Haipeng Xing Department of Applied Mathematics and Statistics Outline 1 An one-step Bionomial model and a no-arbitrage argument 2 Risk-neutral valuation 3 Two-step Binomial trees 4 Delta 5 Matching volatility with u and d 6 Girsanov s Theorem
An one-step Bionomial model and a no-arbitrage argument Consider a stock whose price is S 0 and an option on the stock whose current price is f. Suppose that the option lasts for time T and that during the life of the option the stock price can either move up from S 0 to S 0 u (u >1), or down from S 0 to S 0 d (d <1). The corresponding payo s from the option are f u and f d,respectively. Figure 1: Stock and option prices in a one-step tree (Hull, 2014; Figure 13.2). Consider a portfolio consisting of a long position in short position in one option. shares and a If there is an up movement in the stock price, the value of the portfolio at the end of the life of the option is S 0 u f u.ifthereis a down movement in the stock price, the value becomes S 0 d f d. To makes the portfolio riskless, the values of the portfolio must be equal, so that S 0 u f u = S 0 d f d, or = f u f d S 0 (u d). Suppose there are no arbitrage opportunities in the market. Then the riskless portfolio must earn the risk-free interest rate.
If we denote the risk-free interest rate by r, the present value of the portfolio is (S 0 u f u )e rt. The sots of setting up the portfolio is f. It follows that S 0 (S 0 u f u )e rt = S 0 f or f = S 0 (1 ue rt )+f u e rt. Substituting where into f, we obtain fu f d f = S 0 (1 ue rt )+f u e rt S 0 (u d) =) f = e rt pf u +(1 p)f d, (1) p := ert d u d. (2) Equations (1) and (2) provide an option pricing formula when stock price movements are given by a one-step binomial tree. The only assumption needed for these equations is the absence of arbitrage opportunities. Risk-neutral valuation When valuing a derivative, we can make the assumption that investors are risk-neutral. A world where investors are risk-neutral is referred to as a risk-neutral world. It is natural to interpret p and 1 p as probabilities of up and down movements. Then the expected stock price E(S T ) at time T is given by E(S T )=ps 0 u +(1 p)s 0 d = S 0 e rt. (3) This shows that the stock price grows, on average, at the risk-free rate when p is the probability of an up movement. A risk-neutral world has two features that simplify the pricing of options: The expected return on a stock is the risk-free rate. The discount rate used for the expected payo on an option is the risk-free rate.
Real world vs. risk-neutral world We note that p is the probability of an up movement in a riskneutral world. In general, this is not the same as the probability of an up movement in the real world. When we valuate an option in terms of the price of the underlying asset, the probability of up and down movements in the real world are irrelevant. Two-step Binomial trees We extend the analysis above to a two-step binomial tree. Assume that the stock price is initially S 0.Duringeachtime step, it either moves up to us 0 or moves down to ds 0. Suppose that the risk-free interest rate is r and the length of the time step is t years. The notation for the value of the option is shown on the tree. Figure 2: Stock and option prices in a one-step tree (Hull, 2014; Figure 13.6).
Since the length of a time step is now (1) and (2) become f = e r t pf u +(1 p)f d p = er t d u d t rather than T, equations (4) (5) Repeated application of equation (4) yields f u = e r t pf uu +(1 p)f ud f d = e r t pf ud +(1 p)f dd (6) (7) Substituting from equations (6) and (7) into (4), we obtain f = e 2r t p 2 f uu +2p(1 p)f ud +(1 p) 2 f dd. (8) This is consistent iwth the principle of risk-neutral valuation discussed earlier. Delta Shares of stocks in the Binomial tree model: = f u f d S 0 (u d) The delta ( ) of a stock option is the ratio of the change in the price of the stock option to the change in the price of the underlying stock. The construction of a riskless portfolio is sometimes referred to as delta hedging. The value of varies from node to node.
Matching volatility with u and d The three parameters necessary to construct a binomial tree with time step t are u, d and p. Given u and d, tomaketheexpectedreturnistherisk-freerater, p must be chosen as p =(e r t d) (u d). The parameters u and d should be chosen to match the volatility of the stock,, which is defined so that the p standard deviation of its return in a short period of time t is t. During a time step of length t, the stock will provide a return of u 1 with probability p and a return of d 1 with probability 1 p, respectively. It follows that volatility is matched if p(u 1) 2 +(1 p)(d 1) 2 p(u 1)+(1 p)(d 1) 2 = 2 t. (9) Substituting for p =(e r t d) (u d), thissimplifiesto e r t (u + d) ud e 2r t = 2 t. (10) When terms in ( t) 2 and higher powers of t are ignored, a solution to equation (10) is u = e p t and d = e p t. These are the values of u and d used by Cox, Ross, and Rubinstein (1979). What happens if instead we match volatility in the real world? Define µ as the expected return in the real world. We must have p u +(1 p )d = e µ t or p = e µ t d (u d). The equation matching the variance is the same as equation (9) except that p is replaced by p. We then obtain an equation that is the same as equation (10) except that r is replaced by µ. Ignoring terms in ( t) 2 and higher powers of t, we obtain the same solution as equation (10).
Girsanov s Theorem When we move from the risk-neutral world to the real world, the expected return from the stock price changes, but its volatility remains the same. Moving from one set of risk preferences to another is referred to as changing the measure. The real-world measure is sometimes referred to as the P -measure, while the risk-neutral world measure is referred to as Q-measure.