Asset Pricing in Continuous Time
MATH0085
MSc Examination
2023
Problem 1. Consider a filtered 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 natural filtration generated by {Wt }t≥0 . We denote by E the expectation under the measure P.
(a) Is the process
{9Wt/9}t≥0
a standard Brownian motion with respect to its natural filtration? Justify your answer. [5]
(b) Assuming that the process
is integrable, argue whether it is a martingale with respect to {Ft }t≥0 and justify your answer. [5]
(c) Suppose that T > 0 is constant. Choose a partition Π with points 0 = t0 < t1 < t2 < . . . < tn = T, and mesh (longest subinterval) denoted by ||Π|| . Find the first variation of W, given by the limit in probability
Hint: Use the fact that the quadratic variation (limit in probability) is
Justify your answer. [5]
(d) Derive the stochastic differential equation satisfied by
{Wt3 }t≥0 .
Using this, calculate E[Wt3], for all t ≥ 0. [5]
(e) Calculate the quadratic variation of the process
{Wt3 }t≥0
and explain whether this is independent of the Brownian motion’s path. [5]
Problem 2. Consider a financial market defined on a filtered probability space (Ω , F, {Ft }t≥0, P) with a risky financial asset with price process {St }t≥0 satisfying
dSt = µStdt + etStdWt , S0 > 0, (1)
where µ ∈ R, {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,
where r > 0. Then, consider a European-type option with payoff 1{ST>K} at the expiration time T > 0, with K > 0 and 1{·} denoting the indicator function.
(a) Derive the solution {St }t≥0 to the stochastic differential equation (1). [4]
(b) Derive the risk-neutral stochastic differential equation of the price process {St }t≥0 . Explain in detail all steps. [6]
(c) Derive the explicit formula for the risk-neutral price V (t, St ) of the option at any time t ∈ [0, T]. Explain in detail all steps. [10]
(d) Construct a hedging portfolio for the option and derive the explicit expres- sions of the hedging positions at any time t ∈ [0, T]. Explain in detail all steps. [5]
Problem 3. Consider a filtered probability space (Ω , F, {Ft }t≥0, P) equipped with a standard (P, {Ft }t≥0)-Brownian motion {Wt }t≥0 and the stochastic integral process {Bt }t≥0 given by
where
Throughout this question, you may use without proof that all stochastic integrals against W are martingales.
(a) Is the process {Bt }t≥0 a standard (P, {Ft }t≥0)-Brownian motion? Justify your answer. [5]
(b) What is the stochastic differential equation satisfied by the process {BtWt }t≥0? As a consequence, calculate its expected value. Justify your answers. [5]
(c) What is the stochastic differential equation satisfied by the process {BtWt2 }t≥0? As a consequence, calculate its expected value. Justify your answers. [9]
(d) Are the processes {Bt }t≥0 and {Wt }t≥0
(i) correlated?
(ii) independent?
Justify your answers. [6]
Problem 4. Consider a financial market defined on a risk-neutral filtered prob- ability space (Ω , F, {Ft }t≥0, Q) with a risky asset with price process {St }t≥0 satis-fying
dSt = rtStdt + σStdWt(1) , S0 > 0, (2)
where σ > 0, {Wt(1) }t≥0 is a standard (Q, {Ft }t≥0)-Brownian motion and {rt }t≥0 is the interest rate process.
Consider a long position in a forward contract with expiration time T > 0, when the investor pays the forward price F0,T and receives the aforementioned asset. Recall that, F0,T is F0-measurable (i.e. known at time t = 0).
Suppose that the interest rate is constant rt = r > 0 for all t ∈ [0, T].
(a) Using a replicating portfolio, derive the explicit expression of the no-arbitrage value of the forward price F0,T . [5]
(b) What is the risk-neutral value of the aforementioned forward contract at any intermediate time t ∈ [0, T]? Justify your answer. [5]
Suppose for the remainder of this question that the interest rate {rt }t≥0 satisfies drt = (a − rt )dt + bdWt(2) , r0 ∈ R,
where a,b > 0 and {Wt(2) }t≥0 is a standard (Q, {Ft }t≥0)-Brownian motion. A zero-coupon bond which pays 1 at its maturity time T has value B0(*),T at time 0.
(c) Derive the explicit expression of the no-arbitrage value of the forward price F0,T in terms of B0(*),T and S0 . Explain in detail all steps. [7]
Finally, consider a long position in a futures contract with maturity time T > 0. This is an agreement to receive as a cash flow the changes in the futures price
ft,T = EQ [ST |Ft] (under no-arbitrage)
of the asset during the times t ∈ [0, T].
(d) Derive an explicit expression for the forward–futures spread in terms of B0(*),T .
Based on that, give an example of a positive spread and another example of a zero spread. [8]