Constant Elasticity Of Variance Process Research Articles

Finite maturity caps and floors on continuous flows under the constant elasticity of variance process

This paper offers novel analytical solutions for evaluating perpetual caps and floors on continuous flows under the constant elasticity of variance (CEV) model. We demonstrate that the inclusion of a perpetual bubble value is required to avoid arbitrage opportunities in the case of the CEV process with upward-sloping volatility skews. We then extend the previous literature on caps and floors arrangements by providing new analytical formulae for valuing finite maturity caps and floors that are contingent on continuous flows. We discuss the impact of the finite-lived solutions on the optimal behavior of a firm, relative to the perpetual case. We also show the implications of the correct specification of the underlying state variable process for the valuation of caps and floors by comparing the CEV results with the ones obtained when assuming a lognormal diffusion. Practical applications of these contractual agreements arising within the context of executive management decisions are also discussed.

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.

Simple Closed-Form Formulas for Conditional Moments of Inhomogeneous Nonlinear Drift Constant Elasticity of Variance Process

The stochastic differential equation (SDE) has been used to model various phenomena and investigate their properties. Conditional moments of stochastic processes can be used to price financial derivatives whose payoffs depend on conditional moments of underlying assets. In general, the transition probability density function (PDF) of a stochastic process is often unavailable in closed form. Thus, the conditional moments, which can be directly computed by applying the transition PDFs, may be unavailable in closed form. In this work, we studied an inhomogeneous nonlinear drift constant elasticity of variance (IND-CEV) process, which is a class of diffusions that have time-dependent parameter functions; therefore, their sample paths are asymmetric. The closed-form formulas for conditional moments of the IND-CEV process were derived without having a condition on eigenfunctions or the transition PDF. The analytical results were examined through Monte Carlo simulations.

Basket Credit Default Swap Pricing with Two Defaultable Counterparties

In this paper, we study the basket CDS pricing with two defaultable counterparties based on the reduced-form model. The default jump intensities of the reference firms and counterparties are all assumed to follow the mean-reverting constant elasticity of variance (CEV) processes. Taking the Vasicek process which is a special case of CEV process as an example, the approximate analytic solutions of the joint survival probability density, the probability densities of the first default and the first two defaults can be solved by using PDE method. In addition, we also extend the Vasciek process to the Vasciek process with cojumps and obtain the approximate closed-form solutions of the relevant default probability densities. Then with the expressions of the probability densities, we can get the formula of the basket CDS price with two defaultable counterparties. In the numerical analysis, we do sensitivity analysis and compare the basket CDS prices under our model with that with only one defaultable counterparty. The numerical results show that our model can be applied into practice.

An Investor’s Investment Plan with Stochastic Interest Rate under the CEV Model and the Ornstein-Uhlenbeck Process

The aim of this paper is to maximize an investor’s terminal wealth which exhibits constant relative risk aversion (CRRA). Considering the fluctuating nature of the stock market price, it is imperative for investors to study and develop an effective investment plan that considers the volatility of the stock market price and the fluctuation in interest rate. To achieve this, the optimal investment plan for an investor with logarithm utility under constant elasticity of variance (CEV) model in the presence of stochastic interest rate is considered. Also, a portfolio with one risk free asset and two risky assets is considered where the risk free interest rate follows the Ornstein-Uhlenbeck (O-U) process and the two risky assets follow the CEV process. Using the Legendre transformation and dual theory with asymptotic expansion technique, closed form solutions of the optimal investment plans are obtained. Furthermore, the impacts of some sensitive parameters on the optimal investment plans are analyzed numerically. We observed that the optimal investment plan for the three assets give a fluctuation effect, showing that the investor’s behaviour in his investment pattern changes at different time intervals due to some information available in the financial market such as the fluctuations in the risk free interest rate occasioned by the O-U process, appreciation rates of the risky assets prices and the volatility of the stock market price due to changes in the elasticity parameters. Also, the optimal investment plans for the risky assets are directly proportional to the elasticity parameters and inversely proportional to the risk free interest rate and does not depend on the risk averse coefficient.

Optimal portfolio selection for a defined-contribution plan under two administrative fees and return of premium clauses

We study a defined contribution pension system with the return of premium clauses, which is embedded with two administrative fees: the charge on balance and the charge on flow. The fund wealth is invested in financial market with a riskless asset and a risky asset modeled by a constant elasticity of variance process. We use the Weibull model to characterize the force of mortality. The dynamic programming principle is applied to derive optimal investment strategies explicitly when the pension member has constant absolute risk aversion and constant relative risk aversion. We illustrate how the certainty equivalent of the expected utility works for comparing the two fees. Finally, we give a numerical analysis to highlight our results.

Valuation of options under a constant elasticity of variance process and stochastic volatility

In this paper, we price European call options under a constant elasticity of variance process for the asset price and stochastic volatility. In particular, we derive an analytic approximation formula in the form of asymptotic expansions, which is valid for European options with short times to expiry. Further, we examine the performance of our formula on a market sample of short-tenor crude oil call options (traded on the International Commodities Exchange) and find that the formula provides an excellent fit to market prices.

On The Closed Form Strategies of an Investor under the CEV and CIR Processes

In this paper, the explicit solutions of the optimal investment plans of an investor with exponential utility function exhibiting constant absolute risk aversion (CARA) under constant elasticity of variance (CEV) and stochastic interest rate is studied. A portfolio comprising of a risk-free asset modelled by the Cox-Ingersoll-Ross (CIR) process and two risky assets modelled by the CEV process is considered, where the instantaneous volatilities of the two risky assets form a 2 x 2 matrix n = {np,q}2x2 such that nnT is positive definite. Using the power transformation and change of variable approach with asymptotic expansion technique, explicit solutions of the optimal investment plans are found. Moreover, numerical simulations are used to study the effects of the interest rate, elasticity parameter, correlation coefficient and the risk averse coefficient on the optimal investment plans.

CEV model equipped with the long-memory

In this paper, we define the mixed fractional Constant Elasticity of Variance (CEV) process exploiting a transfer equation. This transformation enables us to shift non-linearity from the volatility coefficient into the drift one and accordingly, the problem can be more easily studied. We confirm the existence of a unique positive solution to the transfer equation and verify the conditions under which the probability that, on any fixed finite interval, the mixed fractional CEV process does not hit zero tends to 1 (by some simulation, we illustrate more). Next, some parameters of the process are estimated and finally, we examine the ability of this process to model a financial market.

Efficient exponential timestepping algorithm using control variate technique for simulating a functional of exit time of one-dimensional Brownian diffusion with applications in finance

The exponential timestepping Euler algorithm with a boundary test is adapted to simulate an expected of a function of exit time, such as the expected payoff of barrier options under the constant elasticity of variance (CEV) model. However, this method suffers from a high Monte Carlo (MC) statistical error due to its exponentially large exit times with unbounded samples. To reduce this kind of error efficiently and to speed up the MC simulation, we combine such an algorithm with an effective variance reduction technique called the control variate method. We call the resulting algorithm the improved Exp algorithm for abbreviation. In regard to the examples we consider in this paper for the restricted CEV process, we found that the variance of the improved Exp algorithm is about six times smaller than that of the Jansons and Lythe original method for the down-and-out call barrier option. It is also about eight times smaller for the up-and-out put barrier option, indicating that the gain in efficiency is significant without significant increase in simulation time.

Optimal Investment Strategy under the CEV Model with Stochastic Interest Rate

Interest rate is an important macrofactor that affects asset prices in the financial market. As the interest rate in the real market has the property of fluctuation, it might lead to a great bias in asset allocation if we only view the interest rate as a constant in portfolio management. In this paper, we mainly study an optimal investment strategy problem by employing a constant elasticity of variance (CEV) process and stochastic interest rate. The assets of investment for individuals are supposed to be composed of one risk-free asset and one risky asset. The interest rate for risk-free asset is assumed to follow the Cox–Ingersoll–Ross (CIR) process, and the price of risky asset follows the CEV process. The objective is to maximize the expected utility of terminal wealth. By applying the dual method, Legendre transformation, and asymptotic expansion approach, we successfully obtain an asymptotic solution for the optimal investment strategy under constant absolute risk aversion (CARA) utility function. In the end, some numerical examples are provided to support our theoretical results and to illustrate the effect of stochastic interest rates and some other model parameters on the optimal investment strategy.

Pricing European options under a diffusion model with psychological barriers and leverage effect

ABSTRACTThis paper derives closed-form formulae for European options under a diffusion model when there exist psychological barriers and leverage effect in underlying dynamics. The state space of the proposed model is divided into different subregions by psychological barriers, and in each subregion, the model behaves like a constant elasticity of variance process. Within this framework, we derive both Laplace transform- and spectral expansion-based analytical solutions, which allow fast and accurate calculation of option prices. Numerical results are presented to explore the nontrivial properties of our model.

A note on options and bubbles under the CEV model: implications for pricing and hedging

The discounted price process under the constant elasticity of variance (CEV) model is not a martingale (wrt the risk-neutral measure) for options markets with upward sloping implied volatility smiles. The loss of the martingale property implies the existence of (at least) two option prices for the call option: the price for which the put-call parity holds and the (risk-neutral) price representing the lowest cost of replicating the call payoff. This article derives closed-form solutions for the Greeks of the risk-neutral call option pricing solution that are valid for any CEV process exhibiting forward skew volatility smile patterns. Using an extensive numerical analysis, we conclude that the differences between the call prices and Greeks of both solutions are substantial, which might yield significant errors of analysis for pricing and hedging purposes.

Option pricing in a subdiffusive constant elasticity of variance (CEV) model

In this paper, we extend the classical constant elasticity of variance (CEV) model to a subdiffusive CEV model, where the underlying CEV process is time changed by an inverse [Formula: see text]-stable subordinator. The new model can capture the subdiffusive characteristics of financial markets. We find the corresponding fractional Fokker–Planck equation governing the PDF of the new process. We also derive the analytical formula for option prices in terms of eigenfunction expansion. This method avoids the evaluation of PDF of an inverse [Formula: see text]-stable variable and also eliminates the need for numerical integration to calculate the option prices. We numerically investigate the sensitivities of the option prices to the key parameters of the newly developed model.

Valuing American-style options under the CEV model: an integral representation based method

This article derives a new integral representation of the early exercise boundary for valuing American-style options under the constant elasticity of variance (CEV) model. An important feature of this novel early exercise boundary characterization is that it does not involve the usual (time) recursive procedure that is commonly employed in the so-called integral representation approach well known in the literature. Our non-time recursive pricing method is shown to be analytically tractable under the local volatility CEV process and the numerical experiments demonstrate its robustness and accuracy.

Appendix: Recursive Marginal Quantization of Higher-Order Schemes

This document serves as the online appendix to McWalter et al. [2017]. It provides efficient matrix implementations for both vector quantization and the generalized recursive marginal quantization, which allows for the implementation of the higher-order discretization schemes presented in the main paper. It includes graphical examples of how the higher-order discretization schemes reduce the error in the implied marginal distributions for both the geometric Brownian motion process (GBM) and its generalization, the constant elasticity of variance (CEV) process. It concludes with an example of the effect of modelling the CEV process with an absorbing or reflecting boundary at zero. The full-text version of this paper can be found at http://ssrn.com/abstract=2894753.

On the Estimation of the SABR Model’s Beta Parameter:The Role of Hedging in Determining the Beta Parameter

The stochastic alpha-beta-rho (SABR) model has become one of the most popular models for real-world interest rate behavior, but not all of its four parameters are equally easy to estimate. The elasticity of the variance beta parameter determines the shape of the returns density, with a beta of 0 implying normal shocks; 1 implying lognormal shocks; and a value between 0 and 1 implying a constant elasticity of variance process. But fitting the model to market returns is not easy, because the data do not discriminate well among different beta values even though the differences are important for hedge design. This article proposes optimizing on a different metric for goodness of fit: hedge performance. Minimizing hedging error as a function of beta in addition to minimizing the error in calibrating to the market volatility smile substantially improves the ability to pin down the beta value. The method can be tuned to reflect the user’s desired trade-off between hedging versus pricing accuracy. <b>TOPICS:</b>Derivatives, quantitative methods

Mean–variance asset–liability management under constant elasticity of variance process

This paper investigates a mean–variance asset–liability management (ALM) problem under the constant elasticity of variance (CEV) process. The company can invest in n+1 assets: one risk-free bond and n risky stocks. The uncontrollable liability process is modelled by a geometric Brownian motion. The feasibility is studied and potential optimal portfolio is proven to be admissible. We derive the efficient frontier and efficient feedback portfolio in terms of the solutions of two backward stochastic differential equations (BSDEs), which do not admit analytical solutions in general. The closed form solutions are obtained under some special cases. Applying the Monte Carlo simulation, we provide several numerical examples to demonstrate how the efficient frontier is influenced by the relevant parameters.

Variance Swaps on Defaultable Assets and Market Implied Time-Changes

We compute the value of a variance swap when the underlying is modeled as a Markov diffusion process time changed by a Lévy subordinator. In this framework, the underlying may exhibit jumps with a state-dependent Lévy measure and local stochastic volatility and have a local stochastic default intensity. Moreover, the Lévy subordinator that drives the underlying can be obtained directly by observing European call/put prices. To illustrate our general framework, we provide an explicit formula for the value of a variance swap when the underlying is modeled as a Lévy subordinated jump-to-default constant elasticity of variance process (see [Carr and V. Linetsky, Finance Stoch., 10, pp. 303--330, 2005]). In this example, we extend the results of [Mendoza-Arriaga, Carr, and Linetsky, Math. Finance, 20, pp. 527--569, 2010], by allowing for joint valuation of credit and equity derivatives as well as variance swaps.

A Discontinuous Galerkin Method for Pricing American Options Under the Constant Elasticity of Variance Model

AbstractThe pricing of option contracts is one of the classical problems in Mathematical Finance. While useful exact solution formulas exist for simple contracts, typically numerical simulations are mandated due to the fact that standard features, such as early-exercise, preclude the existence of such solutions. In this paper we consider derivatives which generalize the classical Black-Scholes setting by not only admitting the early-exercise feature, but also considering assets which evolve by the Constant Elasticity of Variance (CEV) process (which includes the Geometric Brownian Motion of Black-Scholes as a special case). In this paper we investigate a Discontinuous Galerkin method for valuing European and American options on assets evolving under the CEV process which has a number of advantages over existing approaches including adaptability, accuracy, and ease of parallelization.