Xin-Jiang He

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.

An alternative form used to calibrate the Heston option pricing model

This paper presents an alternative form of the Heston model that preserves an essential advantage of the Heston model, its analytic tractability, by imposing the necessary and sufficient conditions for the existence of a solution in affine form, while it is in a different form so that it offers certain advantages in parameter determination. To demonstrate this, we conducted some empirical studies, exploring if this new form does have certain advantages over the original version under certain market conditions.

A closed-form pricing formula for European options under the Heston model with stochastic interest rate

Abstract In this paper, a closed-form pricing formula for European options in the form of an infinite series is derived under the Heston model with the interest rate being another random variable following the CIR (Cox–Ingersoll–Ross) model. One of the main advantages for the newly derived series solution is that we can provide a radius of convergence, which is complemented by some numerical experiments demonstrating its speed of convergence. To further verify our formula, option prices calculated through our formula are also compared with those obtained from Monte Carlo simulations. Finally, a set of pricing formulae are derived with the series expanded at different points so that the entire time horizon can be covered by converged solutions.

A new closed-form formula for pricing European options under a skew Brownian motion

ABSTRACT In this paper, we present a new pricing formula based on a modified Black–Scholes (B-S) model with the standard Brownian motion being replaced by a particular process constructed with a special type of skew Brownian motions. Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options.” The European Journal of Finance 13 (6): 523–544] have worked on this model, the results they obtained are incorrect. In this paper, not only do we identify precisely where the errors in Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options”. The European Journal of Finance 13 (6): 523–544] are, we also present a new closed-form pricing formula based on a newly proposed equivalent martingale measure, called ‘endogenous risk neutral measure’, by which only endogenous risks should and can be fully hedged. The newly derived option pricing formula takes the B-S formula as a special case and it does not induce any significant additional burden in terms of numerically computing option values, compared with the effort involved in computing the B-S formula.

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.

A Monte-Carlo based approach for pricing credit default swaps with regime switching

Abstract This paper considers the valuation of a CDS (credit default swap) contract. To find out a more accurate CDS price, we work on an extended Merton’s model by assuming that the price of the reference asset follows a regime switching Black–Scholes model, and moreover, the reference asset can default at any time before the expiry time. A general pricing formula for the CDS containing the unknown no default probability is derived first. It is then subsequently shown that the no default probability is equivalent to the price of a down-and-out binary option written on the same reference asset. By simulating the Markov chain with the Monte-Carlo technique, we obtain an approximation formula for the down-and-out binary option, with the availability of which, the calculation of the CDS price becomes straightforward. Finally, some numerical experiments are conducted to examine the accuracy of the approximation approach as well as the impacts of the introduction of the regime switching mechanics on the CDS price.

A series-form solution for pricing variance and volatility swaps with stochastic volatility and stochastic interest rate

Abstract In this paper, we present analytical pricing formulae for variance and volatility swaps, when both of the volatility and interest rate are assumed to be stochastic and follow a CIR (Cox–Ingersoll–Ross) process, forming a Heston–CIR hybrid model. The solutions are written in a series form with a theoretical proof of their convergence, ensuring the accuracy of the determined swap prices. The application of the formulae in practice is also demonstrated through the designed numerical experiments.

A modified Black–Scholes pricing formula for European options with bounded underlying prices

In this paper, a modified Black–Scholes (B–S) model is proposed, based on a revised assumption that the range of the underlying price varies within a finite zone, rather than being allowed to vary in a semi-infinite zone as presented in the classical B–S theory. This is motivated by the fact that the underlying price of any option can never reach infinity in reality; a trader may use our new formula to adjust the option price that he/she is willing to long or short. To develop this modified option pricing formula, we assume that a trader has a view on the realistic price range of a particular asset and the log-returns follow a truncated normal distribution within this price range. After a closed-form pricing formula for European call options has been successfully derived, some numerical experiments are conducted. To further demonstrate the meaning of the proposed model, empirical studies are carried out to compare the pricing performance of our model and that of the B–S model with real market data.