Constant Elasticity Of Variance Diffusion Research Articles

Pricing levered warrants under the CEV diffusion model

Much of the work on the valuation of levered (and unlevered) warrants assumes that the volatility of the underlying state variable is constant. This paper extends the literature on warrant pricing to a more general assumption for the state variable process, the so-called constant elasticity of variance (CEV) process. The CEV model is well-known for its ability to capture some empirical observations found in the financial economics literature, namely the asymmetry between equity returns and volatility and the implied volatility skew. Using the CEV process, we are able to reduce pricing bias as the volatility becomes a function of the underlying state variable. We price European-style call warrants without restrictions on the debt maturity. When warrants have the same maturity as debt, it is possible to obtain closed-form solutions for warrants prices. When the maturity of warrants is different from the maturity of debt, prices can be computed numerically through very efficient and simple to implement valuation methodologies.

Closed-form formula for conditional moments of generalized nonlinear drift CEV process

This paper studied a generalized case of the constant elasticity of variance diffusion (CEV) process whereas the drift term is substantially nonlinear in the short rate. Well-known instances deduced by this process are the extended Cox–Ingersoll–Ross (ECIR) process and the extended inverse Feller (EIF) process or 3/2-stochastic volatility model (SVM). We found particular sufficient conditions of existence and uniqueness of a positive pathwise strong solution for time-dependent parameter functions, and obtained closed-form formulas for conditional moments based on Feynman–Kac theorem. The accuracy and validity of the formulas were further investigated based on Monte Carlo simulations.

Hedging with Liquidity Risk under CEV Diffusion

We study a discrete time hedging and pricing problem in a market with the liquidity risk. We consider a discrete version of the constant elasticity of variance (CEV) model by applying Leland’s discrete time replication scheme. The pricing equation becomes a nonlinear partial differential equation, and we solve it by a multi scale perturbation method. A numerical example is provided.

The Binomial CEV Model and the Greeks

AbstractThis article compares alternative binomial approximation schemes for computing the option hedge ratios studied by Chung and Shackleton (2002), Chung, Hung, Lee, and Shih (2011), and Pelsser and Vorst (1994) under the lognormal assumption, but now considering the constant elasticity of variance (CEV) process proposed by Cox (1975) and using the continuous‐time analytical Greeks recently offered by Larguinho, Dias, and Braumann (2013) as the benchmarks. Among all the binomial models considered in this study, we conclude that an extended tree binomial CEV model with the smooth and monotonic convergence property is the most efficient method for computing Greeks under the CEV diffusion process because one can apply the two‐point extrapolation formula suggested by Chung et al. (2011). © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:90–104, 2017

General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options

We propose an accurate method for pricing arithmetic Asian options on the discrete or continuous average in a general model setting by means of a lower bound approximation. In particular, we derive analytical expressions for the lower bound in the Fourier domain. This is then recovered by a single univariate inversion and sharpened using an optimization technique. In addition, we derive an upper bound to the error from the lower bound price approximation. Our proposed method can be applied to computing the prices and price sensitivities of Asian options with fixed or floating strike price, discrete or continuous averaging, under a wide range of stochastic dynamic models, including exponential Lévy models, stochastic volatility models, and the constant elasticity of variance diffusion. Our extensive numerical experiments highlight the notable performance and robustness of our optimized lower bound for different test cases.

A recursive pricing formula for a path-dependent option under the constant elasticity of variance diffusion

In this paper, we consider a path-dependent option in finance under the constant elasticity of variance diffusion. We use a perturbation argument and the probabilistic representation (the Feynman–Kac theorem) of a partial differential equation to obtain a complete asymptotic expansion of the option price in a recursive manner based on the Black–Scholes formula and prove rigorously the existence of the expansion with a convergence error.

PRICING STEP OPTIONS UNDER THE CEV AND OTHER SOLVABLE DIFFUSION MODELS

We consider a special family of occupation-time derivatives, namely proportional step options introduced by [18]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.

Efficient and high accuracy pricing of barrier options under the CEV diffusion

Binomial and trinomial lattices are popular techniques for pricing financial options. These methods work well for European and American options, but for barrier options, the need to place a tree node very close to a barrier brings difficulties in their implementation and a large number of time steps are usually required when the barrier is close to the current asset price. A finite difference implementation is simpler and we propose a fourth-order numerical scheme for continuously and discretely monitored barriers. We demonstrate the superior performance of our technique over existing procedures for the Black–Scholes model and we then price barriers under constant elasticity of variance (CEV) diffusion. Continuously monitored barriers under CEV admit an analytical solution but evaluation via this formula is not straightforward. Furthermore, discretely monitored barriers have to be priced numerically. Our main contribution is therefore a highly accurate and efficient numerical scheme for barrier options under CEV and we provide several numerical examples to illustrate the merit of the new technique.

Structural Models of the Firm Under State-Dependent Volatility and Jump Process Asset Dynamics

We generalize the asset dynamics assumptions of Leland (1994b) and Leland and Toft (1996) to a much richer class of models. By assuming a stationary corporate debt structure with constant principal, coupon payment and average maturity through continuous retirement and refinancing as long as the firm remains solvent, we obtain analytical solutions with the state dependent diffusion volatility following the constant elasticity of variance (CEV) process for the variables of interest, including corporate debt value, total levered firm value and equity value. We also develop an efficient numerical algorithm with mixed jump diffusion asset dynamics, by adopting a restricted structure of default times, and derive numerical solutions for several variables of interest. We study the impact of state dependent volatility and jumps on the optimal capital structure, the debt capacity, the term structure of credit spread, the duration and convexity of risky debt, the equity volatility, the asset substitution impacts and the cumulative default probabilities (CDP). We also find that the term structure of CDP generated by the CEV diffusion process with a negative relationship between asset value and asset volatility could explain the CDP puzzle of structural models for short term debt by examining Moody’s historical CDP data.

Pricing real options under the constant elasticity of variance diffusion

AbstractMuch of the work on real options assumes that the underlying state variable follows a geometric Brownian motion with constant volatility. This paper uses a more general assumption for the state variable process that better captures the empirical regularities found in commodity markets. We use the constant elasticity of variance diffusion, where volatility is a function of underlying asset prices, and we provide analytic solutions for perpetual American options. We show that a firm that uses the standard lognormal assumption is exposed to significant errors of analysis, which may lead to nonoptimal investment and disinvestment decisions. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:230–250, 2011

Static hedging and pricing American options

This paper utilizes the static hedge portfolio (SHP) approach of Derman et al. [Derman, E., Ergener, D., Kani, I., 1995. Static options replication. Journal of Derivatives 2, 78–95] and Carr et al. [Carr, P., Ellis, K., Gupta, V., 1998. Static hedging of exotic options. Journal of Finance 53, 1165–1190] to price and hedge American options under the Black-Scholes (1973) model and the constant elasticity of variance (CEV) model of Cox [Cox, J., 1975. Notes on option pricing I: Constant elasticity of variance diffusion. Working Paper, Stanford University]. The static hedge portfolio of an American option is formulated by applying the value-matching and smooth-pasting conditions on the early exercise boundary. The results indicate that the numerical efficiency of our static hedge portfolio approach is comparable to some recent advanced numerical methods such as Broadie and Detemple [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies 9, 1211–1250] binomial Black-Scholes method with Richardson extrapolation (BBSR). The accuracy of the SHP method for the calculation of deltas and gammas is especially notable. Moreover, when the stock price changes, the recalculation of the prices and hedge ratios of the American options under the SHP method is quick because there is no need to solve the static hedge portfolio again. Finally, our static hedging approach also provides an intuitive derivation of the early exercise boundary near expiration.

Distribution of occupation times for constant elasticity of variance diffusion and the pricing of α-quantile options

The main results of this paper are the derivation of the distribution functions of occupation times under the constant elasticity of variance process. The distribution functions can then be used to price α-quantile options. We also derive the fixed-floating symmetry relation for α-quantile options when the underlying asset price process follows a geometric Brownian motion.

PRICING PATH-DEPENDENT OPTIONS ON STATE DEPENDENT VOLATILITY MODELS WITH A BESSEL BRIDGE

This paper develops bridge sampling path integral algorithms for pricing path-dependent options under a new class of nonlinear state dependent volatility models. Path-dependent option pricing is considered within a new (dual) Bessel family of semimartingale diffusion models, as well as the constant elasticity of variance (CEV) diffusion model, arising as a particular case of these models. The transition p.d.f.s or pricing kernels are mapped onto an underlying simpler squared Bessel process and are expressed analytically in terms of modified Bessel functions. We establish precise links between pricing kernels of such models and the randomized gamma distributions, and thereby demonstrate how a squared Bessel bridge process can be used for exact sampling of the Bessel family of paths. A Bessel bridge algorithm is presented which is based on explicit conditional distributions for the Bessel family of volatility models and is similar in spirit to the Brownian bridge algorithm. A special rearrangement and splitting of the path integral variables allows us to combine the Bessel bridge sampling algorithm with either adaptive Monte Carlo algorithms, or quasi-Monte Carlo techniques for significant numerical efficiency improvement. The algorithms are illustrated by pricing Asian-style and lookback options under the Bessel family of volatility models as well as the CEV diffusion model.

Predicting the value of foreign currency call options with the Constant Elasticity of Variance diffusion process

This paper examines the ability of the Constant Elasticity of Variance (CEV) model of Cox (1975) to explain observed behavior of foreign currency call option prices. A method using a derivative-free nonlinear regression is developed for jointly estimating the parameters of the class of CEV diffusion processes, and is applied to the valuation of foreign currency call options. The results indicate that the elasticity of variance is nonstationary and significantly less than unity. The CEV model is shown to reduce prediction errors with respect to time to maturity, degree of in-or out-of-the-money, and future volatility of exchange rates.

The Constant Elasticity of Variance Model and Its Implications For Option Pricing

where the instantaneous variance of the percentage price change is equal to U2/S2and hence is a direct inverse function of the stock price. In the traditionally used lognormal model, which corresponds to the limiting case a = 2, the variance rate is not a function of the stock price itself. Both casual empiricism and economic rationale tend to support the inverse relationship. If this relationship is borne out by the empirical data, an option pricing formula based on the constant elasticity of variance diffusion could fit the actual market prices better than the Black-Scholes model. In this article we empirically investigate the relationship between the stock price level and its variance of return and perform a comparative statics analysis of the Black-Scholes prices and those based on two special cases of the constant elasticity of variance class (a = 1 and a = 0).

The Constant Elasticity of Variance Model and Its Implications For Option Pricing

where the instantaneous variance of the percentage price change is equal to U2/S2and hence is a direct inverse function of the stock price. In the traditionally used lognormal model, which corresponds to the limiting case a = 2, the variance rate is not a function of the stock price itself. Both casual empiricism and economic rationale tend to support the inverse relationship. If this relationship is borne out by the empirical data, an option pricing formula based on the constant elasticity of variance diffusion could fit the actual market prices better than the Black-Scholes model. In this article we empirically investigate the relationship between the stock price level and its variance of return and perform a comparative statics analysis of the Black-Scholes prices and those based on two special cases of the constant elasticity of variance class (a = 1 and a = 0).