at the coupon times T 1,, T n plus a ZCB with maturity T. Example Consider a fixed rate bond with coupon c and notional N with maturity M years from now and annual coupon payments, the last one at maturity. Show that the value at present time t is V t = cn 1 (1 + r A) M r A + N(1 + r A ) M, assuming constant annually compounded zero rates of r A. 37

41 Foreign Exchange Rates Example The current euro (EUR) to US dollar (USD) exchange rate is 0.89 EUR/USD. Therefore, the USD to EUR exchange rate is Then USD/EUR EUR/USD 150 USD = 150USD 0.89 EUR = 150(0.89) EUR = EUR. USD 38

42 Forward Contracts

43 Derivative Contracts A derivative contract or derivative is a financial contract between two entities whose value is a function of (derives from) the value of another variable. The two entities in the contract are called counerparties. Example A weather derivative: A contract where one counterparty pays either 100 or 0 to the other counterparty one year from now depending on whether the total snowfall in Boston over the year is greater than 50 inches. 39

44 Derivative Contracts We will consider only financial derivatives, which are derivatives of financial variables like stock prices or interest rates. Over-the-counter (OTC) derivatives are direct contracts between two counterparties. Most derivatives we consider are OTC. The interest rate derivatives we study later are examples of OTC derivatives. 40

45 Derivative Contracts Exchange-traded derivatives are contracts where an exchange matches two counterparties together and each counterparty faces the exchange on the contract. An exchange traded derivative can be viewed as a pair of corresponding contracts where each contract is between one counterparty and the exchange. The futures we study later are examples of exchange-traded derivatives. 41

46 Forward Contracts Our first derivative is the forward contract. In a forward contract or forward, two counterparties agree to trade a specific asset (like a stock) at a certain future time T and a certain price K. At current time t T, one counterparty agrees to buy the asset at time T and price K, and the other counterparty agrees to sell the asset at time T and price K. We say the counterparty agreed to buy is long the forward contract, and the counterparty that agreed to sell is short the forward contract. K is the called the delivery price. T is called the maturity or delivery date. 42

47 Value and Payoff Fix an asset. Let S t be its price at time t. Consider a forward on the asset with delivery price K and maturity T. K and T are fixed at the time the forward contract is agreed to, but the value of the forward contract may change over time. V K (t, T) denotes the value of the forward to the long counterparty at time t T. 43

48 Value and Payoff At time T, we know the counterparty long the forward must pay K to buy the asset whose value is S T. Therefore V K (T, T) = S T K. This is the value at maturity long the forward. It is also called the payoff or payout long the forward. 44

49 Value and Payoff The function g(x) = x K is called the forward payoff function. Then the payoff (payout, value at maturity) long the forward can be written as V K (T, T) = g(s T ) = S T K. The payoff (payout, value at maturity) short the forward is g(s T ) = V K (T, T) = K S T. In later sections, we will study V K (t, T) for general t < T. 45

50 Value and Payoff Example Consider a forward with delivery price 100 and maturity T = 2 years. If the underlying asset has price at maturity, find value of the forward to the long party at maturity. We have K = 100, T = 2, S T = Therefore V K (T, T) = S T K = =

51 Value and Payoff Example Consider a forward with delivery price 100 and maturity T. Suppose underlying asset has price { 110 with probability 0.6 S T = 90 with probability 0.4. Find the expected payoff long the forward. 47

52 Value and Payoff The payoff at maturity long the forward is { with probability 0.6 g(s T ) = S T 100 = with probability 0.4 { 10 with probability 0.6 = 10 with probability 0.4 The expected payoff at maturity long the forward is E(g(S T )) = k R(g(S T )) kp(g(s T ) = k) = (10)(0.6)+( 10)(0.4) = 2 48

53 Forward Price Fix an asset. Let S t be its price at time t. For t T, the forward price F(t, T) of the asset is the delivery price K such that V K (t, T) = 0. In other words, F(t, T) is the special delivery price for which the forward with maturity T has value zero at time t. Warning: F(t, T) is NOT the price of a forward contract! To take at time t the long position in a forward contract with maturity T and delivery price K = F(t, T), there is no initial cost, i.e. V F(t,T) (t, T) = 0. Since V K (T, T) = S T K, we have F(T, T) = S T. 49

54 Forward Price for Asset Paying No Income Result Suppose an asset pays no income (no dividends, no coupons, no payment at maturity, no rent, etc). Then F(t, T) = S t Z(t, T) = S te r(t t), where S t is the price of asset at time t. 50

55 Forward Price for Asset Paying No Income Proof Consider two portfolios. Then A: At time t, one unit of the asset. B: At time t, one long forward contract with maturity T and delivery price K plus K ZCBs. V A (T) = S T, and V B (T) = (S T K) + K = S T. So V A (T) = V B (T). 51

56 Forward Price for Asset Paying No Income By replication, V A (t) = V B (t). This means S t = V K (t, T) + KZ(t, T). (1) The forward price F(t, T) is the delivery price of K such that V K (t, T) = 0. Setting K = F(t, T) and noting that V F(t,T) (t, T) = 0 in (1) leads to F(t, T) = S t Z(t, T). 52

57 Forward Price for Asset Paying No Income Example The current price of a certain stock paying no income is 20. Assume the continuous zero rate will be 2.1% for the next 6 months. Find the forward price with maturity in 6 months. T = 6 months = 0.5 years. F(t, T) = S t Z(t, T) = e (0.021)(0.5) 53

58 Forward Price for Asset Paying No Income If the price of an asset does not match the no-arbitrage price (i.e., the price implied by the no-arbitrage assumption), then we can construct an arbitrage portfolio. Example At current time t, a certain stock paying no income has price 45, the forward price with maturity T on the stock is 48, and the price of a zero coupon bond with maturity T is Determine whether there is an arbitrage opportunity. Construct an arbitrage portfolio if possible. Explain the transactions needed to realize the arbitrage. Verify the portfolio you construct is an arbitrage portfolio. 54

59 Forward Price for Asset Paying No Income At current time t: S t = 45, F(t, T) = 48, and Z(t, T) = Since we have S t Z(t, T) = , F(t, T) > S t Z(t, T). This violates the previous result, so there must be an arbitrage portfolio. 55

60 Forward Price for Asset Paying No Income Let s build an arbitrage portfolio A as follows: At current time t, borrow S t cash, buy one share of stock at price S t, and go short on one forward contract with maturity T and delivery price equal to the forward price F(t, T) (which we can do at no cost). That is, the portfolio A at current t includes S t cash (i.e., S t cash borrowed) one share of stock S t one short forward contract with maturity T and delivery price equal to the forward price F(t, T) 56

61 Forward Price for Asset Paying No Income The value of portfolio A is V A (t) = S t + S t V F(t,T) (t, T) = 0. At time T, the portfolio A includes: S t e r(t t) cash one share of stock S T F(t, T) cash sell one share of stock S T The value of A is V A (T) = S t e r(t t) +S T +(F(t, T) S T ) = F(t, T) S t e r(t ) > 0 (with probability one). 57

62 Forward Price for Asset Paying Known Income Result Suppose an asset pays a known amount of income (for example, dividends, coupons, payment at maturity, rent) during the life of the forward contract. Then F(t, T) = S t I t Z(t, T), where S t is the price of asset at time t and I t is the present value of the income at time t. If the continuous zero rate for period t to T is a constant r, then F(t, T) = S t I t Z(t, T) = (S t I t )e r(t t). 58

63 Forward Price for Asset Paying Known Income Proof Consider the following portfolio A and portfolio B: A: At time t, one unit of the asset plus I t cash B: At time t, one long forward contract with maturity T and delivery price K plus K ZCBs. Then we have V A (T) = S T I t e r(t t) + I t e r(t t) = S T, V B (T) = S T K + K = S T. So V A (T) = V B (T) with probability 1. 59

64 Forward Price for Asset Paying Known Income By replication, V A (t) = V B (t). This means S t I t = V K (t, T) + KZ(t, T). (2) The forward price F(t, T) is the delivery price of K such that V K (t, T) = 0, i.e., V F(t,T) (t, T) = 0 Setting K = F(t, T) in (2) leads to F(t, T) = S t I t Z(t, T). By the definition of continuous zero rate, Z(t, T) = e r(t t). Question How to prove it by No Arbitrage Assumption directly? 60

65 Forward Price for Asset Paying Known Income Example Assume the continuously compounded interest rate is a constant r. Consider a stock that pays a dividend equal to d at times T 1, T 2,..., T n. Find the forward price F(t, T) when t T 1 and T = T n. Assume the dividends are deposited as cash. 61

66 Forward Price for Asset Paying Known Income I t = present value of the income = present value of the stream of dividend payments. The dividend payment at T i is equivalent to d ZCBs with maturity T i. Therefore I t = d n Z(t, T i ) = d i=1 n i=1 e r(t i t) Hence F(t, T) = S t I t Z(t, T) = S t d n i=1 e r(t i t) e r(t t). 62

67 Forward Price for Stock Paying Known Dividend Yield Result Suppose a stock pays dividends equal to a percentage q of the stock price. q is called the dividend yield. Suppose the dividends are paid on a continuously compounded per-year basis. Suppose the dividends are automatically reinvested in the stock. Then F(t, T) = S te q(t t) Z(t, T) = S t e (r q)(t t) 63

68 Forward Price for Stock Paying Known Dividend Yield Proof Consider two portfolios as follows: Then A: e q(t t) units of stock B: one forward with delivery price K and maturity T, K ZCBs with maturity T V A (t) = e q(t t) S t V B (t) = V K (t, T) + KZ(t, T) V A (T) = e q(t t) e q(t t) S T = S T Why? V B (T) = (S T K) + K = S T 64

69 Forward Price for Stock Paying Known Dividend Yield Question If we have N s units of an asset at time s, how many more units of the asset can we buy with the dividends? Let s consider the time interval [s, s + ds]. dn s - infinitesimal change in the total number of units. Then dn s = qn s dt or dn s ds = qn s, t s T. Solving this ordinary differential equation gives us N T = N t e q(t t). 65

70 Forward Price for Stock Paying Known Dividend Yield Since V A (T) = V B (T) with probability one, replication gives V A (t) = V B (t). Therefore e q(t t) S t = V K (t, T) + KZ(t, T). Setting K = F(t, T) gives V K (t, T) = 0 and leads to F(t, T) = S te q(t t). Z(t, T) 66

71 Forward Price for Currency Result Suppose X t is the price at time t in USD of one unit of foreign currency. (For example, 1 CAD = 0.77 USD.) Let r $ be the zero rate for USD. Let r f be the zero rate for foreign currency, both constant and compounded continuously. Then the forward price for one unit of foreign currency is F(t, T) = X t e (r $ r f )(T t). Hint: The foreign currency is analogous to a stock paying known dividend yield. The foreign interest rate corresponds to the dividend yield. 67

72 Forward Price for Asset Paying No Income vs Known Income Remark Notice the forward price for an asset paying known income is lower than it would be if the asset paid no income: F(t, T) = S t Z(t, T) v.s. F(t, T) = S t I t Z(t, T) or S te q(t t). Z(t, T) Until maturity of the forward, the short counterparty holds the asset and collects any income it pays, while the long counterparty gets no income. Thus, for an asset paying income, there is an advantage to buying it spot (i.e., buying it immediately at time t) rather than buying it forward. So the forward price is lower to compensate. 68

73 Value of Forward and Forward Price The next result connects the value of a forward for an asset, the delivery price, and the forward price for the asset. Result For any asset, V K (t, T) = (F(t, T) K)Z(t, T) = (F(t, T) K)e r(t t), assuming the continuous zero rate for period t to T is a constant r. 69

74 Value of Forward and Forward Price Proof Consider two portfolios at time t. A: one long forward with maturity T and delivery price K, one short forward with maturity T and delivery price F(t, T); B: (F(t, T) K) ZCBs with maturity T. Then we have V A (t) = V K (t, T) + 0 = V K (t, T) V B (t) = (F(t, T) K)Z(t, T) V A (T) = (S T K) + (F(t, T) S T ) = F(t, T) K V B (T) = F(t, T) K. 70

75 Value of Forward and Forward Price So V A (T) = V B (T) with probability one. By replication, V A (t) = V B (t). But V A (t) = V K (t, T), V B (t) = (F(t, T) K)Z(t, T). Therefore V K (t, T) = (F(t, T) K)Z(t, T). 71

76 Value of Forward and Forward Price Example Assume V K (t, T) < (F(t, T) K)Z(t, T). Find an arbitrage portfolio. Recall the portfolio A from the proof: one long forward with maturity T and delivery price K, one short forward with maturity T and delivery price F(t, T) Portfolio C is A from the proof minus V K (t, T) cash (or V K (t, T)/Z(t, T) ZCBs with maturity T). Then V C (t) = V K (t, T) + 0 V K (t, T) = 0, and V C (T) = (S T K) + (F(t, T) S T ) V K(t, T) Z(t, T) > 0. 72

77 Value of Forward and Forward Price Example (a) Find a formula for the value at time t of a forward contract on a stock paying no income if the delivery price is K and maturity is T. (b) The current price of a certain stock paying no income is 20. Assume the continuous zero rate will be 2.1% for the next 6 months. Use the formula from part (a) to find the value of a forward contract on the stock if the delivery price is 25 and maturity is in 6 months. 73

78 Value of Forward and Forward Price Solution (a) From previous results, we have [ ] S t V K (t, T) = (F(t, T) K)Z(t, T) = Z(t, T) K Z(t, T) = S t KZ(t, T) = S t Ke r(t t) where r is the continuous zero rate for period t to T. (b) T t = 0.5 years, S t = 20, K = 25, r = V K (t, T) = 20 25e (0.021)(0.5) 74

79 Forward Rates and Libor

80 Forward Zero Coupon Bond Prices For T 1 T 2. Consider a forward contract with maturity T 1 on a ZCB with maturity T 2. That is, the underlying asset price at current time t is Z(t, T 2 ) (rather than S t ). Question Suppose the delivery price here is K. What is the pay off from this long forward at maturity T 1? Answer Z(T 1, T 2 ) K (compared to S T K). 75

81 Forward Zero Coupon Bond Prices Question What about the forward price F(t, T 1, T 2 ) of this underlying asset Z(t, T 2 )? Recall F(t, T 1, T 2 ) is the delivery price such that this forward contract has zero value at time t. In short, V F(t,T1,T 2 )(t, T 1, T 2 ) = 0, where V K (t, T 1, T 2 ) is the current value of the forward with delivery price K. 76

82 Forward Zero Coupon Bond Prices Recall The forward price on asset paying no income is F(t, T) = S t Z(t, T). Result The forward price with maturity T 1 on a ZCB with maturity T 2 is F(t, T 1, T 2 ) = Z(t, T 2) Z(t, T 1 ), for t T 1. 77

83 Forward Zero Coupon Bond Prices Proof Consider two portfolios. A: At time t, one unit of the asset, or one ZCB with maturity T 2 ; B: At time t, one long forward contract with maturity T 1 and delivery price K plus K ZCBs with maturity T 1. Then and So V A (T 1 ) = Z(T 1, T 2 ), V B (T 1 ) = (Z(T 1, T 2 ) K) + K = Z(T 1, T 2 ). V A (T 1 ) = V B (T 1 ). 78

84 Forward Zero Coupon Bond Prices By replication, V A (t) = V B (t) for t T 1. This means Z(t, T 2 ) = V K (t, T 1, T 2 ) + KZ(t, T 1 ). (3) The forward price F(t, T 1, T 2 ) is the delivery price of K such that V K (t, T 1, T 2 ) = 0. Setting K = F(t, T 1, T 2 ) and noting that V F(t,T1,T 2 )(t, T 1, T 2 ) = 0 in (3) leads to F(t, T 1, T 2 ) = Z(t, T 2) Z(t, T 1 ). 79

85 Forward Interest Rates Consider t T 1 T 2. Definition The forward (interest) rate at current time t for period T 1 to T 2, is the rate agreed at t at which one can borrow or lend money from T 1 to T 2. In this section, we use the simplified notation f 12 for the forward rate, and we suppose that r 1 and r 2 are the current zero rates for terms T 1 and T 2, respectively. Remark In general, f = f(t, T 1, T 2, m), where m is the compounding frequency of the rate, possibly. 80

86 Forward Interest Rates Result If the rates are continuously compounded, then the forward rate f 12 at current time t for period T 1 to T 2 is f 12 = r 2(T 2 t) r 1 (T 1 t) T 2 T 1. Observations (a) When T 1 = t, f 12 = r 2. (b) r 2 is a weighted average of r 1 and f 12, since r 2 = r 1(T 1 t) + f 12 (T 2 T 1 ). T 2 t 81

87 Forward Interest Rates Forward and zero rates r 1 - zero rate for period t to T 1 ; r 2 - zero rate for period t to T 2 ; f 12 - forward rate at current time t for period T 1 to T 2. 82

88 Forward Interest Rates Proof Consider two portfolios: A: e r 2(T 2 t) cash with zero rate r 2 from t to T 2. B: e f 12(T 2 T 1 ) e r 1(T 1 t) cash with zero rate r 1 from t to T 1 and forward rate f 12 from T 1 to T 2. Apparently, V A (T 2 ) = V B (T 2 ), with probability one. 83

89 Forward Interest Rates By replication, V A (t) = V B (t), and thus, e r 2(T 2 t) = e f 12(T 2 T 1 ) e r 1(T 1 t), or equivalently, e r 2(T 2 t) = e f 12(T 2 T 1 ) e r 1(T 1 t). This means r 2 (T 2 t) = f 12 (T 2 T 1 ) + r 1 (T 1 t). 84

90 Forward Interest Rates Remark Note that Z(t, T i ) = e r i(t i t), for i = 1, 2, we see that the forward ZCB price is related to the forward rate by F(t, T 1, T 2 ) = e f 12(T 2 T 1 ). 85

91 Forward Interest Rates Similarly, we have Result If the rates are annually compounded, we have (1 + r 1 ) (T 1 t) (1 + f 12 ) (T 2 T 1 ) = (1 + r 2 ) (T 2 t), and thus [ (1 + r 2 ) (T 2 t) f 12 = (1 + r 1 ) (T 1 t) ] 1/(T2 T 1 ) 1. Question What is the relation between the forward ZCB price and the forward rate in this case? 86

92 Forward Interest Rates Example One year from now, your business plan requires a loan of 100, 000 to purchase new equipment. You plan to repay the loan one year after that. You want to arrange the interest rate of the loan today, rather than gamble on the interest rate one year from now. The current time is t = 0. The interest rate for period 0 to 1 is 8%. The interest rate for period 0 to 2 is 9%. Assuming (as usual) no-arbitrage, what must the interest rate on the loan be? Assume all rates are for continuous compounding. 87

93 Forward Interest Rates We have t = 0, T 1 = 1, T 2 = 2, r 1 = 0.08, r 2 = What is f 12, the forward rate for the period 1 to 2? f 12 = r 2(T 2 t) r 1 (T 1 t) T 2 T (2 0) 0.08(1 0) = 2 1 = 0.1 = 10% 88

94 Forward Interest Rates Example Assume t T 1 T 2, t = current time. What can you say about interest rates between T 1 and T 2 if (i) Z(t, T 1 ) = Z(t, T 2 ). (ii) Z(t, T 1 ) > 0 and Z(t, T 2 ) = 0. 89

95 Forward Interest Rates For annual compounding: F(t, T 1, T 2 ) = Z(t, T 2) Z(t, T 1 ) = (1 + f 12) (T 2 T 1 ). (i) 1 = (1 + f 12 ) (T 2 T 1 ), so the forward rate f 12 between T 1 and T 2 is 0. (ii) 0 = (1 + f 12 ) (T 2 T 1 ), so the forward rate f 12 between T 1 and T 2 is. The same is true for any compounding frequency. 90

96 Forward Interest Rates For continuous compounding: F(t, T 1, T 2 ) = Z(t, T 2) Z(t, T 1 ) = e f 12(T 2 T 1 ). (i) 1 = e f 12(T 2 T 1 ), so the forward rate f 12 between T 1 and T 2 is 0. (ii) 0 = e f 12(T 2 T 1 ), so the forward rate f 12 between T 1 and T 2 is. 91

97 Forward Interest Rates Definition The m-year forward n-year rate (sometimes called m-year n-year forward rate) is the forward rate for the period starting m years from now and ending n years later. It is the forward rate at current time t for period T 1 = t + m to T 2 = T 1 + n = t + m + n. T 1 t = m T 2 T 1 = n T 2 t = m + n Confusingly, sometimes this forward rate is denoted f mn. 92

98 Forward Interest Rates Example Assume all rates are annually compounded. The one-year and two-year zero rates are 1% and 2%, respectively. (a) What is the one-year forward one-year rate? f 11 = 1y1y forward rate = one-year forward one-year rate = forward rate for the period starting one year from now and ending one year later 93

99 Forward Interest Rates Assume current time is t. Then T 1 = t + 1, T 2 = T = t + 2. Given r 1 = zero rate for t to T 1 = 0.01 r 2 = zero rate for t to T 2 2 = 0.02 Then (1 + r 1 ) T 1 t (1 + f 11 ) T 2 T 1 = (1 + r 2 ) T 2 t. Substituting and solving for f 12 gives [ (1 + r2 ) T 2 t f 11 = (1 + r 1 ) T 1 t ] 1/(T2 T 1 ) 1 = = % 94

100 Forward Interest Rates Example Assume all rates are annually compounded. The one-year and two-year zero rates are 1% and 2%, respectively. (b) If the two-year forward one-year rate is 3%, what is the three-year zero rate? 95

101 Forward Interest Rates We can assume current time is t. f 21 = 2y1y forward rate = two-year forward one year rate = forward rate for the period starting two years from now and ending one year later = forward rate for T 2 = t + 2 to T 3 = T = t + 3 r 2 = zero rate for t to T 2 = 0.02 r 3 = zero rate for t to T 3 =??? Then (1 + r 2 ) T 2 t (1 + f 21 ) T 3 T 2 = (1 + r 3 ) T 3 t Substituting and solving for r 3 gives r 3 = = %. 96

102 Libor LIBOR (London InterBank Offered Rate) is the rate at which banks borrow or lend to each other. one of the most widely used reference rates. matures at α = 1, call it twelve-month libor, or 12mL. matures at α = 0.5, call it 6mL, etc. L t [t, t + α]: the libor rate at current time t for period t to t + α. For T > t, L T [T, T + α] is a random variable! It is sometimes called the libor fix at T. 97

103 Libor Suppose the libor rate at current time t for period t to t + α is L t [t, t + α]. Then banks can deposit (or borrow) N at time t and receive (or pay back) N(1 + αl t [t, t + α]), at time t + α. Remark All interest is paid at the maturity or term of the deposit, and there is no interim compounding. Most interest rate derivatives typically reference 3mL or 6mL. 98

104 Libor current USD LIBOR rate

105 Libor Example The USD 6mL rate on 09/26/2016 is %. Suppose Citibank can borrow 100 million USD from Bank of America on 09/26/2016 for 6 months, how much should Citibank pay back on 03/26/2017? 100( ) million USD. 100

106 Forward Rate Agreement and Forward Libor A forward rate agreement (FRA) is a contract to exchange two cashflows. Three parameters: T - maturity; K - deliver price or fixed rate; α - term length or dayout fraction. The buyer of the FRA agrees at t T to pay αk and receive αl T [T, T + α] at time T + α. The seller agrees to do the opposite. 101

107 Forward Rate Agreement and Forward Libor The value or payout at time T + α of the FRA (for the buyer or long position) is αl T [T, T + α] αk. Remark The FRA is a new type of a derivative contract. But we cannot say that the underlying asset is the libor rate since αl T [T, T + α] is not the price of an asset that is tradable. 102

108 Forward Rate Agreement and Forward Libor Recall: The forward price F(t, T) of an asset is the number such that the forward contract on the asset with maturity T and delivery price K = F(t, T) has value 0 at time t. Denote by V α,k (t, T) the value of the FRA at current time t T. Definition The forward libor rate L t [T, T + α] is the special number such that the value at t of the FRA with maturity T, term length α, and delivery price K = L t [T, T + α] is V α,k (t, T) =

109 Forward Rate Agreement and Forward Libor Summary L t [t, t + α] = libor rate at current time t for period t to t + α. Banks can deposit (or borrow) N at time t and receive (or pay back) N(1 + αl t [t, t + α]) at time t + α. L T [T, T + α] = libor rate at future time T for period T to T + α. Banks can deposit (or borrow) N at time T and receive (or pay back) N(1 + αl T [T, T + α]) at time T + α. L t [T, T + α] = forward libor rate at t for period T to T + α. It is the number such that the value at t of the FRA with maturity T, delivery price K = L t [T, T + α], and term length α is V K (t, T) = 0. L t [t, t + α] and L t [T, T + α] are known (non-random) at time t. L T [T, T + α] is a random variable. 104

110 Forward Rate Agreement and Forward Libor Result The value at current time t of an FRA with maturity T, delivery price K, and term length α is V α,k (t, T) = Z(t, T) (1 + αk)z(t, T + α). The forward libor rate at time t for maturity T and term length α is Z(t, T) Z(t, T + α) L t [T, T + α] =. αz(t, T + α) 105

111 Forward Rate Agreement and Forward Libor Proof Our strategy is to find a portfolio that replicates the FRA. A: 1 FRA with maturity T, delivery price K, and term length α. B: At time t, 1 ZCB with maturity T and (1 + αk) ZCBs with maturity T + α. At time T, put the 1 from the ZCB maturing at T in a deposit with libor rate L T [T, T + α]. (Any cost? Why?) At time T + α: V A (T + α) = α(l T [T, T + α] K) V B (T+α) = (1+αK)+(1+αL T [T, T+α]) = α(l T [T, T+α] K) Since V A (T + α) = V B (T + α) with probability one, the replication theorem gives V A (t) = V B (t). 106

112 Forward Rate Agreement and Forward Libor At time t: V A (t) = value of the FRA = V α,k (t, T). V B (t) = Z(t, T) (1 + αk)z(t, T + α). Therefore: V α,k (t, T) = Z(t, T) (1 + αk)z(t, T + α). 107

113 Forward Rate Agreement and Forward Libor The forward libor rate L t [T, T + α] is the value of K such that V α,k (t, T) = 0. Setting K = L t [T, T + α] gives V α,k (t, T) = 0 and so 0 = Z(t, T) (1 + αl t [T, T + α])z(t, T + α). Rearranging gives L t [T, T + α] = Z(t, T) Z(t, T + α). αz(t, T + α) 108

114 Forward Rate Agreement and Forward Libor Rearranging the previous formula gives 1 Z(t, T + α) = Z(t, T) 1 + αl t [T, T + α]. Discounting by ZCBs and the forward libor rate 109

115 Forward Rate Agreement and Forward Libor Example A bank needs to borrow 1 at future time T until T + α. Show that the bank can at current time t lock in the interest cost for the period T to T + α by combining an FRA trade today with a libor loan at T. What is the interest cost? 110

116 Forward Rate Agreement and Forward Libor At time t: (FRA trade) go long one FRA with maturity T, term length α, and delivery price K equal to the forward libor rate L t [T, T + α]. There is no cost to do this. At time T: (libor loan) borrow 1 at the random libor rate L T [T, T + α]. At time T + α: Execute the FRA, which means receiving αl T [T, T + α] in exchange for paying αk = αl t [T, T + α] Pay back the libor loan: 1 + αl T [T, T + α]. 111

117 Forward Rate Agreement and Forward Libor The value of the portfolio at T + α: 1 + (αl T [T, T + α] αk) (1 + αl T [T, T + α]). The total interest cost: αk = αl t [T, T + α]. 112

118 Valuing Floating and Fixed Cashflows Question What is the value at t of an agreement to receive the fixed (i.e., known, non-random) payment αk at T + α? Answer αkz(t, T + α). 113

119 Valuing Floating and Fixed Cashflows Question. What is the value at t of an agreement to receive the floating (i.e., unknown, random variable) libor payment αl T [T, T + α] at T + α? Answer This is an FRA with K = 0. By the previous result, its value is V α,0 (t, T) = Z(t, T) Z(t, T + α) = αl t [T, T + α]z(t, T + α). Notice the value is non-random and does not depend on the distribution of the random variable L T [T, T + α]. 114

120 Valuing Floating and Fixed Cashflows Remark Receiving an unknown libor interest payment from T to T + α on a unit of cash has the same value as receiving the unit of cash at T and then paying the unit back at T + α. This is intuitively clear, since one can invest the unit of cash in a libor deposit in the interim and receive the libor interest payment. 115

121 Valuing Floating and Fixed Cashflows Example Fixed rate annuity revisited. Suppose annually compounded zero rates for all maturities are a constant r, so Z(0, j) = (1 + r) j, for j = 1, 2,. (a) What is the value today t = 0 of a fixed annuity that pays 1 each year from T 1 to T n? (b) Find the value of the infinite fixed annuity as n. 116

122 Valuing Floating and Fixed Cashflows (a) What is the value today t = 0 of a fixed annuity that pays 1 each year from T 1 to T n? V 0 = n Z(0, j) = j=1 n (1 + r) j = j=1 1 (1 + r) n. r (b) Find the value of the infinite fixed annuity as n. V 0 1 r as n. 117

123 Valuing Floating and Fixed Cashflows Example Floating rate annuity (a) Let T 0, T 1,, T n be a sequence of times, with T i+1 = T i + α for a constant α > 0. Show that a floating rate annuity, i.e., receiving payments αl Ti [T i, T i + α] at times T i+1, i = 0, 1,, n 1, has value at time t T 0 equal to a simple linear combination of ZCBs prices. 118

124 Valuing Floating and Fixed Cashflows (a) Recall that: a derivative contract pays αl T [T, T + α] at time T + α, and the value at t T of the derivative contract is Z(t, T) Z(t, T + α). Then the current value of the floating rate annuity is n 1 V t = (Z(t, T i ) Z(t, T i+1 )). i=0 119

125 Valuing Floating and Fixed Cashflows (b) Find the value of a spot-starting infinite stream of libor payments, that is, when t = T 0 = 0 and as n. Note that V t = Z(t, T 0 ) Z(t, T n ), thus, V 0 = Z(0, 0) Z(0, T n ) = 1 Z(0, nα) 1 as n. 120

126 Interest Rate Swaps

127 Swap Definition Interest rate swaps are the most widely traded and most liquid of all over-the-counter derivative contracts. An (interest rate) swap is a sequence of forward rate agreements (FRAs). It is an agreement between two counterparties to exachange a sequence of cashflows. Motivation - trading (for example, speculation) - risk management 121

128 Swap Definition Parameters: T 0 - start date; T n - maturity date; T i, i = 1,, n - payment dates. We assume the length of time interval is a constant α, i.e. T i+1 = T i + α, in which case, we all it a standard or vanilla swap. Remark In practice, α may differ for each period, so that T i+1 = T i + α i. 122

129 Swap Definition The floating leg of the swap consists of payments αl Ti [T i, T i + α] at T i + α, i.e., libor fixing at T i for the period T i to T i + α, paid at T i + α The fixed leg of the swap consists of payments αk at T i + α, i.e., a fixed rate K accrued from T i to T i + α, paid at T i + α. 123

130 Swap Definition How does a swap really work? floating leg, fixed leg, and swap One counterparty (the payer, buyer or long position ) of a swap pays the fixed leg to, and receives the floating leg from, the other counterparty (the receiver, seller or short position ) of the swap. 124

131 Value of Swap Consider a swap from T 0 to T n with fixed rate K. Notations For t T 0, Result V SW K (t) = value of the swap at t V FL (t) = value of the floating leg at t V FXD K (t) = value of the fixed leg t V SW K (t) = VFL (t) V FXD (t). K Note that V SW K (t) is the value to the buyer. The value to the seller is the V SW K (t). 125

132 Value of Swap The fixed leg is a sequence of ZCBs with maturities at T 1,..., T n, so Result where we define V FXD K (t) = n αkz(t, T i ) = KP t [T 0, T n ], i=1 P t [T 0, T n ] = n αz(t, T i ). We call P t [T 0, T n ] the pv01 of the swap, the present value of receiving 1 times α at each payment date. i=1 126

133 Value of Swap The floating leg is a sequence of libor payments at T 1,..., T n, so Result V FL (t) = = n (value at t of payment αl Ti 1 [T i 1, T i ] i=1 paid at T i = T i 1 + α) n n αl t [T i 1, T i ]Z(t, T i ) = [Z(t, T i 1 ) Z(t, T i )] i=1 = Z(t, T 0 ) Z(t, T n ) i=1 127

134 Value of Swap Remark Previous result V FL (t) = Z(t, T 0 ) Z(t, T n ), says the value of receiving a stream of libor interest payments on an investment of 1 is equal to the value of receiving one dollar at the beginning of the stream and paying it back at the end. We simply take the dollar and repeatedly invest in a sequence of libor deposits. 128

135 Value of Swap In summary, we can write V SW K (t) as a linear combination of ZCB prices. Result V SW K (t) = VFL (t) V FXD (t) K = Z(t, T 0 ) Z(t, T n ) αk n Z(t, T i ). i=1 129

136 Value of Swap Example Consider a swap starting now with fixed rate 3%, quarterly payment frequency, and ending in 2 years. Suppose the quarterly compounded zero rates for all payment times are 2%. Find the present value of (a) the floating leg (b) the swap 130

137 Value of Swap (a) Given t = T 0, T n t = 2, α = 0.25, r 4 = V FL (t) = Z(t, T 0 ) Z(t, T n ) = 1 (1 + r 4 /4) 4(Tn t) = 1 ( /4) 4(2) =

138 Value of Swap (b) Since V SW (t) = V FL (t) V FXD (t), we just need to compute V FXD (t). Have K = Have T i t = T i T 0 = iα = i0.25. Using T n = T 0 + nα, we get n = 8. Then V FXD (t) = αk = αk = αk n Z(t, T i ) i=1 n (1 + r 4 /4) 4(T i t) i=1 n (1 + r 4 /4) i i=1 132

139 Value of Swap We can use the formula n i=1 to compute the sum. Then (1 + r 4 /4) i = 1 (1 + r 4/4) n r 4 /4 V FXD (t) = αk 1 (1 + r 4/4) n r 4 /4 1 ( /4) 8 = 0.25(0.03) 0.02/4 =

140 Value of Swap Therefore V SW (t) = V FL (t) V FXD (t) = 1 ( /4) 4(2) 1 ( /4) (0.03) 0.02/4 =

141 Forward Swap Rate The forward swap rate y t [T 0, T n ] is the special number such that the value at t of the swap from T 0 to T n with fixed rate K = y t [T 0, T n ] has value V SW (t) = 0, i.e., K V SW y t[t 0,T n] (t) = 0. Result The forward swap rate at t T 0 for a swap from T 0 to T n is y t [T 0, T n ] = Z(t, T 0) Z(t, T n ) n i=1 = L t[t i 1, T i ]αz(t, T i ) P t [T 0, T n ] n i=1 αz(t, T. i) (4) Remark The forward swap rate is a weighted average of forward libors. 135

142 Forward Swap Rate Proof Setting K = y t [T 0, T n ] gives V SW K (t) = 0 V FXD K (t) = V FL (t) KP t [T 0, T n ] = Z(t, T 0 ) Z(t, T n ) (5) n n K αz(t, T i ) = αl t [T i 1, T i ]Z(t, T i ) (6) i=1 i=1 Solving (5) for K gives the first equality in (4). Solving (6) for K gives the second equality in (4). 136

143 Value of Swap in Terms of Forward Swap Rate Result The value at t T 0 of a swap from T 0 to T n with fixed rate K is V SW K (t) = (y t[t 0, T n ] K)P t [T 0, T n ]. Remark Compare to the value of a forward contract: V K (t, T) = (F(t, T) K)Z(t, T). 137

144 Value of Swap in Terms of Forward Swap Rate Proof Since V SW K (t) = VFL (t) V FXD K (t) = Z(t, T 0 ) Z(t, T n ) KP t [T 0, T n ]. By the previous result, we have y t [T 0, T n ]P t [T 0, T n ] = Z(t, T 0 ) Z(t, T n ). 138

145 Par or Spot-Starting Swaps When t = T 0, we call y T0 [T 0, T n ] the par swap rate or spot-starting swap rate of maturity T n T 0. Given par swap rates y T0 [T 0, T i ] for all T i, we can recover ZCB prices Z(T 0, T i ) for all T i by using y t [T 0, T i ] = Z(t, T 0) Z(t, T i ). P t [T 0, T i ] This process is known as bootstrapping and will be explored in the exercises. It is used frequently in practice. 139

146 Par or Spot-Starting Swaps Result The par swap rate y T0 [T 0, T n ] is the fixed rate at which one can invest 1 from T 0 until T n to receive fixed payments of αy T0 [T 0, T n ] at times T i and notional 1 back at T n. Remark This is basically a fixed rate bond with notional 1, paying αy T0 [T 0, T n ] (a constant) at time T i, i = 1,, n and the notional 1 at time T n. 140

147 Par or Spot-Starting Swaps Proof The value of this fixed rate bond at time T 0 is n αy T0 [T 0, T n ]Z(T 0, T i ) + Z(T 0, T n ). i=1 Now we replicate this bond in the following way: At time T 0, deposit 1 at the rate L T0 [T 0, T 1 ]; At time T 1, receives 1 + αl T0 [T 0, T 1 ], and then reinvest (deposit) 1 at the rate L T1 [T 1, T 2 ]; At time T 2, receives 1 + αl T1 [T 1, T 2 ], and then reinvest (deposit) 1 at the rate L T2 [T 2, T 3 ]; 141

148 Par or Spot-Starting Swaps In general, for i = 1,, n 1: At time T i, receives 1 + αl Ti 1 [T i 1, T i ], and then reinvest (deposit) 1 at the rate L Ti [T i, T i+1 ]; At time T n, receives 1 + αl Tn 1 [T n 1, T n ]. Therefore, the value of this investment at time T 0 is n αl Ti 1 [T i 1, T i ]Z(T 0, T i ) + Z(T 0, T n ), i=1 which by the result on the forward swap rate is n αy T0 [T 0, T n ]Z(T 0, T i ) + Z(T 0, T n ). i=1 142

149 Swaps as Difference Between Bonds A fixed rate bond with notional N and coupon c is an asset that pays N at time T n and coupon payments αnc at times T i for i = 1,..., n, where T i+1 = T i + α. If N = 1, the price at t of the fixed rate bond is denoted B FXD c (t). A floating rate bond with notional N is an asset that pays N at time T n and coupon payments αnl Ti 1 [T i 1, T i ] at times T i for i = 1,..., n, where T i+1 = T i + α. If N = 1, the price at t of the floating rate bond is denoted B FL (t). 143

150 Swaps as Difference Between Bonds Consider a swap from T 0 to T n with fixed rate K. Result For t T 0, V SW K (t) = BFL (t) B FXD (t). K 144

151 Swaps as Difference Between Bonds Proof The fixed rate bond with notional 1 and coupon K equals the fixed leg of the swap plus a payment of 1 at T n. The floating rate bond with notional 1 equals the floating leg of the swap plus a payment of 1 at T n. Therefore B FL (t) B FXD K (t) = (V FL (t) + Z(t, T n )) (V FXD K (t) + Z(t, T n )) = V FL (t) V FXD K (t) = V SW K (t). 145

152 Swaps as Difference Between Bonds Result Consider the fixed rate bond with notional 1, coupon c, start date T 0, maturity T n, and term length α. It pays αnc at times T 1,..., T n (where T i+1 = T i + α) and pays 1 at time T n. Its value at time T 0 is B FXD c (T 0 ). We have B FXD c (T 0 ) = 1 if and only if c = y T0 [T 0, T n ] = par swap rate 146

153 Swaps as Difference Between Bonds Remark Because of this result, the par-swap rate y T0 [T 0, T n ] is sometimes called the coupon rate. In other words, this result says we can invest 1 at time T 0, receive 1 back at time T n, and receive fixed payments of αy T0 [T 0, T n ] at times T 1,..., T n in between. And this isn t true if y T0 [T 0, T n ] is replaced by any other coupon c. 147

154 Swaps as Difference Between Bonds Proof We know B FL (T 0 ) = V FL (T 0 )+Z(T 0, T n ) = Z(T 0, T 0 ) Z(T 0, T n )+Z(T 0, T n ) = 1 Then B FXD c (T 0 ) = 1 B FL (T 0 ) = B FXD c (T 0 ) V SW c = B FL (T 0 ) B FXD c (T 0 ) = 0 c = y T0 [T 0, T n ]. The last equivalence is a consequence of the definition of forward swap rate. 148

155 Futures Contracts

156 Futures Definition Fix an asset. Let S t be its price at time t. Recall A forward contract on the asset with maturity T and delivery price equal to the forward price F(t, T) is an agreement to trade the asset. At time t, the contract has no cost to enter. At maturity T, the long counterparty receives (pays if negative) value S T F(t, T) = F(T, T) F(t, T). There are no payments in between. 149

157 Futures Definition A futures contract (or future) on the asset with maturity T and delivery price K is an agreement to trade the asset. Unlike a forward contract, there are payments every day until the maturity date T. We describe the payments below. If K equals the futures price Φ(t, T), the contract has no cost to enter (the value is zero) at time t. 150

158 Futures Definition Now we describe the payments of a futures contract below. Let = 1/365. Since we measure time in years, day 1 is time t + and day i is time t + i. We define day n to be maturity T, so that T = t + n. On day 1, the long counterparty receives (pays if negative) the amount Φ(t +, T) Φ(t, T). 151

159 Futures Definition On each day 1 < i < n, the long counterparty receives (pays if negative) the amount Φ(t + i, T) Φ(t + (i 1), T) On day n, the long counterparty receives (pays if negative) the amount Φ(T, T) Φ(t + (n 1), T) The payment amount on each day is called the mark-to-market change (or variation margin). For the short counterparty, the payments are, of course, the negatives of these amounts. 152

160 Futures Definition Remark 1. Φ(t + i, T) is the future price on day i. It costs nothing to enter a futures contract at t + i, if the delivery price is Φ(t + i, T) and maturity is T. 2. At current time t, Φ(t + i, T) is unknown (random) for i

161 Futures Definition The future price at maturity T is defined to be Φ(T, T) = S T. Over the life of the contract, the long counterparty receives mark-to-market payments that total S T K = Φ(T, T) Φ(t, T) = Φ(t + n, T) Φ(t + (n 1), T) + + Φ(t + i, T) Φ(t + (i 1), T) + + Φ(t +, T) Φ(t, T) However, each payment is made at a different time, so the value of the payments at T will not in general equal S T K. 154

162 Futures Definition If two investors get in touch with each other directly and agree to trade an asset in the future for a certain price, there are obvious risks. One of the investors may regret the deal and try to back out. Alternatively, the investor simply may not have the financial resources to honor the agreement. This is one of the key reasons that virtually all futures contracts are traded on exchanges rather than OTC. This is also where margin accounts come in. 155

163 Futures Definition In practice, at t each counterparty in a futures contract makes a deposit called an initial margin (or performance bond) in a margin account at an exchange. Each day, the exchange transfers the appropriate variation margin amount from one counterparties account to the other. The initial margin needs to be large enough to cover several days of likely variation margin transfers. If the account balance of a counterparty drops below a maintenance margin, the exchange may make a margin call, which is a demand for the counterparty to replenish the account. The initial margin and variation margin accrue interest. 156

164 Futures Definition Example Consider an investor who contacts his or her broker to buy two December gold futures contracts on the New York Commodity Exchange (COMEX) We suppose that the current futures price is 1, 450 per ounce. Contract size: 100 ounces The investor has contracted to buy a total of 200 ounces at this price, i.e., long two contracts. The initial margin: 6, 000 per contract, or 12, 000 in total. At the end of each trading day, the margin account is adjusted to reflect the investor s gain or loss. This practice is referred to as daily settlement or marking to market. 157

165 Futures Definition 2 2 this is from the book Options, Futures and Other Derivatives by John Hull. 158

166 Futures Prices vs Forward Prices-Part I Result If interest rates are constant, then for all t T. Φ(t, T) = F(t, T) 159

167 Futures Prices vs Forward Prices-Part I Before we give the proof, here is some easy consequences. Example If interest rates are constant, then for all t T, i = 0, 1,..., n. Example Φ(t + i, T) = F(t + i, T) If interest rates are constant and if the underlying asset is a stock paying no income, then Φ(t, T) = S t e r(t t). 160

168 Futures Prices vs Forward Prices-Part I Proof For simplicity assume t = 0. Then T = n. Let the constant continuously compounded interest rate be r. Note that we can make trades in a portfolio, as long as they have no cost. For example, we can t just add or subtract cash to a portfolio. Consider the portfolio having the following strategy. At time 0, go long e r(n 1) futures contracts with maturity T at deliver price equal to the futures price Φ(0, T). By the definition of the futures prices, we can do this at no cost. 161

169 Futures Prices vs Forward Prices-Part I At time, increase position to e r(n 2) futures at futures price Φ(, T). That is, we go long on e r(n 2) e r(n 1) futures with maturity T at delivery price equal to the futures price Φ(, T). Again, by the definition of the futures prices, we can do this at no cost. At time i (for i = 2,..., n 2), increase position to e r(n i 1) futures at futures price Φ(i, T). At time (n 1), increase position to 1 futures contract at futures price Φ((n 1), T). 162

170 Futures Prices vs Forward Prices-Part I With this strategy we receive the following amounts. At time, we receive mark-to-market gain/loss (Φ(, T) Φ(0, T))e r(n 1). This will be invested at rate r. So by time T = n (after time T = (n 1) has passed) it becomes (Φ(, T) Φ(0, T))e r(n 1) e r(n 1) = Φ(, T) Φ(0, T). 163

171 Futures Prices vs Forward Prices-Part I At time (i + 1), we receive mark-to-market gain/loss (Φ((i + 1), T) Φ(i, T))e r(n i 1). This will be invested at rate r. So by time T = n (after time T (i + 1) = (n i 1) has passed) it becomes (Φ((i + 1), T) Φ(i, T))e r(n i 1) e r(n i 1) = Φ(i, T) Φ((i 1), T). 164

172 Futures Prices vs Forward Prices-Part I Therefore the value at time T = n of the portfolio is n 1 Φ((i + 1), T) Φ(i, T) = Φ(n, T) Φ(0, T) i=0 = S T Φ(0, T). 165

173 Futures Prices vs Forward Prices-Part I Let A be the above portfolio plus Φ(0, T) ZCBs maturing at T (or e rt cash). Then V A (T) = S T. Let B be the portfolio consisting at time 0 of one long forward contract on the asset with maturity T and delivery price equal to the forward price F(0, T) plus F(0, T) ZCBs maturing at T (or e rt cash). Then also V B (T) = S T. 166

174 Futures Prices vs Forward Prices-Part I By replication, V A (0) = V B (0), which means Divide by Z(0, T) to conclude. Φ(0, T)Z(0, T) = F(0, T)Z(0, T). 167

175 Futures Prices vs Forward Prices-Part II In general, (theoretically) Φ(t, T) F(t, T). The difference Φ(t, T) F(t, T) is called the futures convexity correction. Question What can we say about this futures convexity correction? Answer It is related to the covariance between the price of the underlying asset and the interest rates 168

176 Futures Prices vs Forward Prices-Part II The covariance of random variables X and Y is Cov(X, Y) = E[(X E(X))(Y E(Y))]. 169

177 Futures Prices vs Forward Prices-Part II If (X E(X))(Y E(Y)) > 0, then X and Y are either both above or both below their means E(X) and E(Y). If Cov(X, Y) > 0, then X and Y are, on average, on the same side of their means (both above or both below). In this case, we say X and Y are positively correlated. If (X E(X))(Y E(Y)) < 0, then X and Y are on opposite sides of their means E(X) and E(Y). If Cov(X, Y) < 0, then X and Y are, on average, on opposite sides of their means. In this case, we say X and Y are negatively correlated. If Cov(X, Y) = 0, we say X and Y are uncorrelated. 170

178 Futures Prices vs Forward Prices-Part II Example 1. If X and Y are the height and weight of a randomly selected person, then X and Y are positively correlated. Height and weight tend to move together. 2. If X is the snowfall and Y is the temperature in Rochester on a randomly selected day, then X and Y are negatively correlated. On average, there is more snowfall on days with low temperature than on days with high temperature. 3. If X and Y are the numbers shown on two consecutive rolls of a die, then X and Y are uncorrelated. The number shown on the first roll tells us nothing about the second roll. 171

179 Futures Prices vs Forward Prices-Part II Now back to the futures convexity correction: Φ(t, T) F(t, T). Suppose the price of the underlying asset S is positively correlated with interest rates, or formally they tend to increase or decrease together. When S increases, an investor who holds a long futures position makes an immediate gain because of variation margin. The positive correlation indicates that it is likely that interest rates have also increased. The gain will therefore tend to be invested at a higher than average rate of interest. 172

180 Futures Prices vs Forward Prices-Part II Similarly, when S decreases, the investor will incur an immediate loss. This loss will tend to be financed (by borrowing) at a lower than average rate of interest. However, an investor holding a forward contract rather than a futures contract is not affected in this way by interest rate movements. It follows that a long futures contract will be slightly more attractive than a similar long forward contract, and hence Φ(t, T) > F(t, T), or the futures convexity correction is positive. 173

181 Futures Prices vs Forward Prices-Part II When S is negatively correlated with interest rates, a similar argument shows that forward prices will tend to be slightly higher than futures prices, or the futures convexity correction is negative. 174

182 Futures Prices vs Forward Prices-Part II Result If the asset price S and the interest rate r are positively correlated, then Φ(t, T) F(t, T) > 0. If the asset price S and the interest rate r are negatively correlated, then Φ(t, T) F(t, T) < 0. If the asset price S and the interest rate r are uncorrelated, then Φ(t, T) = F(t, T). The proof is beyond our scope in this course. 175

183 Options

184 Option Definitions Recall A long forward contract is the obligation to buy the asset for K at time T. A European call option on the asset with strike (or exercise price) K and maturity (or exercise date) T is the right but not the obligation to buy the asset for K at time T. Using the right to trade the asset is called exercising the option. 176

185 Option Definitions Since we would only choose to pay K for an asset worth S T if S T K, the payout or value at maturity of a European call option is { S T K if S T K 0 if S T K = max{s T K, 0} = (S T K) +. We can also write the payout as g(s T ), where g(x) = (x K) + = max{x K, 0} is the payout function. Terminology: call with strike K = call struck at K = K-strike call = K call 177

186 Option Definitions A European put option is the right to sell the asset for K at time T. The payout or value at maturity of a European put option is { K S T if S T K 0 if S T K = max{k S T, 0} = (K S T ) +. Notice that the payout of a put is not the negative of the payout of a call. 178

187 Option Definitions A European straddle option is a European call plus a European put. It has payout S T K = { S T K if S T K K S T if S T K. This is because x = x + + x, where x = max{ x, 0}. 179

188 Option Definitions A European option allows exercise only at maturity T. An American option allows exercise at any time t T. More precisely, an American call option on the asset with strike (or exercise price) K and maturity (or exercise date) T is the right to buy the asset for K at any time t T. For an American put option, replace buy with sell. There are many other, less common types of options. We skip here. Note that an option is automatically exercised at maturity if the payout is positive. 180

189 Option Definitions At time t T, a call option with strike K is said to be: in-the-money (ITM) if S t > K (the call option will have positive value at maturity if the asset price remains unchanged) out-of-the-money (OTM) if S t < K (the call option will be worthless if the asset price remains unchanged) at-the-money (ATM) if S t = K in-the-money-forward (ITMF) if F(t, T) > K out-of-the-money-forward (OTMF) if F(t, T) < K at-the-money-forward (ATMF) if F(t, T) = K For a put option, we reverse these inequalities. Here F(t, T) is the forward price on the asset at t for maturity T. 181

190 Option Definitions The intrinsic value of an option is its payout if we were able to exercise it now. The intrinsic value at t of a call is max{s t K, 0}. The intrinsic value at t of a put is max{k S t, 0}. Thus an option is in-the-money if its intrinsic value is positive. 182

191 Option Prices Fix an asset. Let S t be its price at time t. Notation. C K (t, T) = price (value) at time t of a European call with strike K and maturity T. P K (t, T) = price (value) at time t of a European put with strike K and maturity T. C K (t, T) = price (value) at time t of an American call with strike K and maturity T. P K (t, T) = price (value) at time t of an American put with strike K and maturity T. 183

192 Option Prices By definition, C K (T, T) = C K (T, T) = max{s T K, 0}. and P K (T, T) = P K (T, T) = max{k S T, 0}. Result for all t T. C K (t, T) C K (t, T) and P K (t, T) P K (t, T), This is intuitively clear because an American option gives the same rights as an European option, and more. As an exercise, give a proof using the no-arbitrage principle or the monotonicity theorem. 184

193 Option Prices Result C K (t, T) 0 and P K (t, T) 0 Proof We have C K (T, T) = max{s T K, 0} 0 P K (t, T) = max{k S T, 0} 0. Then the result is a consequence of the monotonicity theorem. 185

194 Put-Call Parity Fix an asset. Let S t be its price at t. Recall that the value of a forward is related to the forward price and delivery price by V K (t, T) = (F(t, T) K)Z(t, T). The next result relates the value of a forward to the price of a call and the price of a put. Result V K (t, T) = C K (t, T) P K (t, T) 186

195 Put-Call Parity Proof Consider two portfolios. A: long one forward with delivery price K and maturity T. B: long one call and short one put (i.e., +1 call and 1 put), both with strike K and maturity T. We have V A (T) = S T K { } V B (S T K) 0 if S T K (T) = = S T K. 0 (K S T ) if S T K 187

196 Put-Call Parity By replication, V A (t) = V B (t) which means V K (t, T) = C K (t, T) P K (t, T). 188

197 Put-Call Parity In words, put-call parity says: V K (t, T) = C K (t, T) P K (t, T) long one forward equals long one call and short one put C K (t, T) = V K (t, T) + P K (t, T) long one call equals long one forward and long one put P K (t, T) = V K (t, T) + C K (t, T) long one put equals short one forward and long one call Here equals means has the same value as and the options are European. 189

198 Put-Call Parity Result C K (t, T) = P K (t, T) if and only if K = F(t, T) if and only if the call and put are both at-the-money-forward (ATMF) at t. Proof Use put-call parity and the definition of forward price. 190

199 Call Prices for Assets Paying No Income Result For an asset paying no income, the European call price satisfies max{s t KZ(t, T), 0} C K (t, T) S t. 191

200 Call Prices for Assets Paying No Income Proof We have To prove the upper bound C K (T, T) = max{s T K, 0} S T. C K (t, T) S t, apply the monotonicity theorem to the following portfolios. A: buy one call with maturity T and strike K. B: buy one stock. 192

201 Call Prices for Assets Paying No Income We have S T K max{s T K, 0} = C K (T, T). To prove the lower bound S t KZ(t, T) C K (t, T), apply the monotonicity theorem to the portfolios A: buy one stock, short K ZCBs with maturity T; B: buy one call with maturity T and strike K. 193

202 Call Prices for Assets Paying No Income Result For an asset paying no income, the price of an American or European call is at least the intrinsic value of the call: C K (t, T) C K (t, T) max{s t K, 0}. Proof We know C K (t, T) C K (t, T) and C K (t, T) 0. So we just need to observe that by the previous result. C K (t, T) S t KZ(t, T) S t K 194

203 Call Prices for Assets Paying No Income Result For an asset paying no income, an American call will never be exercised before maturity if interest rates are positive. Proof Consider an American call at t < T. By the previous result, we have C K (t, T) S t KZ(t, T). If the interest rates for period t to T are positive, then Z(t, T) < 1, and so C K (t, T) > S t K. We would receive value S t K if we exercise the call at t. But we would receive a strictly larger value C K (t, T) if we sell the call instead. So we would never choose to exercise at t < T. 195

204 Call Prices for Assets Paying No Income Result If an American call is never exercised before the maturity, then its value is the same as a European call, i.e, for all t T. Proof We consider two cases. C K (t, T) = C K (t, T), First, we assume the American call is exercised at T. Then the payout is the same as the payout of a European call. Second, we assume the American call is never exercised. Then C K (T, T) = 0. But also notice that this implies that S T K. In this case, the actual payout for a European call is 0 as well. We finish the proof by replication. 196

205 Call Prices for Assets Paying No Income Result For an asset paying no income, the price of an American call and the price of a European call (with the same strike and maturity) are always equal, i.e., for all t T. C K (t, T) = C K (t, T) 197

206 Call Prices for Assets Paying No Income Proof We know C K (t, T) C K (t, T). To prove C K (t, T) C K (t, T), consider two portfolios at t: A: buy one American call with maturity T, and strike K; B: buy one European call with maturity T, and strike K. Case 1. The American option is never exercised before maturity T. Then by the previous result, V A (T) = V B (T). 198

207 Call Prices for Assets Paying No Income Case 2. The American option is exercised at time T 0 with t T 0 < T. Portfolio A at T 0 consists of one unit of the asset and K cash (i.e., a debt of K cash). Portfolio A at T consists of one unit of the asset and Ke r(t T 0) = K/Z(T 0, T) cash, where r is the continuous zero rate for T 0 to T. Therefore V A (T) = S T Ke r(t T 0) S T K max{s T K, 0} = V B (T). 199

208 Call Prices for Assets Paying No Income Combining both cases, V A (T) V B (T) with probability one. By the monotonicity theorem, V A (t) V B (t), which means C K (t, T) C K (t, T). Remark The equality between American and European calls does not hold for calls on a dividend paying stock. This is because, by exercising early, we may receive dividends from the stock which we would not have received otherwise. 200

209 Put Prices for Assets Paying No Income Result For an asset paying no income, max{0, KZ(t, T) S t } P K (t, T) KZ(t, T). Proof Like the bounds on European call prices, we can prove this by replication. But it s more interesting to prove it using put-call parity. 201

210 Put Prices for Assets Paying No Income We first prove the upper bound P K (t, T) KZ(t, T). According to the previous result on calls, C K (t, T) S t. By put-call parity, C K (t, T) = V K (t, T) + P K (t, T). Therefore Hence V K (t, T) + P K (t, T) S t. P K (t, T) S t V K (t, T). But, for an asset paying no income, V K (t, T) = (F(t, T) K)Z(t, T) = S t KZ(t, T). Therefore P K (t, T) KZ(t, T). 202

211 Put Prices for Assets Paying No Income To prove the lower bound KZ(t, T) S t P K (t, T), we use the result C K (t, T) max{0, S t KZ(t, T)}. Then by put-call parity, P K (t, T) = C k (t, T) V K (t, T) max{0, S t KZ(t, T)} (S t KZ(t, T)) = max{0, KZ(t, T) S t }. 203

212 Put Prices for Assets Paying No Income Remark American puts are usually worth strictly more than European puts (with the same strike and maturity), since by exercising early we receive the strike price K early and can invest this amount. 204

213 Put Prices for Assets Paying No Income Example Consider an American put expiring in one year with strike 80. Suppose that the stock price is S t = 10 and the one-year annually compounded interest rate is 16%. Exercising now, we can gain = 70, which can be invested to become after one year. 70( ) = A European put expiring in one year with strike 80 will payout only max{80 S T, 0} 80 after one year. 205

214 Put Prices for Assets Paying No Income Result For an asset paying no income, for all t T. max{0, K S t } P K (t, T) K, Proof The lower bound is easy. If we exercise the American put at t, we receive value K S t, so the price P K (t, T) must be at least this much. (Easy Exercise: Prove this with an arbitrage argument.) 206

215 Put Prices for Assets Paying No Income To prove the upper bound P K (t, T) K, we assume P K (t, T) > K and deduce a contradiction. Consider portfolio C with the following strategy. At time t, the portfolio is empty. Write and sell one American put. So the portfolio has 1 American put and P K (t, T) cash. It is not necessary to invest the cash, we can just hold it. Case 1. The American put expires without being exercised. Then V C (T) = 0 + P K (t, T) P K (t, T) K, which is strictly positive by assumption. 207

216 Put Prices For Assets Paying No Income Case 2. The American put is exercised at time T 0 with t T 0 T. To fulfill our obligation to the holder of the put, we pay K and receive one unit of the asset at T 0. The portfolio is now P K (t, T) K non-invested cash and one unit of the asset. Nothing else happens until time T. Then V C (T) = P K (t, T) K + S T, which is strictly positive by assumption. Combining both cases, V C (T) > 0 with probability one and V C (t) = 0. This contradicts the no-arbitrage assumption. 208

217 Call and Put Prices for Stocks Paying Known Dividend Yield Result For a stock paying dividends at continuous yield q with automatic reinvestment, S t e q(t t) KZ(t, T) C K (t, T) S t e q(t t) KZ(t, T) S t e q(t t) P K (t, T) KZ(t, T) S t K C K (t, T) S t K S t P K (t, T) K Remark Recall that for a stock paying no income, American and European calls have the same value. For a stock paying dividends, an American call is often worth more than a European call. This is because, by exercising early, we may receive dividends from the stock which we would not have received otherwise. 209

218 Call and Put Prices for Stocks Paying Known Dividend Yield Example Consider a stock paying dividends at continuous yield q = 5% with automatic reinvestment. The current price of the stock is S t = 10. The one-year continuously compounded interest rate is r = 5%. 210

219 Example Consider an American call expiring in one year with strike K = 20. Exercising now, we get the stock and a debt of K = 20. If the stock price in one year is S T = 30, then we will have gained after one year. S T e q(t t) Ke r(t t) = 30e e A European call expiring in one year with strike K = 20 would payout only S T K = = 10 after one year. 211

220 Call and Put Spreads Definition A (K 1, K 2 ) call spread is a portfolio consisting of long one call option with strike K 1 and short one call option with strike K 2, both with maturity T, where K 1 < K 2. At time t, the call spread has value C K1 (t, T) C K2 (t, T). At time T, it payout equals 0, if S T K 1 (neither exercised) S T K 1, if K 1 S T K 2 (only K 1 call exercised) K 2 K 1, if S T K 2 (both exercised) 212

221 Call and Put Spreads Graph of the payout of a call as a function of the price of underlying asset. Payout of a call option with K =

222 Call and Put Spreads Graph of the payout of a put as a function of the price of underlying asset. Payout of a put option with K =

223 Call and Put Spreads Graph of the payout of a call spread as a function of the price of underlying asset. Payout of a call spread with K 1 = 80, K 2 =

224 Call and Put Spreads Result If K 1 < K 2, then the value of a call spread at t satisfies 0 C K2 (t, T) C K1 (t, T) (K 2 K 1 )Z(t, T). Proof Recall that the payout at T is 0, if S T K 1 (neither exercised) S T K 1, if K 1 S T K 2 (only K 1 call exercised) K 2 K 1, if S T K 2 (both exercised) which is always nonegative. Therefore, by the monotonicity theorem, the value at t is also non-negative, and C K1 (t, T) C K2 (t, T). 216

225 Call and Put Spreads Now we prove the second inequality: C K2 (t, T) C K1 (t, T) (K 2 K 1 )Z(t, T). Note that the payout at T is at most K 2 K 1 (Why?), whose present value at t is (K 2 K 1 )Z(t, T). Then the inequality follows from the monotonicity theorem. 217

226 Call and Put Spreads Definition A (K 1, K 2 ) put spread is a portfolio consisting of short one put option with strike K 1 and long one put option with strike K 2, both with maturity T, where K 1 < K

227 Call and Put Spreads Result For a (K 1, K 2 ) put spread: The value at t T is P K2 (t, T) P K1 (t, T). The payout at maturity T is K 2 K 1, if S T K 1 (both exercised) K 2 S T, if K 1 S T K 2 (only K 2 put exercised) 0, if S T K 2 (neither exercised) The value at t satisfies 0 P K2 (t, T) P K1 (t, T) (K 2 K 1 )Z(t, T). Question What is the payout profile of a put spread? 219

228 Call and Put Spreads The previous two results imply Result If K 1 K 2, C K1 (t, T) C K2 (t, T) and P K1 (t, T) P K2 (t, T). In words, the price of a European call is a decreasing function of the strike K, and the price of a European put is an increasing function of the strike K. 220

229 Butterflies and Convexity of Option Prices Fix an asset. Let S t be its price at t. Let K 1 < K < K 2. Then there is a unique λ (0, 1) such that The portfolio consisting of K = λk 1 + (1 λ)k 2. +2λ calls with strike K 1, 2 calls with strike K, +2(1 λ) calls with strike K 2, all with the same maturity, is called a (K 1, K, K 2 ) call butterfly. If λ 1/2, it is called an asymmetric call butterfly. If λ = 1/2, it is called an symmetric call butterfly. 221

230 Butterflies and Convexity of Option Prices For a symmetric call butterfly, the portfolio is +1 call with strike K 1, 2 calls with strike K = 1 2 (K 1 + K 2 ), +1 call with strike K 2. Result For a (K 1, K, K 2 ) symmetric call butterfly, at maturity T the payout is 0 if S T K 1, S T K 1 if K 1 S T K, K 2 S T if K S T K 2, 0 if S T K 2. We will give a more general result later. 222

231 Butterflies and Convexity of Option Prices Payout of a (90, 105, 120) call butterfly. 223

232 Butterflies and Option Prices Result For a (K 1, K, K 2 ) asymmetric or symmetric call butterfly: The value at t T is 2λC K1 (t, T) + 2(1 λ)c K2 (t, T) 2C K (t, T). The payout at maturity T is 0 if S T K 1 2λ(S T K 1 ) if K 1 S T K 2(1 λ)(k 2 S T ) if K S T K 2 0 if S T K 2 The payout at T is 0, so (by the monotonicity theorem) the value at t is also

233 Butterflies and Option Prices Proof The first and third points are clear. For the second point, the cases S T K 1 and K 1 S T K are clear. If K S T K 2, the payout is 2λ(S T K 1 ) 2(S T K ) = 2λ(S T K 1 ) 2(S T λk 1 (1 λ)k 2 ) = 2(1 λ)(k 2 S T ). If S T K 2, the payout is 2λ(S T K 1 ) 2(S T K ) + 2(1 λ)(s T K 1 ) =

234 Butterflies and Option Prices The third point of the previous reslut implies Result If K 1 < K < K 2, then where C K (t, T) λc K1 (t, T) + (1 λ)c K2 (t, T) K = λk 2 + (1 λ)k 2. Geometrically, this means that the graph of C K (t, T) versus K is always below the secant line between any two points on the graph. In other words, C K (t, T) is a concave up (convex) function of K. We also recall that C K (t, T) is a decreasing function of K. Question What does the graph of C K (t, T) versus K look like? 226

235 Digital Options Fix an asset. Let S t be its price at t. Definition A digital call option with strike K and maturity T has payout at T 1, if S T K 0, if S T < K. A digital put option with strike K and maturity T has payout at T 1, if S T K 0, if S T > K. 227

236 Digital Options Digital options are also called binary options or all-or-nothing options. 228

237 Digital Options Result The price/value at time t of a digital call option with strike K and maturity T is which is equal to C K h (t, T) C K (t, T) lim, h 0 + h K C K(t, T) when the European call price C K (t, T) is differentiable with respect to K. Notation means the partial derivative with respect to K (by K freezing other variables). 229

238 Digital Options Proof For h > 0, consider a family of portfolios: long 1/h calls with strike K h, short 1/h calls with strike K with maturity T. Consider another portfolio: B: a digital call with strike K and maturity T. The value of A(h) at t is V A(h) (t) = C K h(t, T) C K (t, T). h 230

239 Digital Options If S T K, then all the calls in A(h) are exercised and for all h > 0. V A(h) (T) = 1 h (S T (K h) + K S T ) = 1 If S T < K, then S T < K h for all small enough h > 0, so nothing in A(h) is exercised and V A(h) (T) = 0. Therefore V A(h) (T) = { 1 if S T K 0 if S T < K } = V B (T) where portfolio B is a digital call with strike K and maturity T. 231

240 Digital Options Then with probability one. lim h 0+ VA(h) (T) = V B (T), By a no-arbitrage argument (non trivial), we have that C K h (t, T) C K (t, T) lim h 0 VA(h) (t) = lim = V B (t). + h 0 + h 232

241 Digital Options If C K (t, T) is differentiable with respect to K, then K C C K+h (t, T) C K (t, T) K(t, T) = lim h 0 h C K+( h) (t, T) C K (t, T) = lim h 0 h C K h (t, T) C K (t, T) = lim h 0 + h = V B (t), where V B (t) is the value at t of a digital call with strike K and maturity T. 233

242 Probability Theory: Conditioning

243 Conditional Probability Fix a sample space Ω and probability measure P. Let A and B be events (i.e., subsets of Ω). The conditional probability of A given B is P(A B) = P(A B). P(B) It is undefined if P(B) = 0. Out of the probability assigned to the outcomes in B, P(A B) is the fraction assigned to those outcomes that are in both A and B. 234

244 Conditional Probability Interpretation: P(A B) is the probability that event A occurs assuming event B occurs. Ω A B 235

245 Conditional Probability Experiment: Flip a fair coin three times. Sample space: Ω = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} Probability Measure: P({ω}) = 1 8, for all ω Ω X = number of heads Y = 1 if the first flip heads, 0 otherwise Z = 1 if second flip heads, 0 otherwise Find P(X = 2), P(X = 2 Y = 1), and P(X = 2 Y = 1, Z = 0). Notation: {Y = 1, Z = 0} = {Y = 1 and Z = 0}. 236

246 Conditional Probability P(X = 2) = P({HHT, HTH, THH}) = 3 8 P(X = 2 Y = 1) = P(X = 2, Y = 1) P(Y = 1) {Y = 1} = {HHH, HHT, HTH, HTT}, and {X = 2, Y = 1} = {HHT, HTH} P(X = 2 Y = 1) = P(X = 2, Y = 1) P(Y = 1) = 2/8 4/8 =

247 Conditional Probability P(X = 2 Y = 1, Z = 0) = P(X = 2, Y = 1, Z = 0), P(Y = 1, Z = 0) {Y = 1, Z = 0} = {HTH, HTT}, and {X = 2, Y = 1, Z = 0} = {HHT}, P(X = 2 Y = 1, Z = 0) = P(X = 2, Y = 1, Z = 0) P(Y = 1, Z = 0) = 1/8 2/8 =

248 Conditional Expectation Let X and Y be discrete random variables. Recall the expected value of X is E(X) = where R(X) is the range of X. x R(X) xp(x = x), It is a weighted average of the possible outputs of X, with the weights being the probability of each output. Terminology: expectation = expected value = average value = mean = first moment 239

249 Conditional Expectation The conditional expectation of X given Y = y is E(X Y = y) = xp(x = x Y = y). x R(X) Interpretation: E(X Y = y) is the expected value of X assuming that Y = y occurs. 240

250 Conditional Expectation Example Roll two fair six-sided dice. X: sum of the dice Y: number on first die Z: number on second die E(X) = E(Y + Z) = E(Y) + E(Z) = = 7 E(X Y = 6) = E(Y + Z Y = 6) = =

251 Conditional Expectation as Random Variable Let X and Y be discrete random variables. Recall that if g : R R is any function, then g(y) is a random variable. It is defined by for all ω Ω. Example g(y)(ω) = g(y(ω)) If g(x) = (x K) + and S T is the price of an asset at a future time T, then g(s T ) = (S T K) + is a random variable representing the payout of a call option. 242

252 Conditional Expectation as Random Variable If we define h by h(y) = E(X Y = y), the conditional expectation of X given Y is the random variable E(X Y) = h(y). In other words, if y ω stands for the real number Y(ω), then for all ω Ω. E(X Y)(ω) = E(X Y = y ω ) Remark: E(X Y = y) is a real number and E(X Y) is a random variable. 243

253 Properties of Conditional Expectation Let X, Y, Z be random variables and let c R. Linear Properties: E(c Z = z) = c and E(c Z) = c E(cX Z = z) = ce(x Z = z) and E(cX Z) = ce(x Z) E(X + Y Z = z) = E(X Z = z) + E(Y Z = z) E(X + Y Z) = E(X Z) + E(Y Z) and 244

254 Properties of Conditional Expectation When conditioning on Y = y, replace Y by y in the expectation: for any function g(x, y). E(g(X, Y) Y = y) = E(g(X, y) Y = y) Conditioning with respect to Y means that Y, hence g(y), should be interpreted as known or constant, so that g(y) can be moved outside the expectation: for any function g(y). E(Xg(Y) Y) = g(y)e(x Y) 245

255 Properties of Conditional Expectation We conclude with the tower law or the law of iterated expectations. Basic form: E(E(X Y)) = E(X) and E(E(X) Y) = E(X). General Form: E(E(X Y, Z) Z) = E(X Z) and E(E(X Z) Y, Z) = E(X Z). 246

256 Independence Discrete random variables X and Y are called independent if P(X = x, Y = y) = P(X = x)p(y = y) for all x, y R. Result: The following statements are equivalent: X and Y are independent P(X = x, Y = y) = P(X = x)p(y = y) P(X = x Y = y) = P(X = x) P(Y = y X = x) = P(Y = y) for all x, y R for all x, y R for all x, y R Intuitively, X and Y are independent if knowing one of them gives no information about the other. 247

257 Independence Roll two fair six-sided dice. X = sum of the dice, Y = number on first die, Z = number on second die. Y and Z are independent. Indeed, it s intuitively clear (and we can easily check) that P(Y = y Z = z) = P(Y = y) for all y, z R. X and Y are not independent because (for example) P(X = 2 Y = 6) = 0 1 = P(X = 2)

258 Independence Random variables X 1,, X n are independent if P(X 1 = x 1,, X n = x n ) = P(X 1 = x 1 ) P(X n = x n ) for all x 1,, x n. An infinite sequence of random variables X 1, X 2, is independent if any finite sub-collection is independent. If X 1, X 2,, are independent, then we have, for example, P(X 1 = x 1 X 2 = x 2 ) = P(X 1 = x 1 ) P(X 1 = x 1 X 2 = x 2,, X n = x n ) = P(X 1 = x 1 ) P(X 1 = x 1 X X n = y) = P(X 1 = x 1 ). 249

259 Fundamental Theorem of Asset Pricing and Binomial Tree

260 Fundamental Theorem of Asset Pricing In the previous chapter, we were only able to derive upper and lower bounds for option prices. Finding exact formulas for option prices (and other derivatives) requires additional theory. We develop that theory in this and the following chapters. 250

261 Fundamental Theorem of Asset Pricing Recall that the prices of assets at future times are random variables. These random variables are defined on a sample space Ω, which we have mostly ignored. Until now, we have mostly ignored the sample space Ω on which these random variables are defined and the probability measure P defined on Ω. Not anymore! Remember that Ω is the set of all possible outcomes or all possible states of the world. And P is a function that assigns a number 0 P(A) 1 to every event (set of outcomes). 251

262 Fundamental Theorem of Asset Pricing Recall that a portfolio A is an arbitrage portfolio if the following conditions are satisfied: At current time t, V A (t) 0. At some future time T, P(V A (T) 0) = 1 P(V A (T) > 0) > 0 Notice how the definition of an arbitrage portfolio depends on the probability measure P. 252

263 Fundamental Theorem of Asset Pricing Consider an asset. It may be a stock, a derivative, or something else. We want to find its price/value D t at time current t. Replication Method: Let A be the portfolio containing the asset at time t. If we can find a portfolio B whose value at t is known and which has the same value as A at some future time T, then the no-arbitrage principle implies (via the replication theorem) that A and B have the same value at t. Hence D t = V B (t). Problem: If the value D T of the asset at T is even slightly complicated, it may be difficult to find a suitable portfolio B. This difficulty occurs when the asset is a European call option on some stock and D T = max {S T K, 0}. 253

264 Fundamental Theorem of Asset Pricing Naive Discount Method: Suppose the random variable D T has a known distribution. This means P(D T = k) is known for all k. Without using the no-arbitrage principle, a natural approach to find D t is: 1. Compute the expected value of the asset at T: E(D T ) = kp(d T = k), k R(D T ) 2. Discount the expected value back to today: Z(t, T)E(D T ), 3. Use the discounted expected value as the current asset price D t = Z(t, T)E(D T ). (7) 254

265 Fundamental Theorem of Asset Pricing Problem I: The pricing formula D t = Z(t, T)E(D T ) ignores any income or payout at times other than T. For example, consider an asset that pays 1 at time (T + t)/2 and 1 at time T. The value of the asset at T is D T = 1, so the formula would say D t = Z(t, T). But the correct price at t is clearly D t = Z(t, (T + t)/2) + Z(t, T). 255

266 Fundamental Theorem of Asset Pricing As another example, consider European and American put options with strike K and maturity T on a stock with price S T at t. They have the same price at T: P K (T, T) = P K (T, T) = max {K S T, 0}. So (7) would assign the same price at t: P K (T, T) = P K (T, T) = Z(t, T)E(max {K S T, 0}). However, American puts are usually worth strictly more than European puts, since by exercising early we receive the strike price K early and can invest this amount. 256

267 Fundamental Theorem of Asset Pricing Solution I: Restrict to assets that pay nothing except possibly at T. Alternatively, consider portfolios rather than assets because the value of a portfolio accounts for things like dividends, exercises, and trades. For example, if A at t contains the asset paying 1 at (t + T)/2 and 1 at T, then V A (T) D T. In fact, V A (T) > 2 > 1 = D T. If portfolio A at t contains the European put, and portfolio B at t contains the American put, then V A (T) and V B (T) are not equal, in general., and so (7) will assign different prices V A (t) and V B (t). 257

268 Fundamental Theorem and Asset Pricing Problem II: The pricing formula (7) can lead to an arbitrage portfolio, as the next example illustrates. (In fact, this example already appeared as Exercise 3 of HW 8. Example Consider a one-year European call option on a stock with strike 40. Suppose: Current stock price = S t = 35 There are three possible states of the world at maturity T: ω 1, ω 2, ω 3 P({ω 1 }) = 1/2, P({ω 2 }) = 1/3, P({ω 3 }) = 1/6 S T (ω 1 ) = 50, S T (ω 2 ) = 55, S T (ω 3 ) = 30 Z(t, T) =

269 Fundamental Theorem and Asset Pricing (a) Find the discounted present value of the expected payout: Z(t, T)E(S T K) +. Since = 10, with probability 1/2, (S T K) + = = 15, with probability 1/3, 0 with probability 1/6, we have E(S T K) + = = 10. Therefore Z(t, T)E(S T K) + = =

270 Fundamental Theorem and Asset Pricing (b) Suppose we take the call price to be discounted present value of the expected payout: C K (t, T) = Z(t, T)E(max {S T K, 0}) = 9. Does this represent an arbitrage opportunity? If so, build an arbitrage portfolio. Verify it is an arbitrage portfolio. Yes, there is an arbitrage opportunity. Start with portfolio C empty at t. short sell one K = 40 call for 9 cash, borrow additional 26 cash, buy 1 stock for 35 cash. 260

271 Fundamental Theorem and Asset Pricing Portfolio C at t: short one K = 40 call, one stock, 26 cash. V C (t) = = 0. V C (T) = S T (S T K) + 26 Z(t, T) /0.9 = , with probability 1/2, = /0.9 = , with probability 1/3, /0.9 = , with probability 1/6. Therefore V C (T) > 0 with probability one. So C is an arbitrage portfolio. 261

272 Fundamental Theorem and Asset Pricing The pricing formula D t = Z(t, T)E(D T ) can lead to an arbitrage portfolio. However, by replacing the probability measure P by another probability measure P (i.e., by adjusting how we assign probabilities), this pricing method can be made compatible with the no-arbitrage principle. In fact, the no-arbitrage principle is equivalent to the existence of a P for which this pricing method works. 262

273 Fundamental Theorem and Asset Pricing Result Fundamental Theorem of Asset Pricing: There are no arbitrage portfolios if and only if there is a probability measure P equivalent to P such that for every T-payout asset we have D t = Z(t, T)E (D T ), (8) where D t and D T are the asset prices at current time t and future time T. Here E means expectation with respect to P : E (D T ) = k R(D T ) kp (D T = k). 263

274 Fundamental Theorem of Asset Pricing Definitions of Terms and Notation. Probability measures P and P are called equivalent if the events that have probability one according to P are exactly the events that have probability one according to P. In other words, the arbitrage portfolios with respect to P are exactly the arbitrage portfolios with respect to P. An asset is called a T-payout asset if it has no income or payout except possibly at time T. Examples: Stock paying no income; European option with maturity T. Non-examples: Stock paying dividends; American option with maturity T. 264

275 Fundamental Theorem of Asset Pricing Definitions of Terms and Notation - Cont. E means expectation with respect to P. For example, E (X) = kp (X = x), E (X Y = y) = k R(X) k R(X) kp (X = x Y = y). If P is a probability measure satisfying the two conditions of the theorem, then P is called a risk-neutral probability measure (with respect to Z(t, T) and P). The formula (8) is called risk-neutral pricing (with respect to Z(t, T)). 265

276 Fundamental Theorem of Asset Pricing Remark. Technically, the formula D t = Z(t, T)E (D T ) should be D t = Z(t, T)E (D T I t ), where I t stands for all relevant information at t. For a derivative that depends on a stock price, I t = S t = stock price at t. For a derivative that depends on interest rates, I t stands for the interest rates at t. 266

277 Fundamental Theorem of Asset Pricing We can restate the fundamental theorem in terms of portfolios. Result: Fundamental Theorem of Asset Pricing (Portfolios) There are no arbitrage portfolios if and only if there is a probability measure P satisfying P is equivalent to P, for any times t T and for any portfolio A, V A (t) = Z(t, T)E (V A (T)). (9) 267

278 Fundamental Theorem of Asset Pricing We will prove only the easy half of the fundamental theorem: Existence of P implies No-Arbitrage. Proof Assume there is a probability measure P satisfying the two conditions in the theorem. Let A be any portfolio satisfying (9). We want to show A cannot be an arbitrage portfolio. We will assume A is an arbitrage portfolio and deduce a contradiction. So we have V A (t) 0, P(V A (T) 0) = 1, P(V A (T) > 0) >

279 Fundamental Theorem of Asset Pricing Since P is equivalent to P, we must have V A (t) 0, P (V A (T) 0) = 1, P (V A (T) > 0) > 0. Since Z(t, T)E (V A (T)) = V A (t) 0, we must have E (V A (T)) = kp (V A (T) = k) 0. (10) k R(V A (T)) 269

280 Fundamental Theorem of Asset Pricing Since P (V A (T) 0) = 1, we must have P (V A (T) = k) = 0 for all k < 0. Therefore the sum in (10) has only non-negative terms. But since the sum is 0, all the terms must be zero. That is, kp (V A (T) = k) = 0 for all k. Therefore, P (V A (T) = k) = 0 for all k 0. So This contradicts that P (V A (T) = 0) = 1. P (V A (T) > 0) >

281 Fundamental Theorem of Asset Pricing Remark. According to the fundamental theorem, if we can find P, then we can find the price of any asset. We will soon study the binomial tree model and the Black-Scholes model, which are natural models in which we can find P. 271

282 Binomial Tree Consider a stock paying no income with price S t at time t. A binomial tree for the stock is a model of the stock price at discrete times. For simplicity, we assume the times are 0, 1, 2,, n,, so the time step is T = 1. Branch of a Binomial Tree 272

283 Binomial Tree It has parameters r, u, d, and p, where r > 0, d < u, and 0 < p < 1. r = constant annually compounded interest rate. u = percentage of stock price going up d = percentage of stock price going down p = the probability that stock price goes up 273

284 Binomial Tree At time 0 the stock price S 0 is known. At time n, the stock price goes from S n 1 up to S n = (1 + u)s n 1 with probability p, or down to S n = (1 + d)s n 1 with probability (1 p). Whether the price goes up or down at time n is independent of whether the price goes up or down at any other time. 274

285 Binomial Tree One-Step Binomial Tree with d = 0.1 and u =

286 Binomial Tre Two-Step Binomial Tree with d = 0.1 and u =

287 Binomial Tree Here is an equivalent way to define how the price changes at each time: S n = ξ n S n 1 for n = 1, 2,..., (11) where ξ 1, ξ 2,..., is a sequence of independent random variables with { (1 + u) with probability p ξ i = (1 + d) with probability 1 p By applying (11) repeatedly, we see S n = ξ n S n 1 = ξ n ξ n 1 S n 2 =... = ξ n ξ 1 S

288 Binomial Tree Example From time 0 to time n, the stock price changes n times. If S n = (1 + u) k (1 + d) n k S 0, it means the stock price moves up exactly k times out of the n possible times. If n = 5 and k = 3, a typical path with exactly k up movements is up-down-up-up-down. 278

289 Binomial Tree There are ( ) n = k n! k!(n k)! possible paths with exactly k up movements: Out of the n changes, choose k of them to be up and the rest down. Each such path has probability p k (1 p) n k. Therefore for 0 k n. P(S n = (1 + u) k (1 + d) n k S 0 ) = ( ) n p k (1 p) n k k 279

290 Arbitrage-Free Binomial Tree Consider a binomial tree (parameters r,u,d and p are given) for a stock paying no income. Example Suppose the binomial tree has no arbitrage portfolios. By the fundamental theorem of asset pricing, there is a risk-neutral probability measure P such that S 0 = Z(0, 1)E (S 1 ). Question What is this P? 280

291 Arbitrage-Free Binomial Tree Let p denote the probability of an up movement in the stock price with respect to P. We have S 0 = Z(t, T)E (S 1 ) = (1 + r) 1 E (S 1 ) = (1 + r) 1 (p (1 + u)s 0 + (1 p )(1 + d)s 0 ) = (1 + r) 1 (p (1 + u) + (1 p )(1 + d)) S 0. Solving for p gives p = r d u d. 281

292 Arbitrage-Free Binomial Tree The previous example proves Result If the binomial tree has no arbitrage portfolios, then p = r d u d, where p is the probability of an up movement in the stock price with respect to the risk-neutral probability measure P. 282

293 Arbitrage-Free Binomial Tree Question How to compute E(S 2 )? By the tower law of expectation, we have E(S 2 ) = E(E(S 2 S 1 )). Question What is E(S 2 S 1 )? E(S 2 S 1 ) = p(1 + u)s 1 + (1 p)(1 + d)s 1 = (p(1 + u) + (1 p)(1 + d))s

294 Arbitrage-Free Binomial Tree Therefore, E(S 2 ) = E(E(S 2 S 1 )) = E((p(1 + u) + (1 p)(1 + d))s 1 ) = (p(1 + u) + (1 p)(1 + d))e(s 1 ) = (p(1 + u) + (1 p)(1 + d)) 2 S 0. In general, for any 0 m n. E(S n S m ) = (p(1 + u) + (1 p)(1 + d)) n m S m 284

295 Arbitrage-Free Pricing on Binomial Tree Consider a binomial tree (parameters r,u,d and p are given) for a stock paying no income. Example Find the price at 0 of European call with maturity T = n and strike K. 285

296 Arbitrage-Free Pricing on Binomial Tree We have C K (T, T) = max {S T K, 0} = max {S n K, 0} = g(s n ). and Z(0, T) = Z(0, n) = (1 + r) n. By the fundamental theorem of asset pricing, C K (0, T) = Z(0, T)E (C K (T, T)) = (1 + r) n E (g(s n )). 286

297 Arbitrage-Free Pricing on Binomial Tree By LOTUS, C K (0, T) = (1 + r) n g(s)p (S n = s). s R(S n) The possible value of S n are (1 + u) k (1 + d) n k S 0, for k = 0, 1,, n. Hence n C K (0, T) = (1 + r) n g((1 + u) k (1 + d) n k S 0 ) k=0 P (S n = (1 + u) k (1 + d) n k S 0 ). 287

298 Arbitrage-Free Pricing on Binomial Tree By the previous result, p = r d u d is the probability of an up movement in the stock price with respect to the risk-neutral probability measure P. Therefore, Hence P (S n = (1 + u) k (1 + d) n k S 0 ) = C K (0, T) = (1+r) n k=0 ( ) n p k (1 p ) n k. k n ( ) n g((1+u) k (1+d) n k S 0 ) p k (1 p ) n k. k 288

299 Arbitrage-Free Pricing on Binomial Tree Since we have g(x) = max {x K, 0}, and p = r d u d, C K (0, T) = (1 + r) n ( n k n k=0 ) ( r d u d { } max (1 + u) k (1 + d) n k S 0 K, 0 ) k ( ) u r n k. u d It s complicated, but it s exact. 289

300 Arbitrage-Free Pricing on Binomial Tree With minor changes in notation, the previous example proves Result The price at 0 of a derivative of the stock with payout g(s T ) at T = n (and no payout otherwise) is where D(0, T) = Z(0, T)E (g(s T )) n = (1 + r) n g((1 + u) k (1 + d) n k S 0 ) k=0 ( ) n p k (1 p ) n k, k p = r d u d. 290

301 Arbitrage-Free Pricing on Binomial Tree Example The constant annually compounded interest rate is 10% At current time 0, a stock paying no income has price 50. Suppose that at each time point, the stock price can go up by 30% or down by 20%. Find the price at 0 of a European put with strike 47 and maturity

302 Arbitrage-Free Pricing on Binomial Tree Example The constant annually compounded interest rate is 10% At current time 0, a stock paying no income has price 50. Suppose that at each time point, the stock price can go up by 30% or down by 20%. Find the price at 0 of a European put with strike 47 and maturity 2. Previously, we use the exact formula to compute the price, but next we will find the price backward in time. 292

303 Arbitrage-Free Pricing on Binomial Tree Have K = 47, T = n = 2, u = 0.3, d = 0.2, r = 0.1, and p = r d u d =

304 Arbitrage-Free Pricing on Binomial Tree u = 0.3, d = (1 + u)50 = 65 S 0 = (1 + d)50 =

305 Arbitrage-Free Pricing on Binomial Tree D B P K (0, 2) =? E C F D: P K (2, 2) = max{k S 2, 0} = max{ , 0} = 0; E: P K (2, 2) = max{k S 2, 0} = max{47 52, 0} = 0; F: P K (2, 2) = max{k S 2, 0} = max{47 32, 0} =

306 Arbitrage-Free Pricing on Binomial Tree 0 B P K (0, 2) =? 0 C 15 B: P K (1, 2) = Z(1, 2)E (P K (2, 2) B) = (1 + r) 1 (p 0 + (1 p ) 0) = 0; C: P K (1, 2) = Z(1, 2)E (P K (2, 2) C) = (1 + r) 1 (p 0 + (1 p ) 15) =

307 Arbitrage-Free Pricing on Binomial Tree 0 0 P K (0, 2) =? P K (0, 2) = Z(0, 1)E (P K (1, 2)) = (1 + r) 1 (p 0 + (1 p )5.45) =

308 Arbitrage-Free Pricing on Binomial Tree Now let s do the example again, but with an American put. Example The constant annually compounded interest rate is 10% At current time 0, a stock paying no income has price 50. Suppose that at each time point, the stock price can go up by 30% or down by 20%. Find the price at 0 of an American put with strike 47 and maturity 2. We use the same idea to find the price backward in time. Key: To test at each node to see whether early exercise is optimal. 298

309 Arbitrage-Free Pricing on Binomial Tree u = 0.3, d = (1 + u)50 = 65 S 0 = (1 + d)50 =

310 Arbitrage-Free Pricing on Binomial Tree D B P K (0, 2) =? E C F D: P K (2, 2) = max{k S 2, 0} = max{ , 0} = 0; E: P K (2, 2) = max{k S 2, 0} = max{47 52, 0} = 0; F: P K (2, 2) = max{k S 2, 0} = max{47 32, 0} =

311 Arbitrage-Free Pricing on Binomial Tree 0 B P K (0, 2) =? 0 C 15 At time t = 1, the owner of the put decides between exercising the put or holding it until time t = 2, given on the information of time t = 1. The owner will choose the one with greater value. 301

312 Arbitrage-Free Pricing on Binomial Tree Suppose we are on the state B, i.e. S 1 = 65. If we hold the put till t = 2, then we can use the risk neutral valuation formula to obtain P K (1, 2) = Z(1, 2)E ( P K (2, 2) B) = (1 + r) 1 (p 0 + (1 p ) 0) = 0. If we choose to exercise the put, then the value is max{47 65, 0} = 0. Therefore, on the state B, the value of the put is max{0, 0} =

313 Arbitrage-Free Pricing on Binomial Tree Suppose we are on the state B, i.e., S 1 = P K (0, 2) =? 0 C

314 Arbitrage-Free Pricing on Binomial Tree Suppose we are on the state C, i.e., S 1 = 40. If we hold the put till t = 2, then we can use the risk neutral valuation formula to obtain P K (1, 2) = Z(1, 2)E ( P K (2, 2) C) = (1 + r) 1 (p 0 + (1 p ) 15) = If we choose to exercise the put, then the value is max{47 40, 0} = 7. Therefore, on the state C, the value of the put is max{5.45, 7} =

315 Arbitrage-Free Pricing on Binomial Tree 0 0 P K (0, 2) =? At time t = 0, we do the same trick as before. 305

316 Arbitrage-Free Pricing on Binomial Tree If we hold the put (may exercise at t = 1 or t = 2), then the value is ( ) 1 ( ) = If we choose to exercise the put, then the value is max{47 50, 0} = 0. Therefore, P K (0, 2) = Remark These examples above show that an American put can have price strictly greater than a European put with the same strike and maturity. 306

317 Arbitrage-Free Pricing on Binomial Tree Example The constant annually compounded interest rate is 10% At current time 0, a stock paying no income has price 50. Suppose that at each time point, the stock price can go up by 30% or down by 20%. Find the price at 0 of a European put with strike 47 and maturity 2. Previously, we use the exact formula to compute the price, but next we will find the price backward in time. 307

318 Arbitrage-Free Pricing on Binomial Tree Have K = 47, T = n = 2, u = 0.3, d = 0.2, r = 0.1, and p = r d u d =

319 Arbitrage-Free Pricing on Binomial Tree u = 0.3, d = (1 + u)50 = 65 S 0 = (1 + d)50 =

320 Arbitrage-Free Pricing on Binomial Tree D B P K (0, 2) =? E C F D: P K (2, 2) = max{k S 2, 0} = max{ , 0} = 0; E: P K (2, 2) = max{k S 2, 0} = max{47 52, 0} = 0; F: P K (2, 2) = max{k S 2, 0} = max{47 32, 0} =

321 Arbitrage-Free Pricing on Binomial Tree 0 B P K (0, 2) =? 0 C 15 B: P K (1, 2) = Z(1, 2)E (P K (2, 2) B) = (1 + r) 1 (p 0 + (1 p ) 0) = 0; C: P K (1, 2) = Z(1, 2)E (P K (2, 2) C) = (1 + r) 1 (p 0 + (1 p ) 15) =

322 Arbitrage-Free Pricing on Binomial Tree 0 0 P K (0, 2) =? P K (0, 2) = Z(0, 1)E (P K (1, 2)) = (1 + r) 1 (p 0 + (1 p )5.45) =

323 Arbitrage-Free Pricing on Binomial Tree Now let s do the example again, but with an American put. Example The constant annually compounded interest rate is 10% At current time 0, a stock paying no income has price 50. Suppose that at each time point, the stock price can go up by 30% or down by 20%. Find the price at 0 of an American put with strike 47 and maturity 2. We use the same idea to find the price backward in time. Key: To test at each node to see whether early exercise is optimal. 313

324 Arbitrage-Free Pricing on Binomial Tree u = 0.3, d = (1 + u)50 = 65 S 0 = (1 + d)50 =

325 Arbitrage-Free Pricing on Binomial Tree D B P K (0, 2) =? E C F D: P K (2, 2) = max{k S 2, 0} = max{ , 0} = 0; E: P K (2, 2) = max{k S 2, 0} = max{47 52, 0} = 0; F: P K (2, 2) = max{k S 2, 0} = max{47 32, 0} =

326 Arbitrage-Free Pricing on Binomial Tree 0 B P K (0, 2) =? 0 C 15 At time t = 1, the owner of the put decides between exercising the put or holding it until time t = 2, given on the information of time t = 1. The owner will choose the one with greater value. 316

327 Arbitrage-Free Pricing on Binomial Tree Suppose we are on the state B, i.e. S 1 = 65. If we hold the put till t = 2, then we can use the risk neutral valuation formula to obtain P K (1, 2) = Z(1, 2)E ( P K (2, 2) B) = (1 + r) 1 (p 0 + (1 p ) 0) = 0. If we choose to exercise the put, then the value is max{47 65, 0} = 0. Therefore, on the state B, the value of the put is max{0, 0} =

328 Arbitrage-Free Pricing on Binomial Tree Suppose we are on the state B, i.e., S 1 = P K (0, 2) =? 0 C

329 Arbitrage-Free Pricing on Binomial Tree Suppose we are on the state C, i.e., S 1 = 40. If we hold the put till t = 2, then we can use the risk neutral valuation formula to obtain P K (1, 2) = Z(1, 2)E ( P K (2, 2) C) = (1 + r) 1 (p 0 + (1 p ) 15) = If we choose to exercise the put, then the value is max{47 40, 0} = 7. Therefore, on the state C, the value of the put is max{5.45, 7} =

330 Arbitrage-Free Pricing on Binomial Tree 0 0 P K (0, 2) =? At time t = 0, we do the same trick as before. 320

331 Arbitrage-Free Pricing on Binomial Tree If we hold the put (may exercise at t = 1 or t = 2), then the value is ( ) 1 ( ) = If we choose to exercise the put, then the value is max{47 50, 0} = 0. Therefore, P K (0, 2) = Remark These examples above show that an American put can have price strictly greater than a European put with the same strike and maturity. 321

332 Normal Distribution and Central Limit Theorem

333 Normal Distribution Let X be a random variable. If there are constants µ R and σ > 0 such that P(a X b) = b for all real numbers a b, then we write a 1 2πσ 2 e (x µ)2 /2σ 2 dx X N (µ, σ 2 ) and we say that X has normal distribution (with respect to P). Note that X is non-discrete with R(X) = R. 322

334 Normal Distribution Suppose X N (µ, σ 2 ). Then: For any function g(x), the expectation of g(x) is E(g(X)) = The expectation of X is E(X) = µ. The variance of X is 1 2πσ 2 e (x µ)2 /2σ 2 g(x)dx Var(X) = E((X µ) 2 ) = E(X 2 ) µ 2 = σ 2. If c is a constant, then c + X N (c + µ, σ 2 ). 323

335 Standard Normal Distribution If X N (0, 1), we say that X has standard normal distribution and the function Φ(t) = P(X t) = t 1 2π e x2 /2 dx is called the standard normal cumulative distribution function or standard normal cdf. 324

336 Central Limit Theorem The central limit theorem is the reason the normal distribution is so important. Basically, it says that any sum of many independent random effects is approximately normally distributed. 325

337 Central Limit Theorem Central Limit Theorem Suppose X 1, X 2, is a sequence of independent random variables all having the same distribution. Suppose Then EX 1 = µ and VarX 1 = σ 2. 1 n n (X i µ) N (0, σ 2 ), i=1 which is an abbreviation for lim P(a 1 n n n (X i µ) b) = i=1 b a 1 2πσ 2 e x2 /2σ 2 dx. 326

338 Central Limit Theorem Central Limit Theorem-triangular arrays Suppose that for each fixed value of n, we have a sequence of n random variables X (n) 1, X(n) 2,, X(n) n that are independent and all have the same distribution. Suppose Then n n lim E( X i (n)) = µ and lim Var( X i (n)) = σ 2. n n i=1 n X i (n) N (µ, σ 2 ), i=1 which is an abbreviation for n lim P(a X i (n) b) = n i=1 b a i=1 1 2πσ 2 e (x µ)2 /2σ 2 dx. 327

339 Continuous-Time Limit and Black-Scholes Formula

340 Black-Scholes Model Black-Scholes model is a mathematical model, from which one can deduce the Black Scholes formula, giving a theoretical estimate of the price of European-style options. Fischer Black ( ) Myron Scholes (1941-) Robert Merton (1944-) 328