Seminar in Financial Product Innovations

Question One
Suppose we are in the midst of the financial crisis in October 2008. Your firm is considering the purchase of a ten year put option on the S&P 500 Index. You are analyzing the pricing of this option and you would like to incorporate different deterministic assumptions for the interest rate, the volatility and the dividend yield for each future year. Let us use the following notation: (15 marks)
a) Explain how you would use market data to estimate the future interest rates, future Index volatilities and dividend yields. Describe how to use Monte Carlo simulation to estimate the price of this option. Assume the initial index level is S0 and the strike price is K. Assume you are writing the instructions for a programmer who will implement the program.
b) Describe in detail an efficient variance reduction technique to improve the efficiency of the algorithm.
c) Describe carefully how you could numerically estimate the delta of this put option.

Question Two (5 marks)
Hedge funds are short two types of funding options. Explain in detail what these options are.
Explain why these options become more valuable during a financial crisis.
During the recent 2008 financial crisis describe how these options contributed to a dry up of liquidity and increased correlation among security prices.
Question Three (10 marks)
a) Describe what is meant by a positive coefficient discretization in the context of valuing options using numerical PDE methods. What is the main advantage of using a positive coefficient discretization? Are there any disadvantages? (You should include some discussion related to the computation of option prices, deltas, and gammas, as well as the rate of convergence for both fully implicit and Crank-Nicolson time stepping schemes.)
b) Explain what "Rannacher smoothing" is and why it is used.

Question Four (10 marks)
Suppose that two assets have jointly normally distributed monthly returns R1 and R2. R1 has mean μ1 and standard deviation σ1 and R2 has mean μ2 and standard deviation σ2, and the correlation between R1 and R2 is ρ. Assume μ1>μ2. Investors observe N i.i.d. observations of (R1,R2), and estimate μ1 and μ2 by the sample means, and declare that asset one has a higher expected return if μ1 >μ2, and that asset two has a higher expected return otherwise.
a) Determine the probability that investors will be correct.
b) For a prescribed confidence level α, determine the number of observations required to make the probability that investors are correct equal to α.
c) Supposing that μ1 = 0.015, μ2 = 0.01, σ1 = σ2 = 0.07, find how many observations are required for an investor to have a 90% probability of being correct when ρ = 0.1, ρ = 0.5, and ρ = 0.8. In determining your answer, you may use Z0.9 ≈ 1.28.
Question Five (5 marks)
Consider applying antithetic variates to E[f(U)], where U ∼ Unif(0, 1). Under what condition on f is the variance of the antithetic variate estimator zero?

Question Six
Consider the problem of estimating E[G(Z)] where G(·) is a function depending on random variable Z = (Z1,...,Zm)T∼ MVN(0, Im). Here 0 is a zero vector of dimension m, Im is the identity matrix of dimension m, and MVN(μ, Σ) is the multivariate normal density with mean vector μ and covariance matrix Σ. Now consider applying the importance sampling technique with importance density MVN(μ, Im), where μ =(μ1,...,μm)T. (10 marks)
a) Show that the expression
exp(-μTZ+0.5μTμ)
is the corresponding likelihood ratio. Hence describe the resulting importance sampling estimator.
b) Show that an optimal choice of μ can be defined as one that satisfies the following
condition: F(μ) = μT, (1)
where F(μ) = ln G(μ) and F (μ) denotes the gradient of F at μ. Provide a justification for the above choice of μ. (Hint: Consider a first order expansion to F(μ+Z)).

Question Seven
Assume that m independent samples {𝜇𝑖}𝑖=1𝑚 are generated for mean return 𝜇∈ℜ𝑛. Let Q be the given n-by-n covariance matrix and assume that the confidence level 𝛽 is specified. Consider the following CVaR robust portfolio optimization problem:
min𝑥𝑥𝑇𝑄𝑥𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑒𝑇𝑥=1𝑥≥0CVaR𝛽𝜇 (−𝜇𝑇𝑥)≤ 𝛬 (2)
Here Λ is a given constant, e is the vector of all ones, and the superscript μ in CVaR𝛽𝜇 represents the fact that the CVaR risk is with respect to the uncertainty in μ. Derive the sample CVaR optimization problem for (2). Explain your derivation.

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