A Shannon Wavelet Method for Pricing American Options under Two-Factor Stochastic Volatilities and Stochastic Interest Rate

In the paper, the pricing of the American put options under the double Heston model with Cox–Ingersoll–Ross (CIR) interest rate process is studied. The characteristic function of the log asset price is derived, and thereby Bermuda options are well evaluated by means of a state-of-the-art Shannon wavelet inverse Fourier technique (SWIFT), which is a robust and highly efficient pricing method. Based on the SWIFT method, the price of American option can be approximated by using Richardson extrapolation schemes on a series of Bermudan options. Numerical experiments show that the proposed pricing method is efficient, especially for short-term American put options.

Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities

Pricing multi-asset options has always been one of the key problems in financial engineering because of their high dimensionality and the low convergence rates of pricing algorithms. This paper studies a method to accelerate Monte Carlo (MC) simulations for pricing multi-asset options with stochastic volatilities. First, a conditional Monte Carlo (CMC) pricing formula is constructed to reduce the dimension and variance of the MC simulation. Then, an efficient martingale control variate (CV), based on the martingale representation theorem, is designed by selecting volatility parameters in the approximated option price for further variance reduction. Numerical tests illustrated the sensitivity of the CMC method to correlation coefficients and the effectiveness and robustness of our martingale CV method. The idea in this paper is also applicable for the valuation of other derivatives with stochastic volatility.

An efficient pricing algorithm for American options with double stochastic volatilities and double jumps

The purpose of the paper is to provide an efficient pricing algorithm for American options with stochastic volatilities and jumps. This paper extends the double Heston model with double exponential jumps and derives the characteristic function of the model by Feynman–Kac theorem. With the obtained characteristic function, this paper also extends the Fourier-cosine expansion method for pricing Bermudan options to the model. Based on the COS method, this paper approximates American options by using Richardson extrapolation schemes on a series of Bermudan options and provides a pricing algorithm for American put options. Numerical results show that the proposed pricing algorithm is efficient, especially for short-term American put options.

On risk measuring in the variance-gamma model

In this paper, we discuss the problem of calculating the primary risk measures in the variance-gamma model. A portfolio of investments in a one-period setting is considered. It is supposed that the investment returns are dependent on each other. In terms of the variance-gamma model, we assume that there are relations in both groups of the normal random variables and the gamma stochastic volatilities. The value at risk, the expected shortfall and the entropic monetary risk measures are discussed. The obtained analytical expressions are based on values of hypergeometric functions.

FINANCIAL MARKETS WITH NO RISKLESS (SAFE) ASSET

We study markets with no riskless (safe) asset. We derive the corresponding Black–Scholes–Merton option pricing equations for markets where there are only risky assets which have the following price dynamics: (i) continuous diffusions; (ii) jump-diffusions; (iii) diffusions with stochastic volatilities, and; (iv) geometric fractional Brownian and Rosenblatt motions. No-arbitrage and market-completeness conditions are derived in all four cases.