Jeong Hoon Kim

Pricing vulnerable options under a stochastic volatility model

Abstract In this paper, we consider the pricing of vulnerable options when the underlying asset follows a stochastic volatility model. We use multiscale asymptotic analysis to derive an analytic approximation formula for the price of the vulnerable options and study the stochastic volatility effect on the option price. A numerical experiment result is presented to demonstrate our findings graphically.

Homotopy analysis method for option pricing under stochastic volatility

Abstract In this paper, the homotopy analysis method, whose original concept comes from algebraic topology, is applied to connect the Black–Scholes option price (the good initial guess) to the option price under general stochastic volatility environment in a recursive manner. We obtain the homotopy solutions for the European vanilla and barrier options as well as the relevant convergence conditions.

Multiscale analysis of a perpetual American option with the stochastic elasticity of variance

Abstract A perpetual American option is considered under a generalized model of the constant elasticity of variance model where the constant elasticity is perturbed by a small fast mean-reverting Ornstein–Uhlenbeck process. By using a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on option prices as well as optimal exercise prices. Our results improve the existing option price structure in view of flexibility and applicability through the market price of risk. The revealed results may provide useful information on real option problems.

Portfolio optimization under the stochastic elasticity of variance

Based on the observation that the elasticity of variance of risky assets is randomly varying around a constant, we take an underlying asset model in which the averaged constant elasticity of variance is perturbed by a small fast fluctuating process and study the Merton type portfolio optimization problem using dynamic programming as well as asymptotic expansions. The Hamilton–Jacobi–Bellman equation for each of the power and exponential utility functions leads to an optimal trading strategy as a perturbation around the well known one. We reveal the impact of both the constant elasticity of variance upon the Merton investment optimal control under the Black–Scholes model and the stochastic elasticity of variance upon the investment optimal control under the constant elasticity of variance model. The concavity of the investment policy with respect to the excess return is characteristic of a market economy with the constant or stochastic elasticity of variance.

A closed-form analytic correction to the Black–Scholes–Merton price for perpetual American options

Abstract This is a complementary study of a recent work by Yoon etxa0al. (2013) [1] [J.-H. Yoon, J.-H. Kim, S.-Y. Choi, Multiscale analysis of a perpetual American option with the stochastic elasticity of variance, Appl. Math. Lett. 26 (7) (2013)] which excludes a certain level of the elasticity of variance. A second-order correction to the Black–Scholes option price and optimal exercise boundary for a perpetual American put option is made under the stochastic elasticity of variance of a risky asset. Contrary to the case of Yoon etxa0al. (2013) [1] , it is given by an explicit closed-form analytic expression so that one can access clearly the sensitivity of the option price and the optimal exercise boundary to changes in model parameters as well as the impact of the presence of a stochastic elasticity term on the option price and the optimal time to exercise.

Turbo Warrants under Hybrid Stochastic and Local Volatility

This paper considers the pricing of turbo warrants under a hybrid stochastic and local volatility model. The model consists of the constant elasticity of variance model incorporated by a fast fluctuating Ornstein-Uhlenbeck process for stochastic volatility. The sensitive structure of the turbo warrant price is revealed by asymptotic analysis and numerical computation based on the observation that the elasticity of variance controls leverage effects and plays an important role in characterizing various phases of volatile markets.

Stochastic volatility asymptotics of defaultable interest rate derivatives under a quadratic Gaussian model

Stochastic volatility of underlying assets has been shown to affect significantly the price of many financial derivatives. In particular, a fast mean-reverting factor of the stochastic volatility plays a major role in the pricing of options. This paper deals with the interest rate model dependence of the stochastic volatility impact on defaultable interest rate derivatives. We obtain an asymptotic formula of the price of defaultable bonds and bond options based on a quadratic term structure model and investigate the stochastic volatility and default risk effects and compare the results with those of the Vasicek model.

The Term Structure of Interest Rates under Heath-Jarrow-Morton Models with Fast Mean-Reverting Stochastic Volatility

This paper is a study of the term structure of interest rates based on the Heath–Jarrow–Morton (HJM) models with Hull–White volatility function. Under fast mean-reverting stochastic volatility, we obtain an analytic formula for an approximate bond price with estimated error using a Markovian transform method combined with a singular perturbation method. The stochastic volatility correction effect against time-to-maturity is revealed so that it can capture more of the complexities of the interest rate term structure.