Pricing Options on an Asset with Bernoulli Jump-Diffusion Returns

The price movements of certain assets can be modeled by stochastic processes that combine continuous diffusion with discrete jumps. This paper compares values of options on assets with no jumps, jumps of fixed size, and jumps drawn from a lognormal distribution. It is shown that not only the magnitude but also the direction of the mispricing of the Black-Scholes model relative to jump models can vary with the distribution family of the jump component. This paper also discusses a methodology for the numerical valuation, via a backward induction algorithm, of American options on a jump-diffusion asset whose early exercise may be profitable. These cannot, in general, be accurately priced using analytic models. The procedure has the further advantage of being easily adaptable to nonanalytic, empirical distributions of period returns and to nonstationarity in the underlying diffusion process.     

Path-dependent game options: a lookback case

The game option, which is also known as Israel option, is an American option with callable features. The option holder can exercise the option at any time up to maturity. This article studies the pricing behaviors of the path-dependent game option where the payoff of the option depends on the maximum or minimum asset price over the life of the option (i.e., the game option with the lookback feature). We obtain the explicit pricing formula for the perpetual case and provide the integral expression of pricing formula under the finite horizon case. In addition, we derive optimal exercise strategies and continuation regions of options in both floating and fixed strike cases.     

Exchange options under clustered jump dynamics

Exchange options are one of the most popular exotic options, and have important implications for many common financial arrangements and for implied beta as a measure of systematic risk. In this study, we extend the existing literature on exchange options to allow for clustered jump contagion dynamics in each single asset, as well as across assets, using the Hawkes jump-diffusion model. We derive the analytical pricing formulae, the Greeks, and the optimal hedging strategy via Fourier transforms. Using an illustrative numerical analysis, we present the relationship between the exchange option price and clustered jump intensities and jump sizes in the underlying assets. We discuss the managerial insights on financial arrangements with exchange option characteristics. Furthermore, we discuss the implications of incorporating clustered jumps into the estimation of implied beta with exchange options, in which the applications can be insightful and useful in finance practice.     

Estimating Early Exercise Premiums on Gold and Copper Options Using a Multifactor Model and Density Matched Lattices

We use the standard geometric Brownian motion augmented by jumps to describe the spot underlying and mean regressive models of interest rates and convenience yields as state variables for gold and copper prices. Estimates of parameters of the diffusion processes are obtained by the Kalman filter. Using these estimates, jump parameters are estimated in the second stage by least squares. Early exercise premia on puts and calls are computed using a lattice with probabilities assigned by the density matching technique. We find that while deep in the money options have greater absolute early exercise premiums, the early exercise premium is roughly constant as a percent of option price. Our findings also confirm that gold behaves like an investment asset and copper behaves like a commodity.     

The performance of model based option trading strategies

This paper analyzes returns to trading strategies in options markets that exploit information given by a theoretical asset pricing model. We examine trading strategies in which a positive portfolio weight is assigned to assets which market prices exceed the price of a theoretical asset pricing model. We investigate portfolio rules which mimic standard mean-variance analysis is used to construct optimal model based portfolio weights. In essence, these portfolio rules allow estimation risk, as well as price risk to be approximately hedged. An empirical exercise shows that the portfolio rules give out-of-sample Sharpe ratios exceeding unity for S&P 500 options. Portfolio returns have no discernible correlation with systematic risk factors, which is troubling for traditional risk based asset pricing explanations.     

Optimal portfolios when volatility can jump

We consider an asset allocation problem in a continuous-time model with stochastic volatility and jumps in both the asset price and its volatility. First, we derive the optimal portfolio for an investor with constant relative risk aversion. The demand for jump risk includes a hedging component, which is not present in models without volatility jumps. We further show that the introduction of derivative contracts can have substantial economic value. We also analyze the distribution of terminal wealth for an investor who uses the wrong model, either by ignoring volatility jumps or by falsely including such jumps, or who is subject to estimation risk. Whenever a model different from the true one is used, the terminal wealth distribution exhibits fatter tails and (in some cases) significant default risk.     

Maturity and exercise price of executive stock options

Using a simple three-period model in which a manager can gather information before making an investment decision, this paper studies optimal contracts with various stock options. In particular, we show how the exercise price of executive stock options is related to a base salary, the size of the option grant, leverage, and the riskiness of a desired investment policy. The optimal exercise price increases in the size of grant and the base salary and decreases in leverage and the riskiness of a desired investment policy. Other things equal, the optimal exercise price of European options with a longer maturity should increase more for an increase in the base salary and the size of grant and decrease more for an increase in leverage than the one with a shorter maturity. The optimal exercise price of American options is determined by the optimal exercise prices of European options with different maturities. Given the fixed exercise price, the size of the option grant does not decrease in the face value of debt.     

Option pricing in the presence of natural boundaries and a quadratic diffusion term

This paper uses a probabilistic change-of-numeraire technique to compute closed-form prices of European options to exchange one asset against another when the relative price of the underlying assets follows a diffusion process with natural boundaries and a quadratic diffusion coefficient. The paper shows in particular how to interpret the option price formula in terms of exercise probabilities which are calculated under the martingale measures associated with two specific numeraire portfolios. An application to the pricing of bond options and certain interest rate derivatives illustrates the main results.     

Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps

This paper considers a robust optimal excess-of-loss reinsurance-investment problem in a model with jumps for an ambiguity-averse insurer (AAI), who worries about ambiguity and aims to develop a robust optimal reinsurance-investment strategy. The AAI’s surplus process is assumed to follow a diffusion model, which is an approximation of the classical risk model. The AAI is allowed to purchase excess-of-loss reinsurance and invest her surplus in a risk-free asset and a risky asset whose price is described by a jump-diffusion model. Under the criterion for maximizing the expected exponential utility of terminal wealth, optimal strategy and optimal value function are derived by applying the stochastic dynamic programming approach. Our model and results extend some of the existing results in the literature, and the economic implications of our findings are illustrated. Numerical examples show that considering ambiguity and reinsurance brings utility enhancements.     

Super-Exponential RE bubble model with efficient crashes

We propose a dynamic Rational Expectations (RE) bubble model of prices, combining a geometric random walk with separate crash (and rally) discrete jump distributions associated with positive (and negative) bubbles. Crashes tend to efficiently bring back excess bubble prices close to a “normal” process. Then, the RE condition implies that the excess risk premium of the risky asset exposed to crashes is an increasing function of the amplitude of the expected crash, which itself grows with the bubble mispricing: hence, the larger the bubble price, the larger its subsequent growth rate. This positive feedback of price on return is the archetype of super-exponential price dynamics. We use the RE condition to estimate the real-time crash probability dynamically through an accelerating probability function depending on the increasing expected return. After showing how to estimate the model parameters, we obtain a closed-form approximation for the optimal investment that maximizes the expected log of wealth (Kelly criterion) for the risky bubbly asset and a risk-free asset. We demonstrate, on seven historical crashes, the promising outperformance of the method compared to a 60/40 portfolio, the classic Kelly allocation, and the risky asset, and how it mitigates jumps, both positive and negative.     

Price trends and patterns in technical analysis: A theoretical and empirical examination

While many technical trading rules are based upon patterns in asset prices, we lack convincing explanations of how and why these patterns arise, and why trading rules based on technical analysis are profitable. This paper provides a model that explains the success of certain trading rules that are based on patterns in past prices. We point to the importance of confirmation bias, which has been shown to play a key role in other types of decision making. Traders who acquire information and trade on the basis of that information tend to bias their interpretation of subsequent information in the direction of their original view. This produces autocorrelations and patterns of price movement that can predict future prices, such as the “head-and-shoulders” and “double-top” patterns. The model also predicts that sequential price jumps for a particular stock will be positively autocorrelated. We test this prediction and find that jumps exhibit statistically and economically significant positive autocorrelations.     

Pricing average options under time-changed Lévy processes

This paper presents an approximate formula for pricing average options when the underlying asset price is driven by time-changed Lévy processes. Time-changed Lévy processes are attractive to use for a driving factor of underlying prices because the processes provide a flexible framework for generating jumps, capturing stochastic volatility as the random time change, and introducing the leverage effect. There have been very few studies dealing with pricing problems of exotic derivatives on time-changed Lévy processes in contrast to standard European derivatives. Our pricing formula is based on the Gram–Charlier expansion and the key of the formula is to find analytic treatments for computing the moments of the normalized average asset price. In numerical examples, we demonstrate that our formula give accurate values of average call options when adopting Heston’s stochastic volatility model, VG-CIR, and NIG-CIR models.     

A closed-form solution to American options under general diffusion processes

This paper investigates American option pricing under general diffusion processes. Specifically, the underlying asset price is assumed to follow a diffusion process in which both the dividend yield and volatility are functions of time and the underlying asset price. Using the generalized homotopy analysis method, the determination of the early exercise boundary is separated from the valuation procedure of American options. Then, an exact and explicit solution for American options on a dividend-paying stock is derived as a Maclaurin series. In addition, the corresponding optimal early exercise boundary and the Greeks are obtained in closed-form solutions. A nonlinear sequence transformation, the Padé technique, is used to effectively accelerate the convergence of the partial sums of the infinite series. As the homotopy constructed in this paper is based on a generalized deformation with a shape parameter and kernel function, the error of the homotopic approximation could be reduced further for a fixed order. Numerical examples demonstrate the validity, effectiveness, and flexibility of the proposed approach.     

Expected returns, risk premia, and volatility surfaces implicit in option market prices

This article presents a pure exchange economy that extends Rubinstein (1976) to show how the jump-diffusion option pricing model of Merton (1976) is altered when jumps are correlated with diffusive risks. A non-zero correlation between jumps and diffusive risks is necessary in order to resolve the positively sloped implied volatility term structure inherent in traditional jump diffusion models. Our evidence is consistent with a negative covariance, producing a non-monotonic term structure. For the proposed market structure, we present a closed form asset pricing model that depends on the factors of the traditional jump-diffusion models, and on both the covariance of the diffusive pricing kernel with price jumps and the covariance of the jumps of the pricing kernel with the diffusive price. We present statistical evidence that these covariances are positive. For our model the expected stock return, jump and diffusive risk premiums are non-linear functions of time.     

Optimal Reinsurance and Investment for a Jump Diffusion Risk Process under the CEV Model

Abstract

We consider an optimal reinsurance-investment problem of an insurer whose surplus process follows a jump-diffusion model. In our model the insurer transfers part of the risk due to insurance claims via a proportional reinsurance and invests the surplus in a “simplified” financial market consisting of a risk-free asset and a risky asset. The dynamics of the risky asset are governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity. The objective of the insurer is to choose an optimal reinsurance-investment strategy so as to maximize the expected exponential utility of terminal wealth. We investigate the problem using the Hamilton-Jacobi-Bellman dynamic programming approach. Explicit forms for the optimal reinsuranceinvestment strategy and the corresponding value function are obtained. Numerical examples are provided to illustrate how the optimal investment-reinsurance policy changes when the model parameters vary.     

Two-dimensional risk-neutral valuation relationships for the pricing of options

The Black–Scholes model is based on a one-parameter pricing kernel with constant elasticity. Theoretical and empirical results suggest declining elasticity and, hence, a pricing kernel with at least two parameters. We price European-style options on assets whose probability distributions have two unknown parameters. We assume a pricing kernel which also has two unknown parameters. When certain conditions are met, a two-dimensional risk-neutral valuation relationship exists for the pricing of these options: i.e. the relationship between the price of the option and the prices of the underlying asset and one other option on the asset is the same as it would be under risk neutrality. In this class of models, the price of the underlying asset and that of one other option take the place of the unknown parameters.      

Two counters of jumps

This paper introduces a class of two counters of jumps option pricing models. The stock price follows a jump-diffusion process with price jumps up and price jumps down, where each type of jumps can have different means and standard deviations. Price jumps can be negatively autocorrelated as it has been observed in practice. We investigate the volatility surfaces generated by this class of two counters of jumps option pricing models. Our formulae, like the jump-diffusion models with a single counter of jumps, are able to generate smiles, and skews   with similar shapes to those observed in the options markets. More importantly, unlike the jump-diffusion models with a single counter of jumps, our formulae are able to generate term structures of implied volatilities of at-the-money options with ∩$\cap$-shaped patterns similar to those observed in the marketplace.