Closed-form Option Pricing Formula Research Articles

Closed-form option pricing for exponential Lévy models: a residue approach

Exponential Lévy processes provide a natural and tractable generalization of the classic Black–Scholes–Merton model which account for several stylized features of financial markets, including jumps and kurtosis. In the existing literature, closed-form option pricing formulas are sparse for exponential Lévy models, outside of special cases such as Merton's jump diffusion, and complex numerical techniques are required even to price European options. To bridge the gap, this work provides a comprehensive and unified pricing framework for vanilla and exotic path independent options under the Variance Gamma (VG), Finite Moment Log Stable (FMLS), one-sided Tempered Stable (TS), and Normal Inverse Gaussian (NIG) models. We utilize the Mellin Transform and residue calculus to obtain closed-form series representations for the price of several options, including vanillas (European), digitals, power, and log options. These formulas provide nice theoretical representations, but are also efficient to evaluate in practice, as numerous numerical experiments demonstrate. The closed-form nature of these option pricing formulas makes them ideal for adoption in practical settings, as they do not require complicated pricing methods to achieve high-accuracy prices, and the resulting pricing error is reliably controllable.

Volatility Bursts: A Discrete-Time Option Model with Multiple Volatility Components

Abstract I propose an affine discrete-time model, called Vector Autoregressive Gamma with volatility Bursts (VARG-B) in which volatility experiences, in addition to frequent and small changes, periods of sudden and extreme movements generated by a latent factor which evolves according to the Autoregressive Gamma Zero process. A key advantage of the discrete-time specification is the possibility of estimating the model using Kalman Filter techniques. Moreover, the VARG-B model leads to a fully analytic conditional Laplace transform which boils down to a closed-form option pricing formula. When estimated on S&P500 index options and returns, the new model provides more accurate option pricing and modeling of the IV surface compared with some alternative models.

Volatility Bursts: A Discrete-time Option Model with Multiple Volatility Components

I propose an affine discrete-time model, called Vector Autoregressive Gamma with volatility Bursts (VARG-B) in which volatility experiences, in addition to frequent and small changes, periods of sudden and extreme movements generated by a latent factor which evolves according to the Autoregressive Gamma Zero process. A key advantage of the discrete-time specification is that it makes it possible to estimate the model via the Extended Kalman Filter. Moreover, the VARG-B model leads to a fully analytic conditional Laplace transform, resulting in a closed-form option pricing formula. When estimated on S&P500 index options and returns the new model provides more accurate option pricing and modelling of the IV surface compared with some alternative models.

The closed-form option pricing formulas under the sub-fractional Poisson volatility models

A new fractional process called the sub-fractional Poisson process NH(t) is proposed, which has continuous sample paths, long- memory, leptokurtosis and heavy tail distribution, is of convenience to price options and simulate the variance process of risk asset return. Based on the sub-fractional Poisson process NH(t) the new fractional variance processes have been proposed, which may capture the skewness and the long-memory as well as mean-reverting in the stock price volatilities. In particular, the characteristic function method for option pricing is given, and the analytical formulas for European option price C(t,St) have been obtained under the risk-neutral probability measure.

Closed-form option pricing for exponential Lévy models: a residue approach

Exponential L\'evy processes provide a natural and tractable generalization of the classic Black-Scholes-Merton model which are capable of capturing observed market implied volatility skews. In the existing literature, closed-form option pricing formulas are sparse for exponential L\'evy models, outside of special cases such as Merton's jump diffusion, and complex numerical techniques are required even to price European options. To bridge the gap, this work provides a comprehensive and unified pricing framework for vanilla and exotic European options under the Variance Gamma (VG), Finite Moment Log Stable (FMLS), one-sided Tempered Stable (TS), and Normal Inverse Gaussian (NIG) models. We utilize the Mellin Transform and residue calculus to obtain closed-form series representations for the price of several European options, including vanillas, digitals, power, and log options. These formulas provide nice theoretical representations, but are also efficient to evaluate in practice, as numerous numerical experiments demonstrate. The closed-form nature of these option pricing formulas makes them ideal for adoption in practical settings, as they do not require complicated pricing methods to achieve high accuracy prices, and the resulting pricing error is reliably controllable.

Option Pricing: Channels, Target Zones and Sideways Markets

After a market downturn, especially in an uncertain economic environment such as the current state, there can be a relatively long period with a sideways market, where indexes, stocks, etc., move in channels with support and resistance levels. We discuss option pricing in such scenarios, in both cases of unattainable as well as attainable boundaries, and obtain closed-form option pricing formulas. Our results also apply to FX rates in target zones without interest rate pegging (USD/HKD, digital currencies, etc.).

Option Pricing: Channels, Target Zones and Sideways Markets

After a market downturn, especially in an uncertain economic environment such as the current state, there can be a relatively long period with a sideways market, where indexes, stocks, etc., move in channels with support and resistance levels. We discuss option pricing in such scenarios, in both cases of unattainable as well as attainable boundaries, and obtain closed-form option pricing formulas. Our results also apply to FX rates in target zones without interest rate pegging (USD/HKD, digital currencies, etc.).

A general closed form option pricing formula

A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed form pricing formula for European options is obtained by employing a modified Gram-Charlier series expansion, known as the Gauss-Hermite expansion. This expansion converges for fat-tailed distributions commonly encountered in the study of financial returns. The expansion coefficients can be calibrated from observed option prices and can also be computed, for example, in models with the probability density function or the characteristic function known in closed form. We investigate the properties of the new option pricing model by calibrating it to both real-world and simulated option prices and find that the resulting implied volatility curves provide an accurate approximation for a wide range of strike prices. Based on an extensive empirical study, we conclude that the new approximation method outperforms other methods both in-sample and out-of-sample.

Volatility Bursts: A Discrete-Time Option Model with Multiple Volatility Components

I propose an affine discrete-time model, called Vector Autoregressive Gamma with volatility Bursts (VARG-B) in which volatility experiences, in addition to frequent and small changes, periods of sudden and extreme movements generated by a latent factor which evolves according to the Autoregressive Gamma Zero process. A key advantage of the discrete-time specification is that it makes it possible to estimate the model via the Extended Kalman Filter. Moreover, the VARG-B model leads to a fully analytic conditional Laplace transform, resulting in a closed-form option pricing formula. When estimated on S&P500 index options and returns the new model provides more accurate option pricing and modelling of the IV surface compared with some alternative models.

An Analytical Approximation for Option Price under the Affine GARCH Model - A Comparison with the Closed-Form Solution of Heston-Nandi

In the option pricing theory, two important approaches have been developed to evaluate the prices of a European option. The first approach develops an almost closed-form option pricing formula under a specific GARCH process (Heston & Nandi, 2000). The second approach develops an analytical approximation for computing European option prices with more widespread NGARCH models (Duan, Gauthier & Simonato, 1999). The analytical approximation was also developed under GJR-GARCH and EGARCH models by Duan, Gauthier, Sasseville & Simonato (2006). However, no empirical work was performed to study the comparative performance of these two formulas (closed-form solution and analytical approximation). Also, it is possible to develop an analytical approximation under the specific GARCH model of Heston & Nandi (2000). In this paper, we have filled up those gaps. We started with the development of an analytical approximation, for computing European option prices, under Heston-Nandis GARCH model. In the second step, we carried out a comparative analysis of the three formulas using CAC 40 index returns from 31 December 1987 to 31 December 2013.

Herding and Stochastic Volatility

In this paper we develop a one-factor non-affine stochastic volatility option pricing model where the dynamics of the underlying is endogenously determined from micro-foundations. The interaction and herding of the agents trading the underlying asset induce an amplification of the volatility of the asset over the volatility of the fundamentals. Although the model is non-affine, a closed form option pricing formula can still be derived by using a Gauss-Hermite series expansion methodology. The model is calibrated using S&P 500 index options for the period 1996-2013. When its results are compared to some benchmark models we find that the new non-affine one-factor model outperforms the affine one-factor Heston model and it is competitive, especially out-of-sample, with the affine two-factor double Heston model.

Pricing and hedging quanto options in energy markets

In energy markets, the use of quanto options has increased significantly in recent years. The payoff from such options are typically written on an underlying energy index and a measure of temperature. They are suited to managing the joint price and volume risk in energy markets. Using a Heath-Jarrow-Morton approach, we derive a closed-form option pricing formula for energy quanto options under the assumption that the underlying assets are lognormally distributed. Our approach encompasses several interesting cases, such as geometric Brownian motions and multifactor spot models. We also derive Delta and Gamma expressions for hedging. Further, we illustrate the use of our model by an empirical pricing exercise using NewYork Mercantile Exchange-traded natural gas futures and Chicago Mercantile Exchange-traded heating degree days futures for New York.

The pricing of basket-spread options

Since the pioneering paper of Black and Scholes was published in 1973, enormous research effort has been spent on finding a multi-asset variant of their closed-form option pricing formula. In this paper, we generalize the Kirk [Managing Energy Price Risk, 1995] approximate formula for pricing a two-asset spread option to the case of a multi-asset basket-spread option. All the advantageous properties of being simple, accurate and efficient are preserved. As the final formula retains the same functional form as the Black–Scholes formula, all the basket-spread option Greeks are also derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark results obtained by numerical integration or Monte Carlo simulation with 10 million paths. An implicit correction method is further applied to reduce the pricing errors by factors of up to 100. The correction is governed by an unknown parameter, whose optimal value is found by solving a non-linear equation. Owing to its simplicity, the computing time for simultaneous pricing and hedging of basket-spread option with 10 underlying assets or less is kept below 1 ms. When compared against the existing approximation methods, the proposed basket-spread option formula coupled with the implicit correction turns out to be one of the most robust and accurate methods.

A General Closed Form Option Pricing Formula

A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed form pricing formula for European options is obtained by employing a modified Gram-Charlier series expansion, known as the Gauss-Hermite expansion. This expansion converges for fat-tailed distributions commonly encountered in the study of financial returns. The expansion coefficients can be calibrated from observed option prices and can also be computed, for example, in models with the probability density function or the characteristic function known in closed form. We investigate the properties of the new option pricing model by calibrating it to both real-world and simulated option prices and find that the resulting implied volatility curves provide an accurate approximation for a wide range of strike prices. Based on an extensive empirical study, we conclude that the new approximation method outperforms other methods both in-sample and out-of-sample.

Pricing and Hedging Quanto Options in Energy Markets

In energy markets, the use of quanto options has increased significantly in recent years. The payoff from such options are typically written on an underlying energy index and a measure of temperature. They are suited to managing the joint price and volume risk in energy markets. Using a Heath-Jarrow-Morton approach, we derive a closed-form option pricing formula for energy quanto options under the assumption that the underlying assets are lognormally distributed. Our approach encompasses several interesting cases, such as geometric Brownian motions and multifactor spot models. We also derive Delta and Gamma expressions for hedging. Further, we illustrate the use of our model by an empirical pricing exercise using NewYork Mercantile Exchange-traded natural gas futures and Chicago Mercantile Exchange-traded heating degree days futures for New York.

Enhanced Valuation of European Options Under Jump Processes and Innovative Characterization of Implied Volatility Smile

An enhanced option pricing framework that makes use of both continuous and discontinuous time paths based on a geometric Brownian motion and Poisson-driven jump processes respectively is performed in order to better fit with real-observed stock price paths while maintaining the analytical trackability of the Black-Scholes model. The main advantage of this model is to lie on a consistent framework that does not make stringent assumptions in order to derive a closed-form option pricing formula and to capture rare events such as major political changes or catastrophic events through the use of discontinuous stochastic processes. Moreover, this model does not much face any calibration issue given that little dependence with non-observable parameters is introduced. Moreover, an innovative quantification of pricing differences is proposed between the in-house (i.e. the own made pricing formula including jumps) and the classic Black-Scholes model through implied volatility and its curve resembles a smile, meaning that the introduction of jumps is quantified via a smile according to implied volatility. In order to derive such an implied volatility smile, an iterative search procedure referred to as the Newton-Raphson algorithm is proposed. Numerical experiments of both the in-house pricing formula and its implied volatility recursive algorithm are presented and results show that both the two are fast computing via computers through the use of coding languages.

Path integral approach to closed-form option pricing formulas with applications to stochastic volatility and interest rate models

We present a path integral method to derive closed-form solutions for option prices in a stochastic volatility model. The method is explained in detail for the pricing of a plain vanilla option. The flexibility of our approach is demonstrated by extending the realm of closed-form option price formulas to the case where both the volatility and interest rates are stochastic. This flexibility is promising for the treatment of exotic options. Our analytical formulas are tested with numerical Monte Carlo simulations.

Pricing freight rate options

In this paper we set up the theoretical framework for the valuation of the Asian-style options traded in the freight derivatives market. Assuming lognormal spot freight dynamics, we show that Forward Freight Agreements (FFA) are also lognormal prior to the settlement period, but that this lognormality subsequently breaks down. We suggest approximate dynamics in the settlement period for the FFA that leads to closed-form option pricing formulas for Asian call and put options written on the spot freight rate indices in the Black [Black, F., 1976. The pricing of commodity contracts. Journal of Financial Economics 3, 167–179] framework. In a Monte Carlo experiment we show that our formula gives very accurate prices, in particular for forward-starting freight options.

A Closed-Form GARCH Option Pricing Model

This paper develops a closed-form option pricing formula for a spot asset whose variance follows a GARCH process. The model allows for correlation between returns of the spot asset and variance and also admits multiple lags in the dynamics of the GARCH process. The single factor (one lag) version of this model contains Heston's (1993) stochastic volatility model as a diffusion limit and therefore unifies the discrete GARCH and continuous-time stochastic volatility literature of option pricing. The new model provides the first option formula for a random volatility model that is solely a function of observables; all the parameters can be easily estimated from the history of asset prices, observed at discreteintervals. Empirical analysis on S&P500 index options shows the single factor version of the GARCH model to be a substantial improvement over the Black-Scholes (1973) model. The GARCH model continues to substantially outperform the Black-Scholes model even when the Black-Scholes model is updated every period while the parameters of the GARCH model are held constant. The improvement is due largely to the ability of the GARCH model to describe the correlation of volatility with spot returns. This allows the GARCH model to capture strike price biases in the Black-Scholes model that give rise to the skew in implied volatilities in the index options market.

The pricing of dollar-denominated yen/DM warrants

This paper investigates the pricing of dollar-denominated cross-currency warrants (options) traded in US markets. In a Black-Scholes setting, we obtain closed-form option pricing formulas for European warrants. We describe the intuition behind the differences in the properties of these securities versus standard foreign currency options, concentrating on parameter sensitivity as well as hedging aspects. We show that US cross-currency warrants can be hedged using three underlying assets although the fundamental PDE involves only one asset. This analytical framework is applied to observed prices for AT&T's NYSE-traded US yen/DM cross-currency warrant. We examine and attempt to explain the deviations between model and observed prices uncovered by the empirical work. We find evidence of misspecification in standard currency option-pricing models. (JEL F31).