International Journal of Theoretical and Applied Finance

Recently, an empirically-validated one-factor model with a 3/4-power diffusion term was introduced in the literature to model oil prices and value futures contracts on oil. In this paper, we provide an exact and analytic approximation for European call option prices on futures under a 3/4-power futures model. The analytic approximation, valid for short times to expiry is then calibrated to market prices. Results from the calibration show that the analytic approximation formula outperforms current popular options on futures formulae in capturing market prices.

This paper presents a method for approximating the underlying stock’s distribution by using a Log–Skew–Normal mixture distribution. The basic properties of a mixture of Skew–Normal distributions are reviewed in this paper. We provide a formula for the European option price by assuming that the log price follows a Skew–Normal mixture distribution. We also calculate the “Greeks”, such as delta, gamma and vega. We compare the proposed model with other existing models and consider an example of calibration to real market option data.

In this work, we consider a stock price process subjected to idiosyncratic Lévy jumps and global structural changes attributed to interventions due to a semi-Markov process. The semi-Markov process decomposes both the time and state domains of the price process into sub-intervals and price state sub-domains respectively, where a Lévy–Ito process operates. The Lévy jumps decompose the space domain of the currently operating Lévy process. We derive an infinitesimal generator for a stock price process and a closed form expression for the conditional characteristic function of a log price. The former result is used to derive a PIDE satisfied by option prices, while the latter could be used to retrieve risk neutral densities via Fourier transform and price European vanilla options. In the sequel, we derive the characteristic function of the residence time of a semi-Markov process. Incompleteness of the market is exhibited through a general change of measure. For pricing purpose, the minimum entropy martingale measure is defined as an Esscher transform.

In this paper, we study the valuation of variable annuities for an insurer. We concentrate on two types of these contracts, namely guaranteed minimum death benefits and guaranteed minimum living benefits that allow the insured to withdraw money from the associated account. Here, the price of variable annuities corresponds to a fee, fixed at the beginning of the contract, that is continuously taken from the associated account. We use a utility indifference approach to determine the indifference fee rate. We focus on the worst case for the insurer, assuming that the insured makes the withdrawals that minimize the expected utility of the insurer. To compute this indifference fee rate, we link the utility maximization in the worst case for the insurer to a sequence of maximization and minimization problems that can be computed recursively. This allows to provide an optimal investment strategy for the insurer when the insured follows the worst withdrawal strategy and to compute the indifference fee. We finally explain how to approximate these quantities via the previous results and give numerical illustrations of parameter sensitivity.

A log-normal random walk with parameters that are functions of market stress naturally accounts for volatility clustering and fat-tailed return distributions. Fitting this model to a stock and a bond index we find no evidence of significant misspecification despite the fact that the model has no adjustable parameters. This model can be interpreted as a stochastic volatility model without latent variables. We obtain a closed-form expression for the Value at Risk (VaR) that accommodates returns of any magnitude and discuss several other applications.

In this paper, we introduce a new form of asymptotic arbitrage, which we call a partial asymptotic arbitrage, half-way between those of Föllmer & Schachermayer (2007) [Mathematics and Financial Economics 1 (34), 213–249] and Kabanov & Kramkov (1998) [Finance and Stochastics 2, 143–172]. In the context of the Heston model, we establish a precise link between the set of equivalent martingale measures, the ergodicity of the underlying variance process and this partial asymptotic arbitrage. In contrast to Föllmer & Schachermayer (2007) [Mathematics and Financial Economics 1 (34), 213–249], our result does not assume a suitable condition on the stock price process to allow for (partial) asymptotic arbitrage.

In this paper, we present an innovative hybrid model for the valuation of equity options. Our approach includes stochastic volatility according to Heston (1993) [Review of Financial Studies 6 (2), 327–343] and features a stochastic interest rate that follows a three-factor short rate model based on Hull and White (1994) [Journal of Derivatives 2 (2), 37–48]. Our model is of affine structure, allows for correlations between the stock, the short rate and the volatility processes and can be fitted perfectly to the initial term structure. We determine the zero bond price formula and derive the analytic solution for European type options in terms of characteristic functions needed for fast calibration. We highlight the flexibility of our approach, by comparing the price and implied volatility surfaces of our model with the Heston model, where we in particular focus on the correlation structure. Our approach forms a comprehensive market model with an intuitive correlation structure that connects both the equity and interest market to the market volatility.