Lecture 10. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 7

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1 Lecture 10 Sergei Fedotov Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) / 7

2 Lecture 10 1 Binomial Model for Stock Price 2 Option Pricing on Binomial Tree 3 Matching Volatility σ with u and d Sergei Fedotov (University of Manchester) / 7

3 Binomial model for the stock price Continuous random model for the stock price: ds = µsdt + σsdw Sergei Fedotov (University of Manchester) / 7

4 Binomial model for the stock price Continuous random model for the stock price: ds = µsdt + σsdw The binomial model for the stock price is a discrete time model: The stock price S changes only at discrete times t, 2 t, 3 t,... Sergei Fedotov (University of Manchester) / 7

5 Binomial model for the stock price Continuous random model for the stock price: ds = µsdt + σsdw The binomial model for the stock price is a discrete time model: The stock price S changes only at discrete times t, 2 t, 3 t,... The price either moves up S Su or down S Sd with d < e r t < u. Sergei Fedotov (University of Manchester) / 7

6 Binomial model for the stock price Continuous random model for the stock price: ds = µsdt + σsdw The binomial model for the stock price is a discrete time model: The stock price S changes only at discrete times t, 2 t, 3 t,... The price either moves up S Su or down S Sd with d < e r t < u. The probability of up movement is q. Sergei Fedotov (University of Manchester) / 7

7 Binomial model for the stock price Continuous random model for the stock price: ds = µsdt + σsdw The binomial model for the stock price is a discrete time model: The stock price S changes only at discrete times t, 2 t, 3 t,... The price either moves up S Su or down S Sd with d < e r t < u. The probability of up movement is q. Let us build up a tree of possible stock prices. The tree is called a binomial tree, because the stock price will either move up or down at the end of each time period. Each node represents a possible future stock price. Sergei Fedotov (University of Manchester) / 7

8 Binomial model for the stock price Continuous random model for the stock price: ds = µsdt + σsdw The binomial model for the stock price is a discrete time model: The stock price S changes only at discrete times t, 2 t, 3 t,... The price either moves up S Su or down S Sd with d < e r t < u. The probability of up movement is q. Let us build up a tree of possible stock prices. The tree is called a binomial tree, because the stock price will either move up or down at the end of each time period. Each node represents a possible future stock price. We divide the time to expiration T into several time steps of duration t = T/N, where N is the number of time steps in the tree. Sergei Fedotov (University of Manchester) / 7

9 Binomial model for the stock price Continuous random model for the stock price: ds = µsdt + σsdw The binomial model for the stock price is a discrete time model: The stock price S changes only at discrete times t, 2 t, 3 t,... The price either moves up S Su or down S Sd with d < e r t < u. The probability of up movement is q. Let us build up a tree of possible stock prices. The tree is called a binomial tree, because the stock price will either move up or down at the end of each time period. Each node represents a possible future stock price. We divide the time to expiration T into several time steps of duration t = T/N, where N is the number of time steps in the tree. Example: Let us sketch the binomial tree for N = 4. Sergei Fedotov (University of Manchester) / 7

10 Stock Price Movement in the Binomial Model We introduce the following notations: S m n is the n-th possible value of stock price at time-step m t. Sergei Fedotov (University of Manchester) / 7

11 Stock Price Movement in the Binomial Model We introduce the following notations: S m n is the n-th possible value of stock price at time-step m t. Then Sn m = u n d m n S0 0, where n = 0,1,2,...,m. S0 0 is the stock price at the time t = 0. Note that u and d are the same at every node in the tree. Sergei Fedotov (University of Manchester) / 7

12 Stock Price Movement in the Binomial Model We introduce the following notations: S m n is the n-th possible value of stock price at time-step m t. Then Sn m = u n d m n S0 0, where n = 0,1,2,...,m. S0 0 is the stock price at the time t = 0. Note that u and d are the same at every node in the tree. For example, at the third time-step 3 t, there are four possible stock prices: S 3 0 = d3 S 0 0, S3 1 = ud2 S 0 0, S3 2 = u2 ds 0 0 and S3 3 = u3 S 0 0. At the final time-step N t, there are N + 1 possible values of stock price. Sergei Fedotov (University of Manchester) / 7

13 Call Option Pricing on Binomial Tree We denote by C m n the n-th possible value of call option at time-step m t. Sergei Fedotov (University of Manchester) / 7

14 Call Option Pricing on Binomial Tree We denote by C m n the n-th possible value of call option at time-step m t. Risk Neutral Valuation (backward in time): C m n = e r t ( pc m+1 n+1 Here 0 n m and p = er t d u d. ) + (1 p)cm+1 n. Sergei Fedotov (University of Manchester) / 7

15 Call Option Pricing on Binomial Tree We denote by C m n the n-th possible value of call option at time-step m t. Risk Neutral Valuation (backward in time): C m n = e r t ( pc m+1 n+1 Here 0 n m and p = er t d u d. ) + (1 p)cm+1 n. Final condition: C N n = max ( S N n E,0 ), where n = 0,1,2,...,N, E is the strike price. Sergei Fedotov (University of Manchester) / 7

16 Call Option Pricing on Binomial Tree We denote by C m n the n-th possible value of call option at time-step m t. Risk Neutral Valuation (backward in time): C m n = e r t ( pc m+1 n+1 Here 0 n m and p = er t d u d. ) + (1 p)cm+1 n. Final condition: C N n = max ( S N n E,0 ), where n = 0,1,2,...,N, E is the strike price. The current option price C0 0 is the expected payoff in a risk-neutral world, discounted at risk-free rate r: C0 0 = e rt E p [C T ]. Example: N = 4. Sergei Fedotov (University of Manchester) / 7

17 Matching volatility σ with u and d We assume that the stock price starts at the value S 0 and the time step is t. Let us find the expected stock price, E [S], and the variance of the return, var [ ] S S, for continuous and discrete models. Sergei Fedotov (University of Manchester) / 7

18 Matching volatility σ with u and d We assume that the stock price starts at the value S 0 and the time step is t. Let us find the expected stock price, E [S], and the variance of the return, var [ ] S S, for continuous and discrete models. Expected stock price: Continuous model: E [S] = S 0 e µ t. Sergei Fedotov (University of Manchester) / 7

19 Matching volatility σ with u and d We assume that the stock price starts at the value S 0 and the time step is t. Let us find the expected stock price, E [S], and the variance of the return, var [ ] S S, for continuous and discrete models. Expected stock price: Continuous model: E [S] = S 0 e µ t. On the binomial tree: E [S] = qs 0 u + (1 q)s 0 d. Sergei Fedotov (University of Manchester) / 7

20 Matching volatility σ with u and d We assume that the stock price starts at the value S 0 and the time step is t. Let us find the expected stock price, E [S], and the variance of the return, var [ ] S S, for continuous and discrete models. Expected stock price: Continuous model: E [S] = S 0 e µ t. On the binomial tree: E [S] = qs 0 u + (1 q)s 0 d. First equation: qu + (1 q)d = e µ t. Sergei Fedotov (University of Manchester) / 7

21 Matching volatility σ with u and d We assume that the stock price starts at the value S 0 and the time step is t. Let us find the expected stock price, E [S], and the variance of the return, var [ ] S S, for continuous and discrete models. Expected stock price: Continuous model: E [S] = S 0 e µ t. On the binomial tree: E [S] = qs 0 u + (1 q)s 0 d. First equation: qu + (1 q)d = e µ t. Variance of the return: Continuous model: var [ S S ] = σ 2 t (Lecture2) Sergei Fedotov (University of Manchester) / 7

22 Matching volatility σ with u and d We assume that the stock price starts at the value S 0 and the time step is t. Let us find the expected stock price, E [S], and the variance of the return, var [ ] S S, for continuous and discrete models. Expected stock price: Continuous model: E [S] = S 0 e µ t. On the binomial tree: E [S] = qs 0 u + (1 q)s 0 d. First equation: qu + (1 q)d = e µ t. Variance of the return: Continuous model: var [ ] S S = σ 2 t (Lecture2) On the binomial tree: var [ ] S S = q(u 1) 2 + (1 q)(d 1) 2 [q(u 1) + (1 q)(d 1)] 2 = qu 2 + (1 q)d 2 [qu + (1 q)d] 2. Recall: var [X] = E [ X 2] [E (X)] 2. Sergei Fedotov (University of Manchester) / 7

23 Matching volatility σ with u and d We assume that the stock price starts at the value S 0 and the time step is t. Let us find the expected stock price, E [S], and the variance of the return, var [ ] S S, for continuous and discrete models. Expected stock price: Continuous model: E [S] = S 0 e µ t. On the binomial tree: E [S] = qs 0 u + (1 q)s 0 d. First equation: qu + (1 q)d = e µ t. Variance of the return: Continuous model: var [ ] S S = σ 2 t (Lecture2) On the binomial tree: var [ ] S S = q(u 1) 2 + (1 q)(d 1) 2 [q(u 1) + (1 q)(d 1)] 2 = qu 2 + (1 q)d 2 [qu + (1 q)d] 2. Recall: var [X] = E [ X 2] [E (X)] 2. Second equation: qu 2 + (1 q)d 2 [qu + (1 q)d] 2 = σ 2 t. Sergei Fedotov (University of Manchester) / 7

24 Matching volatility σ with u and d We assume that the stock price starts at the value S 0 and the time step is t. Let us find the expected stock price, E [S], and the variance of the return, var [ ] S S, for continuous and discrete models. Expected stock price: Continuous model: E [S] = S 0 e µ t. On the binomial tree: E [S] = qs 0 u + (1 q)s 0 d. First equation: qu + (1 q)d = e µ t. Variance of the return: Continuous model: var [ ] S S = σ 2 t (Lecture2) On the binomial tree: var [ ] S S = q(u 1) 2 + (1 q)(d 1) 2 [q(u 1) + (1 q)(d 1)] 2 = qu 2 + (1 q)d 2 [qu + (1 q)d] 2. Recall: var [X] = E [ X 2] [E (X)] 2. Second equation: qu 2 + (1 q)d 2 [qu + (1 q)d] 2 = σ 2 t. Third equation: u = d 1. Sergei Fedotov (University of Manchester) / 7

25 Matching volatility σ with u and d From the first equation we find q = eµ t d u d. This is the probability of an up movement in the real world. Sergei Fedotov (University of Manchester) / 7

26 Matching volatility σ with u and d From the first equation we find q = eµ t d u d. This is the probability of an up movement in the real world. Substituting this probability into the second equation, we obtain e µ t (u + d) ud e 2µ t = σ 2 t. Sergei Fedotov (University of Manchester) / 7

27 Matching volatility σ with u and d From the first equation we find q = eµ t d u d. This is the probability of an up movement in the real world. Substituting this probability into the second equation, we obtain Using u = d 1, we get e µ t (u + d) ud e 2µ t = σ 2 t. e µ t (u + 1 u ) 1 e2µ t = σ 2 t. This equation can be reduced to the quadratic equation. (Exercise sheet 4, part 5). Sergei Fedotov (University of Manchester) / 7

28 Matching volatility σ with u and d From the first equation we find q = eµ t d u d. This is the probability of an up movement in the real world. Substituting this probability into the second equation, we obtain Using u = d 1, we get e µ t (u + d) ud e 2µ t = σ 2 t. e µ t (u + 1 u ) 1 e2µ t = σ 2 t. This equation can be reduced to the quadratic equation. (Exercise sheet 4, part 5). One can obtain u e σ t 1 + σ t and d e σ t. These are the values of u and d obtained by Cox, Ross, and Rubinstein in Recall: e x 1 + x for small x. Sergei Fedotov (University of Manchester) / 7

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