ECON 643 Financial Economics II Assignment 2 Due On 06/04

ECON 643: Financial Economics II Assignment 2: Due on 06/04/2020 Winter 2020–24/03/2020 For simulation exercises, you may use any software or programming language of your choice, but Matlab or R should be easier to use.

This assignment comprises four questions focused on financial derivatives, including options pricing, portfolio insurance strategies, GARCH model simulations, and barrier options. Students are expected to perform a combination of analytical derivations, numerical simulations, and comparative analyses, leveraging appropriate models and methodologies. The tasks involve using real-world data, stochastic modeling, and risk-neutral valuation techniques. Implementation should preferably be in MATLAB or R, but other programming environments are acceptable.

Paper For Above instruction

The present assignment deals comprehensively with advanced topics in financial derivatives and portfolio management, embodying a blend of theoretical derivations and practical simulations. Each question emphasizes key concepts such as option valuation approaches, model-based pricing under stochastic volatility, and complex derivative structures like barrier options. Effectively addressing these questions requires a solid understanding of financial mathematics, stochastic calculus, and numerical methods, along with proficiency in programming for simulations and computations.

Question 1: Portfolio Insurance and Derivative Strategies

A fund manager seeks to hedge a $360 million diversified portfolio tracking the S&P 500 against a decline exceeding 5% within six months. The current S&P 500 level is 1,200, with a dividend yield of 3%, a risk-free rate of 6%, and an index volatility of 30% per annum. The manager considers multiple strategies involving options and futures to implement this hedge.

(a) Calculating the cost of insurance via traded European put options involves determining the appropriate strike price and applying the Black-Scholes formula to derive the option premium. The strike corresponds to a 5% decline, i.e., 1,140 (1,200 × 0.95). The number of options required aligns with the total portfolio value and the multiplier effect. Using the Black-Scholes model and considering the continuous dividend yield, the price of a single at-the-money European put can be calculated. The total insurance cost is obtained by multiplying the per-unit option price by the number of units needed.

(b) Alternatively, the manager can construct a hedge using European call options, which theoretically leads to the same effective protection through a combination of options and dynamic rebalancing. By establishing a synthetic position—combining puts and calls—one can replicate the payoff of the protective put. The payoff equivalence relies on put-call parity in a continuous dividend environment, ensuring that either method yields identical cost and effectiveness when executed correctly.

(c) Providing insurance by allocating part of the portfolio to risk-free securities entails calculating the initial amount that must be invested at the risk-free rate to match the downside protection. This involves determining the present value of the targeted loss and aligning the risk exposure accordingly, typically resulting in a static hedge that balances the potential shortfall over the hedging period without derivative transactions.

(d) Using index futures to hedge the portfolio involves establishing an initial short position in futures contracts. The hedge ratio is derived from the beta of the portfolio to the index, which, assuming perfect correlation, simplifies to a 1:1 hedge adjusted for the portfolio value and futures contract specifications. The initial position is computed by multiplying the desired insurance amount by the portfolio value and dividing by the futures contract size, ensuring that the hedge offset matches the potential loss.

Question 2: GARCH Model Simulation and Option Pricing

The second question involves simulating the price process of an underlying asset following a GARCH(1,1) model with specified parameters. The goal is to compute the price of a European call option with a strike of 100 and 20 days to maturity, using a risk-neutral measure derived from the GARCH dynamics. The process begins with specifying the model's parameters, including the constants \(\omega, \alpha, \beta, \theta, \lambda\), and the initial variance. The risk-neutral dynamics are derived by adjusting the drift component to incorporate the market price of risk, aligning the stochastic process with no-arbitrage conditions.

Simulation entails generating numerous paths of the underlying price under the risk-neutral measure, calculating the payoff at maturity for each path, and discounting back. The GARCH model's conditional variance process influences the stochastic volatility, adding realism to the price dynamics. A comparison with the Black-Scholes (BS) price requires calculating the BS formula under constant volatility \(\sigma = \sqrt{0.00016}\) and comparing results to analyze the impact of stochastic volatility on option valuation.

Question 3: Pricing Outperformance Options in a Bivariate GBM Framework

This question explores the valuation of a European outperformance option based on two indices, S&P 500 (S1) and MSCI (S2), with a payoff at maturity \(T\) given by \(\max(a S_1(T) - b S_2(T), 0)\). The derivation involves choosing an appropriate numeraire to re-express the payoff, revealing that it can be viewed as a standard call or put option on a transformed underlying asset with a specific strike. This transformation simplifies the valuation problem into a standard Black-Scholes framework.

Assuming both assets follow correlated geometric Brownian motions under the risk-neutral measure, the pricing formula involves calculating the joint distribution with correlated variances. Applying risk-neutral valuation yields an analytical expression similar to the Bivariate Black-Scholes formula, which accounts for the correlation and individual asset volatilities. Analyzing the formula highlights the influence of the risk-free rate, especially when expressed in the discounting component, and the impact of correlation on the option's price.

The Monte Carlo simulation with specified parameters enables empirical estimation of the option's value. By simulating multiple paths of \(S_1\) and \(S_2\), computing their payoffs, and discounting, the convergence of the simulation to the analytical price can be examined. Variance reduction techniques such as antithetic variates or control variates could accelerate convergence and improve accuracy.

Question 4: Barrier Options under Different Dynamics and Numerical Estimation

The final question assesses barrier options (down-and-in puts and up-and-out calls) on an underlying asset with stochastic dynamics incorporating stochastic volatility and correlation, specified by the given SDEs. The payoff at maturity depends on whether the barrier levels are breached before expiration, which can be modeled via path-dependent features. The challenge lies in accurately simulating the paths, handling the barrier conditions, and computing the option prices.

Numerical simulation involves generating many paths of the underlying asset price and volatility, tracking barrier breaches over the path. For the model with stochastic volatility, special discretization schemes, such as Euler-Maruyama or more advanced methods, are used, ensuring the correct correlation structure. For each simulated path, the barrier conditions determine whether the payoff is active at maturity; then, the average discounted payoff across simulations yields the option price.

Comparing prices under stochastic volatility-driven dynamics versus classic geometric Brownian motion highlights how volatility clustering, mean reversion, and correlation influence barrier option valuation. Such insights inform risk management and hedging strategies, showcasing the importance of accurate modeling of the underlying's dynamics.

References

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• Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654.
• Engle, R. F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50(4), 987–1007.
• Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6(2), 327–343.
• Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson.
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• Zoolis, A. (2020). Monte Carlo Methods in Financial Engineering. Springer.