In this paper, we introduce novel analytical solutions for valuating volatility derivatives, including volatility options and capped/floored volatility swaps, employing discrete sampling within the ...framework of the Merton jump-diffusion model, which is driven by a nonhomogeneous Poisson process. The absence of a comprehensive understanding of the probability distribution characterizing the realized variance has historically impeded the development of a robust analytical valuation approach for such instruments. Through the application of the cumulative distribution function of the realized variance conditional on Poisson jumps, we have derived explicit expectations for the derivative payoffs articulated as functions of the extremum values of the square root of the realized variance. We delineate precise pricing structures for an array of instruments, encompassing variance and volatility swaps, variance and volatility options, and their respective capped and floored variations, alongside establishing put-call parity and relationships for capped and floored positions. Complementing the theoretical advancements, we substantiate the practical efficacy and precision of our solutions via Monte Carlo simulations, articulated through multiple numerical examples. Conclusively, our analysis extends to the quantification of jump impacts on the fair strike prices of volatility derivatives with nonlinear payoffs, facilitated by our analytic pricing expressions.
•This paper addresses valuation gaps in volatility derivatives under Merton's jump-diffusion, offering novel analytical pricing solutions.•We provide explicit payoff expectations based on the log-return realized variance, enhancing valuation of variance and volatility derivatives.•Our analysis quantifies jump impacts on fair strike prices of nonlinear volatility derivatives using analytic pricing expressions.
This paper derives a closed-form expansion for the conditional expectation of a continuous-time stochastic process, given by Vt,T:=e−∫tTg(vs)dsf(vT) for 0≤t≤T, where vt evolves according to the ...extended Cox-Ingersoll-Ross process, for any C∞ functions f and g. We apply the Feynman-Kac theorem to state a Cauchy problem associated with Vt,T and solve the problem by using the reduction method. Furthermore, we extend our method to any piecewise C∞ function f; demonstrating our method can be applied to price options in financial derivative markets. In numerical study, we employ Monte Carlo simulations to demonstrate the performance of the current method.
ECIR process; Conditional expectation; Closed-form expansion; Piecewise smooth function
In this paper, we present an analytical probabilistic approach for pricing nonlinear payoff volatility derivatives with discrete sampling by assuming that the underlying asset price evolves according ...to the Black–Scholes model with time-varying volatility. A major difficulty to solve the pricing problem analytically is the volatility of the underlying asset price is time-varying, resulting the realized variance is distributed according to a mixture distribution, the probability density function of which is unknown. By utilizing the properties of a linear combination of noncentral chi-square random variables, we can calculate the expectation of square root of the realized variance analytically and provide formulas for pricing volatility swaps, volatility options, and variance options, including put–call parity relationships. Furthermore, we demonstrate an interesting application of our formulas by constructing simple closed-form approximate formulas for pricing the volatility derivatives for the constant elasticity of variance model. Finally, Monte Carlo simulations are conducted to illustrate the performance our approach, and effects of price volatility on the fair strike prices of the volatility derivatives are investigated and analyzed through several numerical experiments.
•This paper presents a novel analytical probabilistic approach using a mixture distribution to obtain the first analytical pricing formulas for volatility swaps, volatility options, and variance options with discrete sampling under the Black–Scholes (B–S) model with time-varying volatility.•We extend our results the constant elasticity of variance (CEV) model by constructing simple closed-form approximate formulas for pricing volatility swaps, volatility options, and variance options as well as demonstrating the accuracy and efficiency of this approach by comparing the approximated prices against those obtained with Monte Carlo simulations.•We investigate and analyze the effects of price volatility on the fair strike prices of volatility swaps, volatility options, and variance options with discrete sampling under the B–S model with time-varying volatility.
This paper focuses mainly on the problem of computing the γth, γ>0, moment of a random variable Yn:=∑i=1nαiXi in which the αi’s are positive real numbers and the Xi’s are independent and distributed ...according to noncentral chi-square distributions. Finding an analytical approach for solving such a problem has remained a challenge due to the lack of understanding of the probability distribution of Yn, especially when not all αi’s are equal. We analytically solve this problem by showing that the γth moment of Yn can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the γth moment of Yn when Xi is a combination of statistically independent Zi2 and Gi in which the Zi’s are distributed according to normal or Maxwell–Boltzmann distributions and the Gi’s are distributed according to gamma, Erlang, or exponential distributions. Our paper has an immediate application in interest rate modeling, where we can explicitly provide the exact transition probability density function of the extended Cox–Ingersoll–Ross (ECIR) process with time-varying dimension as well as the corresponding γth conditional moment. Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests.
In this paper, we derive a simple closed-form formula for the γth conditional moment of a variance process, based on the extended Cox–Ingersoll–Ross (ECIR) process, for any real number γ. The ...closed-form formula presented here is a result of successfully finding an exact solution for the partial differential equation, based on the ECIR process. We conduct Monte Carlo simulations to illustrate the accuracy of the current closed-form formula.
•This paper presents the first analytical pricing formulas for volatility swaps and volatility options with discrete sampling under the standard Black-Scholes (BS) model with time varying risk-free ...interest rate.•We extend our results to the BS model with time varying parameters and to the Heston stochastic volatility model in which the variance process is assumed to follow the extended Cox-Ingersoll-Ross process by constructing approximate formulas for pricing volatility swaps and demonstrate the accuracy and efficiency of this approach by comparing the approximated prices against those obtained with Monte Carlo simulations.
This paper presents the first analytical pricing formulas for volatility swaps and volatility options with discrete sampling under the Black-Scholes model with time varying risk-free interest rate. Despite numerous analytical works on the pricing of variance swaps with discrete sampling under different models of asset prices, an analytical pricing formula for volatility swaps as well as volatility options had not been well addressed until now. The main challenge in pricing volatility swaps and volatility options is that payoff functions contain a square root operator, making their expectations nonlinear. By utilizing properties of noncentral chi random variables, we can compute expectations of payoff functions analytically and obtain formulas for pricing volatility swaps and volatility options, including variance swaps and variance options. Furthermore, we investigate the accuracy of the well-known convexity correction formula. Most interestingly, we extend our results to the Black-Scholes model with time varying parameters and to the Heston stochastic volatility model in which the variance process is assumed to follow the extended Cox-Ingersoll-Ross process by constructing simple closed-form approximate formulas for pricing volatility swaps and demonstrate the accuracy and efficiency of this approach by comparing the approximated prices against those obtained with Monte Carlo simulations.
•Closed-form formula for conditional moments of generalized NLD-CEV process.•Closed-form formulas for conditional and unconditional moments of a nonlinear drift CEV process are presented in ...simplified form.•Well-known instances deduced by the nonlinear drift CEV process are the Cox-Ingersoll-Ross and inverse Feller processes or 3/2-stochastic volatility model.•Sufficient conditions of existence and uniqueness of a positive pathwise strong solution for time-dependent parameter functions are provided.
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.
This paper investigates solutions of a recurrence differential equation (RDE) of the form: ' 1 1 1 ' 2 2 2 2 1 ' 2 2 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), k k k k k k k A t a A t A t a A t b A ...t A t a A t b A t c A t + + + + + + = = + = + + ' 1 1 1 ' 2 2 2 2 1 ' 2 2 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), k k k k k k k A t a A t A t a A t b A t A t a A t b A t c A t + + + + + + = = + = + + ' 1 1 1 ' 2 2 2 2 1 ' 2 2 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), k k k k k k k A t a A t A t a A t b A t A t a A t b A t c A t + + + + + + = = + = + + for k K 1,2,..., 2 and any positive integer K ³ 3 subject to the initial conditions (0) i i A R = Î R for i K = 1,2,..., where , , i i i b c a Î Î C R and i j a a ¹ for i j ¹ . Firstly, we apply Laplace transform to the RDE to obtain a difference equation in Laplace space. Our success in performing Laplace inverse transform leads to an explicit solution of the RDE. Finally, we present an application of our results by deriving closed-form formulas for the conditional moment, variance, covariance, and correlation of quadratic variance diffusion processes which are commonly used for studying model variance or interest rate processes in financial engineering.
