By Jianwei Zhu
This booklet presents a complete, up to date therapy of the appliance of Fourier analyses to pricing regular and unique ideas, and discusses 3 various factors: stochastic volatility, stochastic rate of interest and random leap. The modeling of volatility and rate of interest falls into 4 assorted possible choices: consistent, mean-reverting Ornstein-Uhlenbeck strategy, mean-reverting sq. root strategy and mean-reverting double sq. root strategy, whereas random jumps are specific as natural jumps, lognormal jumps and Pareto jumps. This framework referred to as Modular Pricing of thoughts comprises lots of the current strategies pricing formulation as specific situations.
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Extra info for Modular Pricing of Options: An Application of Fourier Analysis
The FeynmanKac formula is crucial in deriving the final form of the CFs. In the following sections, it will be used repeatedly to help us arrive at closedform solutions. The Heston model is reasonable because some necessary properties of stochastic volatility, for instance, mean-reversion, negative correlation and persistence of shocks to volatility, can be fulfilled in this model in spite of the fact that the modelling is not directly connected with volatility, but with squared volatility (variance).
Consequently, both BS1 and B 8 2 might (not) be a right benchmark for comparison with the exact option prices. 3, we choose the parameters suggested by S&S. 0. Some observations are in order. First of all, options with different moneyness have different sensitivity to the correlation p. The values of at-the-money (ATM) options do not change remarkably overall. However, the sensitivity of out-of-the-money (OTM) options top is more conspicuous than of in-the-money (ITM) options. 635. Secondly, a comparison of Panels A, B and C shows that the meanreversion level 0 is important for the pricing of options.
While Heston (1993), Bates (1994) and BCC (1997) got IJ(