MAT00028 Stochastic Calculus and Black-Scholes Theory 2020/21
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MMath and MSc Examinations 2020/21
Stochastic Calculus and Black-Scholes Theory
1 (of 3). Let (Ω , F, P) be a probability space, let (Ft )t≥0 be a ﬁltration on this space and let W be a Brownian Motion with respect to (Ft )t≥0 .
(a) Show whether ξ , η deﬁned by
,．5, if t e [0, 5) ,．一3, if t e [0, 5)
ξ(t) = , η(t) =．．． 2(2)
．0, if t 2 15 ．0, if t 2 11
are random step processes. Give reasons for your answer. If any of these processes is indeed a random step process, then calculate the mean and
variance of its stochastic integral I(．). 
(b) Let n e N. Deﬁne a process X by setting
Xt = exp ╱ 一Wtn + !0 t ┌
n (n 2 一 1) Ws(n) -2 一 Wn-1) ┐ ds、一 1, t 2 0.
Prove that X can be written as a stochastic Itˆo integral and determine its integrand. 
(c) Deﬁne the space Mlo(2)c (0, o) and determine if the processes A and B deﬁned by
At = cos(Wt ) and Bt = exp ╱ Wt4 、, t 2 0
are in Mlo(2)c (0, o). Carefully justify your answer. 
2 (of 3). Let (Ω , F, P) be a probability space, let (Ft )t≥0 be a ﬁltration on this space and let W be a Brownian Motion with respect to (Ft )t≥0 .
(a) Deﬁne a stochastic process X by setting
Xt = sin (Wt ) + !0 t sin (Ws ) ds.
Find the quadratic variation of the process X. Carefully justify your answer. In particular, ensure that your deﬁnition of the quadratic vari- ation applies. 
(b) Suppose f : R - R is a continuously diﬀerentiable strictly positive function. Consider the the stochastic diﬀerential equation
dXt = f (Xt ) f> (Xt ) dt + f (Xt ) dWt , X0 = 0.
Solve the stochastic diﬀerential equation. Carefully justify your answer. Hint: You may wish to consider the function
g (x) = !0 x dy.
3 (of 3). Suppose that σ > 0, r > 0, T > 0 and µ e R are ﬁxed. Let (Ω , F, P) be a probability space and let W be a Brownian motion deﬁned on this space. Consider the Black-Scholes model, i.e. a market consisting of a stock and a bond with prices given by stochastic processes S and B respectively. Assume that S and B are solutions to the following (stochastic) diﬀerential equations:
dSt = µSt dt + σSt dWt , S0 > 0, t 2 0,
dBt = rBt dt, t 2 0.
For this question you may use the Black-Scholes formula without proof pro- vided you state it clearly.
(a) Does there exist a measure Q absolutely continuous with respect to P such that the stock price S in the Black-Scholes model is a Brownian motion on the probability space (Ω , F, Q)? Carefully justify your answer.
(b) Suppose that C (t, S, r, σ) denotes the price at time t e [0, T] of a Euro- pean call with strike K and exercise time T in the Black-Scholes model. Calculate the functions and . Carefully justify your answer. 
(c) Suppose a trader sells at time 0 a European call option with strike K > 0 and exercise time T at implied volatility σ 1 , i.e. the price is the Black- Scholes value C (0, S0 ) computed with σ = σ 1 . The trader then sets up a self-ﬁnancing replication strategy based on his assumption that σ = σ 1 with initial investment X0 = C (0, S0 ) and ϕt = (t, St ) . Suppose that
the actual price process for the stock price St satisﬁes the equation dSt = µSt dt + σ2 St dWt .
Show that the trader will make a proﬁt (with probability one) as long
as σ 1 > σ2 . Carefully justify your answer.