Section A:
1. Y (t) = wX1(t) + √(1 - w2X2(t)) is a model to construct a process from two uncorrelated Brownian Motions.
The processes X1(t), X2(t) have drifts in addition to the increment dW of Wiener process:
dX1(t) = μ1dt + σ1dW1(t)
dX2(t) = μ2dt + σ2dW2(t)
Brownian Motion represents a source of risk (factor) and w is called 'factor loading'.
(a) Construct a Stochastic Differential Equation for Y (t). Which value of w would make Y (t) a martingale?
(b) Evaluate the variance V[Y (t)] and provide the distribution for the increment of Y (t). Does Y (t) keep the properties of the Brownian Motion?
2. Assume that an asset price S evolves according to the SDE
dS/S = (μ - D) dt + σdX
where μ and σ are constants. In addition S pays out a continuous dividend stream equal to DSdt during the infinitesimal time interval dt, where D the dividend yield is constant.
Now suppose a European style derivative security V (S, t) is written on this asset with the properties that at expiry the holder receives the asset and prior to expiry the derivative pays a continuous cash flow C (S, t) dt during each time interval of length dt:
(a) Show that the option price satisfies
∂V/∂t + 1/2 σ2S2 ∂2V/∂S2 + (r - D) S ∂V/∂t -rV = -C(S,t).
(You are required to derive this PDE)
(b) Suppose that the cash flow C (S, t) in part a. has the form C (S, t) = f (t) S: By writing V = ψ(t) S and assuming a final condition at time T given by
V (S, T ) = S,
show that the delta of the derivative security is
Δ(S, t) = e-D(T -t) + t∫T e-D(T -t) f (τ )dτ :
(You are required to solve the PDE together with the final condition)
3. Consider the function f (y) = e-y and Y ~ N (0, 1): Show that the variance of f (y) is
σ2f = e (e - 1).
4. The "Speed" of an option C (S, t) is given by
Speed = ∂3C/∂S3
If S = nδS and t = mδt , by obtaining 3 suitable Taylor expansions for the option price Cmn show that a Finite Difference Approximation for the Speed is given by
∂3C/∂S3 ≈ 1/∂S3(a1Cmn+2 + a2Cmn+1 + a3Cmn + a4Cmn-1,)
where the values of the constants ai for i = 1, 2, 3, 4 must be obtained.
5. The Black-Scholes formula for the value of a put option P (S, t) is
P (S, t) = E exp(-r(T - t))N (-d2) - SN(-d1)
From this expression, find the Black-Scholes value of the put option in the following limits:
(a) (time tends to expiry) t → T , σ > 0,
(b) (volatility tends to zero) σ → 0, t < T ,
(c) (volatility tends to infinity) σ → ∞, t < T
Section B
6. Find the implied volatility of the following European call option. The call has four months until expiry and an exercise price of $100. This call is worth $6.51 and the underlying trades at $101.5, discount using a short- term risk-free continuously compounding interest rate of 8% per annum. You can use Excel or Matlab.
7. Consider evaluating the Gaussian integral
J = 1/√2Π. -∞∫∞ x2 exp(-x2/2)dx.
by writing this as
J = E[X2] ~ JN = 1/N . n=1ΣNx2
xn follows a standard normal distribution, i.e. xn ~ N (0, 1):
Implement this method in VBA, C++ or Matlab and examine JN for different values of N . Plot an error graph (difference between exact and numerical values).
8. Describe in detail a Monte Carlo simulation algorithm for estimating
θ = 0∫∞ e-x2 dx.
Hint: You are required to first obtain a suitable transformation to convert this to an integral problem over (0, 1) : Implement your algorithm in VBA, C++ or matlab for different numbers of random numbers generated and comment on how accuracy is achieved.