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Unformatted text preview: Math 425 Dr. DeTurck Notes and exercises on Black-Scholes April 2010 On Thursday we talked in class about how to derive the Black-Scholes differential equation, which is used in mathematical finance to assign a value to a “financial deriva- tive”. The latter is usually the option to buy (a “call” option) or sell (a “put” option) a share of stock at a price specified in advance at some date in the future. We’re only going to discuss what are called “European” options, which are options that can be exercised only at the “expiry date” in the future. There are also “American” options, which can be exercised at any time up to the expiry date. These lead to interesting PDE problems as well (so-called “free boundary problems”), but we’ll save that for some later time. There are lots of assumptions, some realistic and some not, that go into the derivation. For instance, we assume that the risk-free interest rate r (what you could get by putting your money in the bank or in government-backed securities) is a constant, so that if you knew that the value of the stock at time T was going to be S , then the value at time t < T will be Se- r ( T- t ) , the standard present-value computation. A second assumption is that the return on the stock follows a special kind of random walk called a Wiener process – this is the same process one observes in Brownian motion. Basically, it says that if you know a particle’s position at time t = 0, then all you can say about its position in the future is that it is a normally-distributed random variable with mean 0 and variance proportional to the time t , say σ 2 = 2 kt . If we remember from stat class that the probability density function for the normal distribution with mean 0 and variance σ 2 is f ( x ) = 1 σ √ 2 π e- x 2 / (2 σ 2 ) , then the probability density function for the position of our particle at time t is F ( x,t ) = 1 √ 4 πkt e- x 2 / 4 kt for t > 0, which we recognize as the fundamental solution of the heat equation. So we should be on the lookout for something related to the heat equation. There are two complications: • Since it’s the return and not the price of the stock that changes according to the Wiener process, it’s the logarithm of the price that is normally distributed, and we’ll have to be alert for an appropriate change of variable • When we think about our stock options, the only time we actually know their value as a function of the stock price is at the future time when the option is to be exercised, and so we’ll have to solve for the current value of the option by working backwards in time....
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