•We price swaps and options under stochastic volatility models with jumps.•We employ a continuous-time Markov chain approximation for the underlying.•Our results are relevant in: finance, insurance, ...and operational research.
After the recent financial crisis, the market for volatility derivatives has expanded rapidly to meet the demand from investors, risk managers and speculators seeking diversification of the volatility risk. In this paper, we develop a novel and efficient transform-based method to price swaps and options related to discretely-sampled realized variance under a general class of stochastic volatility models with jumps. We utilize frame duality and density projection method combined with a novel continuous-time Markov chain (CTMC) weak approximation scheme of the underlying variance process. Contracts considered include discrete variance swaps, discrete variance options, and discrete volatility options. Models considered include several popular stochastic volatility models with a general jump size distribution: Heston, Scott, Hull–White, Stein–Stein, α-Hypergeometric, 3/2 and 4/2 models. Our framework encompasses and extends the current literature on discretely sampled volatility derivatives, and provides highly efficient and accurate valuation methods. Numerical experiments confirm our findings.
•General framework for approximating time-changed Markov processes.•Flexible approach, including three Markov chain approximation strategies.•Accommodates time-changed (subordinated) Levy and ...diffusion processes as special cases.•Pricing for American/Bermudan options, European options, and variance swaps.
In this paper, we propose a general approximation framework for the valuation of (path-dependent) options under time-changed Markov processes. The underlying background process is assumed to be a general Markov process, and we consider the case when the stochastic time change is constructed from either discrete or continuous additive functionals of another independent Markov process. We first approximate the underlying Markov process by a continuous time Markov chain (CTMC), and derive the functional equation characterizing the double transforms of the transition matrix of the resulting time-changed CTMC. Then we develop a two-layer approximation scheme by further approximating the driving process in constructing the time change using an independent CTMC. We obtain a single Laplace transform expression. Our framework incorporates existing time-changed Markov models in the literature as special cases, such as the time-changed diffusion process and the time-changed Lévy process. Numerical experiments illustrate the accuracy of our method.
•CTMC approximation for hybrid short-rate and equity models.•Closed-form equity swap pricing.•Semi-closed-form pricing of equity cap/floor contracts.•Accurate and efficient pricing of bonds and bond ...options.•Numerical analyses and reference prices.
Hybrid equity-rate derivatives are commonly traded between financial institutions, but are challenging to price with traditional methods. Especially challenging are those contracts which involve an explicit interest rate (fixing) dependence in the cashflows, which stretches typical measure-change approaches beyond their practical limit. We introduce a framework for pricing equity swaps, equity cap/floors, and other hybrid derivatives under general stochastic short-rate models with a correlated equity. By utilizing the machinery of Continuous Time Markov Chain (CTMC) approximation, and a decoupled representation of the equity-rate model, we derive semi-closed-form approximations for the hybrid contract prices based on a regime-switching model and prove theoretical convergence. The numerical implementation of the method is fast and very accurate, achieving superquadratic convergence in numerical experiments. The framework also provides a practical alternative to traditional approaches such as trees for pricing bonds and bond options under short-rates models which lack closed-form solutions.
Utilizing frame duality and a FFT-based implementation of density projection we develop a novel and efficient transform method to price Asian options for very general asset dynamics, including regime ...switching Lévy processes and other jump diffusions as well as stochastic volatility models with jumps. The method combines continuous-time Markov chain approximation, with Fourier pricing techniques. In particular, our method encompasses Heston, Hull-White, Stein-Stein, 3/2 model as well as recently proposed Jacobi,
α
-Hypergeometric, and 4/2 models, for virtually any type of jump amplitude distribution in the return process. This framework thus provides a ‘
unified
’ approach to pricing Asian options in stochastic jump diffusion models and is readily extended to alternative exotic contracts. We also derive a characteristic function recursion by generalizing the Carverhill-Clewlow factorization which enables the application of transform methods in general. Numerical results are provided to illustrate the effectiveness of the method. Various extensions of this method have since been developed, including the pricing of barrier, American, and realized variance derivatives.
A history of TOPMODEL Beven, Keith J; Kirkby, Mike J; Freer, Jim E ...
Hydrology and earth system sciences,
02/2021, Letnik:
25, Številka:
2
Journal Article
Recenzirano
Odprti dostop
The theory that forms the basis of TOPMODEL (a topography-based hydrological model) was first outlined by Mike Kirkby some 45 years ago. This paper recalls some of the early developments, the ...rejection of the first journal paper, the early days of digital terrain analysis, model calibration and validation, the various criticisms of the simplifying assumptions, and the relaxation of those assumptions in the dynamic forms of TOPMODEL. A final section addresses the question of what might be done now in seeking a simple, parametrically parsimonious model of hillslope and small catchment processes if we were starting again.
•A consistent framework for option pricing and risk-measurment.•Applicable to case of limited option quotes.•General approach, requires only an observable underlying time series.•Fast and accurate ...computational procedures for practical application.
In this paper, we propose a general data-driven framework that unifies the valuation and risk measurement of financial derivatives, which is especially useful in markets with thinly-traded derivatives. We first extract the empirical characteristic function from market-observable time series for the underlying asset prices, and then utilize Fourier techniques to obtain the physical nonparametric density and cumulative distribution function for the log-returns process, based on which we compute risk measures. Then we risk-neutralize the nonparametric density and distribution functions to model-independently valuate a variety of financial derivatives, including path-independent European options and path-dependent exotic contracts. By estimating the state-price density explicitly, and utilizing a convenient basis representation, we are able to greatly simplify the pricing of exotic options all within a consistent model-free framework. Numerical examples, and an empirical example using real market data (Brent crude oil prices) illustrate the accuracy and versatility of the proposed method in handling pricing and risk management of multiple financial contracts based solely on observable time series data.
In this paper, we develop a novel and efficient transform-based method to price equity-linked annuities (ELAs), including equity-indexed annuities (EIAs) and cliquet-style payoff structures popular ...in the insurance market under a general class of stochastic volatility models with jumps. We utilize frame duality and density projection combined with a continuous-time Markov chain (CTMC) weak approximation scheme and spectral filtering. Contracts considered include EIAs with return guarantees of a cliquet style. Models considered include exponential Lévy processes, regime-switching Lévy processes, and stochastic volatility models with a general jump size distribution including Heston, Scott’s, Hull–White, Schöbel–Zhu, and the 3/2 models. We also consider some recently proposed stochastic volatility models in the literature such as the α-Hypergeometric model, and the 4/2 model. Our framework encompasses and extends the current literature on EIAs with highly efficient and accurate valuation methods. Numerical experiments confirm our findings.
•A novel state-space discretization scheme for simulating SDE by CTMC approximation.•Applicable to a class of generalized SABR stochastic local volatility (SLV) models.•A single-step Fourier sampler ...for European options and multi-step scheme for exotics.•Exact sampling from the approximating integrated variance process.•Efficient and accurate compared with biased and unbiased competitors.
We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms both biased and unbiased simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory.
In this paper, we analyze a form of equity-linked Guaranteed Minimum Death Benefit (GMDB), whose payoff depends on a dollar cost averaging (DCA) style periodic investment in the risky index, with ...rider premiums paid at regular intervals. This rider is a very natural insurance vehicle for equity-linked variable annuities, and the DCA feature has a tendency to reduce the uncertainty associated with the final payoff, beyond the minimum benefit guaranteed to the beneficiary upon death of the insured. This makes the insurer risk easier to manage as the contract ages. From the policyholder's perspective, the protection is cheaper than a standard GMDB whose payoff depends solely on the risky index value upon death, but still offers the upside potential from an investment made at regular intervals. We derive closed-form valuation formulas under the fairly broad class of exponential Lévy models for the risky index, which includes Black-Scholes as a special case. Closed-form valuation is provided by a fundamental link between this contract and a series of Asian options, for which valuation is well established. We provide several valuation strategies, and demonstrate the soundness of the framework.
Many financial assets, such as currencies, commodities, and equity stocks, exhibit both jumps and stochastic volatility, which are especially prominent in the market after the financial crisis. Some ...strategic decision making problems also involve American-style options. In this paper, we develop a novel, fast and accurate method for pricing American and barrier options in regime switching jump diffusion models. By blending regime switching models and Markov chain approximation techniques in the Fourier domain, we provide a unified approach to price Bermudan, American options and barrier options under general stochastic volatility models with jumps. The models considered include Heston, Hull–White, Stein–Stein, Scott, the 3/2 model, and the recently proposed 4/2 model and the α-Hypergeometric model with general jump amplitude distributions in the return process. Applications include the valuation of discretely monitored contracts as well as continuously monitored contracts common in the foreign exchange markets. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed method.
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