11-Black-Scholes.doc - Lecture 11 A Brief Introduction to...

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Lecture 11: A Brief Introduction to Continuous Time Option Pricing Readings: o Ingersoll Chapters 14 – 17 o Cochrane Chapter 17 o Shimko – Finance in Continuous Time: A Primer (from which these notes are largely drawn) 1
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We will spend some time here building up the tools we need to develop the Black- Scholes Partial Differential Equation. This will be done in a relatively informal way and you should consult other texts if you wish to pursue these issues in more depth. First we need to introduce an “Ito Process.” I’ll build this idea up slowly so bear with me if you are already familiar with the concept. Definition: A stochastic process, defined by B(0) = 0 (more generally B(0) = B 0 a fixed starting point) and B(t + 1) = B(t) + (t + 1) t {0, 1, 2, …} where the innovations in B are independent standard normal random variables: (t + 1) ~ iid N(0,1) t , is a special version of a random walk – special in that it has normally distributed increments. This is a simple example of a discrete time stochastic process where we see a new realization of the process B(t) at each point in time, i.e. at each time t . The realization at any time t of the process can be arbitrarily high or low. At each time t the innovations in the process B are unpredictable (and normally distributed). In other words, as with all random walks, the expected value of a future realization of the process as of date t is simply B(t). The expected change in the process is always zero and the variance of the change depends on how far into the future you are trying to forecast. Over one period the variance is 1. Over five periods (you know B(11) and are forecasting B(16)) the variance is 5 (the expectation is still: E 11 (B(16)) = B(11)). 2
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Now suppose we “observe” the process more frequently than at each fixed time interval. Let = 1/ n for some arbitrary integer n > 1. We want to describe a process with the same characteristics as the random walk described above but observed more frequently: B(t + ) = B(t) + (t + ), with B(0) = 0 and B = B(t + ) – B(t) = (t + ) ~ iid N(0, ) Over n periods of length Δ this new process has the same expected change (or “drift” – in this example there is none) and the same variance as the original has over one fixed time “interval” or period. Finally let dt, a very small increment of time (so n is very large). Define “small” heuristically by letting dt be the smallest positive real number such that dt = 0 whenever > 1. Then: B(0) = 0 B(t + dt) = B(t) + (t + dt), t [0, T] where (t + dt) ~ iid N(0, dt) Define dB(t) = B(t + dt) – B(t) = (t + dt), as the increments in the process B(t). dB(t) may be thought of as a normally distributed random variable with mean 0 and variance dt. It is often referred to as white noise. The process B(t) is a standard Wiener process.
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