Stochastic Calculus for Finance

Stochastic Calculus for Finance Albert Cohen Actuarial Sciences Program Department of Mathematics Department of Statistics and Probability A336 Wells Hall Michigan State University East Lansing MI 48823 albert@math.msu.edu acohen@stt.msu.edu Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 1 / 249

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 https://www.stt.msu.edu/academics/classpages/ Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 2 / 249

Course Information Many examples within these slides are used with kind permission of Prof. Dmitry Kramkov, Dept. of Mathematics, Carnegie Mellon University. Course Textbook: Stochastic Calculus for Finance (Springer Finance.) 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 Springer. 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) STT 888: Stochastic Calculus for Finance MSU 2016 3 / 249

What are financial securities? Traded Securities - price given by market. For example: Stocks Commodities Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 4 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 4 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 4 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 4 / 249

How does one fairly price non-traded securities? Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 5 / 249

How does one fairly price non-traded securities? By eliminating all unfair prices Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 5 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 5 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 5 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 6 / 249

And What About... Does the replicating strategy and price computed reflect uncertainty in the market? Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 7 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 7 / 249

Notation Forward Contract: Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 8 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 8 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 8 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 9 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 9 / 249

An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 10 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 10 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 10 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 10 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 11 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 11 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 11 / 249

An Example of Replication: Solution Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 12 / 249

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 = 3.667 (4) 1 + r B Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 12 / 249

Outline 1 Discrete Multiperiod Model Arbitrage Risk Neutral Probability American Options Exotic Options Valuation via Simulation 2 Continuous Model-Ito Calculus Brownian Motion BSM Examples Options on Futures Path Dependent Options 3 Advanced Topics Heat Equation General Solution of Heat Equation Application to B-S-M PDE 4 Continuous Model-Probability Expected Values Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 13 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 14 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 14 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 14 / 249

Arbitrage Assume that S 0 (1 + r) > us 0 Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 15 / 249

Arbitrage Assume that S 0 (1 + r) > us 0 Where is the risk involved with investing in the asset S? Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 15 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 15 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 15 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 15 / 249

Derivative Pricing Let S 1 (ω) be the price of an underlying asset at time 1. Define the following instruments: Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 16 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 16 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 16 / 249

Put-Call Parity Can we replicate a forward contract using zero coupon bonds and put and call options? Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 17 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 17 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 17 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 17 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 17 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 17 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 18 / 249

Replicating Strategy Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 19 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 19 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 19 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 19 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 20 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 20 / 249

If we constrain 0 = pu + qd (1 + r) 1 = p + q 0 p 0 q Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 21 / 249

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 = 1 1 + 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) STT 888: Stochastic Calculus for Finance MSU 2016 21 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 22 / 249

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 = 1 1 + r Ẽ[S 1 F ] F = Ẽ[S 1 ] (13) Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 22 / 249

This leads to the explicit price F = pus 0 + qds 0 = ( p)(1.25)(400) + (1 p)(0.75)(400) = 500 p + 300 300 p = 300 + 200 p 1 + 0.05 0.75 = 300 + 200 = 300 + 200 3 1.25 0.75 5 = 420 Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 23 / 249

This leads to the explicit price F = pus 0 + qds 0 = ( p)(1.25)(400) + (1 p)(0.75)(400) = 500 p + 300 300 p = 300 + 200 p 1 + 0.05 0.75 = 300 + 200 = 300 + 200 3 1.25 0.75 5 = 420 Homework Question: What is the price of a call option in the case above,with strike K = 375? Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 23 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 24 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 25 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 25 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 25 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 25 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 26 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 26 / 249

Example: Risk Neutral measure for trinomial case Homework: Try our first example with (S 0, u, d, r) = (400, 1.25.0.75, 0.05) V digital 1 (ω) = 1 {S1 (ω)>450}(ω). Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 27 / 249

Example: Risk Neutral measure for trinomial case Homework: Try our first example with (S 0, u, d, r) = (400, 1.25.0.75, 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 ] = 0.25. (14) 1 + r Use this extra information to price a call option with strike K = 420. Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 27 / 249

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: 1 1 1 1 M = 500 400 300 420 (15) 1 0 0 0.25(1.05) which results in 1 0 0 0.2625 rref (M) = 0 1 0 0.675. (16) 0 0 1 0.0625 Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 28 / 249

Solution: Risk Neutral measure for trinomial case It follows that ( p 1, p 2, p 3 ) = (0.2625, 0.675, 0.0625), and so V C 0 = 1 1.05Ẽ[(S 1 420) + S 0 = 400] = 0.2625 1.05 (500 420) = 20. (17) Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 29 / 249

Solution: Risk Neutral measure for trinomial case It follows that ( p 1, p 2, p 3 ) = (0.2625, 0.675, 0.0625), and so V C 0 = 1 1.05Ẽ[(S 1 420) + S 0 = 400] = 0.2625 1.05 (500 420) = 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) STT 888: Stochastic Calculus for Finance MSU 2016 29 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 30 / 249

Exchange one stock for another In this case, our matrix M is such that 1 1 1 1 M = 110 100 80 101 (21) 105 100 95 101 which results in It follows that 3 1 0 0 10 rref (M) = 6 0 1 0 10. (22) 1 0 0 1 10 W 0 = Ẽ[Y 1] Ẽ[X 1 ] = Y 0 X 0 = 0 1.01 V 0 = 1 1.01 (15 p 3) = 1.5 1.01 = 1.49. (23) Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 31 / 249

Existence of Risk Neutral measure Let P be a probability measure on a finite space Ω. The following are equivalent: Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 32 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 32 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 32 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 32 / 249

Complete Markets A market is complete if it is arbitrage free and every non-traded asset can be replicated. Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 33 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 33 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 33 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 34 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 35 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 36 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 37 / 249

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, 1.25.0.75, 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 = 1 1 + 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) STT 888: Stochastic Calculus for Finance MSU 2016 38 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 39 / 249

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 0.5 6 5 ) = 0.875. 4 (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) STT 888: Stochastic Calculus for Finance MSU 2016 40 / 249

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 = 0.875 1 + ˆr 1 (T ) = (1 ˆπ 0 )(1 + r) + ˆπ 0 d = 1.3125. (37) Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 41 / 249

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 ) = 1 1 2 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) STT 888: Stochastic Calculus for Finance MSU 2016 42 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 43 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 44 / 249

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, 0.3 0.35, 0.2 ). (46) 0.05 Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 45 / 249

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 6 5 0.5 + 7 6 0.3 + 1 0.2 = 0.6 + 0.35 + 0.05 = 1. (48) 4 Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 46 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 47 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 48 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 49 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 50 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 50 / 249

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 1 2 3 4 5 6 7 8 9 10 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) STT 888: Stochastic Calculus for Finance MSU 2016 51 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 52 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 53 / 249

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 10 10 S 9 S 9 = 0.03 sample standard deviation σ = 0.0319 estimated return deviation σ 0.1001 Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 54 / 249

Calibration Exercise: Linear Approximation We estimate, therefore, that under the Futures-Cox model (u, d) = (e 0.0319, e 0.0319 ) = (1.0324, 0.9686) p = e0.002 e 0.0319 e 0.0319 e 0.0319 = 0.5234. (53) Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 55 / 249

Calibration Exercise: Linear Approximation For the one-period digital option: V 0 = e rh Ẽ 0 [1 {S1 10}] = e 0.002 p = 0.5224. (54) Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 56 / 249

Calibration Exercise: Linear Approximation For the one-period digital option: V 0 = e rh Ẽ 0 [1 {S1 10}] = e 0.002 p = 0.5224. (54) For the two-period digital option: V 0 = e 2rh Ẽ 0 [1 {S2 10}] = e 0.004 [ p 2 + 2 p q ] = 0.7698. (55) Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 56 / 249

Calibration Exercise: No Approximation Without the linear approximation, we can directly estimate σ Y h = 0.03172 (u, d) = (e 0.03172, e 0.03172 ) = (1.0322, 0.9688) For the one-period digital option: p = e0.002 e 0.03172 e 0.03172 = 0.5246. e 0.03172 (56) V 0 = e rh Ẽ 0 [1 {S1 10}] = e 0.002 p = 0.5236. (57) Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 57 / 249

Calibration Exercise: No Approximation Without the linear approximation, we can directly estimate σ Y h = 0.03172 (u, d) = (e 0.03172, e 0.03172 ) = (1.0322, 0.9688) For the one-period digital option: p = e0.002 e 0.03172 e 0.03172 = 0.5246. e 0.03172 (56) V 0 = e rh Ẽ 0 [1 {S1 10}] = e 0.002 p = 0.5236. (57) For the two-period digital option: V 0 = e 2rh Ẽ 0 [1 {S2 10}] = e 0.004 [ p 2 + 2 p q ] = 0.7721. (58) Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 57 / 249

1- and 2-period pricing We can solve for 2-period problems Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 58 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 58 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 58 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 59 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 59 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 59 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 59 / 249

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 3 + 1 = 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) STT 888: Stochastic Calculus for Finance MSU 2016 60 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 61 / 249

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) = 9.50. (63) (64) Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 62 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 63 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 64 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 65 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 66 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 67 / 249

Call Options on Zero-Coupon Bonds Our general recursive formula is B n (r) = 1 1 + r Ẽ[B n+1(r n+1 ) r n = r] C n (r) = 1 1 + r Ẽ[C n+1(r n+1 ) r n = r]. (70) Iterating backwards, we see that at t = 2, B 2 (r) = 1 1 1 + 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) STT 888: Stochastic Calculus for Finance MSU 2016 68 / 249

Call Options on Zero-Coupon Bonds Our associated Bond and Call Option values at time 2: r 2 B 2 C 2 0.08 92.59 0 0.04 96.15 0 0.02 98.04 1.04 0.01 99.01 2.01 0.005 99.50 2.50 Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 69 / 249

Call Options on Zero-Coupon Bonds Our associated Bond and Call Option values at time 1: r 1 B 1 C 1 0.04 91.92 0.33 0.02 95.82 1.00 0.01 97.87 1.83 Our associated Bond and Call Option values at time 0: r 0 B 0 C 0 0.02 93.34 1.03 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) STT 888: Stochastic Calculus for Finance MSU 2016 70 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 71 / 249

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) STT 888: Stochastic Calculus for Finance MSU 2016 72 / 249

Capital Structure Model Table of return on assets for Company X, with h = 0.25. 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 10 10 A 9 A 9 = 0.03 sample standard deviation σ = 0.0319 estimated return deviation σ 0.0638 Albert Cohen (MSU) STT 888: Stochastic Calculus for Finance MSU 2016 73 / 249

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 0.0319, e 0.0319 ) = (1.0324, 0.9686) p = e0.005 e 0.0319 e 0.0319 e 0.0319 = 0.5706. (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 2 9.5 + 2 p q 9.5 + q 2 9.38 ] = 9.38 > 9.00 S 0 = 10 9.38 = 0.62 < 1.00. (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) STT 888: Stochastic Calculus for Finance MSU 2016 74 / 249