Forward Start Options Research Articles

The Jacobi Stochastic Volatility Model

We introduce a novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process. It contains the Heston model as a limit case. We show that the the joint distribution of any finite sequence of log returns admits a Gram-Charlier A expansion in closed-form. We use this to derive closed-form series representations for option prices whose payoff is a function of the underlying asset price trajectory at finitely many time points. This includes European call, put, and digital options, forward start options, and forward start options on the underlying return. We derive sharp analytical and numerical bounds on the series truncation errors. We illustrate the performance by numerical examples, which show that our approach offers a viable alternative to Fourier transform techniques.

Acceptability bounds for forward starting options using disciplined convex programming

Acceptability pricing is formulated, for the dual problem, as a disciplined convex program solvable by the software CVXOPT. Forward starting options are used to illustrate the procedures for acceptability defined by positive expectation under the concave distortion minmaxvar. The measures used are required to reprice the liquid assets and must belong to the support set of acceptability. The bounds obtained bounds are substantially tighter than they would have been by leaving out either of the repricing constraints or the support sets for acceptability.

The forward smile in local–stochastic volatility models

We introduce an asymptotic expansion for forward start options in a multi-factor local-stochastic volatility model. We derive explicit approximation formulas for the so-called forward implied volatility which can be useful to price complex path-dependent options, as cliquets. The expansion involves only polynomials and can be computed without the need for numerical procedures or special functions. Recent results on the exploding behaviour of the forward smile in the Heston model are confirmed and generalized to a wider class of local-stochastic volatility models. We illustrate the effectiveness of the technique through some numerical tests.

Pricing volatility options under stochastic skew with application to the VIX index

In recent years, there has been a remarkable growth of volatility options. In particular, VIX options are among the most actively trading contracts at Chicago Board Options Exchange. These options exhibit upward sloping volatility skew and the shape of the skew is largely independent of the volatility level. To take into account these stylized facts, this article introduces a novel two-factor stochastic volatility model with mean reversion that accounts for stochastic skew consistent with empirical evidence. Importantly, the model is analytically tractable. In this sense, I solve the pricing problem corresponding to standard-start, as well as to forward-start European options through the Fast Fourier Transform. To illustrate the practical performance of the model, I calibrate the model parameters to the quoted prices of European options on the VIX index. The calibration results are fairly good indicating the ability of the model to capture the shape of the implied volatility skew associated with VIX options.

Valuation of forward start options under affine jump-diffusion models

Under the general affine jump-diffusion framework of Duffie et al. [Econometrica, 2000, 68, 1343–1376], this paper proposes an alternative pricing methodology for European-style forward start options that does not require any parallel optimization routine to ensure square integrability. Therefore, the proposed methodology is shown to possess a better accuracy–efficiency trade-off than the usual and more general approach initiated by Hong [Forward Smile and Derivative Pricing. Working paper, UBS, 2004] that is based on the knowledge of the forward characteristic function. Explicit pricing solutions are also offered under the nested jump-diffusion setting proposed by Bakshi et al. [J. Finance, 1997, 52, 2003–2049], which accommodates stochastic volatility and stochastic interest rates, and different integration schemes are numerically tested.

Random Time Forward Starting Options

We introduce a natural generalization of the forward-starting options, first discussed by M. Rubinstein [Rubin]. The main feature of the contract presented here is that the strike-determination time is not fixed ex-ante, but allowed to be random, usually related to the occurrence of some event, either of financial nature or not. We will call these options Random Time Forward Starting (RTFS).We show that, under an appropriate "martingale preserving" hypothesis, we can exhibit arbitrage free prices, which can be explicitly computed in many classical market models, at least under independence between the random time and the assets' prices. Practical implementations of the pricing methodologies are also provided. Finally a credit value adjustment formula for these OTC options is computed for the unilateral counterparty credit risk.

The Forward Smile in Local-Stochastic Volatility Models

We introduce an asymptotic expansion for forward start options in a multi-factor local-stochastic volatility model. We derive explicit approximation formulas for the so-called forward implied volatility which can be useful to price complex path-dependent options, as cliquets. The expansion involves only polynomials and can be computed without the need for numerical procedures or special functions. Recent results on the exploding behaviour of the forward smile in the Heston model are confirmed and generalized to a wider class of local-stochastic volatility models. We illustrate the effectiveness of the technique through some numerical tests. Mathematica codes are freely available on the authors' website.

Pricing Path-Dependent Options with Discrete Monitoring under Time-Changed Lévy Processes

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Lévy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic function of the intertemporal joint distribution of the time-changed Lévy process. Using the derived representation of the characteristic function, we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader and lookback options, all of which are discretely monitored.

Generalized Barndorff-Nielsen and Shephard Model and Discretely Monitored Option Pricing

This paper proposes a generalization of the Generalized Barndorff-Nielsen and Shephard model, in which the log return on an asset price is governed by a Levy process with stochastic volatility modeled by a non-Gaussian Ornstein-Uhlenbeck process. Under the generalized model, we derive a closed-form expression for the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

An Investigation of Model Risk in a Market with Jumps and Stochastic Volatility

The aim of this paper is to investigate model risk aspects of variance swaps and forward start options in a realistic market setup where the underlying asset price process exhibits stochastic volatility and jumps. We devise a general framework in order to provide evidence of the model uncertainty attached to variance swaps even when the popular replication result is accounted for. Then, we consider the model risk of forward-start options and we show how this risk can be reduced by adding variance swaps in the set of calibration instruments. In the adopted framework, variance swaps and forward-start options can be valued by means of analytic methods. We measure model risk using a set of 21 models representing various dynamics with both continuous and discontinuous sample paths. To conduct our empirical analysis, we work with two major equity indices (S&P 500 and Eurostoxx 50) under different market situations.

Pricing Volatility Options Under Stochastic Skew with Application to the VIX Index

In recent years there has been a remarkable growth of volatility options. In particular, VIX options are among the most actively trading contracts at CBOE. These options exhibit upward sloping volatility skew and the shape of the skew is largely independent of the volatility level. To take into account these stylized facts, this article introduces a novel two-factor stochastic volatility model with mean reversion that accounts for stochastic skew consistent with empirical evidence. Importantly, the model is analytically tractable. In this sense, I solve the pricing problem corresponding to standard-start, as well as to forward-start European options through the Fast Fourier Transform.To illustrate the practical performance of the model, I calibrate the model parameters to the quoted prices of European options on the VIX index. The calibration results are fairly good indicating the ability of the model to capture the shape of the implied volatility skew associated with VIX options.

Consistent Pricing of Spot Options and Forward Start Options

The idea of this paper is to present how we can use a specific form of local volatility, namely a forward start local vol, that has already been evoked in different sources, and use it in order to fit a specific Forward Start Smile. A local formula, i.e. a formula similar to a Dupire formula, is available for these models and will be used to fit the Forward Smile Market. These models are therefore particularly adapted for pricing Cliquet Products or generally Forward Starting Products with given frequencies. In particular, it is possible to fit different options like Vanilla options and Forward Start Options using them. We will also consider an extension of these models adapted for the pricing of lookback options.

The Small-Maturity Heston Forward Smile

In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-of-the-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter configurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent analysis shows that the small-maturity limit is well defined for any Ito diffusion. The proofs rely on the theory of sharp large deviations (and refinements) and incidentally we provide an example of degenerate large deviations behaviour.

The Small-Maturity Heston Forward Smile

In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-of-the-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore, the result shows that the square-root behavior of the variance process induces a singularity such that for certain parameter configurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent analysis shows that the small-maturity limit is well defined for any Itô diffusion. The proofs rely on the theory of sharp large deviations [B. Bercu and A. Rouault, Theory Probab. Appl., 46 (2002), pp. 1--19] (and refinements), and incidentally we provide an example of degenerate large deviation behavior.

Asymptotics of Forward Implied Volatility

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Levy models. This expansion applies to both small and large maturities and is based solely on the properties of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques and allows us to recover (in particular) many results for the spot implied volatility smile. In passing we (i) show that the forward-start date has to be rescaled in order to obtain nontrivial small-maturity asymptotics, (ii) prove that the forward-start date may influence the large-maturity behavior of the forward smile, and (iii) provide some examples of models with finite quadratic variation where the small-maturity forward smile does not explode.

Super-Replication of Financial Derivatives Via Convex Programming

We give a method based on convex programming to calculate the optimal super-replicating and sub-replicating prices and corresponding hedging portfolios of a financial derivative in terms of other financial derivatives in a discrete-time setting. Our method produces a model that matches the super-replicating (or sub-replicating) price within an arbitrary precision and is consistent with the other financial derivatives prices. Applications include robust replication in terms of call prices with various strikes and maturities of forward start options, volatility and variance swaps and derivatives, cliquets calls, barrier options, lookback and Asian options. Numerical examples show that, in some cases, the best super-replicating and/or sub-replicating prices are within 10% of the price obtained by a standard model, but considerably differ from it in other cases. Our method can incorporate bid-ask spreads, interest rates and dividends and various limitations to the diffusion model.

NEARLY EXACT OPTION PRICE SIMULATION USING CHARACTERISTIC FUNCTIONS

This paper presents a new approach to perform a nearly unbiased simulation using inversion of the characteristic function. As an application we are able to give unbiased estimates of the price of forward starting options in the Heston model and of continuously monitored Parisian options in the Black-Scholes framework. This method of simulation can be applied to problems for which the characteristic functions are easily evaluated but the corresponding probability density functions are complicated.

Pricing Path-Dependent Options with Discrete Monitoring Under Time-Changed Levy Processes

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Levy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic functions of the intertemporal joint distribution of time-changed Levy processes. Using the derived representation of the characteristic function we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader, and lookback options, all of which are discretely monitored.

Nearly Exact Option Price Simulation Using Characteristic Functions

This paper presents a new approach to perform a nearly unbiased simulation using inversion of the characteristic function. As an application we are able to give unbiased estimates of the price of forward starting options in the Heston model and of continuously monitored Parisian options in the Black-Scholes framework. This method of simulation can be applied to problems for which the characteristic functions are easily evaluated but the corresponding probability density functions are complicated.

Asymptotics of Forward Implied Volatility

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Levy models.This expansion applies to both small and large maturities and is based solely on the properties of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques, and allows us to recover (in particular) many results for the spot implied volatility smile.In passing we (i) show that the forward-start date has to be rescaled in order to obtain non-trivial small-maturity asymptotics,(ii) prove that the forward-start date may influence the large-maturity behaviour of the forward smile,and (iii) provide some examples of models with finite quadratic variation where the small-maturity forward smile does not explode.