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. 1. Introduction Turbo warrants, which appeared first in Germany in late 2001, have experienced a considerable growth in Northern Europe and Hong Kong. They are special types of knockout barrier options in which the rebate is calculated as another exotic option. For one thing, this contract has a low vega so that the option price is less sensitive to the change of the implied volatility of the security market, and for another, it is highly geared owing to the possibility of knockout. Closed form expression for the price has been presented by Eriksson [1] under geometric Brownian motion (GBM) for the underlying security. It is well-known that the assumption of the GBM for the underlying security price in the Black-Scholes model [2] does not capture many empirical lines of evidence appeared in financial markets. Maybe, the two most significant shortcomings of the model's assumption lie in flat implied volatility, whereas the volatility fluctuates depending upon market conditions and the underestimation of extreme moves, yielding tail risk. So, there have been a lot of alternative underlying models developed to extend the GBM and overcome these problems. Local volatility models are one type of them, where the volatility depends on the price level of the underlying security itself. The most well-known local volatility model is the constant elasticity of variance (CEV) model in which the volatility is given by a power function of the underlying security price. It has been proposed by Cox [3] and Cox and Ross [4]. Another version of volatility models has been developed by assuming security's volatility to be a random process governed by another state variables such as the tendency of volatility to revert to some long-run mean value, the variance of the volatility process itself, and so forth. This type of models is called (pure) stochastic volatility models. The models developed by Heston [5] and Fouque et al. [6] are popular ones among others in this category. Also, there is another type of generalized models for the underlying