Constant Elasticity Of Variance Research Articles

The fractional and mixed-fractional CEV model

The continuous observation of the financial markets has identified some ‘stylized facts’ which challenge the conventional assumptions, promoting the born of new approaches. On the one hand, the long-range dependence has been faced replacing the traditional Gauss–Wiener process (Brownian motion), characterized by stationary independent increments, by a fractional version. On the other hand, the CEV model addresses the Leverage effect and smile-skew phenomena, efficiently. In this paper, these two insights are merging and both the fractional and mixed-fractional extensions for the CEV model, are developed. Using the fractional versions of both the Itô’s calculus and the Fokker–Planck equation, the transition probability density function of the asset price is obtained as the solution of a non-stationary Feller process with time-varying coefficients, getting an analytical valuation formula for a European Call option. Besides, the Greeks are computed and compared with the standard case.

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.

On a positivity preserving numerical scheme for jump-extended CIR process: the alpha-stable case

We propose a positivity preserving implicit Euler-Maruyama scheme for a jump-extended Cox-Ingersoll-Ross (CIR) process where the jumps are governed by a compensated spectrally positive $\alpha$-stable process for $\alpha \in (1,2)$. Different to the existing positivity preserving numerical schemes for jump-extended CIR or CEV (Constant Elasticity Variance) process, the model considered here has infinite activity jumps. We calculate, in this specific model, the strong rate of convergence and give some numerical illustrations. Jump extended models of this type were initially studied in the context of branching processes and was recently introduced to the financial mathematics literature to model sovereign interest rates, power and energy markets.

Classes of elementary function solutions to the CEV model I

In the equity markets the stock price volatility increases as the stock price declines. The classical Black–Scholes–Merton (BSM) option pricing model does not reconcile with this association. Cox introduced the constant elasticity of variance (CEV) model in 1975, in order to capture this inverse relationship between the stock price and its volatility. An important parameter in the model is the parameter β , the elasticity of volatility. The CEV model subsumes some of the previous option pricing models. For β = 0 , β = − 1 ∕ 2 , and β = − 1 the CEV model reduces respectively to the BSM model, the square-root model of Cox and Ross, and the Bachelier model. Both in the case of the BSM model and in the case of the CEV model it has become traditional to begin a discussion of option pricing by starting with the vanilla European calls and puts. However, there are simpler solutions to both models than those pertaining to the standard calls and puts. Mathematically, it makes sense to investigate the simpler cases first. In the case of BSM model simpler solutions are the log and power solutions. Similar simple solutions have not been studied so far for the CEV model. We use a group-theoretic method, Kovacic’s algorithm, which has not been used before to problems of Finance or Economics and obtain new classes of elementary function solutions to the CEV model for all half-integer values of β . In particular, when β = ⋯ , − 5 2 , − 2 , − 3 2 , − 1 , 1 , 3 2 , 2 , 5 2 , … , we obtain four new classes of denumerably infinite elementary function solutions, when β = − 1 2 and β = 1 2 we obtain two new classes of denumerably infinite elementary function solutions, whereas, when β = 0 we find two elementary function solutions.

Willow tree algorithms for pricing Guaranteed Minimum Withdrawal Benefits under jump-diffusion and CEV models

This paper presents the willow tree algorithms for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB), where the underlying fund dynamics evolve under the Merton jump-diffusion process or constant-elasticity-of-variance (CEV) process. The GMWB rider gives the policyholder the right to make periodic withdrawals from his policy account throughout the life of the contract. The dynamic nature of the withdrawal policy allows the policyholder to decide how much to withdraw on each withdrawal date, or even to surrender the contract. For numerical valuation of the GMWB rider, we use willow tree algorithms that adopt more effective placement of the lattice nodes based on better fitting of the underlying fund price distribution. When compared with other numerical algorithms, like the finite difference method and fast Fourier transform method, the willow tree algorithms compute GMWB prices with significantly less computational time to achieve a similar level of numerical accuracy. The design of our pricing algorithm also includes an efficient search method for the optimal dynamic withdrawal policies. We perform sensitivity analysis of various model parameters on the prices and fair participating fees of the GMWB riders. We also examine the effectiveness of delta hedging when the fund dynamics exhibit various jump levels.

Optimal reinsurance-investment problem with dependent risks based on Legendre transform

<p style='text-indent:20px;'>This paper investigates an optimal reinsurance-investment problem in relation to thinning dependent risks. The insurer's wealth process is described by a risk model with two dependent classes of insurance business. The insurer is allowed to purchase reinsurance and invest in one risk-free asset and one risky asset whose price follows CEV model. Our aim is to maximize the expected exponential utility of terminal wealth. Applying Legendre transform-dual technique along with stochastic control theory, we obtain the closed-form expression of optimal strategy. In addition, our wealth process will reduce to the classical Cramér-Lundberg (C-L) model when <inline-formula><tex-math id="M1">\begin{document}$ p = 0 $\end{document}</tex-math></inline-formula>, in this case, we achieve the explicit expression of the optimal strategy for Hyperbolic Absolute Risk Aversion (HARA) utility by using Legendre transform. Finally, some numerical examples are presented to illustrate the impact of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategy.

Optimal time-consistent reinsurance-investment strategy with delay for an insurer under a defaultable market

In this paper, we consider a reinsurance-investment problem with delay for an insurer under the mean-variance criterion in a defaultable market. The financial market consists of a risk-free bond, a stock and a defaultable bond. The insurer's surplus process is described by a jump-diffusion risk model and the price process of the stock is assumed to follow a constant elasticity of variance (CEV) model. In particular, we take the delay of feedback time for strategy into account. Applying stochastic control approach, we derive the time-consistent reinsurance-investment strategy in post-default case and pre-default case explicitly via a game theoretic framework, respectively. Finally, numerical examples and sensitivity analyses are provided to show the impact of financial parameters on the optimal strategies.

Optimal reinsurance and investment problem with default risk and bounded memory

This paper considers an insurer with constant absolute risk aversion (CARA) preference and explores a reinsurance-investment optimisation problem where the insurer income is related to the historical wealth performance. The insurer can purchase proportional reinsurance contracts to transfer the claim risk and allocate the remaining liquid wealth in a savings account, a defaultable corporate zero-coupon bond and a risky asset with constant elasticity of variance (CEV) stochastic volatility. The problem is modelled using a stochastic system with delay. Under certain conditions, the analytic optimal reinsurance-investment policy is derived, and the corresponding verification theorem is provided. Finally, a sensitivity analysis is conducted on the optimal reinsurance-investment policies over different memory parameters. The results show that longer memory leads to more prudent reinsurance and investment decisions for the insurer.

Family optimal investment strategy for a random household expenditure under the CEV model

The family optimal investment strategy for a random household expenditure is investigated. Assume that the family is allowed to invest in a financial market consisting of one risk-free asset and one risky asset whose price process satisfies the constant elasticity of variance (CEV) model. The target is to maximize the expected exponential utility of the family terminal wealth and obtain the optimal investment strategy. Employing techniques of stochastic control theory and the dual theory, we derive the Hamilton–Jacobi–Bellman (HJB) equation and obtain an approximate expression for the optimal investment strategy in the slow-fluctuating regime. Numerical examples are presented to illustrate the effects of parameters on the optimal strategies.

Solutions in quadratures to the CEV model

Cox introduced the constant elasticity of variance (CEV) model in 1975, in order to capture this inverse relationship between the stock price and its volatility. An important parameter in the model is the parameter β, the elasticity of volatility. We use Kovacic's algorithm to derive, for all half-integer values of β, all solutions 'in quadratures' of the CEV ordinary differential equation (CEV ODE). These solutions give rise, by separation of variables, to simple solutions to the CEV partial differential equation. In particular, when β = …, …5⁄2, …2, …3⁄2, …1, 1, 3⁄2, 2, 5⁄2, …, we obtain four classes of denumerably infinite elementary function solutions, when β = …½ and β = ½ we obtain two classes of denumerably infinite elementary function solutions, whereas, when β = 0 we find two elementary function solutions.

On the Modified Optimal Investment Strategy for Annuity Contracts under the Constant Elasticity of Variance (CEV) Model

In this work, the optimal pension wealth investment strategy during the decumulation phase, in a defined contribution (DC) pension scheme is constructed. The pension plan member is allowed to invest in a risk free and a risky asset, under the constant elasticity of variance (CEV) model. The explicit solution of the constant relative risk aversion (CRRA) and constant absolute risk aversion (CARA) utility functions are obtained, using Legendre transform, dual theory, and change of variable methods. It is established herein that the elastic parameter, β, say, must not necessarily be equal to one (β ≠ 1). A theorem is constructed and proved on the wealth investment strategy. Observations and significant results are made and obtained, respectively in the comparison of our various utility functions and some previous results in literature.

Subtle is the Lord, but malicious He is not: the calculation of abnormal stock returns in applied research

ABSTRACTWe use the expected logarithmic returns formula for the Geometric Brownian Motion (GBM) in conjunction with the expected logarithmic returns formula for the Feller diffusion to illustrate the nature and magnitude of errors which arise in computed abnormal returns when one applies an expected logarithmic returns formula which is incompatible with the stochastic process that generates a stock’s returns. Empirical analysis based on FTSE 100 stock price data for the five year period ending in 2017 shows that the scale of the errors in computed abnormal returns will hinge on the volatility of the returns generating process but will be particularly pronounced for relatively low stock prices. Although our principal focus is with comparing abnormal returns on the GBM and Feller diffusion, we also simulate logarithmic returns for the Uhlenbeck and Ornstein (1930) process, several interpretations of the Constant Elasticity of Variance (CEV) process and the scaled ‘t’ process of Praetz (1972) and Blattberg and Gonedes (1974). Taken in conjunction with the GBM and the Feller diffusion, these processes underpin virtually every equilibrium based asset pricing model which appears in the literature. However, computing abnormal returns for any of these processes using the expected logarithmic returns formula for the GBM inevitably leads to errors in the abnormal returns. Hence, an important principle which emerges from our analysis is that it is crucially important for researchers and others to test the compatibility of empirically observed returns with the distributional assumptions on which the empirical analysis is based if the complications arising from mis-specified modelling procedures are to be avoided.

Subtle Is the Lord, but Malicious He Is Not: The Calculation of Abnormal Stock Returns in Applied Research

We use the expected logarithmic returns formula for the Geometric Brownian Motion (GBM) in conjunction with the expected logarithmic returns formula for the Feller diffusion to illustrate the nature and magnitude of errors which arise in computed abnormal returns when one applies an expected logarithmic returns formula which is incompatible with the stochastic process that generates a stock’s returns. Empirical analysis based on FTSE 100 stock price data for the five year period ending in 2017 shows that the scale of the errors in computed abnormal returns will hinge on the volatility of the returns generating process but will be particularly pronounced for relatively low stock prices. Although our principal focus is with comparing abnormal returns on the GBM and Feller diffusion, we also simulate logarithmic returns for the Uhlenbeck and Ornstein (1930) process, several interpretations of the Constant Elasticity of Variance (CEV) process and the scaled “t” process of Praetz (1972) and Blattberg and Gonedes (1974). Taken in conjunction with the GBM and the Feller diffusion, these processes underpin virtually every equilibrium based asset pricing model which appears in the literature. However, computing abnormal returns for any of these processes using the expected logarithmic returns formula for the GBM inevitably leads to errors in the abnormal returns. Hence, an important principle which emerges from our analysis is that it is crucially important for researchers and others to test the compatibility of empirically observed returns with the distributional assumptions on which the empirical analysis is based if the complications arising from mis-specified modelling procedures are to be avoided.

Determining Equivalent Administrative Charges for Defined Contribution Pension Plans under CEV Model

In defined contribution pension plan, the determination of the equivalent administrative charges on balance and on flow is investigated if the risk asset follows a constant elasticity of variance (CEV) model. The maximum principle and the stochastic control theory are applied to derive the explicit solutions of the equivalent equation about the charges. Using the power utility function, our conclusion shows that the equivalent charge on balance is related to the charge on flow, risk-free interest rate, and the length of accumulation phase. Moreover, numerical analysis is presented to show our results.

CDS calibration under an extended JDCEV model

ABSTRACTWe propose a new methodology for the calibration of a hybrid credit-equity model to credit default swap (CDS) spreads and survival probabilities. We consider an extended Jump to Default Constant Elasticity of Variance model incorporating stochastic and possibly negative interest rates. Our approach is based on a perturbation technique that provides an explicit asymptotic expansion of the CDS spreads. The robustness and efficiency of the method is confirmed by several calibration tests on real market data.

Portfolio optimization in a defined benefit pension plan where the risky assets are processes with constant elasticity of variance

The paper studies the optimal asset allocation problem of a defined benefit pension plan that operates in a financial market composed of risky assets whose prices are constant elasticity variance processes. The benefits paid to the participants are deterministic. The contributions to the fund are designed by a spread amortization method, which takes into account the size of the unfunded actuarial liability, defined as the difference between the actuarial liability and the fund assets. We address the case where the fund manager wishes to minimize the solvency risk at the final date of the plan when the fund is underfunded, as well as the case where the fund manager wishes to maximize an increasing, constant elasticity utility function of the fund surplus, when the fund is overfunded. The optimal portfolio and contributions are obtained in both scenarios, with the help of the Hamilton–Jacobi–Bellman equation. A numerical illustration shows the evolution of the plan for several values of the elasticity parameter of the CEV price processes and the risk aversion of the manager, yielding some tips on the main properties of the optimal portfolio.

Sovereign CDS Calibration Under a Hybrid Sovereign Risk Model

ABSTRACTThe European sovereign debt crisis, started in the second half of 2011, has posed the problem for asset managers, trades and risk managers to assess sovereign default risk. In the reduced form framework, it is necessary to understand the interrelationship between creditworthiness of a sovereign, its intensity to default and the correlation with the exchange rate between the bond’s currency and the currency in which the Credit Default Swap CDS spread are quoted. To do this, we propose a hybrid sovereign risk model in which the intensity of default is based on the jump to default extended constant elasticity variance model. We analyse the differences between the default intensity under the domestic and foreign measure and we compute the default-survival probabilities in the bond’s currency measure. We also give an approximation formula to CDS spread obtained by perturbation theory and provide an efficient method to calibrate the model to CDS spread quoted by the market. Finally, we test the model on real market data by several calibration experiments to confirm the robustness of our method.

Willow Tree Algorithms for Pricing Guaranteed Minimum Withdrawal Benefits Under Jump-Diffusion and CEV Models

This paper presents the willow tree algorithms for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB), where the underlying fund dynamics evolve under the Merton jump-diffusion process or constant-elasticity-of-variance (CEV) process. The GMWB rider gives the policyholder the right to make periodic withdrawals from his policy account throughout the life of the contract. The dynamic nature of the withdrawal policy allows the policyholder to decide how much to withdraw on each withdrawal date, or even surrender the contract. For numerical valuation of the GMWB rider, we use the willow tree algorithms that adopt more effective placement of the lattice nodes based on better fitting of the underlying fund price distribution. When compared with other numerical algorithms, like the finite difference method and fast Fourier transform method, the willow tree algorithms compute GMWB prices with significantly less computational time to achieve similar level of numerical accuracy. The design of our pricing algorithms also includes an efficient search method for the optimal dynamic withdrawal policies. We perform sensitivity analysis of various model parameters on the prices and fair participating fees of the GMWB riders. We also examine effectiveness of hedging when the fund dynamics exhibit various levels of jump.

SHORT MATURITY ASIAN OPTIONS FOR THE CEV MODEL

We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows the constant elasticity of variance (CEV) model. The leading order short maturity limit of the Asian option prices under the CEV model is obtained in closed form. We propose an analytical approximation for the Asian options prices which reproduces the exact short maturity asymptotics, and demonstrate good numerical agreement of the asymptotic results with Monte Carlo simulations and benchmark test cases for option parameters relevant for practical applications.