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Unformatted text preview: Derivative Securities – Fall 2007– Section 5 Notes by Robert V. Kohn, extended and improved by Steve Allen. Courant Institute of Mathematical Sciences. The Black-Scholes formula and its applications . This Section deduces the Black- Scholes formula for a European call or put, as a consequence of risk-neutral valuation in the continuous time limit. Then we discuss the delta, gamma, vega, theta, and rho of a portfolio, and their significance for hedging. Hedging is a very important topic, and these notes don’t do justice to it; see Chapter 15 of Hull’s 6th edition for further discussion. We’ll do two passes, as usual. First we consider options on a non-dividend-paying underlying with lognormal dynamics. Then we consider options on a forward price with lognormal dynamics. As we’ll see, the latter case is actually simpler and more general, because the forward price is a martingale under the risk-neutral measure no matter what the value of the risk-free rate (and regardless of whether the underlying pays a dividend). Thus we could alternatively have started with options on a forward price, then deduced the results for options on a non-dividend-paying stock price (and options on a dividend-paying underlying, like a foreign currency rate) from that. Steve Allen’s version of these notes follows this alternative route. All the mathematics in this section uses probability and calculus to derive conclusions from the results obtained in Section 4. We won’t be introducing any new financial assumptions or arbitrage arguments (but remember that the results in Section 4 were based on such arguments). Later in the semester we’ll derive results similar to those of Section 4 in other settings. The analysis in this section won’t have to be redone – it will permit us to immediately deduce the prices and hedges of puts and calls in those settings too. ******************* The Black-Scholes formula for a European call or put . The upshot of Section 4 is this: the value at time t of a European option with payoff f ( s T ) is V ( f ) = e- r ( T- t ) E RN [ f ( s T )] . Here E RN [ f ( s T )] is the expected value of the price at maturity with respect to a special probability distribution – the risk-neutral one. For a non-dividend-paying stock with log- normal dynamics, this distribution is determined by the property that s T = s t exp h ( r- 1 2 σ 2 )( T- t ) + σ √ T- tZ i where s t is the spot price at time t and Z is Gaussian with mean 0 and variance 1. Equiv- alently: log[ s T /s t ] is Gaussian with mean ( r- 1 2 σ 2 )( T- t ) and variance σ 2 ( T- t ). The value of the option can be evaluated for any payoff f by numerical integration. But for puts and calls we can do better, by obtaining explicit expressions in terms of the “cumulative distribution function” N ( x ) = 1 √ 2 π Z x-∞ e- u 2 / 2 du....
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- Fall '11
- Finance, Forward price, Mathematical finance