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Cox, John C., Ross, Stephen A., Rubinstein, Mark, (1979) “Option Pricing: A Simplified Approach,” Journal of Financial Economics, vol 7. pp. 229-263. Contention; the CRR binomial model contains the Black-Scholes model as a limiting case, i.e. as the number of binomial steps in a period of calendar time tends to infinity. Calendar time (0-t) divided into increments 0,1,2, ….,t-1,t. We will investigate the properties of the binomial model as the number of increments (binomial steps), n. Black-Scholes model: Given constant r, m, S t S 0 exp(Wt mt) Bt exp( rt ) Given the Black-Scholes assumptions, stock prices are distributed log-normal, continuously compounded returns are distributed normal. ln( S1 S 0 ) ~ N (m, 2 ) ln( S t S 0 ) ~ N (mt, t 2 ) S t S 0 S1 S 0 * S 2 S1 * .... * S t S t 1 t ln( S t S 0 ) ln( S i S i 1 ) i 0 E ln( S t S 0 ) tm VARln( S t S 0 ) t 2 Under the assumptions of the Black-Scholes model the continuously compounded rate of return 0….t is distributed normal with mean, tu, and variance t2. Binomial Model: n = #trials, j = 1(uptick) with probability=q, j = 0 (downtick) with probability =(1-q). For a sequence of n-trials, j = number of upticks is distributed binomial E(j) = nq, VAR(j) = nq(1-q) S t S 0 u j d n j S t S 0 u j d n j ln( S t S 0 ) j ln( u d ) n ln d E ln( S t S 0 ) nln( u d )q ln d ̂ n 1 VARln( S t S 0 ) E ln( S t S 0 ) uˆ n 2 ˆ n2 ˆ n2 E j ln u (n j ) ln d (ln( u d ) E ( j ) n ln d )2 E j ln( u d ) ln( u d ) E ( j )2 ln( u d ) 2 E j E ( j ) 2 ˆ n2 nq(1 q) ln( u d ) 2 The first step in illustrating the contention above is to show that the first two moments of the return distribution of the binomial process match the moments of the continuously compounded return distribution as the number of binomial time steps increases. lim n ˆ n tm ˆ n2 t 2 Select: u exp( t n) d exp( t n) q 1 1 (m ) t n 2 2 Substituting for u, d, q in the mean and variance of the binomial return process using the specified values for u,d,q; ˆ n mt ˆ n2 2 t m 2 t 2 n Hence in the limit as n the first two moments of the Brownian motion and the binomial process match. Alternative specifications of the steps/probability (u,d,q) give rise to models of stock returns (price) with different characteristics. The second step to illustrate the contention is to demonstrate that as n the terminal distribution of returns produced by the proportional binomial process is normal distributed. It is natural to rely on the central limit theorem to make this argument. However the Central Limit Theorem (CLT) that applies to identically and independently distributed random variables is not appropriate in this setting. The CLT applies to a sum of random variables as additional random variables are added to the sum. This is essentially the circumstance for this application. However as n is 2 increased, given the definition of (u,d,q), the distribution of each of the elements of the sum is changed. Lyapunov’s Third Formulation of the Central Limit Theorem; Let x1,x2, . . .,xn be a sequence of independent random variables with means 1,2, . . .,n 3 and variances ai = E(xi - i)2 and third absolute moments d i E xi i . Then if in the limit as n the ratio defined below tends to zero then the distribution of the summation of the standardized x’s tends to the standard normal distribution. If in lim n (d1 d 2 ... d n ) 2 (a1 a 2 ... a n ) 3 0 then x 1 ... xn n 1 t 2 Pr s 1 t exp( x )dx 2(a1 ... a n ) s Two levels of substitutions are necessary to check this condition: xi ln( Si Si 1 ) i ˆ ai ˆ 2 and uˆ q ln( u d ) ln d ˆ ln( u d ) q(1 q) Substituting the specifications for (u,d,q), the ratio reduces with much algebra to checking; n (1 q) 2 q 2 0 which it does. Hence in the limit as n the nq(1 q) binomial proportional process produces normal distributed returns such that If in lim ln( S t S 0 ) ˆ n Pr z N ( z ) ˆ n Each of the previous arguments illustrated a convergence of the terminal distribution of the n-step binomial process with the normal distribution with mean and variance equivalent to the mean and variance produced from the continuous return model (arithmetic Brownian motion). The third step to illustrate the contention is to impose the results derived from the Arbitrage Theorem. 3 In the single period – two state setting of Neftci Chapter 2 we discovered that under the risk neutral measure the expected return on the underlying risky asset is the risk free rate. In the continuous environment of the Black-Scholes Model: ~ dSt St dWt rS t dt then applying Ito’s Lemma ~ S t S 0 exp(Wt (r 12 2 )t ) Hence, under the risk neutral probability measure and the assumptions of the BlackScholes model the share price will have expected value, S 0 (r 12 2 )t at the end of calendar period (0,t). u exp( t n) d exp( t n) p 12 12 ((ln r 12 2 ) ) t n For reference: rˆ r t n Eln( St S0 ) nln( u d ) p ln d ̂ pn VARln( S t S 0 ) np(1 p) ln( u d ) 2 ˆ 2pn Substitute (u,d,p) into definitions for ̂ pn , ˆ 2pn and examine in the limit as n. The p-subscript denotes that the probability of an uptick has been selected to be consistent with the risk neutral measure. lim ˆ lim (ln rˆ 1 2 )t (r 1 2 )t n pn n 2 2 t lim rˆ lim r n e r n n lim ˆ 2 2 t n pn The specifications of (u,d,q) produce the desired risk-neutral results. To illustrate the equivalence between the binomial and Black-Scholes as n start with the binomial model (CRR pg. 239): 4 C Sa; n, p Kr n a; n, p (r d ) p (u ) p r (u d ) ln( K n ) Sd a u ln( ) d The complementary binomial distribution, [a;n,p] = Pr[j >= a], is the probability that the sum of n independent binomial variable with probability of up-tick, p, will be greater than a after n-trials. Illustration of the equivalence between models takes the limit of the complement of [a;n,p], 1 - [a;n,p], to develop the equivalence of [a;n,p] and N(d2) of the Black-Scholes model. p Other binomial approaches – Jarrow, R.A. and A. Rudd, 1983 Option Pricing, Irwin, Englewood Cliffs, NJ. Leisen, D.P.J. amd M. Reimer, 1996, “Binomial Moels for Option Valuation – Examining and Improving Convergence,” Applied Mathematical Finance, 3, 319-346. Risk neutral valuation: We have explored the equivalent martingale method for derivative pricing and also examined the central role of the replicating portfolio. Recall that for a European call option with strike price, k, and expiration date T, and letting t=0, the value of the derivative is equivalent to the expectation; V0= e r (T ) EQ ( Max ( ST k ,0) 0 ) We utilized methods appropriate for martingale processes (justified by the Arbitrage Theorem) that allowed us to specify the distribution of ST under the equivalent martingale measure and take the necessary expectation. A slightly different approach, risk neutral valuation, which essentially carries out the same valuation steps was developed before the equivalent martingale method was well understood. Recall that the equivalent martingale measure is a probability measure that changes the drift of the underlying stochastic processes such that all non-payout assets have an instantaneous expected rate of return equal to the risk free rate. 5 Suppose that we could specify the stochastic process for the primitive asset and conditional on this specification know the type of distribution for the terminal (time T) asset price. This is obviously possible we already know that a stock price that follows a geometric Brownian motion will have a log-normal terminal distribution. For other specifications of the stochastic process the solution to the Kolmogorov backward equation is the probability density function for the terminal price. Ingersoll Chp. 16 – An Introduction to Stochastic Calculus Markov: The conditional distribution of X(t) given information up until < t depends at most on X(). The conditional distribution of X(t) is independent of all other information in the filtration at . If x evolves according to the Markov diffusion process defined above; dx(t) =u(x,t)dt + (x,t)d(t) and f(x,t; x0, t0) is the probability density function for x at time t conditional on x(t0) = x0, then f(x,t) satisfies the partial differential equations of motion. They are the backward Kolmogorov equation 2 f 1 2 ( x , t ) 0 0 2 x02 u ( x0 , t 0 ) f f 0 x0 t 0 and the forward Kolmogorov (Fokker-Planck) equation 1 2 2 x 2 2 ( x, t ) f u( x, t ) f f 0 . x t The backward equation describes how the probability density function, f, changes when the initial time, t0, is allowed to change holding x and terminal time, t, constant. The forward equation describes how the probability density function changes when the terminal time, t, is allowed to vary and t0, x0 are held constant. The risk neutral valuation approach proceeds along lines analogous to the equivalent martingale method but relies on a heuristic argument that was later validated by the more rigorous development of the equivalent martingale method. At the time of its heyday the risk neutral valuation method utilized the then well accepted partial differential equation approach to pricing derivative assets as a point of departure. As we will see the partial differential equation approach is based on the specification of the stochastic process for the primitive assets, specification of the derivative assets stochastic process using Ito’s Lemma and the construction of a portfolio consisting of the derivative, the underlying primitive asset and the risk free asset. Depending on the type of portfolio constructed zero-investment or delta-hedged, the rate of return of the portfolio necessary to prevent arbitrage opportunities is known. The resulting partial 6 differential equation that the derivative asset value must satisfy is solved subject to the boundary conditions appropriate for the derivative contract. Heuristic argument of the risk neutral valuation method: Because no assumptions about risk bearing behavior are required to develop the partial differential equation valuation method the solution obtained by the partial differential equation method must be valid in all preference (risk bearing) environments. One possible preference structure is risk-neutral, i.e. decision makers require no compensation for bearing risk. In a risk-neutral environment all assets (primitive and derivative) will have an expected rate of return equal to the risk free rate. If it is possible to identify the terminal distribution of the primitive assets price in the risk-neutral world, taking the expectation of the derivative contract’s payoffs with respect to this distribution and discounting at the risk free rate will produce the value of the derivative contract. (Sounds suspiciously like the equivalent martingale method without the rigorous proof of the self-financing quality of the replicating portfolio). 7