1. All random variables in the following questions are defined on a probability space (Ω,F,P), and G denotes a σ-algebra contained in F.
(a) Let X and Y be random variables such that E|X| < ∞. Define precisely the conditional expectations E(X|G) and E(X|Y ). [7 marks]
(b) Let X be a random variable with finite second moment, and consider R = X − E(X|G), the difference between the “true value” of X and the “predicted value” of X based on the “information” G. Compute ER and E(R|G). Show that
where Z = E(X|G).
[7 marks]
The solution r of the stochastic differential equation
drt =100(4−rt)dt+(|rt −5||rt −3|)1/2dWt,
r0 =4 is suggested as a suitable model, where W = (Wt)t≥0 is a Wiener process.
ER2 = EX2 − EZ2,
(c) Let X be a random variable with finite second moment, and let L2(G) denote the space of G-measurable random variables with finite second moment. Show that Z = E(X|G) minimises the mean square distance of X from L2(G), i.e.,
E(X−Z)2 =min{E(X−Y)2 :Y ∈L2(G)}.
(d) Let X1 and X2 be independent identically distributed random variables such that
E|X1| = E|X2| < ∞. Set X ̄ = (X1 + X2)/2, and determine E(αX1 + (1 − α)X2|X ̄)
as a function of X ̄ for any constant α ∈ R. [Hint: You may use without proof that due to symmetry, E(X1|X ̄) = E(X2|X ̄).] [5 marks]
2. We want to model the evolution of the instantaneous interest rate by an Itoˆ process r = (rt)t≥0 satisfying following conditions:
(i)r0 =4and3≤rt ≤5almostsurelyforallt≥0; (ii)Ert =4forallt≥0;
(iii) E(|rt − 4|2) ≤ 1/200 for all t ≥ 0.
(a) State precisely a comparison theorem for SDEs, and applying it to suitable SDEs, show that property (i) holds for the solution r. [12 marks]
(b) Prove that r satisfies property (ii). [6 marks]
[6 marks]
(c) Setting Yt := rt − 4 and using Itˆo’s formula, write an expression for the stochastic differential of e100t Yt . Hence, estimating E (|e100t Yt |2 ), or otherwise, deduce that r satisfies property (iii). [7 marks]
3. Consider the standard Black-Scholes market with bond price Bt = ert and stock price St = S0 exp(αt + σWt) at time t ∈ [0, T ], where W is a Wiener process, α is any constant, and S0 and σ are positive constants. Let f be a nonnegative function on R satisfying the polynomial growth condition, and denote by A0 and B0 the prices at time t = 0 of an American type option with pay-off process (f(St))t∈[0,T] and a European type option with pay-off f (ST ) at maturity T , respectively.
(a) Using appropriate formulas define precisely the prices A0 and B0, and hence show that A0 ≥ B0. [7 marks]
(b) State precisely the Main Theorem on Pricing European Type Options, and hence show that the replicating strategy (ψt∗, φ∗t )t∈[0,T ] for the European type option with pay-off f (ST ) is a hedging strategy for the American type option with pay-off process (f(St))t∈[0,T] if and only if
e−r(T−t)EQ (f(ST)|Ft) ≥ f(St) for every t ∈ [0,T], where Q is the risk neutral probability measure.
[7 marks]
(c) Assume that f is a convex function such that f(0) = 0. Then show that λf(x) ≥ f(λx) for every x ∈ R and λ ∈ [0,1], and hence using Part (b) prove that the replicating strategy (ψt∗,φ∗t)t∈[0,T] for the European type option with pay-off f(ST) is a hedging strategy for the American type option with pay-off process (f(St))t∈[0,T].
[7 marks]
(d) Prove that if f is convex such that f(0) = 0 then A0 = B0.
4. Consider again the Black-Scholes market with bond and stock prices as in Question
[4 marks] 3. We want to compute the price V at t = 0 of the European type option with payoff
L if maxt∈[0,T]St≥K h := 0 otherwise
at expiry date T, where L > 0 and K > 0 are some constants.
(a) Using the main theorem on pricing European options, show that
V =Le−rTP max(Wt +at)≥b , t∈[0,T ]
where a := σr − 12σ, b := σ−1 ln(K/S0).
[7 marks]
(b) State precisely the Girsanov theorem.
(c) Using Girsanov’s theorem show that
V =Le−rTE1[maxt∈[0,T]Wt≥b]eaWT−21a2T,
where a and b are the constants defined in Part (a).
[6 marks]
[7 marks] (d) Denote the event [maxt≤T Wt ≥ b, WT ≤ x] by Bx. Knowing that, due to the reflection
principle for the Wiener process W,
P(WT ≥2b−x) if x