MATH 361: Financial Mathematics for Actuaries I

Transcription

1 MATH 361: Financial Mathematics for Actuaries I Albert Cohen Actuarial Sciences Program Department of Mathematics Department of Statistics and Probability C336 Wells Hall Michigan State University East Lansing MI albert@math.msu.edu acohen@stt.msu.edu Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

2 Course Information Syllabus to be posted on class page in first week of classes Homework assignments will posted there as well Page can be found at Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

3 Course Information Many examples within these slides are used with kind permission of Prof. Dmitry Kramkov, Dept. of Mathematics, Carnegie Mellon University. Book for course: Financial Mathematics: A Comprehensive Treatment (Chapman and Hall/CRC Financial Mathematics Series) 1st Edition. Can be found in MSU bookstores now Some examples here will be similar to those practice questions publicly released by the SOA. Please note the SOA owns the copyright to these questions. This book will be our reference, and some questions for assignments will be chosen from it. Copyright for all questions used from this book belongs to Chapman and Hall/CRC Press. From time to time, we will also follow the format of Marcel Finan s A Discussion of Financial Economics in Actuarial Models: A Preparation for the Actuarial Exam MFE/3F. Some proofs from there will be referenced as well. Please find these notes here Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

4 What are financial securities? Traded Securities - price given by market. For example: Stocks Commodities Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

5 What are financial securities? Traded Securities - price given by market. For example: Stocks Commodities Non-Traded Securities - price remains to be computed. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

6 What are financial securities? Traded Securities - price given by market. For example: Stocks Commodities Non-Traded Securities - price remains to be computed. Is this always true? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

7 What are financial securities? Traded Securities - price given by market. For example: Stocks Commodities Non-Traded Securities - price remains to be computed. Is this always true? We will focus on pricing non-traded securities. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

8 How does one fairly price non-traded securities? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

9 How does one fairly price non-traded securities? By eliminating all unfair prices Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

10 How does one fairly price non-traded securities? By eliminating all unfair prices Unfair prices arise from Arbitrage Strategies Start with zero capital End with non-zero wealth Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

11 How does one fairly price non-traded securities? By eliminating all unfair prices Unfair prices arise from Arbitrage Strategies Start with zero capital End with non-zero wealth We will search for arbitrage-free strategies to replicate the payoff of a non-traded security Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

12 How does one fairly price non-traded securities? By eliminating all unfair prices Unfair prices arise from Arbitrage Strategies Start with zero capital End with non-zero wealth We will search for arbitrage-free strategies to replicate the payoff of a non-traded security This replication is at the heart of the engineering of financial products Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

13 More Questions Existence - Does such a fair price always exist? If not, what is needed of our financial model to guarantee at least one arbitrage-free price? Uniqueness - are there conditions where exactly one arbitrage-free price exists? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

14 And What About... Does the replicating strategy and price computed reflect uncertainty in the market? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

15 And What About... Does the replicating strategy and price computed reflect uncertainty in the market? Mathematically, if P is a probabilty measure attached to a series of price movements in underlying asset, is P used in computing the price? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

16 Notation Forward Contract: Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

17 Notation Forward Contract: A financial instrument whose initial value is zero, and whose final value is derived from another asset. Namely, the difference of the final asset price and forward price: V (0) = 0, V (T ) = S(T ) F (1) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

18 Notation Forward Contract: A financial instrument whose initial value is zero, and whose final value is derived from another asset. Namely, the difference of the final asset price and forward price: V (0) = 0, V (T ) = S(T ) F (1) Value at end of term can be negative - buyer accepts this in exchange for no premium up front Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

19 Notation Interest Rate: The rate r at which money grows. Also used to discount the value today of one unit of currency one unit of time from the present Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

20 Notation Interest Rate: The rate r at which money grows. Also used to discount the value today of one unit of currency one unit of time from the present V (0) = 1, V (1) = 1 (2) 1 + r Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

21 An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

22 An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: S B A = 4 is the spot exchange rate - one unit of B is worth S B A of A today (time 0) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

23 An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: S B A = 4 is the spot exchange rate - one unit of B is worth S B A of A today (time 0) r A = 0.1 is the domestic borrow/lend rate Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

24 An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: S B A = 4 is the spot exchange rate - one unit of B is worth S B A of A today (time 0) r A = 0.1 is the domestic borrow/lend rate r B = 0.2 is the foreign borrow/lend rate Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

25 An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: S B A = 4 is the spot exchange rate - one unit of B is worth S B A of A today (time 0) r A = 0.1 is the domestic borrow/lend rate r B = 0.2 is the foreign borrow/lend rate Compute the forward exchange rate FA B. This is the value of one unit of B in terms of A at time 1. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

26 An Example of Replication: Solution At time 1, we deliver 1 unit of B in exchange for FA B currency A. units of domestic Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

27 An Example of Replication: Solution At time 1, we deliver 1 unit of B in exchange for FA B currency A. units of domestic This is a forward contract - we pay nothing up front to achieve this. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

28 An Example of Replication: Solution At time 1, we deliver 1 unit of B in exchange for FA B currency A. units of domestic This is a forward contract - we pay nothing up front to achieve this. Initially borrow some amount foreign currency B, in foreign market to grow to one unit of B at time 1. This is achieved by the initial SA amount B (valued in domestic currency) 1+r B Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

29 An Example of Replication: Solution At time 1, we deliver 1 unit of B in exchange for FA B currency A. units of domestic This is a forward contract - we pay nothing up front to achieve this. Initially borrow some amount foreign currency B, in foreign market to grow to one unit of B at time 1. This is achieved by the initial SA amount B (valued in domestic currency) 1+r B Invest the amount currency) F B A 1+r A in domestic market (valued in domestic Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

30 An Example of Replication: Solution Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

31 An Example of Replication: Solution This results in the initial value V (0) = Since the initial value is 0, this means F B A = S B A F B A 1 + r A S B A 1 + r B (3) 1 + r A = (4) 1 + r B Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

32 Discrete Probability Space Let us define an event as a point ω in the set of all possible outcomes Ω. This includes the events The stock doubled in price over two trading periods or the average stock price over ten years was 10 dollars. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

33 Discrete Probability Space Let us define an event as a point ω in the set of all possible outcomes Ω. This includes the events The stock doubled in price over two trading periods or the average stock price over ten years was 10 dollars. In our initial case, we will consider the simple binary space Ω = {H, T } for a one-period asset evolution. So, given an initial value S 0, we have the final value S 1 (ω), with Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

34 Discrete Probability Space Let us define an event as a point ω in the set of all possible outcomes Ω. This includes the events The stock doubled in price over two trading periods or the average stock price over ten years was 10 dollars. In our initial case, we will consider the simple binary space Ω = {H, T } for a one-period asset evolution. So, given an initial value S 0, we have the final value S 1 (ω), with S 1 (H) = us 0, S 1 (T ) = ds 0 (5) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

35 Discrete Probability Space Let us define an event as a point ω in the set of all possible outcomes Ω. This includes the events The stock doubled in price over two trading periods or the average stock price over ten years was 10 dollars. In our initial case, we will consider the simple binary space Ω = {H, T } for a one-period asset evolution. So, given an initial value S 0, we have the final value S 1 (ω), with S 1 (H) = us 0, S 1 (T ) = ds 0 (5) with d < 1 < u. Hence, a stock increases or decreases in price, according to the flip of a coin. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

36 Discrete Probability Space Let us define an event as a point ω in the set of all possible outcomes Ω. This includes the events The stock doubled in price over two trading periods or the average stock price over ten years was 10 dollars. In our initial case, we will consider the simple binary space Ω = {H, T } for a one-period asset evolution. So, given an initial value S 0, we have the final value S 1 (ω), with S 1 (H) = us 0, S 1 (T ) = ds 0 (5) with d < 1 < u. Hence, a stock increases or decreases in price, according to the flip of a coin. Let P be the probability measure associated with these events: P[H] = p = 1 P[T ] (6) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

37 Arbitrage Assume that S 0 (1 + r) > us 0 Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

38 Arbitrage Assume that S 0 (1 + r) > us 0 Where is the risk involved with investing in the asset S? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

39 Arbitrage Assume that S 0 (1 + r) > us 0 Where is the risk involved with investing in the asset S? Assume that S 0 (1 + r) < ds 0 Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

40 Arbitrage Assume that S 0 (1 + r) > us 0 Where is the risk involved with investing in the asset S? Assume that S 0 (1 + r) < ds 0 Why would anyone hold a bank account (zero-coupon bond)? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

41 Arbitrage Assume that S 0 (1 + r) > us 0 Where is the risk involved with investing in the asset S? Assume that S 0 (1 + r) < ds 0 Why would anyone hold a bank account (zero-coupon bond)? Lemma Arbitrage free d < 1 + r < u Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

42 Derivative Pricing Let S 1 (ω) be the price of an underlying asset at time 1. Define the following instruments: Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

43 Derivative Pricing Let S 1 (ω) be the price of an underlying asset at time 1. Define the following instruments: Zero-Coupon Bond : V B 0 = 1 1+r, V B 1 (ω) = 1 Forward Contract : V F 0 = 0, V F 1 = S 1(ω) F Call Option : V C 1 (ω) = max(s 1(ω) K, 0) Put Option : V P 1 (ω) = max(k S 1(ω), 0) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

44 Derivative Pricing Let S 1 (ω) be the price of an underlying asset at time 1. Define the following instruments: Zero-Coupon Bond : V B 0 = 1 1+r, V B 1 (ω) = 1 Forward Contract : V F 0 = 0, V F 1 = S 1(ω) F Call Option : V C 1 (ω) = max(s 1(ω) K, 0) Put Option : V P 1 (ω) = max(k S 1(ω), 0) In both the Call and Put option, K is known as the Strike. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

45 Derivative Pricing Let S 1 (ω) be the price of an underlying asset at time 1. Define the following instruments: Zero-Coupon Bond : V B 0 = 1 1+r, V B 1 (ω) = 1 Forward Contract : V F 0 = 0, V F 1 = S 1(ω) F Call Option : V C 1 (ω) = max(s 1(ω) K, 0) Put Option : V P 1 (ω) = max(k S 1(ω), 0) In both the Call and Put option, K is known as the Strike. Once again, a Forward Contract is a deal that is locked in at time 0 for initial price 0, but requires at time 1 the buyer to purchase the asset for price F. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

46 Derivative Pricing Let S 1 (ω) be the price of an underlying asset at time 1. Define the following instruments: Zero-Coupon Bond : V B 0 = 1 1+r, V B 1 (ω) = 1 Forward Contract : V F 0 = 0, V F 1 = S 1(ω) F Call Option : V C 1 (ω) = max(s 1(ω) K, 0) Put Option : V P 1 (ω) = max(k S 1(ω), 0) In both the Call and Put option, K is known as the Strike. Once again, a Forward Contract is a deal that is locked in at time 0 for initial price 0, but requires at time 1 the buyer to purchase the asset for price F. What is the value V 0 of the above put and call options? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

47 Put-Call Parity Can we replicate a forward contract using zero coupon bonds and put and call options? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

48 Put-Call Parity Can we replicate a forward contract using zero coupon bonds and put and call options? Yes: The final value of a replicating strategy X has value Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

49 Put-Call Parity Can we replicate a forward contract using zero coupon bonds and put and call options? Yes: The final value of a replicating strategy X has value V C 1 V P 1 + (K F ) = S 1 F = X 1 (ω) (7) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

50 Put-Call Parity Can we replicate a forward contract using zero coupon bonds and put and call options? Yes: The final value of a replicating strategy X has value This is achieved (replicated) by V C 1 V P 1 + (K F ) = S 1 F = X 1 (ω) (7) Purchasing one call option Selling one put option Purchasing K F zero coupon bonds with value 1 at maturity. all at time 0. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

51 Put-Call Parity Can we replicate a forward contract using zero coupon bonds and put and call options? Yes: The final value of a replicating strategy X has value This is achieved (replicated) by V C 1 V P 1 + (K F ) = S 1 F = X 1 (ω) (7) Purchasing one call option Selling one put option Purchasing K F zero coupon bonds with value 1 at maturity. all at time 0. Since this strategy must have zero initial value, we obtain V C 0 V P 0 = F K 1 + r (8) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

52 Put-Call Parity Can we replicate a forward contract using zero coupon bonds and put and call options? Yes: The final value of a replicating strategy X has value This is achieved (replicated) by V C 1 V P 1 + (K F ) = S 1 F = X 1 (ω) (7) Purchasing one call option Selling one put option Purchasing K F zero coupon bonds with value 1 at maturity. all at time 0. Since this strategy must have zero initial value, we obtain V C 0 V P 0 = F K 1 + r Question: How would this change in a multi-period model? (8) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

53 General Derivative Pricing -One period model If we begin with some initial capital X 0, then we end with X 1 (ω). To price a derivative, we need to match X 1 (ω) = V 1 (ω) ω Ω (9) to have X 0 = V 0, the price of the derivative we seek. A strategy by the pair (X 0, 0 ) wherein X 0 is the initial capital 0 is the initial number of shares (units of underlying asset.) What does the sign of 0 indicate? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

54 Replicating Strategy Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

55 Replicating Strategy Initial holding in bond (bank account) is X 0 0 S 0 Value of portfolio at maturity is X 1 (ω) = (X 0 0 S 0 )(1 + r) + 0 S 1 (ω) (10) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

56 Replicating Strategy Initial holding in bond (bank account) is X 0 0 S 0 Value of portfolio at maturity is X 1 (ω) = (X 0 0 S 0 )(1 + r) + 0 S 1 (ω) (10) Pathwise, we compute V 1 (H) = (X 0 0 S 0 )(1 + r) + 0 us 0 V 1 (T ) = (X 0 0 S 0 )(1 + r) + 0 ds 0 Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

57 Replicating Strategy Initial holding in bond (bank account) is X 0 0 S 0 Value of portfolio at maturity is X 1 (ω) = (X 0 0 S 0 )(1 + r) + 0 S 1 (ω) (10) Pathwise, we compute V 1 (H) = (X 0 0 S 0 )(1 + r) + 0 us 0 V 1 (T ) = (X 0 0 S 0 )(1 + r) + 0 ds 0 Algebra yields 0 = V 1(H) V 1 (T ) (u d)s 0 (11) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

58 Risk Neutral Probability Let us assume the existence of a pair ( p, q) of positive numbers, and use these to multiply our pricing equation(s): pv 1 (H) = p(x 0 0 S 0 )(1 + r) + p 0 us 0 qv 1 (T ) = q(x 0 0 S 0 )(1 + r) + q 0 ds 0 Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

59 Risk Neutral Probability Let us assume the existence of a pair ( p, q) of positive numbers, and use these to multiply our pricing equation(s): pv 1 (H) = p(x 0 0 S 0 )(1 + r) + p 0 us 0 qv 1 (T ) = q(x 0 0 S 0 )(1 + r) + q 0 ds 0 Addition yields X 0 (1 + r) + 0 S 0 ( pu + qd (1 + r)) = pv 1 (H) + qv 1 (T ) (12) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

60 If we constrain 0 = pu + qd (1 + r) 1 = p + q 0 p 0 q Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

61 If we constrain 0 = pu + qd (1 + r) 1 = p + q 0 p 0 q then we have a risk neutral probability P where V 0 = X 0 = r Ẽ[V 1] = pv 1(H) + qv 1 (T ) 1 + r p = P[X 1 (ω) = H] = 1 + r d u d u (1 + r) q = P[X 1 (ω) = T ] = u d Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

62 Example: Pricing a forward contract Consider the case of a stock with S 0 = 400 u = 1.25 d = 0.75 r = 0.05 Then the forward price is computed via Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

63 Example: Pricing a forward contract Consider the case of a stock with S 0 = 400 u = 1.25 d = 0.75 r = 0.05 Then the forward price is computed via 0 = r Ẽ[S 1 F ] F = Ẽ[S 1 ] (13) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

64 This leads to the explicit price F = pus 0 + qds 0 = ( p)(1.25)(400) + (1 p)(0.75)(400) = 500 p p = p = = = 420 Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

65 This leads to the explicit price F = pus 0 + qds 0 = ( p)(1.25)(400) + (1 p)(0.75)(400) = 500 p p = p = = = 420 Homework Question: What is the price of a call option in the case above,with strike K = 375? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

66 General one period risk neutral measure We define a finite set of outcomes Ω {ω 1, ω 2,..., ω n } and any subcollection of outcomes A F 1 := 2 Ω an event. Furthermore, we define a probability measure P, not necessarily the physical measure P to be risk neutral if P[ω] > 0 ω Ω X 0 = 1 1+r Ẽ[X 1] for all strategies X. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

67 General one period risk neutral measure The measure is indifferent to investing in a zero-coupon bond, or a risky asset X Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

68 General one period risk neutral measure The measure is indifferent to investing in a zero-coupon bond, or a risky asset X The same initial capital X 0 in both cases produces the same average return after one period. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

69 General one period risk neutral measure The measure is indifferent to investing in a zero-coupon bond, or a risky asset X The same initial capital X 0 in both cases produces the same average return after one period. Not the physical measure attached by observation, experts, etc.. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

70 General one period risk neutral measure The measure is indifferent to investing in a zero-coupon bond, or a risky asset X The same initial capital X 0 in both cases produces the same average return after one period. Not the physical measure attached by observation, experts, etc.. In fact, physical measure has no impact on pricing Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

71 Example: Risk Neutral measure for trinomial case Assume that Ω = {ω 1, ω 2, ω 3 } with S 1 (ω 1 ) = us 0 S 1 (ω 2 ) = S 0 S 1 (ω 3 ) = ds 0 Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

72 Example: Risk Neutral measure for trinomial case Assume that Ω = {ω 1, ω 2, ω 3 } with S 1 (ω 1 ) = us 0 S 1 (ω 2 ) = S 0 S 1 (ω 3 ) = ds 0 Given a payoff V 1 (ω) to replicate, are we assured that a replicating strategy exists? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

73 Example: Risk Neutral measure for trinomial case Homework: Try our first example with (S 0, u, d, r) = (400, , 0.05) V digital 1 (ω) = 1 {S1 (ω)>450}(ω). Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

74 Example: Risk Neutral measure for trinomial case Homework: Try our first example with (S 0, u, d, r) = (400, , 0.05) V digital 1 (ω) = 1 {S1 (ω)>450}(ω). Now, assume you are observe the price on the market to be V digital 0 = 1 digital Ẽ[V1 ] = (14) 1 + r Use this extra information to price a call option with strike K = 420. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

75 Solution: Risk Neutral measure for trinomial case The above scenario is reduced to finding the risk-neutral measure ( p 1, p 2, p 3 ). This can be done by finding the rref of the matrix M: M = (15) (1.05) which results in rref (M) = (16) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

76 Solution: Risk Neutral measure for trinomial case It follows that ( p 1, p 2, p 3 ) = (0.2625, 0.675, ), and so V C 0 = Ẽ[(S 1 420) + S 0 = 400] = ( ) = 20. (17) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

77 Solution: Risk Neutral measure for trinomial case It follows that ( p 1, p 2, p 3 ) = (0.2625, 0.675, ), and so V C 0 = Ẽ[(S 1 420) + S 0 = 400] = ( ) = 20. (17) Could we perhaps find a set of digital options as a basis set { } V1 d1 (ω), V1 d2 (ω), V1 d3 (ω) = {1 A1 (ω), 1 A2 (ω), 1 A3 (ω)} (18) with A 1, A 2, A 3 F 1 to span all possible payoffs at time 1? How about (A 1, A 2, A 3 ) = ({ω 1 }, {ω 2 }, {ω 3 })? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

78 Exchange one stock for another Assume now an economy with two stocks, X and Y. Assume that and (X 0, Y 0, r) = (100, 100, 0.01) (19) (110, 105) : ω = ω 1 (X 1 (ω), Y 1 (ω)) = (100, 100) : ω = ω 2 (80, 95) : ω = ω 3. Consider two contracts, V and W, with payoffs Price V 0 and W 0. V 1 (ω) = max {Y 1 (ω) X 1 (ω), 0} W 1 (ω) = Y 1 (ω) X 1 (ω). (20) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

79 Exchange one stock for another In this case, our matrix M is such that M = (21) which results in It follows that rref (M) = (22) W 0 = Ẽ[Y 1] Ẽ[X 1 ] = Y 0 X 0 = V 0 = (15 p 3) = = (23) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

80 Existence of Risk Neutral measure Let P be a probability measure on a finite space Ω. The following are equivalent: Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

81 Existence of Risk Neutral measure Let P be a probability measure on a finite space Ω. The following are equivalent: P is a risk neutral measure Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

82 Existence of Risk Neutral measure Let P be a probability measure on a finite space Ω. The following are equivalent: P is a risk neutral measure For all traded securities S i, S0 i = 1 1+r Ẽ [ S1 i ] Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

83 Existence of Risk Neutral measure Let P be a probability measure on a finite space Ω. The following are equivalent: P is a risk neutral measure For all traded securities S i, S0 i = 1 1+r Ẽ [ S1 i ] Proof: Homework (Hint: One direction is much easier than others. Also, strategies are linear in the underlying asset.) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

84 Complete Markets A market is complete if it is arbitrage free and every non-traded asset can be replicated. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

85 Complete Markets A market is complete if it is arbitrage free and every non-traded asset can be replicated. Fundamental Theorem of Asset Pricing 1: A market is arbitrage free iff there exists a risk neutral measure Fundamental Theorem of Asset Pricing 2: A market is complete iff there exists exactly one risk neutral measure Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

86 Complete Markets A market is complete if it is arbitrage free and every non-traded asset can be replicated. Fundamental Theorem of Asset Pricing 1: A market is arbitrage free iff there exists a risk neutral measure Fundamental Theorem of Asset Pricing 2: A market is complete iff there exists exactly one risk neutral measure Proof(s): We will go over these in detail later! Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

87 Optimal Investment for a Strictly Risk Averse Investor Assume a complete market, with a unique risk-neutral measure P. Characterize an investor by her pair (x, U) of initial capital x X and utility function U : X R +. Assume U (x) > 0. Assume U (x) < 0. Define the Radon-Nikodym derivative of P to P as the random variable Z(ω) := P(ω) P(ω). (24) Note that Z is used to map expectations under P to expectations under P: For any random variable X, it follows that Ẽ[X ] = E[ZX ]. (25) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

88 Optimal Investment for a Strictly Risk Averse Investor A strictly risk-averse investor now wishes to maximize her expected utility of a portfolio at time 1, given initial capital at time 0: u(x) := max X 1 A x E[U(X 1 )] A x := { all portfolio values at time 1 with initial capital x}. (26) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

89 Optimal Investment for a Strictly Risk Averse Investor Theorem Define ˆX 1 via the relationship ( ) U ˆX 1 := λz (27) where λ sets ˆX 1 as a strategy with an average return of r under P: Ẽ[ ˆX 1 ] = x(1 + r). (28) Then ˆX 1 is the optimal strategy. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

90 Optimal Investment for a Strictly Risk Averse Investor Proof. Assume X 1 to be an arbitrary strategy with initial capital x. Then for f (y) := E[U(yX 1 + (1 y) ˆX 1 )] (29) it follows that [ )] f (0) = E U ( ˆX 1 ) (X 1 ˆX 1 [ = E λz (X 1 ˆX )] 1 )] = λẽ [(X 1 ˆX 1 = 0 [ f (y) = E U (yx 1 + (1 y) ˆX 1 ) (X 1 ˆX ) ] 2 1 < 0 (30) and so f attains its maximum at y = 0. We conclude that E[U(X 1 )] < E[U( ˆX 1 )] for any admissible strategy X 1. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

91 Optimal Investment: Example Assume an investor and economy defined by It follows that U(x) = ln (x) (S 0, u, d, p, q, r) = (400, , 0.5, 0.5, 0.05). Since U (x) = 1 x, we have ( 3 ( p, q) = 5, 2 ) 5 ( 6 (Z(H), Z(T )) = 5, 4 ). 5 ˆX 1 (ω) = 1 1 λ Z(ω) x = X 0 = r Ẽ[ ˆX 1 ] = 1 ( p 1 p 1 + r λ p + q 1 ) q = 1 1 λ q λ 1 + r. (31) (32) (33) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

92 Optimal Investment: Example Combining the previous results, we see that ˆX 1 (ω) = x(1 + r) Z(ω) u(x) = p ln ˆX 1 (H) + (1 p) ln ˆX 1 (T ) ( ) ( ) x(1 + r) x(1 + r) = p ln + (1 p) ln Z(H) Z(T ) ( ) (1 + r) = ln Z(H) p Z(T ) 1 p x = ln (1.0717x) > ln (1.05x). (34) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

93 Optimal Investment: Example In terms of her actual strategy, we see that ˆπ 0 := ˆ 0 S 0 = S 0 X 0 x = 1 + r u d ˆX 1 (H) ˆX 1 (T ) S 1 (H) S 1 (T ) = 1 + r ( 1 u d Z(H) 1 ) Z(T ) ( p p 1 p ) = 1.05 ( 5 1 p ) = (35) Therefore, the optimal strategy is to sell a stock portfolio worth 87.5% of her initial wealth x and invest the proceeds into a safe bank account. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

94 Optimal Investment: Example ( ) In fact, since ˆπ 0 = 1+r u d p p 1 p 1 p, we see that qualitatively, her optimal strategy involves > 0 : p > p ˆπ 0 = = 0 : p = p < 0 : p < p. This links with her strategy via 1 + ˆr 1 (ω) := ˆX 1 (ω) X 0 = (1 ˆπ 0 )(1 + r) + ˆπ 0 S 1 (ω) S 0 (36) and so for our specific case where (r, u, d, ˆπ 0 ) = (0.05, 1.25, 0.75, 0.875), we have 1 + ˆr 1 (H) = (1 ˆπ 0 )(1 + r) + ˆπ 0 u = ˆr 1 (T ) = (1 ˆπ 0 )(1 + r) + ˆπ 0 d = (37) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

95 Optimal Investment: U(x) = x Consider now the same set-up as before, only that the utility function changes to U(x) = x. It follows that Solving for λ returns U ( ˆX 1 ) = ˆX1 ˆX 1 = 1 1 4λ 2 Z 2 (38) x(1 + r) = Ẽ[ ˆX 1 ] = E[Z ˆX 1 ] [ = E Z 1 ] 1 4λ 2 Z 2 = 1 [ ] 1 4λ 2 E. Z (39) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

96 Optimal Investment: U(x) = x Combining the results above, we see that x(1 + r) ˆX 1 = Z 2 E [ ] 1 Z [ ] [ ] x(1 + r) u(x) = E ˆX 1 = E Z 2 E [ ] 1 Z = [ ] 1 x(1 + r) E. Z (40) Question: Is it true for all (p, p) (0, 1) (0, 1) that [ ] 1 p E = 2 (1 p)2 + 1? (41) Z p 1 p Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

97 Optimal Betting at the Omega Horse Track! Imagine our investor with U(x) = ln (x) visits a horse track. There are three horses: ω 1, ω 2 and ω 3. She can bet on any of the horses to come in 1 st. The payoff is 1 per whole bet made. She observes the price of each bet with payoff 1 right before the race to be (B 1 0, B 2 0, B 3 0 ) = (0.5, 0.3, 0.2). (42) Symbolically, B i 1(ω) = 1 {ωi }(ω). (43) Our investor feels the physical probabilities of each horse winning is (p 1, p 2, p 3 ) = (0.6, 0.35, 0.05). (44) How should she bet if the race is about to start? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

98 Optimal Betting at the Omega Horse Track! In this setting, we can assume r = 0. This directly implies that ( p 1, p 2, p 3 ) = (0.5, 0.3, 0.2). (45) Our Radon-Nikodym derivative of P to P is now ( 0.5 (Z(ω 1 ), Z(ω 2 ), Z(ω 3 )) = 0.6, , 0.2 ). (46) 0.05 Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

99 Optimal Betting at the Omega Horse Track! Her optimal strategy ˆX 1 reflects her betting strategy, and satisfies ˆX 1 (ω) = X 0 Z(ω) ( ˆX 1 (ω 1 ), ˆX 1 (ω 2 ), ˆX ) 3 (ω 1 ) = X 0 X 0 X 0 ( 6 5, 7 6, 1 ). 4 (47) So, per dollar of wealth, she buys 6 5 of a bet for Horse 1 to win, 7 6 of a bet for Horse 2 to win, and 1 4 of a bet for Horse 3 to win. The total price (per dollar of wealth) is thus = = 1. (48) 4 Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

100 Dividends What about dividends? How do they affect the risk neutral pricing of exchange and non-exchange traded assets? What if they are paid at discrete times? Continuously paid? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

101 Dividends What about dividends? How do they affect the risk neutral pricing of exchange and non-exchange traded assets? What if they are paid at discrete times? Continuously paid? Recall that if dividends are paid continuously at rate δ, then 1 share at time 0 will accumulate to e δt shares upon reinvestment of dividends into the stock until time T. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

102 Dividends What about dividends? How do they affect the risk neutral pricing of exchange and non-exchange traded assets? What if they are paid at discrete times? Continuously paid? Recall that if dividends are paid continuously at rate δ, then 1 share at time 0 will accumulate to e δt shares upon reinvestment of dividends into the stock until time T. It follows that to deliver one share of stock S with initial price S 0 at time T, only e δt shares are needed. Correspondingly, F prepaid = e δt S 0 F = e rt e δt S 0 = e (r δ)t S 0. (49) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

103 Binomial Option Pricing w/ cts Dividends and Interest Over a period of length h, interest increases the value of a bond by a factor e rh and dividends the value of a stock by a factor of e δh. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

104 Binomial Option Pricing w/ cts Dividends and Interest Over a period of length h, interest increases the value of a bond by a factor e rh and dividends the value of a stock by a factor of e δh. Once again, we compute pathwise, V 1 (H) = (X 0 0 S 0 )e rh + 0 e δh us 0 V 1 (T ) = (X 0 0 S 0 )e rh + 0 e δh ds 0 and this results in the modified quantities 0 = e δh V 1(H) V 1 (T ) (u d)s 0 p = e(r δ)h d u d q = u e(r δ)h u d Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

105 Binomial Models w/ cts Dividends and Interest For σ, the annualized standard deviation of continuously compounded stock return, the following models hold: Futures - Cox (1979) u = e σ h d = e σ h. General Stock Model u = e (r δ)h+σ h d = e (r δ)h σ h. Currencies with r f the foreign interest rate, which acts as a dividend: u = e (r r f )h+σ h d = e (r r f )h σ h. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

106 1- and 2-period pricing Consider the case r = 0.10, δ = 0.05, h = 0.01, σ = 0.1, S 0 = 10 = K. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

107 1- and 2-period pricing Consider the case r = 0.10, δ = 0.05, h = 0.01, σ = 0.1, S 0 = 10 = K. Now price two digital options, using the 1 General Stock Model 2 Futures-Cox Model with respective payoffs V K 1 (ω) := 1 {S1 K}(ω) V K 2 (ω) := 1 {S2 K}(ω). Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

108 Calibration Exercise Assume table below of realized gains & losses over a ten-period cycle. Use the adjusted values (r, δ, h, S 0, K) = (0.02, 0, 0.10, 10, 10). Calculate binary options from last slide using these assumptions. Period Return S 1 S 0 = 1.05 S 2 S 1 = 1.02 S 3 S 2 = 0.98 S 4 S 3 = 1.01 S 5 S 4 = 1.02 S 6 S 5 = 0.99 S 7 S 6 = 1.03 S 8 S 7 = 1.05 S 9 S 8 = 0.96 S 10 S 9 = 0.97 Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

109 Calibration Exercise: Linear Approximation We would like to compute σ for the logarithm of returns ln ( Si S i 1 ). Assume the returns per period are all independent. Q: Can we use a linear (simple) return model instead of a compound return model as an approximation? If so, then for our observed simple return rate values: Calculate the sample variance σ. 2 Estimate that σ σ h. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

110 Calibration Exercise: Linear Approximation Note that if S i S i 1 = 1 + γ for γ 1, then ( ) Si ln γ = S i S i 1. (50) S i 1 S i 1 Approximation: Convert our previous table, using simple interest. Over small time periods h, define linear return values for i th period: X i h := S i S i 1 S i 1. (51) In other words, for simple rate of return X i for period i: S i = S i 1 (1 + X i h). (52) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

111 Calibration Exercise: Linear Approximation Our returns table now looks like Period Return S 1 1 S 0 S 0 = 0.05 S 2 2 S 1 S 1 = 0.02 S 3 3 S 2 S 2 = 0.02 S 4 4 S 3 S 3 = 0.01 S 5 5 S 4 S 4 = 0.02 S 6 6 S 5 S 5 = 0.01 S 7 7 S 6 S 6 = 0.03 S 8 8 S 7 S 7 = 0.05 S 9 9 S 8 S 8 = 0.04 S S 9 S 9 = 0.03 sample standard deviation σ = estimated return deviation σ Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

112 Calibration Exercise: Linear Approximation We estimate, therefore, that under the Futures-Cox model (u, d) = (e , e ) = (1.0324, ) p = e0.002 e e e = (53) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

113 Calibration Exercise: Linear Approximation For the one-period digital option: V 0 = e rh Ẽ 0 [1 {S1 10}] = e p = (54) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

114 Calibration Exercise: Linear Approximation For the one-period digital option: V 0 = e rh Ẽ 0 [1 {S1 10}] = e p = (54) For the two-period digital option: V 0 = e 2rh Ẽ 0 [1 {S2 10}] = e [ p p q ] = (55) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

115 Calibration Exercise: No Approximation Without the linear approximation, we can directly estimate σ Y h = (u, d) = (e , e ) = (1.0322, ) For the one-period digital option: p = e0.002 e e = e (56) V 0 = e rh Ẽ 0 [1 {S1 10}] = e p = (57) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

116 Calibration Exercise: No Approximation Without the linear approximation, we can directly estimate σ Y h = (u, d) = (e , e ) = (1.0322, ) For the one-period digital option: p = e0.002 e e = e (56) V 0 = e rh Ẽ 0 [1 {S1 10}] = e p = (57) For the two-period digital option: V 0 = e 2rh Ẽ 0 [1 {S2 10}] = e [ p p q ] = (58) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

117 1- and 2-period pricing We can solve for 2-period problems Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

118 1- and 2-period pricing We can solve for 2-period problems on a case-by-case basis, or by developing a general theory for multi-period asset pricing. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

119 1- and 2-period pricing We can solve for 2-period problems on a case-by-case basis, or by developing a general theory for multi-period asset pricing. In the latter method, we need a general framework to carry out our computations Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

120 Risk Neutral Pricing Formula Assume now that we have the regular assumptions on our coin flip space, and that at time N we are asked to deliver a path dependent derivative value V N. Then for times 0 n N, the value of this derivative is computed via Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

121 Risk Neutral Pricing Formula Assume now that we have the regular assumptions on our coin flip space, and that at time N we are asked to deliver a path dependent derivative value V N. Then for times 0 n N, the value of this derivative is computed via V n = e rh Ẽ n [V n+1 ] (59) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

122 Risk Neutral Pricing Formula Assume now that we have the regular assumptions on our coin flip space, and that at time N we are asked to deliver a path dependent derivative value V N. Then for times 0 n N, the value of this derivative is computed via and so V n = e rh Ẽ n [V n+1 ] (59) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

123 Risk Neutral Pricing Formula Assume now that we have the regular assumptions on our coin flip space, and that at time N we are asked to deliver a path dependent derivative value V N. Then for times 0 n N, the value of this derivative is computed via and so V n = e rh Ẽ n [V n+1 ] (59) X 0 = Ẽ 0 [X N ] X n := V n e nh. (60) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

124 Computational Complexity Consider the case p = q = 1 2 (61) but now with term n = 3. S 0 = 4, u = 4 3, d = 3 4 There are 2 3 = 8 paths to consider. However, there are = 4 unique final values of S 3 to consider. In the general term N, there would be 2 N paths to generate S N, but only N + 1 distinct values. At any node n units of time into the asset s evolution, there are n + 1 distinct values. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

125 Computational Complexity At each value s for S n, we know that S n+1 = 4 3 s or S n+1 = 3 4 s. Using multi-period risk-neutral pricing, we can generate for v n (s) := V n (S n (ω 1,..., ω n )) on the node (event) S n (ω 1,..., ω n ) = s: v n (s) = e rh[ pv n+1 ( 4 3 s ) + qv n+1 ( 3 4 s )]. (62) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

126 An Example: Assume r, δ, and h are such that It follows that Compute V 0. p = 1 2 = q, e rh = 9 10 S 0 = 4, u = 2, d = 1 2 V 3 := max {10 S 3, 0}. v 3 (32) = 0 v 3 (8) = 2 v 3 (2) = 8 v 3 (0.50) = (63) (64) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

127 Markov Processes If we use the above approach for a more exotic option, say a lookback option that pays the maximum over the term of a stock, then we find this approach lacking. There is not enough information in the tree or the distinct values for S 3 as stated. We need more. Consider our general multi-period binomial model under P. Definition We say that a process X is adapted if it depends only on the sequence of flips ω := (ω 1,..., ω n ) Definition We say that an adapted process X is Markov if for every 0 n N 1 and every function f (x) there exists another function g(x) such that Ẽ n [f (X n+1 )] = g(x n ). (65) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

128 Markov Processes This notion of Markovity is essential to our state-dependent pricing algorithm. Indeed, our stock process evolves from time n to time n + 1, using only the information in S n. We can in fact say that for every f (s) there exists a g(s) such that In fact, that g depends on f : g(s) = Ẽ n [f (S n+1 ) S n = s]. (66) g(s) = e rh[ pf ( 4 ) ( 3 )] 3 s + qf 4 s. (67) So, for any f (s) := V N (s), we can work our recursive algorithm backwards to find the g n (s) := V n (s) for all 0 n N 1 Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

129 Markov Processes Some more thoughts on Markovity: Consider the example of a Lookback Option. Here, the payoff is dependent on the realized maximum M n := max 0 i n S i of the asset. M n is not Markov by itself, but the two-factor process (M n, S n ) is. Why? Let s generate the tree! Homework Can you think of any other processes that are not Markov? Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

130 Call Options on Zero-Coupon Bonds Assume an economy where One period is one year The one year short term interest rate from time n to time n + 1 is r n. The rate evolves via a stochastic process: r 0 = 0.02 r n+1 = Xr n P[X = 2 k ] = 1 3 for k { 1, 0, 1}. (68) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

131 Call Options on Zero-Coupon Bonds Consider now a zero-coupon bond that matures in 3 years with common face and redemption value F = 100. Also consider a call option on this bond that expires in 2 years with strike K = 97. Denote B n and C n as the bond and call option values, respectively. Note that we iterate backwards from the values B 3 (r) = 100 C 2 (r) = max {B 2 (r) 97, 0}. (69) Compute (B 0, C 0 ). Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

132 Call Options on Zero-Coupon Bonds Our general recursive formula is B n (r) = r Ẽ[B n+1(r n+1 ) r n = r] C n (r) = r Ẽ[C n+1(r n+1 ) r n = r]. (70) Iterating backwards, we see that at t = 2, B 2 (r) = r 3 At time t = 2, we have that 1 k= 1 B 3 (2 k r). (71) r 2 {0.08, 0.04, 0.02, 0.01, 0.005}. (72) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

133 Call Options on Zero-Coupon Bonds Our associated Bond and Call Option values at time 2: r 2 B 2 C Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

134 Call Options on Zero-Coupon Bonds Our associated Bond and Call Option values at time 1: r 1 B 1 C Our associated Bond and Call Option values at time 0: r 0 B 0 C Question: What if the delivery time of the option is changed to 3? Symbolically, what if C 3 (r) = max {B 3 (r) 97, 0}? (73) Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

135 Capital Structure Model As an analyst for an investments firm, you are tasked with advising whether a company s stock and/or bonds are over/under-priced. You receive a quarterly report from this company on it s return on assets, and have compiled a table for the last ten quarters below. Today, just after the last quarter s report was issued, you see that in billions of USD, the value of the company s assets is 10. There are presently one billions shares of this company that are being traded. The company does not pay any dividends. Six months from now, the company is required to pay off a billion zero-coupon bonds. Each bond has a face value of 9.5. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

136 Capital Structure Model Assume Miller-Modigliani holds with A t = B t + S t, where the assets of a company equal the sum of its share and bond price. Presently, the market values are (B 0, S 0, r) = (9, 1, 0.02). The Merton model for corporate bond pricing asserts that at redemption time T, B t = e r(t t) Ẽ [min {A T, F }] S t = e r(t t) Ẽ [max {A T F, 0}]. (74) With all of this information, your job now is to issue a Buy or Sell on the stock and the bond issued by this company. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

137 Capital Structure Model Table of return on assets for Company X, with h = Period Return on Assets A 1 1 A 0 A 0 = 0.05 A 2 2 A 1 A 1 = 0.02 A 3 3 A 2 A 2 = 0.02 A 4 4 A 3 A 3 = 0.01 A 5 5 A 4 A 4 = 0.02 A 6 6 A 5 A 5 = 0.01 A 7 7 A 6 A 6 = 0.03 A 8 8 A 7 A 7 = 0.05 A 9 9 A 8 A 8 = 0.04 A A 9 A 9 = 0.03 sample standard deviation σ = estimated return deviation σ Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161

138 Capital Structure Model We scale all of our calculation in terms of billions ($, shares, bonds). Using the Futures- Cox model, we have (u, d) = (e , e ) = (1.0324, ) p = e0.005 e e e = (75) Using this model, the only time the payoff of the bond is less than the face is on the path ω = TT. The price of the bond and stock are thus modeled to be B 0 = e 0.02 (2 0.25) [ p p q q ] = 9.38 > 9.00 S 0 = = 0.62 < (76) It follows that,according to our model, one should Buy the bond as it is underpriced and one should Sell the stock as it is overpriced. Albert Cohen (MSU) Financial Mathematics for Actuaries I MSU Spring / 161