Title: Methods for Pricing Pre-Earnings Equity Options and Leveraged ETF Options
Description: In this thesis, we present several analytical and numerical methods for two financial engineering problems: 1) accounting for the impact of an earnings announcement on the price and implied volatility of the associated equity options, and 2) analyzing the price dynamics of leveraged exchange-traded funds (LETFs) and valuation of LETF options. Our pricing models capture the main characteristics of these options, along with jumps and stochastic volatility in the underlying asset. We illustrate our results through numerical implementation and calibration using market data. In the first part, we model the pricing of equity options around an earnings announcement (EA). Empirical studies have shown that an earnings announcement can lead to an immediate price shock to the company stock. Since many companies also have options written on their stocks, the option prices should reflect the uncertain price impact of an upcoming EA before expiration. To represent the shock due to earnings, we incorporate a random jump on the announcement date in the dynamics of the stock price.

We consider different distributions of the scheduled earnings jump as well as different underlying stock price dynamics before and after the EA date. Our main contributions include analytical option pricing formulas when the underlying stock price follows the Kou model along with a double-exponential or Gaussian EA jump on the announcement date. Furthermore, we derive analytic bounds and asymptotics for the pre-EA implied volatility under various models. The calibration results demonstrate adequate fit of the entire implied volatility surface prior to an announcement. The comparison of the risk-neutral distribution of the EA jump to its historical counterpart is also discussed. Moreover, we discuss the valuation and exercise strategy of pre-EA American options, and present an analytical approximation and numerical results. The second part focuses on the analysis of LETFs. We start by providing a quantitative risk analysis of LETFs with an emphasis on the impact of leverage ratios and investment horizons.

Given an investment horizon, different leverage ratios imply different levels of risk. Therefore, the idea of an {admissible range of leverage ratios} is introduced. For an admissible leverage ratio, the associated LETF satisfies a given risk constraint based on, for example, the value-at-risk (VaR) and conditional VaR. Moreover, we discuss the concept of {admissible risk horizon} so that the investor can control risk exposure by selecting an appropriate holding period. The intra-horizon risk is calculated, showing that higher leverage can significantly increase the probability of an LETF value hitting a lower level. This leads us to evaluate a stop-loss/take-profit strategy for LETFs and determine the optimal take-profit given a stop-loss risk constraint. In addition, the impact of volatility exposure on the returns of different LETF portfolios is investigated. In the last chapter, we study the pricing of options written on LETFs. Since LETFs on the same reference index share the same source of risk, it is important to consider a consistent pricing methodology of these options.

In addition, LETFs can theoretically experience a loss greater than 100\%. In practice, some LETF providers design the fund so that the daily returns are capped both downward and upward. We incorporate these features and model the reference index by a stochastic volatility model with jumps. An efficient numerical algorithm based on transform methods to value options under this model is presented. We illustrate the accuracy of our pricing algorithm by comparing it to existing methods. Calibration using empirical option data shows the impact of leverage ratio on the implied volatility. Our method is extended to price American-style LETF options.