Introduction to Finance

Transcription

1 Introduction to Finance Cours de L2 DEMI2E donné en par Eric Séré Département MIDO, Université Paris-Dauphine Notes prises en cours et mises en forme par Robin Michard

2 Contents Financial markets and instruments 2 2 Interest rates 3 2. Simple interest rate Compound interest Short compounding periods Some useful summation formulas Loans 6 3. Notations Repayment methods Repayment of capital in fine Constant amortization Constant annuity Investment choice 2 4. Present values, future values Present value of a sequence of cash flows The Internal Rate of Return (IRR) Bonds 4 5. Introduction The arbitrage price of a bond Pricing of a bond using zero-coupon bonds Derivatives Futures and forwards Options The binomial model with one time step Multistep binomial model:

3 Chapter Financial markets and instruments See slides 2

4 Chapter 2 Interest rates Note that in this course, all amounts of money and values of assets are expressed in the same unit or currency, for instance the Euro. A unit of time also has to be chosen: unless otherwise stated, it will be a year. Interest is an amount I charged by a lender to a borrower for the use of an amount of money S(0), called the principal, over a period of time T. The amount I depends on the principal and on an interest rate. From the point of view of the lender, the amount of moneygrowswithtime. Indeed,thelenderinvestsS(0)attime0andownsS(T) = S(0)+I at time T. Let S(t) be the value of the lender s investment at time 0 t T. This value is called the accumulated or future value of S(0) at time t. The quotient a(t) = S(t)/S(0) is called accumulation factor or accumulation function. We always have a(0) =. Usually a is a nondecreasing function of time, however in some exceptional situations the interest rate is negative and a is decreasing: 2. Simple interest rate Simple interest rates correspond to affine accumulation functions: a(t) = +tr, 0 t T. So at time T, the borrower has paid the total interest I = TrS(0), and must repay the principal S(0). The number r is the simple interest rate per unit of time. In practice, simple interest is applied for loans over short periods of time T, usually less than a year. 2.2 Compound interest In the case of simple interest, the interest earned is not invested to earn additional interest. In contrast, in the case of compound interest, the duration T of the loan is divided into equal elementary compounding periods T. At the end of each compounding period, the interest earned is reinvested to earn additional interest. As a result, the accumulation factor grows exponentially with time: after n periods of compounding, the accumulation factor is a(n T) = a n where a is the accumulation factor at the end of the first compounding period. As an example, consider a loan with annual interest rate r, compounded annually over N years (taking one year as unit of time, this corresponds to T = ). Assume that this loan is reimbursed in one payment made at the end of the N-th year. At time 0 the lender invests the principal S = S 0. 3

5 At time the interest on S is I = rs. This interest is reinvested, so the basis for calculation of interest over the next period is S = S 0 +I = (+r)s. At time 2 the interest over the period 2 is I 2 = rs = r(+r)s. This interest is reinvested, so the basis for calculation of interest over the next period is S 2 = S +I 2 = (+r)s = (+r) 2 S. At time N the interest over the period N N is I N = rs N = r(+r) N S. The final cash flow received by the investor is S N = S N +I N = (+r)s N = (+r) N S. 2.3 Short compounding periods Notation S will be the principal. r the nominal annual interest rate. m the number of compounding periods per year (for instance m = 2 for monthly compounding, m = 360 for daily compounding). t the number of years (our unit of time is one year). i = r m the interest rate per compounding period. n = mt the number of compounding periods over t years. S n/m the future value (also called accumulated value) of S, after n periods, corresponding to time n/m (in years). r eff the effective annual interest rate. As in 2.2, we see that S n/m = (+i) n S in other words, the future value after t years is S t = (+ r m )mt S We define r eff by the condition S = (+r eff )S 4

6 This means that after year, the accumulated value is the same as for an investment of the principal S with simple annual interest rate r eff. Then, after t years, the accumulated value will be the same as for an investment of the principal with annual interest rate r eff and annual compounding: S t = (+r eff ) t S. In order to determine r eff we have to solve (+r eff )S = (+ r m )m S +r eff = (+ r m )m r eff = (+ r m )m Continuous compounding is a mathematical concept which consists in letting m + : extremely short compounding periods. The result is S t = (+ r m )mt S m ert S So if we have continuous compounding at the nominal annual rate r, the future value after t years is e rt S. In this case r eff = e r. 2.4 Some useful summation formulas arithmetic series: a n = nα+β a k = α k=0 geometric series: g k = r k g 0 with r. k=0 k +(n+)β k= = α n(n+) 2 k=0 +(n+)β ( r g k = g 0 r k n+) = g 0 r g k = krk g 0 with r. g k = rg 0 k=0 k=0 kr k ( ) = rg 0 d r k dr k=0 ( ) = rg 0 d r n+ dr r ( nr n+ (n+)r n ) + = rg 0 ( r) 2 5

7 Chapter 3 Loans A loan takes place between two agents at time t = 0. A : borrower B : lender To reimburse the loan, A will make a sequence of payments to B separated by equal time intervals. Such a sequence is called an annuity. To simplify our notations, the interval of time between successive payments will coincide with the compounding period, and will be our unit of time. 3. Notations S is the principal amount, also called initial capital: the amount borrowed by A at t = 0. n is the maturity, or duration, of the loan. This means the time delay between the beginning of the loan (time 0) and the last payment made by A (at time n). It will be an integer, since our time unit is the interval between successive payments. r is the interest rate of the loan per unit of time. Since we have decided that the compounding period coincides with the unit of time, r is a nominal rate as well as an effective rate, relative to the unit of time which can be a day, a week, a month, a semester, a year... I k is the amount of interest corresponding to the interval of time [k,k] (usually I k is payed at the end of the period k). A k is the amortization: the part of capital refunded at time k. a k = A k +I k is the annuity payed by A at time k. V k is the outstanding capital at time k: the amount of capital still due at the end of the year k, after payment of the annuity a k. We have V k = V k A k At maturity, after payment of the last annuity, we must have V n = 0. 6

8 But V n = V n A n = V n 2 A n A n = S A A 2... A n So S = i= A i Then with S = V 0 The total cost of the loan is I k = rv k Cost = k= I k 3.2 Repayment methods 3.2. Repayment of capital in fine This means that A k = 0 k {0,,...,n } A n = S Then V k = V 0 = S k {0,,...,n } V n = 0 I = I 2 =... = I n = rs = rv 0 We can summarize this in an amortization table k V k I k A k a k 0 V V 0 rv 0 0 rv k V 0 rv 0 0 rv n V 0 rv 0 0 rv 0 n 0 rv 0 V 0 (+r)v 0 Total nrv 0 V 0 (nr+)v Constant amortization This means that A 0 = 0 A k = A k {,2,...,n} 7

9 So So V 0 = na A = V 0 n V k = V k A = V k 2 2A = V ( 0 ka = k ) V 0 n So I k = rv k = r(v 0 (k )A) ( = r k ) V 0 n a k = A+I ( k ) = n +rn k+ V 0 n +(n k +)r = V 0 n k V k I k A k a k 0 ( V 0 ) V n V0 rv 0 0 n. k. ( k n.. n n n 0 V 0 ). ) V0 r ( k n. 2 n rv 0. ) V0 V 0 n. +nr n V 0. +r(n k+) n V 0 V 0 n +2r n V 0 V 0 n V 0 n +r n V 0 Total Cost V 0 Cost+V 0 Cost = I k = k= k=0 = r n V 0 = r n V 0 ( r k ) V 0 n n k+ k= j j= = r n V 0 n(n+) 2 = r(n+)v

10 3.2.3 Constant annuity We borrow an amount of capital S = V 0 at time 0 at an annual rate r > 0. Repayment will take place at the end of each year and annuity will be constant: Question : Find the value of a. The method of present value a = a 2 =... = a n = a Suppose we borrow S = V 0 and the annuities are a,a 2...a n. At the end of the period n I n = rv n a n = I n +A n So So A n = V n a n = rv n +V n = (+r)v n V n = a n (+r) }{{} discount factor Similarly, at the end of period k+. (k n ) I k+ = rv k a k+ = I k+ +A k+ So A k+ = V k V k+ a k+ = rv k +V k V k+ = (+r)v k V k So So V k = discount factor {}}{ (+r) (a k+ +V k+ ) In particular V k = (+r) a k+ +(+r) V k+ = (+r) a k+ +(+r) ((+r) a k+2 +(+r) V k+2 ) = (+r) a k+ +(+r) 2 a k+2 +(+r) 2 V k+2. = (+r) a k (+r) (n k+) a n +(+r) (n k) a n S = V 0 = (+r) k a k k= The interpretation of this formula is that S must equal the present value of the annuity. The notion of present value will be discussed in more detail in the next chapter. One can even allow r >. 9

11 Application to constant annuity a = a 2 =... = a n = a So So Now k {,...,n} ( ) S = a (+r) k = a +r = a +r k= ( n ( ) ) k +r k=0 ( ) n +r +r = a (+r) n r a = rs (+r) n I k = rv k a = I k +A k So We now compute V k : V k = n k i= a k+i (+r) i = a +r So, replacing a by its expression, A k = a rv k n k j=0 ( ) ( j = a ( ) ) n k +r r +r ) n k ( +r V k = S ( +r The above formulas allow us to compute A k : ) n k+ +( +r A k = a rv k = rs ) n ( So A = rs (+r) n = rs = and, for any k n: ( +r ) n +r ) n k+ ( +r ) n rs (+r) n (+r)k A k = (+r) k A We have proved that (A k ) k,...,n is a geometric sequence of multiplier (+r). 0

12 Finally, we compute the cost of the loan: Cost = I k = (a A k ) = na k= k= k= A k So Cost = nrs ( +r ) n S = S nr ( +r ) n Comment : If we assume that payments are made at the end of each year, then r is both the effective and the nominal annual rate. But for a loan with monthly payments and nominal annual rate R, the monthly interest rate is r = R 2, and the number of annuities is n = 2T where T is the maturity of the loan expressed in years.

13 Chapter 4 Investment choice 4. Present values, future values Let A be an actor on the market. Assume that at time t it receives a cash flow a, with the natural convention a > 0 if A receives money, a < 0 if A pays an amount of money. Consider a date s < t. Assume that we are given an effective interest rate r per unit of time; The present value at time s of the future cash flow a is PV(a) = (+r) s t a The factor ( + r) s t is called a discount factor. This notion already appeared in the preceding chapter, in the context of loans. If s > t then the cash flow at time t has a future value at time s > t, given by FV s (a) = (+r) s t a Thefactor (+r) s t is called anaccumulation factor (notion already definedinthechapter on interest rates). 4.2 Present value of a sequence of cash flows If we have an effective interest rate playing the role of a reference, we can evaluate and compare investments using r. Usually, one takes the effective risk-free interest rate per unit of time available on the market. Note, however, that the existence of an interest rate common to all risk-free investments independently of their duration is not very realistic. Givenr,thenetpresentvalueattime0ofasequenceofcashflowsC = (c t0,c t,...,c tn ) occurring at nonnegative times t 0 <... < t N is NPV 0 r (C) = N i=0 c ti (+r) t i This gives us a criterion for comparison between two investments generating the sequences of cash flows C,C 2. We will prefer C if NPV 0 r (C ) > NPV 0 r (C 2 ). If NPV 0 r (C ) = NPV 0 r (C2 ) we say that C and C 2 are indifferent. We will prefer C 2 if NPV 0 r (C ) < NPV 0 r (C 2 ). Limit of the criterion : 2

14 It does not take risk into account. Maturity should play a role. For instance, if NPV 0 r (C ) = NPV 0 r (C2 ) we should prefer the investment with small maturity N. The interest rate r is not easy to choose. 4.3 The Internal Rate of Return (IRR) Definition : Consider a sequence of cash flows C = (c t0,c t,...,c tn ) occurring at nonnegative times t 0 <... < t N. The Internal Rate of Return (IRR) of C is the interest rate r > solution of NPV 0 r (C) = 0 Note that r = IRR(C) is intrinsic: it does not depend on another interest rate r. Unfortunately it can happen that IRR(C) does not exist, or is not unique. Theorem : Consider a sequence of nonzero cash flows C = (c t0,c t,...,c tn ) occurring at nonnegative times t 0 <... < t N. Assume that c t0,c t,...,c tp have the same sign and that c tp+,...,c tn all have the opposite sign to c t0,c t,...,c tp. Then the IRR of C exists and is unique. Proof : We only treat the case c t0,c t,...,c tp < 0 and c tp+,...,c tn > 0 since the other case follows, replacing C by C. Let τ = tp+t p+ 2. Consider the function F(r) = (+r) τ NPV 0 r (C) = p (+r) τ t i c ti + i=0 N i=p+ c ti (+r) t i τ, r >. Obviously, the possible values of the IRR are the solutions of the equation F(r) = 0. Now, the fonction F is well defined and continuous on ],+ [. Moreover r > (+r) τ t i c ti is decreasing for 0 i p. r > ct i (+r) t i τ is decreasing for p+ i N. As a consequence, F is decreasing and continuous on ], + [. Moreover lim r r> lim r + F(r) = + F(r) = So the equation F(r) = 0 has an unique solution, and the theorem is proved. 3

15 Chapter 5 Bonds 5. Introduction Definition A bond is a part of the debt of an institution called the issuer (see slides of Chapter ). It is a debt security, that is, a contract between the owner of the bond and the issuer, which specifies the repayment rules. It has: a face value or nominal value N an annual nominal interest rate z a periodicity of coupon delivery (i.d. payment of interest): if the periodicity is /m year, there are m coupons per year. each year, the amount of interest paid is zn. If the periodicity is /m year, each coupon equals C = z m N. a date of issue (beginning of the contract) and an issue price E. If E = N the bond is issued at par If E < N the bond is issued at a discount If E > N the bond is issued at a premium a maturity date (end of the contract), and a refund price R that the owner receives at maturity. If R = N refund is at par If R < N refund is at a discount If R > N refund is at a premium Sincethe nominal value N, thenominal rate z, theperiodicity of coupons, theduration (also called maturity) and the refund price R are known in advance, bonds are called fixedincome securities. But a bond can be exchanged many times on the secondary market between its date of issue and its date of maturity. Its price on the secondary market varies in an unpredictable way along this period. Question: How can one choose between different bonds? 4

16 Example: Take two bonds A, B each of maturity 2 years, with the same nominal N A = N B = 000 Assume that A delivers two coupons at the end of years, 2, each of value C A = 00. Assume that B delivers only coupon of C B = 000 at the end of year. Assume that both A, B are refunded at par: R A = R B = N = 000 time 2 A B Assume that A is issued at E A = 000, and B at E B = 735 Comparison of A and B Choose an interest rate r, considered as reference(for instance the interest rate of American Treasury Bonds) Then NPV(A) = r + 00 (+r) 2 NPV(B) = r (+r) 2 If NPV(A) > NPV(B), you should prefer A to B (occurs here for r < 0%) NPV(A) < NPV(B), you should prefer B to A (occurs here for r > 0%) NPV(A) = NPV(B), A,B are indifferent (occurs here for r = 0%) 5.2 The arbitrage price of a bond Definition : An arbitrage at time t is a set of financial operations, corresponding to cash flows adding to zero at time t, and which will necessarily generate nonnegative cash flows in the future, with a positive probability that at least one of the future cash flows will be positive. In an ideal market, arbitrage is impossible. This means that the market fixes the prices of assets in such a way that no arbitrage can occur. Our goal is to compute the value of a bond under the no arbitrage hypothesis. This value will be called the arbitrage price of the bond. Another simplifying assumption we make is the perfect liquidity of the market: actors on the market can buy or short-sell arbitrary quantities of securities at any moment without paying trading fees (even non-integer quantities will be allowed, to simplify the arguments). Short-selling means selling a security you do not own. Algebraically, this is equivalent to buying unit of the security.then, in the future, you will have to cancel your short position by buying the same security. If an investor short sells a security that generates an income I in the interval of time before the negative position is closed, he will receive the negative cash flow I (in other words, he will have to pay the amount I). Simplified (and unrealistic) case : Take a market with only two kinds of securities : A risk-free bond (e.g. American Treasury Bonds) at a fixed annual interest rate r, independent of maturity. A (possiblyrisky)bondbthat generates futurecash flow at times t+,...,t+n, F t+,...,f t+n (consisting of coupons and refund at maturity). 5

17 Claim : The arbitrage price of bond B at time t is P arbitrage (B) = i= F t+i (+r) i In other words, P arbitrage (B) is such that NPV r (B,t) = 0 where NPV r (B,t) = P arbitrage (B,t)+ n i= F t+i (+r) i Proof of the claim : we argue by contradiction. Let P = n i= Assume that P B < P. Then we can do the following : F t+i (+r) i At time t, we buy one bond B at price P B, and we sell a fraction f of Treasury Bond A, such that Unit price(a) f = P B (this is equivalent to borrowing the amount P B at interest rate r) At time 0, our net cash flow is P B +P B = 0 at time t+i, we plan to do the following : we receive the amount F i and we reinvest it in Treasury Bonds, for i = t+,...,t+n net cash flow =0. At time t+n, we receive F t+n and we close our line of treasury bonds A. The amount we receive from Treasury Bonds at time t+n is P B (+r) n +F t+ (+r) n +...+F t+i (+r) n i +...+F t+n (+r) So our net cash flow at time t+n is n P B (+r) n + F i (+r) n i +F t+n = (+r) n ( P B +P ) > 0 i= Arbitrage is impossible, so P B P. Similarly, if P B > P we do exactly the opposite operations : we get an arbitrage, which is forbidden. So P B = P, in absence of arbitrage. 5.3 Pricing of a bond using zero-coupon bonds Definition : A zero-coupon bond is a bond that delivers no coupon. At maturity, the owner of the zero-coupon bond receives an amount R equal to its nominal N by convention. So a zero-coupon bond is characterized at time t by: Its date of maturity T t Its nominal N whichis equal toits refundpricer(r is payed at timet). Inthe sequel, we will use the normalization N = R = 00 for one unit of zero-coupon bond. Its price Z(t,T) at time t 6

18 Example : Suppose that on the markets, there are bonds A,B,C,D available at time t = 0. C, D are zero-coupon bonds. Bond t = 0 t = t = 2 A B C D C has price 95 at time 0, it maturity date is and its refund value is 00. D has price 80 at time 0, it maturity date is 2 and its refund value is 00. Using the zero-coupon bonds C and D, we can replicate bond A : We buy unit of bond C + units of bond D, at time 0. We pay = = 975 At time, we receive = 00. At time 2, we receive 00 = 00. Using this replication, we can construct an arbitrage ( free lunch ): Actions t=0 t= t=2 Short-sell unit of A Buy unit of C Buy units of D Invest (+r) 2 Net balance (+r) 2 > 0 Conclusion : considering C and D, the price of A at time 0 is above its arbitrage price P arbitrage (A) = 975. You should not buy bond A, instead you should short-sell it and do the arbitrage. Similarly, we can replicate bond B as follows : Buy 0 C +0 D at time 0. You pay = 750. This is higher than the price of bond B. So we can make an arbitrage: Actions t=0 t= t=2 Buy unit of B Sell 0 unit of C Sell 0 units of D Net balance P arbitrage (B) = 750 > 735 = P B is a market abnormality. General case : on a financial market at time t, we will admit that there are zero-coupon bonds of all maturities T t. We will normalize them so that for each unit of such bonds, the value at maturity is N = R = 00. The price of the zero-coupon bond of maturity date T and nominal N = 00 is Z(t,T) at time t T. 7

19 One defines an interest rate per year z(t,t) of this zero-coupon bond by the formula 00 = (+z(t,t)) T t Z(t,T). In fact, z(t,t) is the I.R.R. of the zero-coupon bond of maturity date T. Indeed, if you buy this bond at time t, the present value at time t of your cash flow with actualization rate z(t,t) is: PV z(t,t) (zero coupon) = Z(t,T)+(+z(t,T)) t T N = 0 Then one can draw the so-called zero-rate curve at time t which is the graph of the function T z(t,t). Usually this function is increasing, due to the preference of investors for short-term investments. But when the market anticipates lower interest rates in the near future, the curve can become flat or even decreasing. The formula for z(t,t) in terms of t,t and Z(t,T) is : z(t,t) = ( ) 00 T t. Z(t,T) Now, take a bond B delivering future cash flows F t,f t2,...,f tn at future times t < t 2 <... < t n = T. Its arbitrage price at time t < t is : P arbitrage (B) = i= F ti (+z(t,t i )) t i t = i= Z(t,t i )F ti 00 Let us prove this equation. To do so, we replicate the bond B using zero-coupon bonds. We remind that F ti is the cash flow at time t i by an owner of one unit of bond B. The replication works as follows: At time t, one buys F t /00 unit of zero-coupon bond of maturity t, F t2 /00 unit of zero-coupon bond of maturity t 2,. F tn /00 unit of zero-coupon bond of maturity t n. For i n, at time t i the only cash flow provided by this replication is coming from step i: F ti /00 units of Z B of maturity t i and nominal N = 00 provide the cash flow (F ti /00) 00 = F ti. This is exactly the cash flow provided by B at time t. This shows that we have built a replication R of B. At time t, the price of R is: P repl = Z(t,t )F t Z(t,t n)f tn = F ti (+z(t,t i )) t i t i= Assume that the price: P of one unit of bond B is smaller than P repl. Then one could : At time t : buy one unit of B, short sell one unit of R. The net cash flow at time t is P(B)+P repl > 0 8

20 At time t i ( i n) : receive F ti from bond B and pay F ti coming from F ti /00 short positions on the Z B of maturity t i NET = 0 This is an arbitrage in the generalized sense : Positive cash flow at time t Nonnegative cash flow in the future Similarly, if we assume that P(B) > P repl we could short sell one unit of B and buy one unit of R. We would get P(B) Prepl at time t, and no cash flow in the future. This shows that P arbitrage (B) = P repl = i= Z(t,t i )F ti 00 = i= F ti (+z(t,t i )) t i t 9

21 Chapter 6 Derivatives Definition : A derivative is a contract on an underlying object, which can be a currency exchange rate, an interest rate, a bond, a stock, or any kind of asset. So the derivative itself is a security, its value at a given time depends on the underlying and on the market. 6. Futures and forwards Definition : in a future/forward contract, two parties A, B decide the following at time 0 : At time T > 0, A will buy from B one unit of a specified underlying asset at a fixed price K. Note that T, the asset and the price K are known at time 0. What is unknown is the price S T of the asset X at time T on the market. By assumption, S T is unpredictable. Note that this contract is symmetric : A will have to buy X from B at price K at time T B will have to sell X to A at price K at time T Forward contracts are traded over the counter, while futures are traded on regulated markets. In their simplest version, forward contracts generate only one cash flow, at time T. In contrast, futures can generate daily cash flows on a margin account, depending on the variation of value of the underlying and the interest rates on the market. The role of these cash flows is to reduce the risk of default of a counterparty at time T. For simplicity, we will focus on forward contracts in the sequel. Notion of payoff : Thepayoff foraat timet willbes T K. SinceP T isunpredictable, Payoff(A) is unpredictable. Payoff(A) is the net cash flow that A will get, buying X from B at price K, and selling it immediately on the market price S T. Similarly, the payoff of B at time T is K S T. This is the cash flow that B will get, buying X on the market at price S T and selling it to A at price K. At time 0, T is known but S T is unknown. So the future payoff is unknown at time 0. Question : can one compute the value of a forward contract for A/B? In other words, should A/B pay a compensation to B/A, at time 0? At time 0, the future value S T of the underlying is unknown, but the present value S 0 of the underlying is known. 20

22 Proposition : In the absence of arbitrage opportunity, the value at time 0 of the forward contract for A is FC 0 = S 0 (+r) T K where r is the interest rate per compounding period (=) for an risk-free investmenton the market (which is assumed to be the same as the interest rate for loans). In other words, if S 0 (+r) T K > 0, then A should pay P 0 = S 0 (+r) T K to B. If S 0 (+r) T K < 0 then B should pay P 0 = (+r) T K S 0 to A (at time 0). Proof : Let P 0 be the algebraic price payed by A to B, in order to sign the contract at time 0. We want to show that P 0 = S 0 (+r) T K. If P 0 < S 0 (+r) T K, A can do the following At t = 0: Sign the contract with B and pay P 0 to B Take a short position on the underlying : cash flow =S 0 Invest the amount (+r) T K (with no risk) at interest rate r The net cash flow of A at time 0 is then : At t = T, A : S 0 P 0 (+r) T K > 0 Buys the underlying from B at price K and sell it immediately : cash flow= K +P T Closes the short position on the underlying (give back the underlying at price P T ) Closes the investment at rate r : cash flow=k This is an arbitrage in the general sense. So we cannot have P 0 < S 0 ( + r) T K. 2. If P 0 > S 0 (+r) T K, B can do the following At t = 0: Sign the contract with A and receive P 0 from A Take a long position on the underlying : cash flow = S 0 Borrow the amount (+r) T K (with no risk) at interest rate r The net cash flow of B at time 0 is then : At t = T, B : S 0 +P 0 +(+r) T K > 0 SellstheunderlyingtoAatpriceK andbuyitimmediately: cashflow=k P T Closes the long position on the underlying(get back the underlying at price P T ) Reimburses the loan at rate r : cash flow=k This is an arbitrage in the general sense. So we cannot have P 0 > S 0 ( + r) T K. 2

23 Definition : The forward price of the underlying X at time 0 is the amount K forward, solution of the equation S 0 (+r) T K = 0 So we have K forward = (+r) T S 0. If the forward contract is such that K = K forward then FC 0 = 0 : there is no cash flow between A and B at time 0. In general, a forward contract does not generate any cash flow between the counterparties at t = 0. This means that K = K forward at time 0. But as t variesintheintervaloftimebetween0andt,theprices t oftheunderlyingandtheinterest rater t (forrisk-freeinvestments andloansofmaturity datet)canvaryinanunpredictable way. This results in a nonzero value FC t of the forward contract, which varies in an unpredictable way according to the no-arbitrage formula FC t = S t ( + r t ) t T K. At time T, this value becomes FC T = S T K : we recognize the payoff of the forward contract. 6.2 Options An option is a contract between two parties A and B, which is not symmetric. As in a forward, there is an underlying asset X. We will only consider European options, for which the future date T of exercise of the option is fixed. There are two kinds of (European) options : Call options : At time 0 A pays an amount P 0 > 0 to B, and gets the right to buy the underlying X from B at time T, at strike price K. B receives P 0 at time 0, but at time T, if A decides to buy X, B will be forced to sell it at price K. In that case, we will say that the option is exercised. We say that the buyer A of the call option is long on the call option, and that the sell B is short on the option. Put options : At time 0, A pays an amount P 0 > 0 to B, and gets the right to sell X to B at time T at the strike price K. B receives P 0 but will be forced to buy X from A at price K at time T. A is long, B is short on the put option. At time T, the value of underlying asset is S T. The strike price is K. Call options : If S T > K we should exercise the option. In this case the call is in the money. If S T = K the call is at the money. If S T < K we should not exercise the option. In this case the call is out of the money. In terms of value: the value of the call at time T is zero if S T K, and equal to S T K > 0 when S T > K. In all case the value or payout of the call at time T is were x + = x if x > 0, x + = 0 if x 0. (S T K) + 22

24 Put options : If S T < K we should exercise the option. In this case the put is in the money. If S T = K the put is at the money. If S T > K we should not exercise the option. In this case the put is out of the money. In terms of value: the value of the put at time T is zero if S T K, and equal to K S T > 0 when S T < K. In all case the value or payout of the put at time T is (K S T ) + Question : How much should one pay for a call? For a put? Let C 0, P 0, V 0 be the value at time 0 of a call, a put and a forward on the same underlying asset X, with same T > 0, same K > 0. Theorem (put-call parity): In the absence of arbitrage opportunity, C 0 P 0 = V 0 Proof: Assume for instance that C 0 P 0 < V 0. At time 0, for the same underlying asset X with same T > 0 and K > 0, you can: Take a long position on a call. Take a short position on a put. Take a short position on a forward. The resulting cash flow is C 0 +P 0 +V 0 > 0 At time T: If S T K you do not exercise the call. But the buyer of the put will decide to sell you X at price K, and you will sell X at price K to the buyer of the forward. Your cash flow is K +K = 0 If S T > K, you exercise the call, but the buyer of the put does not exercise it. You sell X at price K to the buyer of the forward. Your cashflow at time T is then K +K = 0 So, whatever S T is, your cash flow at time T is zero : this is an arbitrage. Similarly, if C 0 P 0 > V 0, at time 0 you can : Take a short position on a call. Take a long position on a put. Take a long position on a forward. The resulting cash flow is C 0 P 0 V 0 > 0 At time T: If S T K you do not exercise the put. But the buyer of the call will decide to buy X from you at price K, and you will buy X at price K to the seller of the forward. Your cash flow is K K = 0 If S T > K, you exercise the put, but the buyer of the call does not exercise it. You buy X at price K to the seller of the forward. Your cash flow at time T is then K K = 0 So, whatever S T is, your cash flow at time T is zero : this is an arbitrage. 23

25 Explanation : behind the proof is the equality In other words : At time T : x + ( x) + = x x R Payoff(call) Payoff(put) = Payoff(forward) This holds for any value of S T. The remaining question is : how can one compute C 0? Naive approach (which fails) Suppose we can only trade the option and the asset X at time t = 0 and t =. One can also lend/borrow cash between t = 0 and t = at interest rate r > 0. At time 0 the asset X has value S 0. At time, it can have three possible values, each with probability 3 : S = 40,70 or 0. We also assume that K = 60 and +r = 0,95 Then +r E((S K) + ) = 9 is the present value of the expected payoff of the call option. A natural idea would be to choose C 0 = 9 as price of the call option at time 0. The problem is that, if C 0 = 9 then an arbitrage is possible. Indeed, at time 0 you could : Sell one call option. Borrow 3 at interest rate r. Buy one underlying asset. Net cash flow at time 0: At time : C 0 +3 S 0 = = 0 If S = 0 or 70 the call option is exercised : you have to sell the asset at price K = 60, and you reimburse 3 (+r) = 3 0,95 = 32,63 Net cash flow = 60 32,63 > 0 Is S = 40, the option is not exercised. You sell your asset on the market at price 40 and you reimburse 32,63. Net cash flow = 40 32;63 > The binomial model with one time step As in the example above, we consider a simplified market with only two dates of trade: t = 0 and t =, and only two assets: a risk-free asset, which is equivalent to the possibility of lending or borrowing cash at a deterministic interest rate r = 0% between t = 0 and t =, and a risky asset X. At time 0 the asset X has value S 0 = 00. The expression binomial model means that at time, the market price S of X can take exactly two possible values, each with nonzero probability : S = 90 or

26 Let K = 0 be the strike price of a call option on the asset. What is the no arbitrage price C(0,) of this option at time 0? It will turn out that one can determine this price without knowing the probability law of S. In order to find it, let us try to replicate the option. Suppose that at time 0 you buy λ units of the underlying asset and you lend µ (euros) without risk at rate r. Here, λ,µ R At time 0 your net cash flow is λs 0 µ = 00λ µ At time, you receive the reimbursement µ(+r) =,µ So : If S = 20 you sell your λ units of asset and receive S λ = 20λ If S = 90 you sell λ assets and receive 90λ If S = 20 your net cash flow is 20λ+,µ If S = 90 your net cash flow is 90λ+,µ If, instead, you buy unit of option at time 0, your cash flow at time 0 is C(0,). At time : If S = 20 you buy the asset at K = 0, you sell it on the market at S = 20: net cash flow = 0. If S = 90 you do nothing : net cash flow = 0. A replication must provide the same future cash flow as the call option, in any situation : { 20λ+,µ = 0 L Subtracting L L 2, we get 90λ+,µ = 0 L 2 30λ = 0 λ = 3 L 2 µ = 90, λ = 30, of asset and in borrowing µ = 30, (euros) So the replication corresponds to buying 3 at interest rate r. General principle : If two investment strategies deliver exactly the same cash flows in the future, then their price at time 0 should be the same; otherwise an arbitrage would be possible. Consequence : the no arbitrage price C(0,) of the option is the price of its replication at time 0: C(0,) = 00λ+µ = , One can give general formulas taking r > 0 arbitrary and S = u S 0 or S = d S 0, with 0 < d < u. Assume that ds 0 < K < us 0. If we buy λ units of asset and lend µ euros at time 0, then at time : If S = us 0 we receive S 0 uλ+(+r)µ 25

27 If S = ds 0 we receive S 0 dλ+(+r)µ So { S0 uλ+(+r)µ = us So : 0 K S 0 dλ+(+r)µ = 0 λ = us 0 K S 0 (u d) µ = d us 0 K u d +r C(0,) = S 0 λ+µ C(0,) = S 0 λ+µ = us 0 K u d = us 0 K u d d us 0 K ( u d +r d ) +r Risk-neutral probability : Let us look for a probability law P on the issues us 0, ds 0 such that +r E P ((S K) + ) = C(0,). In other words, we look for P, probability law on the random variable S, such that the present value of the call s expected payoff is the same as its no arbitrage price. It turns out that this problem has a unique solution P, which is called the risk-neutral probability law. Let us compute P. Let p = P (S = us 0 ). Then So the condition on p is : +r E P ((S K) + ) = p +r (us 0 K) p +r (us 0 K) = us 0 K u d ( d ) +r p = +r ( d ) +r d = u d +r u d In order to guarantee 0 < p < we must have which is equivalent to 0 < +r d < u d d < +r < u Note that the assumption d < +r < u is necessary if we want to forbid arbitrage. Indeed if, for instance, +r d, then at time 0, we can borrow the amount S 0 at rate r and buy one unit of asset at price S 0 net cash flow = S 0 S 0 = 0 At time, we reimburse (+r)s 0 and we sell the asset at price ds 0 net cash flow (+r)s 0 +ds 0 = (d (+r))s 0 Moreover, the price of the asset at time is us 0 with positive probability net cash flow = (u (+r))s 0 > 0 with positive probability (here we refer to the true probability law). 26

28 Similarly, if u +r, we can do the opposite (lend S 0 and short sell one unit of asset) and we obtain an arbitrage. This ends the argument. so Coming back to the risk-neutral probability, we have seen that P (S = us 0 ) = p = +r d u d P (S = ds 0 ) = p = u (+r) u d meaning that the price of the call on the asset at time 0 Now, we compute C(0,) = +r E P ((S K) + ) +r E P (S ) = +r (p u+( p )d)s 0 = S 0 Finally, consider an amount A 0 invested at time 0 at rate r. Since its value at time is deterministic, we have +r E(A ) = +r (+r)a 0 = A 0. So any contract consisting of a linear combination of the underlying asset X, the option and the risk-free asset will have the property V 0 = +r E P (V ) Note that this property implies that no arbitrage is possible. Indeed, an arbitrage would be a linear combination of contracts with value V 0 = 0 at time 0. By the property, its value V at time would satisfy S 0 E P (V ) = 0, so either V is identically zero or it can have two opposite signs, each with positive probability (even if P is not the true probability, an event has nonzero probability for P if and only if it has nonzero true probability). This shows that no arbitrage is possible!!! 6.4 Multistep binomial model: Now we consider a more complex model involving n + dates of trade: 0,,,n but only two assets as before: a riskless asset with is equivalent to the possibility of borrowing or lending cash at a deterministic rate r per unit of time, with compounding period equal to ; and a risky asset X of known value S 0 at time 0. We assume that, if the value S i of X is known at time i, then S i+ can only take two values : us i or ds i, 0 < d < +r < u. After n iterations, the possible values of S n are u k d n k S 0, where k is the number of times i such that the asset price varies from S i to us i. Question : What is the price C(0,n) at time 0 of a European call on one unit of asset X, with exercise date n and strike price K? As often in Mathematics, it is easier to solve a more general question : Take 0 i n. What is the price C(i,n) at time i of a call on one unit of asset X with exercise date n and strike price K? One can determine C(i,n) going backwards, from i = n to i = 0. 27

29 What is C(n,n)? We can consider the one-step binomial model of evolution from the (known) value S n to the random value S n. This amount to consider the risk-neutral probability law of S n conditional to S n. We get C(n,n) = +r E P ((S n K) + S n ) = +r E P (C(n,n) S n ) Similarly, at each step, knowing C(i+,n) in all possible situations occuring at time i+, we can compute C(i,n) = +r E P (C(i+,n) S i ) = p +r uc(i,n)+ p +r dc(i,n) in all situations occuring at time i. In fact there is even a global formula: C(0,n) = (+r) ne P ((S n K) + ) Note that, under P, the event { S n = u k d n k S 0 } has probability C k n pk ( p ) n k so we find the following formula for the no arbitrage price at time 0 of our European call option: C(0,n) = (+r) ne P ((S n K) + ) = (+r) n Cnp k k ( p ) n k (S 0 u k d n k K) + +r d p = u d Now, we take a time unit such that the time delay between step k and k + is T, very small, with T = n T fixed, and n. Note that ln(s k+ ) = ln(s k )+ln(u) or ln(s k )+ln(d) Denote ( ) Sk Y k = ln k n S k Then Y k is a random variable which equals ln(u) with probability p and ln(d) with probability ( p ), and Y,...,Y n are independent. Then ln( Sn S 0 ) = Y +...+Y n hence k=0 C(0,n) = +r E P ((S 0 e Y +...+Y n K) + ) As n, ) it seems reasonable to expect that the mean and variance of the random variable ln( Sn S 0 remain finite. This is achieved if for some fixed σ > 0 and some fixed real number µ, Then ( u = exp µ T +σ ) T ( d = exp µ T +σ ) T { ln(u) = µ T +σ T ln(d) = µ T σ T = +(µ+ σ2 2 ) T +σ T +o( T) = +(µ+ σ2 2 ) T σ T +o( T) 28

30 Moreover, the compounding period is now T, so the interest rate for this period should behave like r T instead of r. This gives a formula for p. We find p = +r T d u d = (r µ σ2 ) 2 T +O( T). σ hence E P (Y k ) = T(r σ2 2 )+o( T) and Var P (Y k ) = σ 2 T +o( T). So, by the central limit theorem, the random variable W n = Y +...+Y n T(r σ 2 ) σ converges in law to N(0,) as n goes to infinity (remarkably, µ has now disappeared from the T formulas). As a result, one finds, for n very large, the approximation ( ) ( ST ln N (r ) 2 σ2 )T,σ 2 T hence, in the limit n, S 0 C(0,T) = e rt E P ((S T K) + ) = e rt σ 2πT + where ν = ln(s 0 )+(r σ2 ) 2 )T. This is the Black-Scholes formula for European option pricing. lnk e (y ν)2 2σ 2 T (e y K)dy 2 29