Consider a filtered probability space (12, F, {Ft}t>0,P), and let {Wt}t>o be a Brow- nian motion. Let a stock price be a Geometric Brownian motion dSt = u Stdt + o StdWt, where ? and o are positive constants. Let denote by r the constant risk-free interest rate. ? — ? We define 0 = and the stochastic process process {St}+20, where? 9 ? St = exp (-ow, - ( -+):) (a.) Show that d?t = -r5tdt – 05tdWt. 1 The constant = ur is often referred to as the market price of risk. o (b.) Denote by {Vt}t>o the value process of an investor's portfolio consisting of stock and bond. Furthermore, denote by at the number of shares of stock held by the investor at time t. As shown dV+ = r Vedt + at(u – r) Stdt +ato StdWt. Show that St Vt is a martingale. (c.) Hint: show that the differential d(G+Vt) has no dt term. Let Vo be the initial capital, and furthermore, let Xt be a Fr-measurable random variable. Show that if the investor wants to perfectly replicate Xt, then the amount of initial capital must be Vo = Ep [STXT]. (As such, the process {St} is usually called the state price density process.) are Consider the function f(x) of x for which the partial derivatives de le, and defined and continuous. one-dimensional Itô rule df (X.) = (X.)dx, + (W) (dx,)?, where {Xt}te(0,1) is an Itô process. Here, the notation (Xt) and (X4) means that the partial derivatives are evaluated at Xt. We consider the special case Xt = W4 under which the above Itô rule becomes af . 1a2f df (W) -(W+)dW+ + (W+)dt. ?? 2 ax2 Integrating the above from t= 0 to t=T gives af (W+) dt. (1) ax ?r2 In this question, you are asked to prove (1). To this end, for fixed T > 0 and fixed positive integer n, with An T/n, let IIn {ti}_o, where t; iAn, be a partition of the interval [0,T]. To simplify the notation, we set AWt; = Wtit1 – Wti, i = 0,..., n - 1. Prove (1) for the case f(x) = 2x2. (Wr) – 5 (W) = 6" (w.) aw. + = a. Hint: For fixed n, use n-1 f (WT) – f (Wo) = (f (Wi+1) – f (W;)), i=0 and then use Taylor's Theorem for each term f (Wi+1) – f(Wi). Finally, find the limit of the sum as n + O. b. Mathematically prove that (1) still holds for an (sufficiently smooth) arbitrary function f(x). Hint: Use the same approach as in part (a). A key difference between part (a) and part (b) is that in part (a), since f(x) = 2x2, it follows that one 1; however, in part (b), this is not necessarily the case. 7 Sme the above grandion only defines the values a x to the right hand site the the left hand side. for us to conduto that they limit sualy exist the purchase the same value notwithstanding the drochon thereg And is because the aforementional is not true. gor This juncher as x approaches o, the lint exist

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