MAT00028 Stochastic Calculus and Black-Scholes Theory 2020/21

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MAT00028M

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(．).                                                  [9]

(b)   Let n e N. Define a process X by setting

Xt  = exp ╱ 一Wtn + !0 t  ┌ n (n2一 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.                                                                                         [9]

(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.                                 [7]



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.                                                                                       [11]

(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.

[14]




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.

[5]

(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.     [7]

(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.                                          [13]

 

 

 

 

 

 

 



2022-05-19