In the Heston stochastic volatility model, the stock price follows the following stochastic differential equation:

where the two Brownian components, W(1)t and W(2)t , might be correlated with correlation ρ. The variable vt represents the mean reverting stochastic volatility, where θ is the long-term variance, κ is the speed of the mean reversion, and σ is the volatility of the variance. The presence of the ˆš vt term in the diffusion component of this equation prevents the volatility from becoming negative by forcing the diffusion component to zero as the volatility approaches zero. Its characteristics function for the log of stock price process under Heston model Hirsa (2012) is given by

where γ = ˆš σ2(u2 + iu) + (κ ˆ’ iρσu)2. Use the FFT method to price a European put using the following parameters: spot price S0 =$100, strike price K =100, maturity T = 2 years, risk free rate r = 1.25%, volatility σ = 20%, κ = 1, θ = 0.025, ρ = ˆ’0.7, ν0 = 0.05. Use various values for B,N, and α.

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at + σ1TdW2) (22,102) (22.103) expl eu n Sol ril esplrathr..ne.l -(112 + iu) CO (22.104)