HOMEWORK 10 MATH 7349 1. Let X be an exponential random variable of parameter λ — that is, P[X ≤ a] = 1 − e−λa, for any a ≥ 0. Let Z(X) = eγX for γ < λ. Let P˜ be the Z-tilted measure of P. Calculate P˜[X ≤ a]. 2. Consider a standard Brownian motion Bt on [0, T ] associated with the measure P, and let Ft be the filtration generated by the Brownian motion. Let Θ(t) be a stochastic process adapted toFt, and define Z(t) = exp(−∫ t 0 Θ(s)dBs − 1 2 ∫ t 0 Θ(s)2ds) and B˜t = Bt + ∫ t 0 Θ(s)ds. Let Z = Z(T ), and define P˜ to be Z-tilted measure of P.● Calculate d ( 1Z(t)).● Let M˜t be a martingale with respect to P˜. Show that Mt = M˜tZt is a P-martingale.● Assume that dMt = Γ(t)dBt for some adapted process Γ. Calculate d(Mt/Zt) and use this to show that d(M˜t) = Γ˜(t)dB˜t for some adapted process Γ˜(t). 3. Let c(0, x) be the solution to the Black–Scholes equation for the zero time cost of a European option with strike price K and maturation T if the initial stock cost S(0) is x. We have seen that we can write this as c(0, x) = E˜ [e−rT (S(T ) −K)+] , where P˜ is the risk-neutral measure, and where the price of the stock S(t) satisfies the stochastic differential equation dSt = rStdt + σStdB˜t.● Show that c(0, x) = E˜⎡⎢⎢⎢⎢⎣e−rT (xeσB˜T+(r− σ2 2 )T −K)+⎤⎥⎥⎥⎥⎦ Hint: calculate d(log(St)).● Note that, if h(x) = (x −K)+, then h′(x) = ⎧⎪⎪⎨⎪⎪⎩0 x K .
Differentiate under the expectation sign to find a formula for ∂c∂x(0, x).● Show that
∂c
∂x
(0, x) = Pˆ(S(T ) >K),
where Bˆt = B˜t−σt is a Brownian motion with respect to Pˆ. Hint: this is Girsanov’s theorem
in disguise!
1
2 HOMEWORK 10 MATH 7349
● Rewrite S(T ) in terms of Bˆt and deduce that
Pˆ(S(T ) >K) = Pˆ(− BˆT√
T
< d+(T,x)) = Φ(d+(T,x)),
where d+(T,x) = 1
σ
√
T
[log ( xK ) + (r + σ22 )T ] and Φ(x) = P[N(0,1) ≤ x].

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