Asset Pricing in Continuous Time
MATH0085
MSc Examination
2022
Problem 1. Consider a probability space (Ω , F, P) equipped with a standard (P, {Ft }t≥0)-Brownian motion {Wt }t≥0 starting at zero, where {Ft }t≥0 is the nat- ural filtration generated by {Wt }t≥0 . We denote by E the expectation under the measure P.
(a) Is the process
{W1-t - W1 }t∈[0;1]
a standard Brownian motion with respect to its natural filtration? Justify your answer. [5]
(b) Is the process
{(4Wt2 - 2t)}t≥0
(d) Is the process
{ Wudu }t≥0
a Markov process with respect to (Ft )t≥0? Justify your answer. [5]
(e) Calculate the expected value of the process
{ W2udu }t≥0
Justify your answer. [5]
Problem 2. Consider a Black-Scholes-Merton market defined on a filtered proba- bility space (Ω , F, {Ft }t≥0, P) with a risky financialasset with price process {St }t≥0 satisfying
dSt = μStdt + σStdWt , S0 > 0,
where μ ∈ R, σ > 0, {Wt }t≥0 is a (P, {Ft }t≥0)-Brownian motion and {Bt }t≥0 is a money market account satisfying
dBt = rBtdt, B0 = 1.
Then consider a European-type financial option with payof at expiration time T > 0 given by
ST(2)1{ST>K} .
We denote the indicator function by 1{·} .
(a) Derive the stochastic diferential equation satisfied by the square price pro-cess {St(2)}t≥0 . [3]
(b) What is the probability of exercising the option at expiration? Give your answer in terms of the model parameters and the standard normal cumulative distribution function Φ . [5]
(c) Derive the risk-neutral stochastic diferential equation of the square price process {St(2)}t≥0 and then solve it. Explain in detail all steps. [10]
(d) Derive the explicit formula for the risk-neutral price V (0, S0 ) of the option at time t = 0. Explain in detail all steps. [7]
Problem 3. Consider a financial market defined on a risk-neutral filtered proba- bility space (Ω , F, {Ft }t≥0, Q) with a risky financialasset with price process {St }t≥0 satisfying
dSt = rtStdt + σStdWt , S0 > 0, (1)
and an interest rate process {rt }t≥0 satisfying
drt = (a - rt )dt + bdBt , r0 ∈ R, (2)
where a,b, σ > 0 and {Wt }t≥0 , {Bt }t≥0 are two independent standard (Q, {Ft }t≥0)- Brownian motions. Then, consider a financial option with payof at expiration time T > 0 given by the integral
Stdt.
(a) Solve the stochastic diferential equation (2) to obtain a closed-form expres- sion for the interest rate rt , for all t ≥ 0. [5]
(b) Derive a Markovian formulation for the arbitrage-free option pricing problem at any time t ∈ [0, T]. Write down explicitly the price of the option at the expiration time T. Explain in detail all steps. [5]
(c) Derive the partial diferential equation and boundary conditions satisfied by the price of the option. Explain in detail all steps. [12]
(d) Comment on whether “delta-hedging” is appropriate for an investor to hedge the risk from selling this option. [3]
Problem 4. Consider a filtered probability space (Ω , F, {Ft }t≥0, P) and the stochastic integral process {Gt }t≥0 given by
Gt := Ks dWs ,
where {Wt }t≥0 is a (P, {Ft }t≥0)-Brownian motion and the process {Kt }t≥0 is de- fined by
(1 0 ≤ t < 1
Kt := , 1 ≤ t < 2
:3, t ≥ 2,
for an F1-measurable random variable ϕ(ω) with mean 2 and variance 1.
(a) Find the expectation and variance of G5 . [5]
(b) Suppose in this part that ϕ(ω) is given by a function h of the Brownian motion’s value at time t = 1, namely ϕ = h(W1 ). Are the increments G5 - G2 , G2 - G1 and G1 - G0 independent? [5]
Consider also the stochastic processes
Xt := X0 + μsds + σsdBs ,
where the processes {μt }t≥0, {σt }t≥0 are {Ft }t≥0-measurable and {Bt }t≥0 is a stan- dard Brownian motion satisfying [W, B]t = pt, for all t ≥ 0, where p ∈ [-1, 1].
(c) Calculate the dynamics of the process {Yt }t≥0 given by
Yt = exp{Gt } + Xt(3) - θt,
where θ > 0. [5]
(d) Express the process {Zt }t≥0 given by Zt = GtXt , for all t ≥ 0, in the forms:
(i) dZt = at dt + btdBt + ctdWt ,
for a (P, {Ft }t≥0)-Brownian motion {Wt }t≥0 independent of {Bt }t≥0 and
find the explicit expressions for at , bt and ct. [5]
(ii) dZt = ft dt + gtdMt ,
for a (P, {Ft }t≥0)-Brownian motion {Mt }t≥0 and find the explicit ex- pressions for ft and gt. [5]