L4 - Lecture 4 From Binomial Trees to the Black-Scholes...

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Unformatted text preview: Lecture 4 From Binomial Trees to the Black-Scholes Option Pricing Formulas In this lecture, we will extend the example in Lecture 2 to a general setting of binomial trees, as an important model for a single risky security. It has been extensively used by practitioners in pricing various kinds of derivatives of stocks or bonds. Historically, the model was proposed independently by Cox/Ross/Rubinstein (1979, J. Fin. Econ. 7 , 229-263) and Rendleman/Bartter (1979, J. Fin. 34 , 1093-1110), although it was often referred to as the CRR model. Furthermore, we will show that the celebrated Black-Scholes formulas in option pricing can be derived from the binomial option pricing formulas through an asymptotic argument, provided the parameters in the binomial model are set appropriately. 4.1 The basic binomial tree model The evolution of a risky security, say stock, is represented by S = { S ( t ) , t = 0 , 1 ,...,T } . Starting from an initial (positive) price S (0), assume in each time period the stock price either goes up by a factor u > 1 with probability p , or goes down by a factor 0 < d < 1 with probability 1- p . The moves over time are iid Bernoulli random variables. For each t , S ( t ) = S (0) u n t d t- n t , where n t represents the number of up moves up to t . The bank account process B is deterministic with B (0) = 1 and a constant interest rate 0 < r < 1. Hence B ( t ) = (1 + r ) t . The filtration FF is taken as the one generated by the history of S . The sample space contains K = 2 T different paths. The underlying probability P is defined by P ( ) = p U ( ) (1- p ) T- U ( ) , where U ( ) represents the total number of up moves in the path . We assume 0 < p < 1 so that P ( ) > . As for EMMs, we have the following Proposition 4.1 There exists a unique EMM Q d < 1 + r < u . In this case, Q ( ) = q U ( ) (1- q ) T- U ( ) , with q = 1 + r- d u- d . (4.1) Proof Let t = n t- n t- 1 . Then for every t , S * ( t ) = S * ( t- 1) (1 + r )- 1 u t d 1- t . Therefore, E Q [ S * ( t ) | F t- 1 ] = S * ( t- 1) u Q ( t = 1 | n t- 1 ) + d [1- Q ( t = 1 | n t- 1 )] = 1 + r Q ( t = 1 | n t- 1 ) = 1 + r- d u- d , 1 where Q ( t = 1 | n t- 1 ) denotes the conditional probability (under Q ) that the next move is up given n t- 1 up moves up to time t- 1. We can denote this (constant) conditional probability by q since it does not depend on...
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L4 - Lecture 4 From Binomial Trees to the Black-Scholes...

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