Lecture7 - LECTURE 7 BLACKSCHOLES THEORY 1 Introduction The...

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LECTURE 7: BLACK–SCHOLES THEORY 1. Introduction: The Black–Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing and hedging of (European) call and put options. In this paper the famous Black-Scholes formula made its debut, and the Itˆ o calculus was unleashed upon the world of finance. 2 In this lecture we shall explain the Black-Scholes argument in its original setting, the pricing and hedging of European contingent claims. In subsequent lectures, we will see how to use the Black–Scholes model in conjunction with the Itˆ o calculus to price and hedge all manner of exotic derivative securities. In its simplest form, the Black–Scholes(–Merton) model involves only two underlying assets, a riskless asset Cash Bond and a risky asset Stock . 3 The asset Cash Bond appreciates at the short rate , or riskless rate of return r t , which (at least for now) is assumed to be nonrandom , although possibly time–varying. Thus, the price B t of the Cash Bond at time t is assumed to satisfy the differential equation (1) dB t dt = r t B t , whose unique solution for the value B 0 = 1 is (as the reader will now check) (2) B t = exp t 0 r s ds . The share price S t of the risky asset Stock at time t is assumed to follow a stochastic differential equation (SDE) of the form (3) dS t = μ t S t dt + σS t dW t , where { W t } t 0 is a standard Brownian motion, μ t is a nonrandom (but not necessarily constant) function of t , and σ > 0 is a constant called the volatility of the Stock . Proposition 1. If the drift coefficient function μ t is bounded, then the SDE (3) has a unique solution with initial condition S 0 , and it is given by (4) S t = S 0 exp σW t - σ 2 ( t/ 2) + t 0 μ s ds Moreover, under the risk–neutral measure, it must be the case that (5) r t = μ t . 1 “The pricing of options and corporate liabilities” in Journal of Political Economy , volume 81, pages 637–654 2 In fact, the use of the Wiener process in financial models dates back to the early years of the 20th century, in Bachelier’s thesis. However, the success of the Black-Scholes model assured that the Itˆ o calculus would have a permanent place in the world of mathematical finance. 3 Keep in mind that the designation of a riskless asset is somewhat arbitrary. Recall that any tradeable asset whose value at the termination time T is strictly positive may be used as numeraire , in which case it becomes the riskless asset, with rate of return r = 0. This is the principle of numeraire invariance . Note, however, that the equilibrium measure depends on the choice of riskless asset. More on this when we talk about the Cameron–Martin–Girsanov theorem later in the course. 1
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Proof. As in many arguments to follow, the magical incantation is “Itˆ o’s formula”. Consider first the formula (4) for the share price of Stock ; to see that this defines a solution to the SDE (3), apply Itˆ o’s formula to the function u ( x, t ) = exp σx - σ 2 ( t/ 2) + t 0 μ s ds .
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