A tale of two regimes: Theory and empirical evidence for a Markov-modulated jump diffusion model of equity returns and derivative pricing implications

We provide closed-form solutions for a continuous time, Markov-modulated jump diffusion model in a general equilibrium framework for options prices under a variety of jump diffusion specifications. We further demonstrate that the two-state model provides the leptokurtic return features, volatility smile, and volatility clustering observed empirically for the Dow Jones Industrial Average (DJIA) and its component stocks. Using 10 years of stock return data, we confirm the existence of jump intensity switching and clustering, illustrate transition probabilities, and verify superior empirical fit over competing Poisson-style models.     

Buy-and-hold mean-variance portfolios with a random exit strategy

We show how buy-and-hold investors can move from horizon uncertainty to profit opportunity. The analysis is conducted under a risk-averse framework rather than the standard Markowitz formulation in the case of i.i.d. asset processes. We make this practical achievement by considering a threshold stopping rule as the strategy to determine when to exit the market. The resulting investment horizon is random and can be correlated with the market. Under this setting, we first provide an analytical approximation to optimal weights, and then identify a class of reference variables associated with the stopping rule that leads to ex-ante improvements in portfolio allocation, vis-a-vis the fixed exit time alternative. The latter conclusion is based on a generalization of the Sharpe ratio, adjusted for horizon uncertainty. The obtained investment suggestion is simple and can be implemented empirically.     

ESO compensation: The roles of default risk, employee sentiment, and insider information

This paper derives a pricing model for employee stock options (ESO) that includes default risk and considers employee sentiment. Using ESO data from 1992 to 2004, the study finds that the average executive's subjective value is about 55% of the Black-Scholes value. Only employees who over-estimate firm returns (or insiders who know that the firm is under-valued) by about 10% per annum will prefer ESOs over cash compensation. Our model also shows that work incentives offered by ESOs may be far lower than those implied by Black-Scholes but that ESOs may induce less risk-taking behavior, contrary to typical moral hazard arguments. Findings may impact relevant accounting regulations as well as compensation decisions.     

Pricing discrete path-dependent options under a double exponential jump–diffusion model

We provide methodologies to price discretely monitored exotic options when the underlying evolves according to a double exponential jump diffusion process. We show that discrete barrier or lookback options can be approximately priced by their continuous counterparts’ pricing formulae with a simple continuity correction. The correction is justified theoretically via extending the corrected diffusion method of Siegmund (1985). We also discuss the jump effects on the performance of this continuity correction method. Numerical results show that this continuity correction performs very well especially when the proportion of jump volatility to total volatility is small. Therefore, our method is sufficiently of use for most of time.     

The Pricing of Risk and Sentiment: A Study of Executive Stock Options

Option pricing models accounting for illiquidity generally imply the options are valued at a discount to the Black‐Scholes value. Our model considers the role of sentiment, which offsets illiquidity. Using executive stock options and compensation data from 1992 to 2004 for S&P 1500 firms, we find that executives value employee stock options (ESOs) at a 48% premium to the Black‐Scholes value. These premia are explained by a sentiment level of 12% in risk‐adjusted, annualized return, suggesting a high level of executive overconfidence. Subjective value relates negatively to illiquidity and idiosyncratic risk, and positively to sentiment in all specifications, consistent with the offsetting roles of sentiment and risk aversion.     

On an automatic and optimal importance sampling approach with applications in finance

Calculating high-dimensional integrals efficiently is essential and challenging in many scientific disciplines, such as pricing financial derivatives. This paper proposes an exponentially tilted importance sampling based on the criterion of minimizing the variance of the importance sampling estimators, and its contribution is threefold: (1) A theoretical foundation to guarantee the existence, uniqueness, and characterization of the optimal tilting parameter is built. (2) The optimal tilting parameter can be searched via an automatic Newton’s method. (3) Simplified yet competitive tilting formulas are further proposed to reduce heavy computational cost and numerical instability in high-dimensional cases. Numerical examples in pricing path-dependent derivatives and basket default swaps are provided.     

Risk Management for Linear and Non-Linear Assets: A Bootstrap Method with Importance Resampling to Evaluate Value-at-Risk

Many empirical studies suggest that the distribution of risk factors has heavy tails. One always assumes that the underlying risk factors follow a multivariate normal distribution that is a assumption in conflict with empirical evidence. We consider a multivariate t distribution for capturing the heavy tails and a quadratic function of the changes is generally used in the risk factor for a non-linear asset. Although Monte Carlo analysis is by far the most powerful method to evaluate a portfolio Value-at-Risk (VaR), a major drawback of this method is that it is computationally demanding. In this paper, we first transform the assets into the risk on the returns by using a quadratic approximation for the portfolio. Second, we model the return’s risk factors by using a multivariate normal as well as a multivariate t distribution. Then we provide a bootstrap algorithm with importance resampling and develop the Laplace method to improve the efficiency of simulation, to estimate the portfolio loss probability and evaluate the portfolio VaR. It is a very powerful tool that propose importance sampling to reduce the number of random number generators in the bootstrap setting. In the simulation study and sensitivity analysis of the bootstrap method, we observe that the estimate for the quantile and tail probability with importance resampling is more efficient than the naive Monte Carlo method. We also note that the estimates of the quantile and the tail probability are not sensitive to the estimated parameters for the multivariate normal and the multivariate t distribution. The research of Shih-Kuei Lin was partially supported by the National Science Council under grants NSC 93-2146-H-259-023. The research of Cheng-Der Fuh was partially supported by the National Science Council under grants NSC 94-2118-M-001-028.