Forward Start Options Research Articles

Forward Starting Option Pricing under Double Fractional Stochastic Volatilities and Jumps

This paper aims to provide an effective method for pricing forward starting options under the double fractional stochastic volatilities mixed-exponential jump-diffusion model. The value of a forward starting option is expressed in terms of the expectation of the forward characteristic function of log return. To obtain the forward characteristic function, we approximate the pricing model with a semimartingale by introducing two small perturbed parameters. Then, we rewrite the forward characteristic function as a conditional expectation of the proportion characteristic function which is expressed in terms of the solution to a classic PDE. With the affine structure of the approximate model, we obtain the solution to the PDE. Based on the derived forward characteristic function and the Fourier transform technique, we develop a pricing algorithm for forward starting options. For comparison, we also develop a simulation scheme for evaluating forward starting options. The numerical results demonstrate that the proposed pricing algorithm is effective. Exhaustive comparative experiments on eight models show that the effects of fractional Brownian motion, mixed-exponential jump, and the second volatility component on forward starting option prices are significant, and especially, the second fractional volatility is necessary to price accurately forward starting options under the framework of fractional Brownian motion.

Forward starting options pricing under a regime-switching jump-diffusion model with Wishart stochastic volatility and stochastic interest rate

Vanilla options become effective immediately after they are entered, while some exotic options will only come to effective some time after they are bought or sold. Forward starting options are one kind of such exotic options started actually at some pre-specified future date. This paper presents an extension of regime-switching jump diffusion model, in which the parameters are driven by a continuous time and stationary Markov chain on a finite state space, by introducing the Wishart process into the instantaneous variance-covariance matrix of the risky asset price and stochastic interest rate. We derive the discounted conditional joint characteristic function and the forward characteristic function of the log-asset price and its the instantaneous variance-covariance process, and thereby the price of forward starting options are well evaluated by the probabilistic approach combined with the Fourier-cosine (COS) method. We also provide efficient Monte Carlo simulation of this proposed model, and simulated solutions to forward starting options pricing within a two-state regime switching framework. Numerical results show that the COS method is accurate and efficient for pricing forward starting options. Finally, we analyse impacts of some main parameters (especially, parameters in the Wishart stochastic volatility) in this proposed model on option prices and Δ values. Also, we consider the forward implied volatility. Furthermore, the forward starting options under the regime-switching jump diffusion model with Wishart stochastic volatility and stochastic interest rate which we derived are more generalized than those recently appeared in the derivatives pricing literature, and thus have wider application.

A Shannon Wavelet Method for Pricing Forward Starting Options under the Double Exponential Jump Framework with Two-Factor Stochastic Volatilities

This paper provides an efficient pricing method for forward starting options based on Shannon wavelet expansions. Specifically, we derive the pricing results under a more realistic stock model that incorporates the double exponential jump, stochastic jump intensity, and two-factor stochastic volatilities to capture various features observed in financial markets. We obtain the characteristic function related to the payoff function; then, the options can be well evaluated by the Shannon wavelet method. Numerical experiments show that this method is fast and accurate compared to the Monte Carlo simulation. Finally, we study the influences of changing some important parameters to further illustrate the robustness and stability of the model.

Forward start options under Heston affine jump-diffusions and stochastic interest rate

This paper presents a generalization of forward start options under jump diffusion framework of Duffie et al. [Duffie, D, J Pan and K Singleton (2000). Transform analysis and asset pricing for affine jump-diffusions, Econometrica 68, 1343–1376.]. We assume, in addition, the short-term rate is governed by the CIR dynamics introduced in Cox et al. [Cox, JC, JE Ingersoll and SA Ross (1985). A theory of term structure of interest rates, Econometrica 53, 385–408.]. The instantaneous volatilities are correlated with the dynamics of the stock price process, whereas the short-term rate is assumed to be independent of the dynamics of the price process and its volatility. The main result furnishes a semi-analytical formula for the price of the Forward Start European call option. It is derived using probabilistic approach combined with the Fourier inversion technique, as developed in Ahlip and Rutkowski [Ahlip, R and M Rutkowski (2014). Forward start foreign exchange options under Heston’s volatility and CIR interest rates, Inspired By Finance Springer, pp. 1–27], Carr and Madan [Carr, P and D Madan (1999). Option valuation using the fast Fourier transform, Journal of Computational Finance 2, 61–73, Carr, P and D Madan (2009). Saddle point methods for option pricing, Journal of Computational Finance 13, 49–61] as well as Levendorskiĩ [Levendorskiĩ, S (2012). Efficient pricing and reliable calibration in the Heston model, International Journal of Applied Finance 15, 1250050].

A closed-form pricing formula for forward start options under a regime-switching stochastic volatility model

Abstract In this paper, we consider the pricing problem of forward start options in the presence of stochastic volatility and regime-switching. By making use of the measure transform technique, with the underlying price as a new numeraire, a closed-form pricing formula is derived in which the only unknown term is the so-called forward characteristic function of the underlying price. The analytical expression of the forward characteristic function under the new measure is subsequently obtained in two steps; the first step treats the regime-switching Heston model as a time-dependent Heston model with the information of the Markov chain beingassumed to be given in advance, while the second step takes the expectation with respect to the Markov chain. Finally, the influence of introducing regime-switching into the Heston model is investigated to show the difference in terms of forward start option pricing between the two models.

Managing Forward Volatility and Skew Risk

The forward start dual volatility swap is introduced. It can be regarded as the analog for volatility of what the entropy contract is for variance. Under the risk neutral measure it is shown that the difference between the forward start volatility swap and its dual is approximately the difference between two specific forward start implied volatilities. A dynamic replicating strategy is given which enables practitioners to approximately but robustly hedge forward start (dual) volatility swaps and the forward start skew using no more than two forward start options.

Pricing of Arithmetic Asian Options under Stochastic Volatility Dynamics: Overcoming the Risks of High-Frequency Trading

This research extended the model developed by Hull and White by integrating Taylor-series expansion into the model for deriving approximate analytical solutions for stochastic volatility forward-starting Asian options. Numerical experiments were performed to compare the proposed model with the Monte Carlo model over numerous simulations and demonstrated that the developed model has a pricing accuracy greater than 99%. Furthermore, the computation time was approximately 10−5 s for each simulation. The model’s outstanding computational performance demonstrates its capability to address the challenges of high-frequency trading.

Numerical techniques for the Heston collocated volatility model

In the collocating volatility (CLV) model, the stochastic collocation technique is used as a convenient representation of the terminal distribution of the market option prices. A specific dynamic is added in the form of a stochastic driver process, which allows more control over the prices of forward starting options. This is reminiscent of the Markov functional models. (Grzelak uses a single-factor Ornstein–Uhlenbeck process as the driver for the CLV model, and Fries uses a single-factor Wiener process with time-dependent volatility in his equity Markov functional model. Van der Stoep et al consider a Heston stochastic volatility driver process and show that it offers more flexibility to capture the forward smile in the context of foreign exchange options.) In this paper, we discuss all aspects of derivative pricing under the Heston– CLV model: calibration with an efficient Fourier method; a Monte Carlo simulation with second-order convergence; and accurate partial differential equation pricing through implicit and explicit finite-difference methods.

On the Forward Smile

Using short-time expansion techniques, we obtain analytic implied volatilities for European and forward starting options for a family of stochastic volatility models with arbitrary local volatility component and time dependent (piecewise constant) parameters. The formulas can be used to efficiently calibrate the model to European options at two expiries and to calculate the spanning forward starting option price.

A spectral approach to pricing of forward starting options

We propose a new class of models for pricing forward starting options. We assume that the asset price is a nonlinear function of a CIR process, time changed by a composition of a Levy subordinator and an absolutely ´ continuous process. The new models introduce the nonlinearity in both drift and diffusion components of the underlying process and can capture jumps and stochastic volatility in a flexible way. By employing the spectral expansion technique, we are able to derive the analytical formulas for the forward starting option prices. We also implement a specific model numerically and test its sensitivity to some of the key parameters of the model.

The implied volatility of Forward-Start options: ATM short-time level, skew and curvature

ABSTRACTUsing Malliavin Calculus techniques, we derive closed-form expressions for the at-the-money behaviour of the forward implied volatility, its skew and its curvature, in general Markovian stochastic volatility models with continuous paths.

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 joint density of any finite sequence of log returns admits a Gram-Charlier A expansion with closed-form coefficients. We derive closed-form series representations for option prices whose discounted payoffs are functions of the asset price trajectory at finitely many time points. This includes European call, put, and digital options, forward start options, and can be applied to discretely monitored Asian options. In a numerical analysis we show that option prices can be accurately and efficiently approximated by truncating their series representations.

Pathwise superhedging for time-dependent barrier options on càdlàg paths—Finite or infinite tradeable European, One-Touch, lookback or forward starting options

We establish pathwise duality using simple predictable trading strategies for the robust hedging problem associated with a barrier option whose payoff depends on the terminal level and the infimum of a càdlàg strictly positive stock price process, given tradeable European options at all strikes at a single maturity. The result allows for a significant dimension reduction in the computation of the superhedging cost, via an alternate lower-dimensional formulation of the primal problem as a convex optimization problem, which is qualitatively similar to the duality which was formally sketched using linear programming arguments in Duembgen and Rogers [10] for the case where we only consider continuous sample paths. The proof exploits a simplification of a classical result by Rogers (1993) which characterizes the attainable joint laws for the supremum and the drawdown of a uniformly integrable martingale (not necessarily continuous), combined with classical convex duality results from Rockefellar (1974) using paired spaces with compatible locally convex topologies and the Hahn–Banach theorem. We later adapt this result to include additional tradeable One-Touch options using the Kertz and Rösler (1990) condition. We also compute the superhedging cost when in the more realistic situation where there is only finite tradeable European options; for this case we obtain the full duality in the sense of quantile hedging as in Soner (2015), where the superhedge works with probability 1−ε where ε can be arbitrarily small), and we obtain an upper bound for the true pathwise superhedging cost. In Section 5, we extend our analysis to include time-dependent barrier options using martingale coupling arguments, where we now have tradeable European options at both maturities at all strikes and tradeable forward starting options at all strikes. This set up is designed to approximate the more realistic situation where we have a finite number of tradeable Europeans at both maturities plus a finite number of tradeable forward starting options.11The author would like to thank Professors Charles Akemann and Teemu Pennanen for useful discussions.

Superreplication of Financial Derivatives via Convex Programming

We give a method based on convex programming to calculate the optimal superreplicating and subreplicating 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 superreplicating (or subreplicating) 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, cliquet calls, barrier options, and lookback and Asian options. Numerical examples show that, in some cases, the best superreplicating and/or subreplicating 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. The supplementary material is available at https://doi.org/10.1287/mnsc.2017.2786 . This paper was accepted by Gustavo Manso, finance.

The Implied Volatility of Forward-Start Options: ATM Short-Time Level, Skew and Curvature

Using Malliavin Calculus techniques, we derive closed-form expressions for the at-the-money behaviour of the forward implied volatility, its skew and its curvature, in general Markovian stochastic volatility models with continuous paths.

RANDOM TIME FORWARD-STARTING OPTIONS

We introduce a natural generalization of the forward-starting options. 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 and in absence of simultaneous jumps between the random time and the assets' prices. Practical implementations of the pricing methodologies are also provided. Finally, a credit value adjustment (CVA) formula for these over the counter (OTC) options is computed for the unilateral counterparty credit risk.

Pricing Options with Complex Fourier Series

This paper develops a complex Fourier series (CFS) expansion pricing formula for evaluating European-type (call/put) options -- asset-or-nothing options, cash-or-nothing options, forward-start options, ratchet options and vanilla options -- when these risky assets are driven by Levy processes, time-changed Levy processes or stochastic volatility with or without jumps. This novel pricing formula is an extension of the original CFS expansion method proposed in Chan (2016). The major contribution of this paper is to derive a CFS pricing formula for both forward-start and ratchet options. In this paper, we also conduct an error-bound analysis of the CFS expansion pricing formula. In most cases, we can obtain an exponential convergence rate when we have an appropriate definite integration range for pricing European-type options. Moreover, based on the pricing formula, we can also derive different option Greeks. Finally, in follow-up papers, we will present the application of this pricing formula to options with early-exercise features (such as American options and barrier options) and path-dependent features (such as Asian options).

Fourier-cosine method for pricing forward starting options with stochastic volatility and jumps

ABSTRACTThis article provides an efficient method for pricing forward starting options under stochastic volatility model with double exponential jumps. The forward characteristic function of the log asset price is derived and thereby forward starting options are well evaluated by Fourier-cosine technique. Based on adaptive simulated annealing algorithm, the model is calibrated to obtain the estimated parameters. Numerical results show that the pricing method is accurate and fast. Double exponential jumps have pronounced impacts on long-term forward starting options prices. Stochastic volatility model with double exponential jumps fits forward implied volatility smile pretty well in contrast to stochastic volatility model.

GENERALIZED BARNDORFF-NIELSEN AND SHEPHARD MODEL AND DISCRETELY MONITORED OPTION PRICING

This paper proposes a generalization of the Barndorff-Nielsen and Shephard model, in which the log return on an asset is governed by a Lévy process with stochastic volatility modeled by a non-Gaussian Ornstein–Uhlenbeck process. Under the generalized model, we derive a closed-form expression of 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 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.