# Lecture 11. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 7

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1 Lecture 11 Sergei Fedotov Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) / 7

2 Lecture 11 1 American Put Option Pricing on Binomial Tree 2 Replicating Portfolio Sergei Fedotov (University of Manchester) / 7

3 American Option An American Option is one that may be exercised at any time prior to expire (t = T). Sergei Fedotov (University of Manchester) / 7

4 American Option An American Option is one that may be exercised at any time prior to expire (t = T). We should determine when it is best to exercise the option. It is not subjective! It can be determined in a systematic way! Sergei Fedotov (University of Manchester) / 7

5 American Option An American Option is one that may be exercised at any time prior to expire (t = T). We should determine when it is best to exercise the option. It is not subjective! It can be determined in a systematic way! The American put option value must be greater than or equal to the payoff function. If P < max(e S,0), then there is obvious arbitrage opportunity. Sergei Fedotov (University of Manchester) / 7

6 American Option An American Option is one that may be exercised at any time prior to expire (t = T). We should determine when it is best to exercise the option. It is not subjective! It can be determined in a systematic way! The American put option value must be greater than or equal to the payoff function. If P < max(e S,0), then there is obvious arbitrage opportunity. We can buy stock for S and option for P and immediately exercise the option by selling stock for E. Sergei Fedotov (University of Manchester) / 7

7 American Option An American Option is one that may be exercised at any time prior to expire (t = T). We should determine when it is best to exercise the option. It is not subjective! It can be determined in a systematic way! The American put option value must be greater than or equal to the payoff function. If P < max(e S,0), then there is obvious arbitrage opportunity. We can buy stock for S and option for P and immediately exercise the option by selling stock for E. E (P + S) > 0 Sergei Fedotov (University of Manchester) / 7

8 American Put Option Pricing on Binomial Tree We denote by P m n the n-th possible value of put option at time-step m t. Sergei Fedotov (University of Manchester) / 7

9 American Put Option Pricing on Binomial Tree We denote by P m n the n-th possible value of put option at time-step m t. European Put Option: P m n = e r t ( pp m+1 n+1 ) + (1 p)pm+1 n. Here 0 n m and the risk-neutral probability p = er t d u d. Sergei Fedotov (University of Manchester) / 7

10 American Put Option Pricing on Binomial Tree We denote by P m n the n-th possible value of put option at time-step m t. European Put Option: P m n = e r t ( pp m+1 n+1 ) + (1 p)pm+1 n. Here 0 n m and the risk-neutral probability p = er t d u d. American Put Option: P m n = max { max(e S m n,0),e r t ( pp m+1 n+1 ) } + (1 p)pm+1 n, where S m n is the n-th possible value of stock price at time-step m t. Sergei Fedotov (University of Manchester) / 7

11 American Put Option Pricing on Binomial Tree We denote by P m n the n-th possible value of put option at time-step m t. European Put Option: P m n = e r t ( pp m+1 n+1 ) + (1 p)pm+1 n. Here 0 n m and the risk-neutral probability p = er t d u d. American Put Option: P m n = max { max(e S m n,0),e r t ( pp m+1 n+1 ) } + (1 p)pm+1 n, where S m n is the n-th possible value of stock price at time-step m t. Final condition: P N n = max ( E S N n,0 ), where n = 0,1,2,...,N, E is the strike price. Sergei Fedotov (University of Manchester) / 7

12 Example: Evaluation of American Put Option on Two-Step Tree We assume that over each of the next two years the stock price either moves up by 20% or moves down by 20%. The risk-free interest rate is 5%. Find the value of a 2-year American put with a strike price of \$52 on a stock whose current price is \$50. Sergei Fedotov (University of Manchester) / 7

13 Example: Evaluation of American Put Option on Two-Step Tree We assume that over each of the next two years the stock price either moves up by 20% or moves down by 20%. The risk-free interest rate is 5%. Find the value of a 2-year American put with a strike price of \$52 on a stock whose current price is \$50. In this case u = 1.2, d = 0.8, r = 0.05, E = 52. Risk-neutral probability: p = e = Sergei Fedotov (University of Manchester) / 7

14 Example: Evaluation of American Put Option on Two-Step Tree P u = e ( ) = Sergei Fedotov (University of Manchester) / 7

15 Example: Evaluation of American Put Option on Two-Step Tree P u = e ( ) = P d = e ( ) = Sergei Fedotov (University of Manchester) / 7

16 Example: Evaluation of American Put Option on Two-Step Tree P u = e ( ) = P d = e ( ) = Payoff: E S = = 12 > Early exercise is optimal! P d = 12 Sergei Fedotov (University of Manchester) / 7

17 Example: Evaluation of American Put Option on Two-Step Tree P u = e ( ) = P d = e ( ) = Payoff: E S = = 12 > Early exercise is optimal! P d = 12 P 0 = e ( ) = Sergei Fedotov (University of Manchester) / 7

18 Example: Evaluation of American Put Option on Two-Step Tree P u = e ( ) = P d = e ( ) = Payoff: E S = = 12 > Early exercise is optimal! P d = 12 P 0 = e ( ) = Payoff: E S = = 2 < Early exercise is not optimal at the initial node Sergei Fedotov (University of Manchester) / 7

19 Replicating Portfolio The aim is to calculate the value of call option C 0. Let us establish a portfolio of stocks and bonds in such a way that the payoff of a call option is completely replicated. Final value: Π T = C T = max(s E,0) Sergei Fedotov (University of Manchester) / 7

20 Replicating Portfolio The aim is to calculate the value of call option C 0. Let us establish a portfolio of stocks and bonds in such a way that the payoff of a call option is completely replicated. Final value: Π T = C T = max(s E,0) To prevent risk-free arbitrage opportunity, the current values should be identical. We say that the portfolio replicates the option. The Law of One Price: Π t = C t. Sergei Fedotov (University of Manchester) / 7

21 Replicating Portfolio The aim is to calculate the value of call option C 0. Let us establish a portfolio of stocks and bonds in such a way that the payoff of a call option is completely replicated. Final value: Π T = C T = max(s E,0) To prevent risk-free arbitrage opportunity, the current values should be identical. We say that the portfolio replicates the option. The Law of One Price: Π t = C t. Consider replicating portfolio of shares held long and N bonds held short. Sergei Fedotov (University of Manchester) / 7

22 Replicating Portfolio The aim is to calculate the value of call option C 0. Let us establish a portfolio of stocks and bonds in such a way that the payoff of a call option is completely replicated. Final value: Π T = C T = max(s E,0) To prevent risk-free arbitrage opportunity, the current values should be identical. We say that the portfolio replicates the option. The Law of One Price: Π t = C t. Consider replicating portfolio of shares held long and N bonds held short. The value of portfolio: Π = S NB. A pair (,N) is called a trading strategy. How to find (,N) such that Π T = C T and Π 0 = C 0? Sergei Fedotov (University of Manchester) / 7

23 Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. Sergei Fedotov (University of Manchester) / 7

24 Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The value of portfolio: Π = S NB. When stock moves up: S 0 u NB 0 e rt = C u. When stock moves down: S 0 d NB 0 e rt = C d. Sergei Fedotov (University of Manchester) / 7

25 Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The value of portfolio: Π = S NB. When stock moves up: S 0 u NB 0 e rt = C u. When stock moves down: S 0 d NB 0 e rt = C d. We have two equations for two unknown variables and N. Sergei Fedotov (University of Manchester) / 7

26 Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The value of portfolio: Π = S NB. When stock moves up: S 0 u NB 0 e rt = C u. When stock moves down: S 0 d NB 0 e rt = C d. We have two equations for two unknown variables and N. Current value: C 0 = S 0 NB 0. Sergei Fedotov (University of Manchester) / 7

27 Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The value of portfolio: Π = S NB. When stock moves up: S 0 u NB 0 e rt = C u. When stock moves down: S 0 d NB 0 e rt = C d. We have two equations for two unknown variables and N. Current value: C 0 = S 0 NB 0. Prove: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. (Exercise sheet 5) Sergei Fedotov (University of Manchester) / 7

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