# Lecture 9. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 8

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1 Lecture 9 Sergei Fedotov Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) / 8

2 Lecture 9 1 Risk-Neutral Valuation 2 Risk-Neutral World 3 Two-Steps Binomial Tree Sergei Fedotov (University of Manchester) / 8

3 Risk-Neutral Valuation Reminder from Lecture 8. Call option price: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. No-Arbitrage Principle: d < ert < u. Sergei Fedotov (University of Manchester) / 8

4 Risk-Neutral Valuation Reminder from Lecture 8. Call option price: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. No-Arbitrage Principle: d < ert < u. In particular, if d > e rt then there exists an arbitrage opportunity. We could make money by taking out a bank loan B 0 = S 0 at time t = 0 and buying the stock for S 0. Sergei Fedotov (University of Manchester) / 8

5 Risk-Neutral Valuation Reminder from Lecture 8. Call option price: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. No-Arbitrage Principle: d < ert < u. In particular, if d > e rt then there exists an arbitrage opportunity. We could make money by taking out a bank loan B 0 = S 0 at time t = 0 and buying the stock for S 0. We interpret the variable 0 p 1 as the probability of an up movement in the stock price. This formula is known as a risk-neutral valuation. Sergei Fedotov (University of Manchester) / 8

6 Risk-Neutral Valuation Reminder from Lecture 8. Call option price: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. No-Arbitrage Principle: d < ert < u. In particular, if d > e rt then there exists an arbitrage opportunity. We could make money by taking out a bank loan B 0 = S 0 at time t = 0 and buying the stock for S 0. We interpret the variable 0 p 1 as the probability of an up movement in the stock price. This formula is known as a risk-neutral valuation. The probability of up q or down movement 1 q in the stock price plays no role whatsoever! Why??? Sergei Fedotov (University of Manchester) / 8

7 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] Sergei Fedotov (University of Manchester) / 8

8 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d Sergei Fedotov (University of Manchester) / 8

9 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d Sergei Fedotov (University of Manchester) / 8

10 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d = S 0 e rt. Sergei Fedotov (University of Manchester) / 8

11 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d = S 0 e rt. This shows that stock price grows on average at the risk-free interest rate r. Since the expected return is r, this is a risk-neutral world. Sergei Fedotov (University of Manchester) / 8

12 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d = S 0 e rt. This shows that stock price grows on average at the risk-free interest rate r. Since the expected return is r, this is a risk-neutral world. In Real World: E [S T ] = S 0 e µt. In Risk-Neutral World: E p [S T ] = S 0 e rt Sergei Fedotov (University of Manchester) / 8

13 Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d = S 0 e rt. This shows that stock price grows on average at the risk-free interest rate r. Since the expected return is r, this is a risk-neutral world. In Real World: E [S T ] = S 0 e µt. In Risk-Neutral World: E p [S T ] = S 0 e rt Risk-Neutral Valuation: C 0 = e rt E p [C T ] The option price is the expected payoff in a risk-neutral world, discounted at risk-free rate r. Sergei Fedotov (University of Manchester) / 8

14 Two-Step Binomial Tree Now the stock price changes twice, each time by either a factor of u > 1 or d < 1. We assume that the length of the time step is t such that T = 2 t. After two time steps the stock price will be S 0 u 2, S 0 ud or S 0 d 2. Sergei Fedotov (University of Manchester) / 8

15 Two-Step Binomial Tree Now the stock price changes twice, each time by either a factor of u > 1 or d < 1. We assume that the length of the time step is t such that T = 2 t. After two time steps the stock price will be S 0 u 2, S 0 ud or S 0 d 2. The call option expires after two time steps producing payoffs C uu, C ud and C dd respectively. Sergei Fedotov (University of Manchester) / 8

16 Option Price The purpose is to calculate the option price C 0 at the initial node of the tree. Sergei Fedotov (University of Manchester) / 8

17 Option Price The purpose is to calculate the option price C 0 at the initial node of the tree. We apply the risk-neutral valuation backward in time: C u = e r t (pc uu + (1 p)c ud ), C d = e r t (pc ud + (1 p)c dd ). Sergei Fedotov (University of Manchester) / 8

18 Option Price The purpose is to calculate the option price C 0 at the initial node of the tree. We apply the risk-neutral valuation backward in time: C u = e r t (pc uu + (1 p)c ud ), C d = e r t (pc ud + (1 p)c dd ). Current option price: C 0 = e r t (pc u + (1 p)c d ). Sergei Fedotov (University of Manchester) / 8

19 Option Price Substitution gives C 0 = e 2r t ( p 2 C uu + 2p(1 p)c ud + (1 p) 2 C dd ), where p 2, 2p(1 p) and (1 p) 2 are the probabilities in a risk-neutral world that the upper, middle, and lower final nodes are reached. Sergei Fedotov (University of Manchester) / 8

20 Option Price Substitution gives C 0 = e 2r t ( p 2 C uu + 2p(1 p)c ud + (1 p) 2 C dd ), where p 2, 2p(1 p) and (1 p) 2 are the probabilities in a risk-neutral world that the upper, middle, and lower final nodes are reached. Finally, the current call option price is C 0 = e rt E p [C T ], T = 2 t. Sergei Fedotov (University of Manchester) / 8

21 Option Price Substitution gives C 0 = e 2r t ( p 2 C uu + 2p(1 p)c ud + (1 p) 2 C dd ), where p 2, 2p(1 p) and (1 p) 2 are the probabilities in a risk-neutral world that the upper, middle, and lower final nodes are reached. Finally, the current call option price is C 0 = e rt E p [C T ], T = 2 t. The current put option price can be found in the same way: P 0 = e 2r t ( p 2 P uu + 2p(1 p)p ud + (1 p) 2 P dd ) or P 0 = e rt E p [P T ]. Sergei Fedotov (University of Manchester) / 8

22 Two-Step Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Risk-free interest rate is 10%. Find the current option price. Sergei Fedotov (University of Manchester) / 8

23 Two-Step Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Risk-free interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Sergei Fedotov (University of Manchester) / 8

24 Two-Step Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Risk-free interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Risk-neutral probability p = er t d u d = e = Sergei Fedotov (University of Manchester) / 8

25 Two-Step Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Risk-free interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Risk-neutral probability p = er t d u d = e = The possible stock prices at final nodes are 40 (1.2) 2 = 57.6, = 38.4, and 40 (0.8) 2 = Sergei Fedotov (University of Manchester) / 8

26 Two-Step Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Risk-free interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Risk-neutral probability p = er t d u d = e = The possible stock prices at final nodes are 40 (1.2) 2 = 57.6, = 38.4, and 40 (0.8) 2 = We obtain P uu = 0, P ud = 0 and P dd = = 6.4. Sergei Fedotov (University of Manchester) / 8

27 Two-Step Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Risk-free interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Risk-neutral probability p = er t d u d = e = The possible stock prices at final nodes are 40 (1.2) 2 = 57.6, = 38.4, and 40 (0.8) 2 = We obtain P uu = 0, P ud = 0 and P dd = = 6.4. Thus the value of put( option is ) P 0 = e ( ) = Sergei Fedotov (University of Manchester) / 8

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