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We consider the stochastic volatility model $dS_t = sigma_t S_t dW_t,dsigma_t = omega sigma_t dZ_t$, with $(W_t,Z_t)$ uncorrelated standard Brownian motions. This is a special case of the Hull-White and the $beta=1$ (log-normal) SABR model, which are widely used in financial practice. We study the properties of this model, discretized in time under several applications of the Euler-Maruyama scheme, and point out that the resulting model has certain properties which are different from those of the continuous time model. We study the asymptotics of the time-discretized model in the $nto infty$ limit of a very large number of time steps of size $tau$, at fixed $beta=frac12omega^2tau n^2$ and $rho=sigma_0^2tau$, and derive three results: i) almost sure limits, ii) fluctuation results, and iii) explicit expressions for growth rates (Lyapunov exponents) of the positive integer moments of $S_t$. Under the Euler-Maruyama discretization for $(S_t,log sigma_t)$, the Lyapunov exponents have a phase transition, which appears in numerical simulations of the model as a numerical explosion of the asset price moments. We derive criteria for the appearance of these explosions.
We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler-Maruyama scheme. We study its asymptotic
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
In this chapter, we consider volatility swap, variance swap and VIX future pricing under different stochastic volatility models and jump diffusion models which are commonly used in financial market. We use convexity correction approximation technique
In the option valuation literature, the shortcomings of one factor stochastic volatility models have traditionally been addressed by adding jumps to the stock price process. An alternate approach in the context of option pricing and calibration of im
The problem of portfolio allocation in the context of stocks evolving in random environments, that is with volatility and returns depending on random factors, has attracted a lot of attention. The problem of maximizing a power utility at a terminal t