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We consider explicit approximations for European put option prices within the Stochastic Verhulst model with time-dependent parameters, where the volatility process follows the dynamics $dV_t = kappa_t (theta_t - V_t) V_t dt + lambda_t V_t dB_t$. Our methodology involves writing the put option price as an expectation of a Black-Scholes formula, reparameterising the volatility process and then performing a number of expansions. The difficulties faced are computing a number of expectations induced by the expansion procedure explicitly. We do this by appealing to techniques from Malliavin calculus. Moreover, we deduce that our methodology extends to models with more general drift and diffusion coefficients for the volatility process. We obtain the explicit representation of the form of the error generated by the expansion procedure, and we provide sufficient ingredients in order to obtain a meaningful bound. Under the assumption of piecewise-constant parameters, our approximation formulas become closed-form, and moreover we are able to establish a fast calibration scheme. Furthermore, we perform a numerical sensitivity analysis to investigate the quality of our approximation formula in the Stochastic Verhulst model, and show that the errors are well within the acceptable range for application purposes.
We consider closed-form approximations for European put option prices within the Heston and GARCH diffusion stochastic volatility models with time-dependent parameters. Our methodology involves writing the put option price as an expectation of a Blac
We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a
In the present paper, a decomposition formula for the call price due to Al`{o}s is transformed into a Taylor type formula containing an infinite series with stochastic terms. The new decomposition may be considered as an alternative to the decomposit
We focus on mean-variance hedging problem for models whose asset price follows an exponential additive process. Some representations of mean-variance hedging strategies for jump type models have already been suggested, but none is suited to develop n
We consider stochastic volatility models under parameter uncertainty and investigate how model derived prices of European options are affected. We let the pricing parameters evolve dynamically in time within a specified region, and formalise the prob