Asset Pricing in Continuous Time
MSc Examination
2021
Problem 1. Let (Ω , F, {Ft }t≥0, P) be some probability space and {Wt }t≥0 and {W(ˆ)t }t≥0 be two (P, {Ft }t≥0)-Brownian motions, where EP [(Wt − Ws )(W(ˆ)t − W(ˆ)s )] = ρ(t − s) for some ρ ∈ (0, 1).
(a) [8 points] [SS] Let {Yt }t≥0 and {Zt }t≥0 be Ito processes governed by
If St = sin(t) + Yt2 + exp(Zt ) for all t ≥ 0, governed by the stochastic differential equation dSt = Kt dt + Lt dWt + Nt dW(ˆ)t , what are Kt , Lt and Nt in terms of Yt , Zt , at , bt , ct and dt for all t ≥ 0?
(b) [8 points] [S]
From above, if Xt = YtZt for all t ≥ 0 such that dXt = µt dt + σt dMt , where {Mt }t≥0 is a (P, {Ft }t≥0)-Brownian motion, what are µt and σt in terms of Yt , Zt , at , bt , ct and dt for all t ≥ 0?
(c) [9 points] [SS]
Define {Pt }0≤t<τ by Pt = sin(Wt )/ cos(Wt ), where τ = inf{t ≥ 0 : |Wt | = 2/1π}. Show that the following holds:
Problem 2.
(a) [8 points] [SS]
Let (Ω , F, {Ft }t≥0, P) be some probability space and {Wt }t≥0 be a (P, {Ft }t≥0)- Brownian motion. For a finite Q > 0, let {Ft }t≥0 , {Gt }t≥0 and {Ht }t≥0 be
for all t ≥ 0, respectively. For each above, prove whether it is a (P, {Ft }t≥0)- martingale or not.
(b) [8 points] [S]
Let {Bt }t≥0 given by Bt = ert model the money-market account for some finite r ≥ 0 and let {Zt }t≥0 be governed by dZt = −λtZt dWt where {λt }t≥0 is the market price of risk. What is the stochastic differential equation of {πt }t≥0 given by πt = Zt /Bt? Also, prove that {πt }t≥0 is a (P, {Ft }t≥0)- supermartingale.
(c) [9 points] [U]
Let Q ≠ 0 be some finite constantan {Et }0≤t<τ be governed by the following:
Provide an explicit solution to the stochastic integral above – that is, what is the function f if the solution is Et = f (t, Wt ). Based on this solution, what should be the definition of τ in terms of Wt and Q?
Problem 3. Let (Ω , F, {Ft }t≥0, P) be some probability space and {Wt }t≥0 be a (P, {Ft }t≥0)-Brownian motion. Let {Bt }t≥0 governed by dBt = rBt dt model the money-market account, where B0 = 1 and r > 0 is finite. Let {Xt }t≥0 model asset price dynamics where dXt = μXt dt + σXt dWt for some finite constants μ ∈ R and σ > 0 with X0 = x.
(a) [8 points] [S]
Prove that {Zt }t≥0 defined by
is a (P, {Ft }t≥0)-martingale if λ = (μ − r)/σ . (b) [8 points] [SS]
Define the risk-neutral probability measure Q as follows:
For some Q > 0 and K ≥ 0, derive the following option price: V0 = e-rTEQ [(XT(α) − K)+]. (Hint: These are called power options)
(c) [9 points] [SS]
Now define the probability measure U as follows:
Show that for any 0 ≤ s ≤ t ≤ T , the following holds:
Problem 4. Let (Ω , F, {Ft }t≥0, Q) be a probability space where Q is the risk- neutral probability measure and {Wt }t≥0 is a (Q, {Ft }t≥0)-Brownian motion. For interest rate r > 0 and volatility σ > 0 (both finite), let {Xt }t≥0 be governed by dXt = rXt dt + σXt dWt.
(a) [9 points] [U] If g(x) is a non-negative convex function for x ≥ 0 with g(0) = 0, prove that {St }t≥0 given by St = e-rtg(Xt ) is a (Q, {Ft }t≥0)- submartingale. (Hint: Since g is convex, g ((1 − Q)x1 + Qx2 ) ≤ (1−Q)g(x1 )+ Qg(x2 ) for 0 ≤ Q ≤ 1 and 0 ≤ x1 ≤ x2 . Set x1 = 0 and x2 = x)
(b) [8 points] [S]
Define the stochastic integral process {Mt }t≥0 as follows:
Compute the expected value EQ [Mt ] and the variance VarQ [Mt ].
(c) [8 points] [SS]
Let {W(ˆ)t }t≥0 be another (Q, {Ft }t≥0)-Brownian motion which is independent of {Wt }t≥0 . Define Yt = QWt/α2 for some Q > 0 and Y(ˆ)t = tW(ˆ)1/t for t ≥ 0, with Y0 = Y(ˆ)0 = 0. If Rt = Yt /Y(ˆ)t whenever Y(ˆ)t 0, what is the distribution Q(R1 ∈ dr)?