Path dependent volatility

Decisions in Economics and Finance, 2013

In this paper we consider a generalisation of the Hobson-Rogers model proposed by Foschi and Pascucci in [10] for financial markets where the evolution of the prices of the assets depends not only on the current value but also on past values. Using differentiability of stochastic processes with respect to the initial condition, we analyse the robustness of such a model with respect to the so-called offset function, which generally depends on the entire past of the risky asset and is thus not fully observable. In doing this, we extend previous results of [3] to contingent claims which are globally Lipschitz with respect to the price of the underlying asset, and we improve the dependence of the necessary observation window on the maturity of the contingent claim, which now becomes of linear type, while in [3] it was quadratic. Finally, in this framework we give a characterisation of the stationarity assumption used in [3], and prove that this model is stationary if and only if it is reduced to the original Hobson-Rogers model. We conclude by calibrating the model to the prices of two indexes using two different volatility shapes.

2003

We compare the dynamic hedging performance of the deterministic local volatility function approach with the implied/constant volatility method. Using an example in which the underlying price follows an absolute diffusion process, we illustrate that hedge parameters computed from the implied/constant volatility method can have significant error even though the implied volatility method is able to calibrate the current option prices of different strikes and maturities. In particular the delta hedge parameter produced by the implied/constant volatility method is consistently significantly larger than that of the exact delta when the underlying price follows an absolute diffusion.

Journal of Mathematical Finance, 2022

Asian options are generally priced using arithmetic or geometric averages of the underlying stock. However, these methods are not suitable when stock's volatilities are very low. The motivation to develop derivative prices based on averaging the underlying asset stems from the robust features associated with Asian options which suggest that they are more suited to African markets where prices can be dormant for long periods resulting in low volatilities in stock prices. We propose the use of the modal average as the measure of the underlying stock price when stocks have low volatilities instead of the more popular arithmetic and geometric averages. In particular, the stock price is assumed to follow Geometric Brownian Motion and using the concept of maximum of a function, a model for the modal average of the underlying stock is derived. A process of obtaining the price of a call option is subsequently developed. Theoretically, we prove further that for very low volatilities the modal average model is a better estimator of the expected average of the stock price and consequently produces cheaper option prices than geometric and arithmetic average models. Using data from the Ghana Stock Exchange and the Nasdaq, the proposed model is used to price options sold on selected stocks on the exchange. The numerical results consistently show that for underlying stocks with volatility less than 3%, the modal average model provides cheaper call options than the arithmetic or geometric averages pricing models.

Available at SSRN 1014172, 2007

We show that it is possible to avoid the discrepancies of continuous path models for stock prices and still be able to hedge options if one models the stock price process as a birth and death process. One needs the stock and another market traded derivative to hedge an option in this setting. However, unlike in continuous models, number of extra traded derivatives required for hedging does not increase when the intensity process is stochastic. We obtain parameter estimates using Generalized Method of Moments and describe the Monte Carlo algorithm to obtain option prices. We show that one needs to use filtering equations for inference in the stochastic intensity setting. We present real data applications to study the performance of our modeling and estimation techniques.

Review of Derivatives Research, 2007

We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying's futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility.

Mathematical Finance, 1996

In the stochastic volatility framework of Hull and White (1987), we characterize the so-called Black and Scholes implied volatility as a function of two arguments: the ratio of the strike to the underlying asset price and the instantaneous value of the volatility. By studying the variations in the first argument, we show that the usual hedging methods, through the Black and Scholes model, lead to an underhedged (resp. overhedged) position for in-the-money (resp. out-of-the-money) options, and a perfect partial hedged position for at-the-money options. These results are shown to be closely related to the smile efect, which is proved to be a natural consequence ofthe stochastic volatility feature. The deterministic dependence of the implied volatility on the underlying volatility process suggests the use of implied volatility data for the estimation of the parameters of interest. A statistical procedure of filtering (of the latent volatility process) and estimation (of its parameters) is shown to be strongly consistent and asymptotically normal.

Mathematical Finance, 1998

We thank Jin Duan for his comments on a first draft of this note and a referree for very useful comments. The first author gratefully acknowledges financial support from the Fonds de la Formation de Chercheurs et à l'Aide à la Recherche du Québec (FCAR) and the PARADI research program funded by the Canadian International Development Agency (CIDA). The second author thanks CIRANO and C.R.D.E. for financial support.

Applied Financial Economics Letters, 2005

A new market-based approach to evaluating options on an asset is offered. The model corresponds to the real situations encountered in the market: option prices are not uniquely determined by their underlying asset but mainly by another factor, namely stochastic market volatility (or simply SMV). To begin constructing SMV, it is assumed that there exists a hedging portfolio which replicates perfectly the value of the underlying option. By 'perfectly', it is meant that the value of the hedging portfolio will always equal exactly to the option. The hedging portfolio takes asset price and SMV as its input, therefore, for a given asset price the correct value of SMV gives the correct value for the option. SMV presents the dynamics of options market. We provide the proof of existence and uniqueness of solutions for SMV.

The Hobson and Rogers model for option pricing is considered. This stochastic volatility model preserves the completeness of the market and can potentially reproduce the observed smile and term structure patterns of implied volatility. A calibration procedure based on ad-hoc numerical schemes for hypoelliptic PDEs is proposed and used to quantitatively investigate the pricing performance of the model. Numerical results based on S&P500 option prices are discussed.

Physica A: Statistical Mechanics and its Applications, 2003

In a seminal paper in 1973, Black and Scholes argued how expected distributions of stock prices can be used to price options. Their model assumed a directed random motion for the returns and consequently a lognormal distribution of asset prices after a finite time. We point out two problems with their formulation. First, we show that the option valuation is not uniquely determined; in particular ,strategies based on the delta-hedge and CAPM (the Capital Asset Pricing Model) are shown to provide different valuations of an option. Second, asset returns are known not to be Gaussian distributed. Empirically, distributions of returns are seen to be much better approximated by an exponential distribution. This exponential distribution of asset prices can be used to develop a new pricing model for options that is shown to provide valuations that agree very well with those used by traders. We show how the Fokker-Planck formulation of fluctuations (i.e., the dynamics of the distribution) can be modified to provide an exponential distribution for returns. We also show how a singular volatility can be used to go smoothly from exponential to Gaussian returns and thereby illustrate why exponential returns cannot be reached perturbatively starting from Gaussian ones, and explain how the theory of 'stochastic volatility' can be obtained from our model by making a bad approximation. Finally, we show how to calculate put and call prices for a stretched exponential density.