 # 2. Stochastic Calculus We begin by recalling the definition of an ...

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2. Stochastic Calculus We begin by recalling the definition of an Itô process: df(t)=(t,f(t)) dt + o (t, f(t)) dW(t), where W (t) is a standard Brownian motion. = + 1829 (df)2 , = at + 2 ) We now have at hand all the tools we need. We postulate a two-asset vector price process: dS0 = roodt, dS1 = ( - y)S1 dt + os, dW(t). We can think of S0 as a riskless bank account or zero-coupon bond; it serves to define the riskless interest rate r and provide a "measuring stick" (numéraire) for S1. S1 is our risky asset (e.g. a stock) price process. 11 is the gross expected rate of return of S1, and o is its proportional volatility, while y is a proportional, continuous dividend or convenience yield: if we own the stock, we will receive a continuous cashflow at a rate equal to yS1dt. An established, convenient convention is to assume that the yield is continuously re- invested in the stock. If we define the re-invested price process S1(t) = exp (Jo dt y(t)) S1 (t), then a simple application of Ito's formula yields: = Henceforth, for brevity, we abbreviate S1(t) by S(t). (a) Assume that we wish to analyze some derivative security whose value can be written as t,S(t)). Use Itô's formula to derive an Itô process for C. (b) Understanding the effect of simple transformations to the forms of C and S plays an important role in the analysis and solution of stochastic differential equations like these. Assuming that u, o, r, and y are all constants, derive Itô processes for: i. S(t)) = ept (t, S(t)); ii. S(t) = eat Stt. For what value of O is S(t) a martingale? What would be the resulting Itô process for C(t,S(t))? iii. r(t) = log (S(t)/K), where K is a constant with price units consistent with S. What would be the resulting Itô process for C(t,z(t))? iv. 2(t) = x(t) + at. For what value of O is 2(t) a martingale? What would be the resulting Itô process for C(t,2(t))? (c) Assume that we are able to calculate A(1,5)=00 = Form a portfolio (of value) II consisting of +1 unit of C and - A units of S. Note that we will need to buy and sell S over time as A changes: this is called "rebalancing" the portfolio. For an infinitesimal dt over which the portfolio composition is constant (i.e., in between rebalancings) use the Itô processes for C and S to write a stochastic differential equation for the change dII in the portfolio value over dt. Do not forget the effect of (reinvestment of) the dividend or convenience yield y! Note: If identifying an appropriate treatment of y proves elusive, simply set y to zero and proceed. (d) Simplify your formula for dll as much as possible, collecting together all terms pro- portional to dt and all those proportional to dW (t). What is the coefficient of the latter? Therefore, what is the (local in t) variance of dII? (e) What is the (expected or deteministic) rate of return of II = C - AS over dt? Combine this with your formula for dll from part (d) and collect terms to obtain a partial differential equation (the Black-Scholes-Merton PDE) for C. Compare it to the process you derived in (a). What has happened to us? What has it been replaced by? What else has changed? If possible, explain how this might suggest interpreting C as a expectation with respect to a modified process for S(t) (and, if so, what is that process)? What role does the term in the PDE proportional to Cplay?

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