Variance and volatility swaps under a two-factor stochastic volatility model with regime switching

In this paper, the pricing problem of variance and volatility swaps is discussed under a two-factor stochastic volatility model. This model can be treated as a two-factor Heston model with one factor following the CIR process and another characterized by a Markov chain, with the motivation originating from the popularity of the Heston model and the strong evidence of the existence of regime switching in real markets. Based on the derived forward characteristic function of the underlying price, analytical pricing formulae for variance and volatility swaps are presented, and numerical experiments are also conducted to compare swap prices calculated through our formulae and those obtained under the Heston model to show whether the introduction of the regime switching factor would lead to any significant difference

A hybrid computational approach for option pricing

In this paper, we propose a novel numerical approach for option pricing with the combination of the MC (Monte Carlo) simulation and the PDE (partial differential equation) approach. Our motivation originates from the fact that within a finite life time of an option contract, the underlying price as well as the range of volatility are expected to vary within a relatively small region centered around the current value of the underlying and the volatility and hence there is no need to compute option prices for the underlying and the volatility values beyond this region. Thus, our hybrid approach takes the advantage of both the MC simulation and PDE approach to form an approach that takes the MC simulation as a special case with the region being extremely small and the PDE approach as another special case with the region being extremely large. Through numerical experiments, we demonstrate that such a hybrid approach enhances computational efficiency, while maintaining the same level of accuracy when either the MC simulation or the PDE approach is used alone for the option prices computed within a suitably chosen interested region

An accurate approximation formula for pricing European options with discrete dividend payments

In this article, two relevant problems related to pricing European options with discrete dividend under the classic Black-Scholes framework are considered. For the case when a discrete dividend payment is proportional to the underlying asset value, we discuss an interesting phenomenon observed; the option price is independent of the dividend payment date. This appears to be at odds with one\u27s intuition that dividend amount, as well as the dividend date, should both affect the price of a European call or put option. We reveal the fundamental reasons, from both mathematical and financial viewpoints, why this occurs. When the amount of the discrete dividend is fixed, we provide an approximation formula for European option prices, with only one-dimensional integrals involved. It should be noted that our formula is a general one since it can not only be applied when there is only a single dividend, but also be suitable for the case of multiple dividends

On the convergence of He and Zhu\u27s new series solution for pricing options with the Heston model

In this paper, a modified formula for European options and a set of complete convergence proofs for the solution that cover the entire time horizon of a European option contact are presented under the Heston model with minimal entropy martingale measure. Although He & Zhu [5] worked on this model, they only provided a converged solution with a condition imposed on the time to expiry. The new solution presented here is only slightly modified in its form. But, it is accompanied with the proof of convergence of the solution for the entire span of the time horizon of an option

How should a local regime-switching model be calibrated?

Local regime-switching models are a natural consequence of combining the concept of a local volatility model with that of a regime-switching model. However, even though Elliott et al. (2015) have derived a Dupire formula for a local regime-switching model, its calibration still remains a challenge, primarily due to the fact that the derived volatility function for each state involves all the state price variables whereas only one market price is available for model calibration, and a direct implementation of Elliott et al.\u27s formula may not yield stable results. In this paper, a closed system for option pricing and data extraction under the classical regime-switching model is proposed with a special approach, splitting one market price into two market-implied state prices . The success of our approach hinges on the recovery of the two local volatility functions being transformed into an optimal control problem, which is solved through the Tikhonov regularization. In addition, an efficient algorithm is proposed to obtain the optimal solution by iteration. Our numerical experiments show that different shapes of local volatility functions can be accurately and stably recovered with the newly-proposed algorithm, and this algorithm also works quite well with real market data

Pricing European options with stochastic volatility under the minimal entropy martingale measure

In this paper, a closed-form pricing formula in the form of an infinite series for European call options is derived for the Heston stochastic volatility model under a chosen martingale measure. Given that markets with the stochastic volatility are incomplete, there exists a number of equivalent martingale measures and consequently investors face a problem of making a choice of appropriate measure when they price options. The one we adopt here is the so-called minimal entropy martingale measure shown to be related to the expected utility maximization theory (Frittelli 2000 Math. Finance 10(1), 39-52) and the financial rationality for choosing this measure will be further illustrated in this paper. A great advantage of our newly-derived pricing formula is that the convergence of the solution in series form can be proved theoretically; such a proof of the convergence is also complemented by some numerical examples to demonstrate the speed of convergence. To further show the validity of our formula, a comparison of prices calculated through the newly derived formula is made with those obtained directly from the Monte Carlo simulation as well as those from solving the PDE (partial differential equation) with the finite difference method

Pricing convertible bonds based on a multi-stage compound option model

In this paper, we introduce the concept of multi-stage compound options to the valuation of convertible bonds. Rather than evaluating a nested high-dimensional integral that has arisen from the valuation of multi-stage compound options, we found that adopting the Finite Difference Method (FDM) to solve the Black-Scholes equation for each stage actually resulted in a better numerical efficiency. By comparing our results with those obtained by solving the Black-Scholes equation directly, we can show that the new approach does provide an approximation approach for the valuation of convertible bonds and demonstrate that it offers a great potential for a further extension to CBs with more complex structures such as those with call and/or put provisions

A new integral equation formulation for American put options

In this paper, a completely new integral equation for the price of an American put option as well as its optimal exercise price is successfully derived. Compared to existing integral equations for pricing American options, the new integral formulation has two distinguishable advantages: (i) it is in a form of one-dimensional integral, and (ii) it is in a form that is free from any discontinuity and singularities associated with the optimal exercise boundary at the expiry time. These rather unique features have led to a significant enhancement of the computational accuracy and efficiency as shown in the examples