MSRI-UP

The Chicago Board Options Exchange introduced the Market's Volatility Index (VIX) in 1993. It is a financial instrument which uses Standard & Poor's 500 Index options to measure market volatility. The first VIX derivatives were introduced in 2004. We are interested in modeling these derivatives. After observing mean regressing characteristics of the VIX derivatives, we use the Ornstein-Uhlenbeck process to model these derivatives. We then implement two different methods to estimate the model's parameters using daily market data. We compare the modeling accuracy of an Ornstein-Uhlenbeck process with a geometric Brownian motion model. Our results confirm the Ornstein-Uhlenbeck process can model VIX derivatives more accurately than a geometric Brownian motion.

We show it is possible to model VIX derivatives with reasonable accuracy looking one month forward. Our results can potentially aid investors using volatility derivatives and influence strategies to hedge against risk.

Many investors would like to invest in the VIX, a measure of implied volatility of stocks in the S&P 500 index, as a way to hedge against downturns in the stock market. While the VIX is not tradable, there exist tradable derivatives on the VIX that investors can use as hedging instruments. Two such derivatives are the VXX and VXZ, both exchange- traded notes (ETNs) based on futures contracts on the VIX. Our project centered around analyzing intraday data on the VIX, VXX and VXZ to better understand these derivatives. We used various methods to model the intraday behavior of the VXX and VXZ in the hopes that our analysis could lead to a successful trading strategy. Implementing these models on future intraday data on the VXX and VXZ confirmed the veracity of our models.

The concepts of hedging and dynamic hedging are important concepts in mathematical finance. A practical use of dynamic hedging is one of reducing the risk of a position in an option. The process of forming a replicating trading strategy that efficiently mimics the value of an option is of great interest and should be examined carefully. One process of forming a replicating trading strategy involves the use of the European call option solution to the Black-Scholes partial differential equation. While this process does a fair job of providing direction as to how to rebalance a replicating trading strategy, it makes the assumption that readjustments can be made in continuous time. This of course is not realistically possible, not only because it is physically impossible to adjust continuously but also because of transaction costs. Once transaction costs are considered, the idea of using a self-financing strategy, or a strategy in which a portfolio is formed and no funds are required after the initial investment, is no longer realistic. Various ways of discretizing this continuous process will be examined in order to find an optimal strategy that both accurately approximates the value of an option and yet remains as close to self-financing as possible. Uniform partitioning will be examined but will most likely not be the sole basis of our optimal strategy. We demonstrate that, with the proper set of conditions, a strategy for readjusting a replicating portfolio can be found such that our goals are properly met. Once this strategy has been developed, the assumption of the absence of transaction costs will be eliminated, and a more realistic way of forming a replicating trading strategy that helps hedge away risk will have been formed.

The trading of volatility derivatives was introduced less than two decades ago; it is a fairly new practice in the world of mathematical finance. When markets crash, stocks are often traded frantically, and volatility imminently increases. Hence, adding volatility derivatives to a portfolio should balance losses in the event of bearish market behavior. We quantify and minimize the risk of a portfolio using Markowitz’s optimization theory. One assumption of this model is that past performance of a portfolio is indicative of its future behavior. The goal of this theory is to minimize the portfolio’s risk given a fixed rate of expected return. We implement this method in MATLAB to plot the minimized portfolio risk versus portfolio returns, producing a curve known as the efficient frontier. We collect market data on stocks from various indices and sectors, as well as data on volatility derivatives. From this data, we construct portfolios with and without volatility derivatives. By comparing the efficient frontier generated by the portfolio with volatility derivatives against the efficient frontier generated without the derivatives, we can quantify the risk reduction induced. Preliminary results confirm our hypothesis, which suggests that volatility derivatives can serve as effective instruments of insurance in an investor’s portfolio.

Financial economists are always interested in improving models to predict expected returns. Previous research indicates that variations in time of expected returns are connected with business cycles. Particularly, investors are less likely to hold risky assets during economic recessions, so expected returns during those times are higher than expected returns in times of economic expansions. Thus, it is implied that variables of the business cycles affect time variations in equity premiums. Supporting this assumption and expanding on previous research, we develop a conditional macroeconomic variable by the Johansen cointegration method, a method used to determine a stationary relationship between multiple non-stationary time series, to measure time variations in risk premiums and incorporate it into the Capital Asset Pricing Model and the Fama and French Three Factor Model. The conditional term includes the following macroeconomic variables: dividend yield, default spread, term spread, short-term interest rate, and implied volatility. Using linear regression our analysis of the models show that the multi-factor models generally predict better than CAPM, but there is room to improve upon the accuracy of all of the models. This paper is heavily based on and will continue the work of Maria Jansen and Maria Casandra Rusti’s “Cointegrating the Capital Asset Pricing Model to Incorporate Macro-Economic Based Variables” from the 2010 WPI REU Program in Industrial Mathematics and Statistics.

Unlike European options, American options can be exercised at any time up to maturity. As a result of the early exercise feature of American options, they are at least as valuable as their European counterparts. This, however, makes them harder to price as the analytical closed form equations used for pricing vanilla European options do not apply. In order to price an American option, each time t prior to maturity T must be considered to determine whether it is optimal for the option holder to exercise the option immediately or to hold on to the option until a more advantageous future time before it expires. We implemented the Longstaff-Schwartz algorithm, which incorporates Monte Carlo methods and regression to price American options. We also used variance reduction techniques and quasi-Monte Carlo methods to improve the convergence and computational speed of the algorithm. We were able to significantly reduce the width of the 95% confidence interval of our estimated price of the option by using control variates, and we determined the exercise boundary that results from applying the stopping rule. We found that the Longstaff-Schwartz algorithm efficiently prices American put options.