ECON 643: Financial Economics II Assignment 2 Due On 27/11/2 ✓ Solved
ECON 643: Financial Economics II Assignment 2 Due on 27/11/2018
Analyze a set of financial scenarios involving portfolio insurance through options, futures, and risk-neutral pricing, as well as the dynamics of asset prices modeled by GARCH processes and stochastic differential equations. Provide detailed calculations, explanations of alternative strategies, simulation methods, and convergence analyses as per the given complex problem set.
Sample Paper For Above instruction
Introduction
This paper comprehensively addresses the multifaceted problems presented in the assignment for ECON 643: Financial Economics II. It explores portfolio insurance strategies using options, futures, and risk-free securities, delves into the GARCH model for option pricing, evaluates a multi-asset outperformance option, and assesses barrier options under stochastic volatility. The analysis combines theoretical derivations with simulation techniques, providing a synthesized understanding of advanced financial modeling.
Question 1: Portfolio Insurance and Derivative Strategies
Part (a): Cost of Protecting the Portfolio through European Put Options
The first part involves calculating the cost of insuring a well-diversified portfolio worth $360 million, which tracks the S&P 500 index, against a drop exceeding 5% over 6 months using European put options. Given the underlying index value of 1,200, the portfolio value, risk-free rate, dividend yield, and volatility are used to determine the option's premium.
Using the Black-Scholes framework, the strike price \(K\) corresponding to a 5% loss on the portfolio is: \( K = 0.95 \times (1,200 \times \text{number of units}) \). As the portfolio mirrors the index, we consider one point of the index, and the number of units held is approximately \( \frac{360\ \text{million}}{1,200} = 300,000 \) units. This implies a total equivalent of $360 million, facilitating the calculation using the index options.
Applying the Black-Scholes formula for a European put, the cost per option is:
P = K exp(-q T) N(-d2) - S exp(-q T) N(-d1)
where
d1 = [ln(S/K) + (r - q + 0.5 σ^2) T] / (σ * sqrt(T))
d2 = d1 - σ * sqrt(T)
and \(S = 1200\), \(K= 1200 \times 0.95 = 1140\), \(σ= 30\%\), \(r= 6\%\), \(q= 3\%\), \(T= 0.5\) years.
Calculations yield the per-unit premium, which scaled by 300,000 units provides the total insurance cost.
Part (b): Alternatives Using Call Options and Their Equivalence
Alternative risk mitigation strategies involve using European call options. Through put-call parity and the concept of replicating payoffs, a protective put can be replicated by holding a long position in a call and short a bond, or vice versa. The parity relation:
C - P = S exp(-qT) - K exp(-rT)
which demonstrates the equivalence of different strategies for insurance. Constructing a synthetic put via a long call and borrowing the strike price amount illustrates the same hedge, affirming the cost equivalence.
Part (c): Insurance via Risk-Free Securities
Implementing insurance by allocating part of the portfolio into risk-free securities involves determining the proportion \( \alpha \) such that the initial hedge replicates the protection. The amount \( \alpha \) is derived from the delta of a put option:
\(\Delta_{P} = - e^{-qT} N(-d1)\)
which indicates the fraction of the portfolio to hedge with a risk-free asset. Calculating \( \alpha \) ensures that the hedge covers the potential loss beyond 5%. The initial position in risk-free securities is then \( \alpha \times 360\ \text{million} \).
Part (d): Using Index Futures for Insurance
In the case of index futures, the hedge ratio is the delta of the option, with the initial position given by:
\(\text{Futures position} = \Delta_{P} \times \text{number of units}\)
which involves no upfront cost apart from margin requirements. The number of futures contracts needed is computed based on the delta and the contract specifications, providing an efficient hedge over the 6-month horizon.
Question 2: GARCH Model Pricing of a European Call Option
Risk-Neutral Dynamics Derivation
The GARCH(1,1) model specifies the evolution of the asset's variance. To price options under this model, the first step is to derive the risk-neutral dynamics.
By assuming the asset price process \(S_t\) follows the distribution:
\[
\ln \frac{S_t}{S_{t-1}} = r + \lambda \sigma_{t-1} - \frac{1}{2} \sigma_{t-1}^2 + \sigma_{t-1} z_t
\]
with \(z_t \sim NID(0,1)\), the risk-neutral measure adjusts \(z_t\) to a standard normal with mean \( - \lambda \), so that:
\[
\mathrm{d}\ln S_t = r dt + \sigma_t dW_t^{\mathbb{Q}}
\]
where \( dW_t^{\mathbb{Q}} \) is a Brownian motion under the risk-neutral measure. The process ensures the discounted stock price is a martingale.
Simulation of GARCH-based Option Price
Using the derived dynamics, simulate paths of the underlying using Monte Carlo with the specified GARCH parameters. Generate 10,000 paths to compute the average payoff of the call option with strike 100 and maturity 20 days, discounting at the risk-free rate. The GARCH model's conditional variances are updated recursively for each path, incorporating the parameters: \(\omega=0.1883\), \(\alpha=0.7162\), \(\beta=0\), \(\theta=0\), and \(\lambda=0.007452\).
Compare the resulting GARCH-based price with the classical Black-Scholes (BS) price calculated assuming constant volatility 0.00016. The BS price is computed as:
\(C_{BS} = S_0 N(d_1) - K e^{-rT} N(d_2)\)
with the respective standard formulas for \(d_1\) and \(d_2\).
Question 3: Pricing Outperformance Option
Payoff Structure and Numeraire Choice
The payoff at maturity for this outperformance option is:
\[
C(S_1(T), S_2(T)) = \max(a S_1(T) - b S_2(T), 0)
\]
By selecting an appropriate numeraire—either \(S_2(T)\) or \(S_1(T)\)—the payoff simplifies to a European call or put option on a transformed underlying. Choosing \(S_2(T)\) as numeraire, define:
\[
X(T) = \frac{a S_1(T)}{S_2(T)}
\]
which converts the payoff to: \(\max(X(T) - \frac{b}{a}, 0)\), a standard call option with strike \(\frac{b}{a}\).
Similarly, with \(S_1(T)\) as numeraire, the payoff converts into a put or call form depending on the algebraic transformation, aligning with the fundamental theorem of asset pricing.
Analytical Pricing Under Black-Scholes Dynamics
Assuming \(S_1\) and \(S_2\) follow geometric Brownian motions with correlation \(\rho\), the joint dynamics under risk-neutral measure involve a bivariate normal distribution for the log-returns. The normalized joint distribution enables the derivation of the option price as:
\[
V_0 = e^{-rT} \mathbb{E}[\max(a S_1(T) - b S_2(T), 0)]
\]
which can be expressed analytically leveraging the correlation and volatilities. The derivation entails calculating the joint probability that \(a S_1(T) > b S_2(T)\) and integrating over the joint distribution, yielding a closed-form solution similar to a bivariate normal cumulative distribution function (CDF).
Implication Regarding Risk-Free Rate
The resulting formula reveals that the risk-free rate appears explicitly in the discount factor but not directly in the probability calculation, emphasizing that in a risk-neutral framework, expected growth rates are adjusted for risk preferences and market prices of risk.
Question 4: Barrier Options Under Stochastic Volatility
Payoff Structure of Barrier Options
Down-and-in put and up-and-out call barrier options payoff at maturity \(T_i\) are:
Down-and-in put: \( V_{DIP} = \max(K - S(T_i), 0) \times \mathbb{I}\{ S(t) \text{ hits } H_1 \text{ before } T_i \} \)
Up-and-out call: \( V_{UOC} = \max(S(T_i) - K, 0) \times \mathbb{I}\{ S(t) \text{ stays below } H_2 \text{ before } T_i \} \)
where \(\mathbb{I}\) is the indicator function reflecting the barrier crossing.
Simulate the paths of \(S_t\) and \(V_t\) using discretized stochastic differential equations under the specified parameters, recording whether barrier crossings occur before maturity. Use Monte Carlo methods to estimate the expected discounted payoff, averaging over a large number of paths (e.g., 10,000).
Compare these prices with models assuming geometric Brownian motion with constant volatility \(\sigma=0.2\). The differences highlight the impact of stochastic volatility modeling versus constant volatility assumptions on barrier option pricing.
Conclusion
The complex problems examined herein demonstrate the integration of advanced quantitative techniques in financial derivative valuation. From portfolio insurance using options and futures, to asset price modeling with GARCH and stochastic differential equations, and the sophisticated structuring of multi-asset options, the analysis underscores the importance of modeling assumptions, simulation accuracy, and strategy equivalence. Proper application of risk-neutral valuation and numerical methods facilitates deeper insight into dynamic hedge strategies and derivative pricing under various market conditions.
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