Compute the mean and variance as functions of time of the

Question

Work 2 -

Part 1 -

1. Determine for a geometric Brownian motion Z_t = Z_oexp{μt + σ W_t} the Ito differential for Z_t and ln(Z_t) by the use of the Ito formula, where W is a standard Wiener process.

2. Consider two transformed Wiener processes with Y_t¹ = a₁t + b₁W_t¹ and Y_t² = a²t + b₂ W_t², where W¹ and W² are two independent standard Wiener processes. What is the Ito differential for Y_t¹ Y_t²?

3. For a process X = {X_t, t ∈ [0, ∞)} with X_t = σW_t + ξN_t, where W is a standard Wiener process and N a Poisson process with intensity λ > 0, characterize the stochastic differential of its exponential when σ, ξ > 0.

4. Compute the mean and variance as functions of time of the standard Ornstein-Uhlenbeck process with initial value X_t0 = 1.

5. Solve explicitly the scalar SDE

dX_t = - ½X_t dt + X_tdW_t¹ + X_tdW_t²,

where W¹ and W² are independent standard Wiener processes.

6. Show for the BS model that the discounted European call option price of the discounted Black-Scholes formula satisfies the discounted Black-Scholes partial differential equation with corresponding terminal condition.

7. Show for the European put option under the BS model that the corresponding P&L process is zero.

8. Show that the discounted European call option price process for the BS model with constant parameters is a martingale under the risk neutral probability measure.

9. Starting from the risk neutral SDE for the stock price verify that the benchmarked stock price for the BS model is an (A, P)-martingale.

10. For a nonnegative portfolio S_t^δ with SDE and the GOP S_t^δ_* satisfying the SDE

dS_t^δ_* = S_t^δ_*(r_tdt+_k=1∑^dθ_t^k(θ_t^kdt + dW_t^k)),

show for the benchmarked portfolio value S^{^}_t^δ = S_t^δ/S_t^δ_* its SDE. If S^^δ is square integrable does this SDE, in general, imply that S^^δ is a sub-martingale, martingale or supermartingale?

11. Compute the maximum expected power utility for the BS model.

Part 2 -

Problem 1: Let W(t) be a standard Brownian Motion,

X_t  = t + (W(4^t)/2^t), t ≥ 0.

(1) Find the covariance function of the process X_t.

(2) Is X_t a stationary process? Explain your answer.

(3) Find the conditional expectation

E(X_t|X₁)

and the expectation

E[(X_t - E(X_t|X₁))²].

Problem 2: A certain town never has two sunny days in a row. Each day is classified as being either sunny, cloudy (but dry), or rainy. If it is sunny one day, then it is equally likely to be either cloudy or rainy the next day. If it is rainy or cloudy one day, then there is one chance in two that it will be the same the next day, and if it changes then it is equally likely to be either of the other two possibilities.

Let X_n denote the corresponding Markov chain with states 1 = sunny, 2 cloudy (but dry) and 3 = rainy.

(4) Find the matrix of transition probabilities for the Markov chain X_n.

(5) Is the Markov chain ergodic? Provide your arguments.

(6) Assume that P {X₀ = i} = 1/3 for i = 1, 2, 3. Find the probability P{X₂ = 2}.

Problem 3: To model credit rating dynamics of a firm, XYZ Bank uses a homogeneous Markov process with continuous time parameter and three states AA, BB, C. The expectation of sojourn times (times between jumps) are 1 year for the states AA and BB and 10 years for the state C. The matrix of state transition probabilities (after jumps) is

(7) Find the generator matrix.

(8) Find a stationary distribution.

Problem 4: Consider a Poisson process N_t, t ≥ 0, with a rate λ = 1.

(9) Calculate

P(N₄ = 4|N₃ = 3, N₅ = 5).

(10) Given a sequence {X_n = 1, 2, ...} of independent identically standard normally distributed random variables, consider the stochastic process

Y_t = t/3 + _n=1∑^N_t X_n.

Find

Var (Y₁).

Problem 5: Let W_t, t ≥ 0, be a standard Brownian Motion and the diffusion process X_t be defined as

X_t = e^-t W_t.

(11) Find a transition distribution function of X_t, i.e. the function

F(x, t|y, s) = P{X_t < x|X_s = y}.

(12) Using Ito formula, find drift and diffusion coefficients of the diffusion process X_t.

(13) Write down Kolmogorov forward equation for

p(x, t | y, s) = (∂/∂x)F(x, t|y, s).

Problem 6: On a filtered probability space (Ω, A, A, P) consider a Black-Scholes financial market model with once source of uncertainty; that is, modelled by a standard Wiener process W = {W(t), t ∈ [0, T]}. It consists of a savings account B = {Bet), t ∈ [0, T]} with

B(t) = exp(rt)

and a non-dividend paying stock with price

X(t) = exp(μt + σW(t))

at time t ∈ [0, 7], where r > 0, μ > 0 and σ > 0.

(14) Write down the stochastic differential equations for the values of both primary security accounts; that is, B(t) and X(t), t ∈ [0, T].

(15) Does this market have a numeraire portfolio (NP)? If yes, then provide the optimal fractions π⁰_{δ_*}(t) and π¹_{δ_*}(t) of wealth to be invested in B(t) and X(t), respectively, at time t ∈ [0, T] for forming the NP.

(16) Does there exist an equivalent risk neutral probability measure P_θ for the above model? If yes, then mention the main property that must be satisfied for this to be the case.

(17) Provide the (potential) risk neutral stochastic differential equations for the primary security account values B(t) and X(t) under the risk neutral probability measure P_θ.

Work 3 -

Please note that there is a collection of selected formulas appended to this exam.

1. Consider a model, of two economies (domestic and foreign), where interest rate dynamics in each economy are given by a Gauss/Markov HJM model, and the two economies are linked by an exchange rate with deterministic relative volatility. Thus, domestic zero coupon bond prices B(t, T) (i.e., observed at time t and maturing at time T) follow the dynamics

dB(t, T) = B(t, T)(r(t)dt - σ*(t,T)dW_β(t))

under the domestic spot risk-neutral measure, and similarly, foreign zero coupon bond prices B^~(t, T) (i.e., observed at time t and maturing at time T) follow the dynamics

dB^~(t, T) = B^~(t, T)(r^~(t)dt - σ^~*(t, T)dW^~_β(t))

under the foreign spot risk-neutral measure, where r and f- denote the domestic and foreign continuously compounded short rates, respectively. The spot exchange rate in units of domestic currency per unit of foreign currency follows the dynamics

dX = X (t)((r(t) - r^~(t))dt + σ_xdW_β(t))

under the domestic spot risk-neutral measure. Furthermore, consider a domestic (non-dividend-paying) equity assets S, which under the domestic spot risk-neutral measure follows the dynamics

d_S(t) = S(t)(r(t)dt + σ_SdW_β(t))

In this model, consider a European option to exchange one unit of S for κ units of foreign zero coupon bonds B maturing at time T^{^} > T. I.e., the time T payoff in domestic currency is

max(0, κX(T)B^~(T, T^{^}) - S(T))

(a) Derive the pricing formula for the option. You do not need to solve the (classical) integrals over volatility functions that may arise in your solution.

(b) Determine the initial (i.e., time t = 0) positions required for a self financing hedging strategy replicating this option. You may assume that the underlying asset S and any number of zero coupon bonds (domestic or foreign) are avail-able to use as hedge instruments.

2. Suppose you have entered into an interest rate swap with a counter party, where you pay floating and receive fixed. Assume that the trade is not collateralised.

(a) Describe the market risk and the counterparty risk to which you are exposed in this situation. What would constitute "wrong way risk" in this context?

(b) How can your counterparty credit risk exposure be interpreted as an option? Describe the option position. What simplifying assumptions do you need to make in order to price the counterparty credit risk as a weighted sum of option prices?

(c) Suppose that the current term structure of interest rates is flat (i.e., equal yields for all maturities). Describe qualitatively the shape of the term structure of the expected future counterparty credit risk exposure. How would this change if the current term structure of interest rates were upward sloping (i.e., higher interest rates for longer maturities than for shorter maturities). Give reasons for your answer.

(d) Suppose that bilateral credit valuation adjustment (CVA) for this trade works out to be equal to zero. For what reasons might bilateral CVA be zero? Does a zero bilateral CVA imply the absence of counterparty credit risk?

3. In the judgment in the case Bathurst Regional Council v Local Government Financial Services Pty Ltd (No 5) [2012] FCA 1200 (5 November 2012), Jagot J states, "Again, a reasonably competent ratings agency could not have assigned the AAA rating to the Rembrandt 2006-2 CPDO in these circumstances." Similarly, she states further, "A reasonably competent ratings agency could not have rated the Rembrandt 2006-3 CPDO AAA in these circumstances."

(a) What role did mathematical modelling play in this case?

(b) What lessons for quantitative analysts might one draw from this case?