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

σ²_f = 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 C^m_n show that a Finite Difference Approximation for the Speed is given by

∂³C/∂S³ ≈ 1/∂S³(a₁C^m_n+2 + a₂C^m_n+1 + a₃C^m_n + a₄C^m_n-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] ~ J_N = 1/N . _n=1Σ^Nx² 

x_n follows a standard normal distribution, i.e. x_n ~ N (0, 1):

Implement this method in VBA, C++ or Matlab and examine J_N 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.