Underlying Asset Price Process Research Articles

Short-time at-the-money skew and rough fractional volatility

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian motion with Hurst parameter less than half. The fractional Brownian motion is correlated with a Brownian motion which drives the asset price process. We derive an asymptotic expansion of the implied volatility as the time to maturity tends to zero. For this purpose, we introduce a new approach to validate such an expansion, which enables us to treat more general models than in the literature. The local-stochastic volatility model is treated as well under an essentially minimal regularity condition in order to show such a standard model cannot be dynamically consistent to the power law.

Pricing Bounds and Approximations for Discrete Arithmetic Asian Options under Time-Changed LLvy Processes

We derive efficient and accurate analytical pricing bounds and approximations for discrete arithmetic Asian options under time-changed Levy processes. By extending the conditioning variable approach, we derive the lower bound on the Asian option price and construct an upper bound based on the sharp lower bound. We also consider the general partially exact and bounded (PEB) approximations, which include the sharp lower bound and partially conditional moments matching approximation as special cases. The PEB approximations are known to lie between a sharp lower bound and an upper bound. Our numerical tests show that the PEB approximations to discrete arithmetic Asian option prices can produce highly accurate approximations when compared to other approximation methods. Our proposed approximation methods can be readily applied to pricing Asian options under the most common types of underlying asset price processes, like the Heston stochastic volatility model nested in time-changed Levy processes with leverage effect.

Pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes

We derive efficient and accurate analytical pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes. By extending the conditioning variable approach, we derive the lower bound on the Asian option price and construct an upper bound based on the sharp lower bound. We also consider the general partially exact and bounded (PEB) approximations, which include the sharp lower bound and partially conditional moment matching approximation as special cases. The PEB approximations are known to lie between a sharp lower bound and an upper bound. Our numerical tests show that the PEB approximations to discrete arithmetic Asian option prices can produce highly accurate approximations when compared to other approximation methods. Our proposed approximation methods can be readily applied to pricing Asian options under most common types of underlying asset price processes, like the Heston stochastic volatility model nested in the class of time-changed Lévy processes with the leverage effect.

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 and forward-start options. In our study, both variance swaps and forward-start options can be valued by means of analytic methods. We measure model risk using a set of 21 models embedding 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. Our results evaluate model risk between 50 and 200 basis points, with an average value slightly above 100 basis points of the contract notional.

A RECOMBINING TREE METHOD FOR OPTION PRICING WITH STATE-DEPENDENT SWITCHING RATES

This paper develops simple and efficient tree approaches for option pricing in switching jump diffusion models where the rates of switching are assumed to depend on the underlying asset price process. The models generalize many existing models in the literature and in particular, the Markovian regime-switching models with jumps. The proposed trees grow linearly as the number of tree steps increases. Conditions on the choices of key parameters for the tree design are provided that guarantee the positivity of branch probabilities. Numerical results are provided and compared with results reported in the literature for the Markovian regime-switching cases. The reported numerical results for the state-dependent switching models are new and can be used for comparison in the future.

An approximation formula for basket option prices under local stochastic volatility with jumps: An application to commodity markets

This paper develops a new approximation formula for pricing basket options in a local-stochastic volatility model with jumps. In particular, the model admits local volatility functions and jump components in not only the underlying asset price processes, but also the volatility processes. To the best of our knowledge, the proposed formula is the first one which achieves an analytical approximation for the basket option prices under this type of the models.Moreover, in numerical experiments, we provide approximate prices for basket options on the WTI futures and Brent futures based on the parameters through calibration to the plain-vanilla option prices, and confirm the validity of our approximation formula.

Pricing options on discrete realized variance with partially exact and bounded approximations

We derive efficient and accurate analytic approximation formulas for pricing options on discrete realized variance (DRV) under affine stochastic volatility models with jumps using the partially exact and bounded (PEB) approximations. The PEB method is an enhanced extension of the conditioning variable approach commonly used in deriving analytic approximation formulas for pricing discrete Asian style options. By adopting either the conditional normal or gamma distribution approximation based on some asymptotic behaviour of the DRV of the underlying asset price process, we manage to obtain PEB approximation formulas that achieve a high level of numerical accuracy in option values even for short-maturity options on DRV.

Pricing and static hedging of European-style double barrier options under the jump to default extended CEV model

This paper develops two novel methodologies for pricing and hedging European-style barrier option contracts under the jump to default extended constant elasticity of variance (JDCEV) model, namely: a stopping time approach based on the first passage time densities of the underlying asset price process through the barrier levels; and a static hedging portfolio approach in which the barrier option is replicated by a portfolio of plain-vanilla and binary options. In doing so, both valuation methodologies are extended to a more general set-up accommodating endogenous bankruptcy, time-dependent barriers and the commonly observed stylized facts of a positive link between default and equity volatility and of a negative link between volatility and stock price. The two proposed numerical methods are shown to be accurate, easy to implement and efficient under both the JDCEV model and the nested constant elasticity of variance model.

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.

Forward Backward Semimartingale Systems for Utility Maximization

We consider the problem of maximizing the expected utility of terminal wealth with a terminal random liability when the underlying asset price process is a continuous semimartingale. The optimal strategy is characterized in terms of a forward backward semimartingale system of equations. The results cover the cases of exponential, logarithmic, and power utilities, which we analyze as illustrative examples.

A NEW LOOK AT SHORT‐TERM IMPLIED VOLATILITY IN ASSET PRICE MODELS WITH JUMPS

We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short‐end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model‐independent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on short‐maturity option prices.

Mean-Variance Hedging on Uncertain Time Horizon in a Market with a Jump

In this work, we study the problem of mean-variance hedging with a random horizon T ^ tau , where T is a deterministic constant and is a jump time of the underlying asset price process. We rst formulate this problem as a stochastic control problem and relate it to a system of BSDEs with jumps. We then provide a veri cation theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from ltration enlargement theory.

Pricing Bermudan options using low-discrepancy mesh methods

This paper proposes a new simulation method for pricing Bermudan derivatives that is applicable to problems where the transition density of the underlying asset price process is known analytically. We assume that the owner can exercise the option at a finite, although possibly large, number of exercise dates. The method is computationally efficient for high-dimensional problems and is easy to apply. Its efficiency stems from our use of quasi-Monte Carlo techniques, which have proven effective in the case of European derivatives. The valuation of a Bermudan derivative hinges on the optimal exercise strategy. The optimal exercise decision can be reduced to the evaluation of a series of conditional expectations with respect to different distributions. These expectations can be approximated by sampling from just a single distribution at each exercise point. We provide a theoretical basis for the selection of this distribution and develop a simple approximation that has good convergence properties. We describe how to implement the method and confirm its efficiency using numerical examples involving Bermudan options written on multiple assets and options on a foreign asset with a stochastic interest rate.

Perpetual American options in a diffusion model with piecewise-linear coefficients

Abstract We derive closed form solutions to the discounted optimal stopping problems related to the pricing of the perpetual American standard put and call options in an extension of the Black–Merton–Scholes model with piecewise-constant dividend and volatility rates. The method of proof is based on the reduction of the initial optimal stopping problems to the associated free-boundary problems and the subsequent martingale verification using a local time-space formula. We present explicit algorithms to determine the constant hitting thresholds for the underlying asset price process, which provide the optimal exercise boundaries for the options.

Pricing Basket Options Under Local Stochastic Volatility with Jumps

This paper develops a new approximation formula for pricing basket options in a local-stochastic volatility model with jumps. In particular, the model admits local volatility functions and jump components in not only the underlying asset price processes, but also the volatility processes. To the best of our knowledge, the proposed formula is the first one which achieves an analytical approximation for the basket option prices under this type of the models. Moreover, some numerical experiments confirm the validity of the method.

Robust Pricing of European Options with Wavelets and the Characteristic Function

We present a novel method for pricing European options based on the wavelet approximation (WA) method and the characteristic function. We focus on the discounted expected payoff pricing formula, and compute it by means of wavelets. We approximate the density function associated to the underlying asset price process by a finite combination of $j$th order B-splines, and recover the coefficients of the approximation from the characteristic function. Two variants for wavelet approximation will be presented, where the second variant adaptively determines the range of integration. The compact support of a B-splines basis enables us to price options in a robust way, even in cases where Fourier-based pricing methods may show weaknesses. The method appears to be particularly robust for pricing long-maturity options, fat tailed distributions, as well as staircase-like density functions encountered in portfolio loss computations.

Robust Pricing of European Options with Wavelets and the Characteristic Function

We present a novel method for pricing European options based on the wavelet approximation method and the characteristic function. We focus on the discounted expected payoff pricing formula and compute it by means of wavelets. We approximate the density function associated to the underlying asset price process by a finite combination of $j$th order B-splines, and recover the coefficients of the approximation from the characteristic function. Two variants for wavelet approximation will be presented, where the second variant adaptively determines the range of integration. The compact support of a B-splines basis enables us to price options in a robust way, even in cases where Fourier-based pricing methods may show weaknesses. The method appears to be particularly robust for pricing long-maturity options, fat-tailed distributions, as well as staircase-like density functions encountered in portfolio loss computations.

Is the Driving Force of a Continuous Process a Brownian Motion or Fractional Brownian Motion?

Ito’s semimartingale driven by a Brownian motion is typically used in modeling the asset prices, interest rates and exchange rates, and so on. However, the assumption of Brownian motion as a driving force of the underlying asset price processes is rarely contested in practice. This naturally raises the question of whether this assumption is really appropriate. In the paper we propose a statistical test to answer the above question using high frequency data. The test can be used to validate the assumption of semimartingale framework and test for the existence of the long run dependence captured by the fractional Brownian motion in a parsimonious way. Asymptotic properties of the test statistics are investigated. Simulations justify the performance of the test. Real data sets are also analyzed.

Pricing and Hedging Contingent Claims Using Variance and Higher-Order Moment Futures

This paper suggests perfect hedging strategies of contingent claims under stochastic volatility and/or random jumps of the underlying asset price. This is done by enlarging the market with appropriate swap contracts whose payoffs depend on higher-order sample moments of the underlying asset price process. It also derives a model-free relation between these higher-order moment swaps and the value of a composite portfolio of European options, which can be employed to perfectly hedge variance swaps. Based on the theoretical results of the paper, and on options and variance swaps rates written on the S&P 500 index, the paper provides clear cut evidence that, first, random jumps are priced in the market and, second, hedging strategies for European options employing variance and higher-order moment swaps considerably improves upon the performance of traditional delta hedging strategies. These results indicate the kind of new financial instruments that can be introduced to perfectly hedge the market.

LOCALLY RISK-NEUTRAL VALUATION OF OPTIONS IN GARCH MODELS BASED ON VARIANCE-GAMMA PROCESS

This study develops a GARCH-type model, i.e., the variance-gamma GARCH (VG GARCH) model, based on the two major strands of option pricing literature. The first strand of the literature uses the variance-gamma process, a time-changed Brownian motion, to model the underlying asset price process such that the possible skewness and excess kurtosis on the distributions of asset returns are considered. The second strand of the literature considers the propagation of the previously arrived news by including the feedback and leverage effects on price movement volatility in a GARCH framework. The proposed VG GARCH model is shown to obey a locally risk-neutral valuation relationship (LRNVR) under the sufficient conditions postulated by Duan (1995). This new model provides a unified framework for estimating the historical and risk-neutral distributions, and thus facilitates option pricing calibration using historical underlying asset prices. An empirical study is performed comparing the proposed VG GARCH model with four competing pricing models: benchmark Black–Scholes, ad hoc Black–Scholes, normal NGARCH, and stochastic volatility VG. The performance of the VG GARCH model versus these four competing models is then demonstrated.