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This is the Exam of Statistical Science which includes Upcrossing Inequality, Understood, Time Series and Monte Carlo Inference Two, Second Order Stationary, Telephone Banking Facility, Headline, Supermarkets etc. Key important points are: Stochastic Proces, Conditions, Stochastic Pr, Martingale, Process, Measurable Denoted, Approximation, Continuous, Stochastic, Martingale Differences
Typology: Exams
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Friday 10 June, 2005 9 to 11
Attempt THREE questions. There are FIVE questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 State the conditions under which discrete time stochastic process M := {Mn : n = 0,... , N } with filtration {Fn} is a P-martingale.
Show that the process V := {Vn : n = 0,... , N } with Vn :=
∑n i=1 φi(Mi^ −^ Mi−^1 ) is a discrete time P-martingale if M is a P-martingale and φ is a bounded previsible process (i.e. φ is Fi− 1 measurable denoted φ ∈ Fi− 1 ).
Using an approximation by such sums of martingale differences, give an heuristic argument to show that for a continuous time stochastic Ito integral
∫ (^) t
0
g(s, Ws)dWs = 0
and
E
(∫ (^) t
0
g(s, Ws)dWs
∫ (^) t
0
g(s, Ws)^2 ds
for a suitable function g with
∫ (^) t 0 Eg(s,^ Ws)
(^2) ds < ∞.
The Ornstein-Uhlenbeck (OU) X process satisfies the SDE
dXt = κ(θ − Xt)dt + σdWt, X 0 = x a.s.,
where κ, θ and σ are constants and W is a Wiener process. Solve this SDE by setting Yt := eκtXt and using Ito’s lemma to show that
Xt = θ + (x − θ)e−κt^ + σ
∫ (^) t
0
e−κ(t−s)dWs.
Find the mean and variance of Xt.
Let C be a Cox-Ingersoll-Ross (CIR) process satisfying the SDE
dCt = a(b − Ct)dt + σ
CtdWt.
By considering Zt :=
Ct derive an SDE for Z. Choosing the parameters so that 2ab = σ^2 , show that Z is an OU process and hence find E(Ct).
3 Consider the pricing of a European call option consistently with a given volatility smile or skew. To this end assume a Black- Scholes-type price satisfying
∂V ∂t
σ^2 (S, t)S^2
− rV = 0, 0 < S, 0 < t < T
with a suitable terminal condition V (S, T ) = (S − K)+^ and a given local volatility function σ satisfying σ− 6 σ(S, t) 6 σ+ (e.g. from a previous calibration step), where σ± are known constants. Suppose also r > 0.
(a) It is convenient to introduce the transformations x := ln S, u(x, t) := er(T^ −t)V (ex, t). Write down the PDE for u and the corresponding SDE for x.
(b) Consider a numerical scheme in which the new variables are restricted to a finite interval (which is large enough not to introduce any significant errors by setting asymptotic boundary values) with an equidistant space grid with n points (mesh width ∆x) and m time steps ∆t. Let uji be the approximation at xi and tj , i = 1 ,... , n, j = 1 ,... , m and write down the discretisation using central differences for the space derivatives and explicit Euler time stepping.
(c) An initial approximate solution is performed on a very coarse grid with given values of ∆x and ∆t. To improve the accuracy, ∆x is repeatedly divided by 2 while monitoring the numerical solution evaluated at the money, which is assumed to coincide with a grid point. Does the solution converge? If so, towards what?
(d) Repeating the same procedure by refining ∆t instead, does the solution converge? If so, towards what?
(e) Finally, the space step is divided by 2 and the time step by 4 during refinements. Comparing the difference between the solutions of two subsequent refinement levels, which (convergence) behaviour is observed? (f) In order to hedge the contract, the previous numerical solution is used to approx- imate the ∆ (as before) by a central difference. What is the order of convergence for ∆? Explain. Can reliable information about Γ be retrieved from this numerical solution? Why (not)? (g) Noting that explicit finite differences can be viewed as a trinomial tree, write the recursive scheme in the form
uj i −^1 = pji,duji− 1 + pji,muji + pji,uuji+
and sketch the corresponding tree, giving the weights pji explicitly in terms of the parameters. Under which conditions (on ∆x and ∆t) can the weights be seen as probabilities? How do these relate to the stability of the finite difference scheme?
(a) In the single factor Heath-Jarrow-Morton (HJM) model, given an initial forward rate curve f (0, ·), the forward rate for each maturity T evolves as
df (t, T ) = α(t, T )dt + σ(t, T )dWt 0 6 t 6 T,
where W is a Wiener process and αt and σt can depend on W and f (·, T ), T > t, up to time t. Give expressions for the bond price P(t, T ), the short rate rt := f (t, t) and, defining the cash bond price Bt := exp
(∫ (^) t 0 rsds
, the discounted bond price Zt(t, T ) := B− t 1 P(t, T ).
(b) Defining Σ(t, T ) := −
t σ(t, u)du, under the risk neutral measure Q,^ P^ has dynamics dP(t, T ) = P(t, T )[rtdt + Σ(t, T )dWt].
If X is the single payoff of a derivative maturing at T > t show that its current price at t is given by V(t, r) = EQ
e−
t rsdsX|Ft
(c) Give an expression for the dynamics of the short rate r under Q. Is it necessarily Markov? Explain your answer.
