Option Pricing Under A Class Of Localstochastic Volatility Models  Posted on:20131203  Degree:Doctor  Type:Dissertation  Country:China  Candidate:L Xie  Full Text:PDF  GTID:1109330395973494  Subject:Basic mathematics  Abstract/Summary:  PDF Full Text Request  Based on noarbitrage principle, Black and Scholes derived the famous BlackScholes option pricing formula. Indeed, the advent of this formula caused a revolution in financial economics and financial industries. However, under the assumption of constant volatility, the model implied volatility in BlackScholesâ€™framework is a constant, which is independent of the strike price K and expiration time T, and does not change over time.In fact, the market implied volatility surface not only depends on the strike price K and expiration time T, but also changes with current time t:(1) For a fixed maturity time T, the market implied volatility curve reflects the volatility smile feature.(2) The implied volatility is a monotonically increasing function of the underlying price.Since these market characteristics have a close link with the option hedge, they are very important. Therefore, the defect of BlackScholes has motivated many improvements from the industry.At present there are mainly two types of models to reflect the characteristics of volatility:one is the stochastic volatility model, which directly models the instantaneous volatility with a new stochastic process; the other one is the local volatility model, whose instantaneous volatility is built as a deterministic function of the underlying asset and time t.In my opinion, local and stochastic volatility depict the internal and external factors, respectively, and they both depict the instantaneous volatility partially. It will be more comprehensive to model the instantaneous volatility as the weighted sum of local and stochastic volatility. Therefore we propose Additive localstochastic volatility model and specify it to SSCEV. We make use of JP Fouqueâ€™s singular perturbation technique to derive the firstorder approximation formula of the European option pricing formula, where the firstorder and zeroorder terms are approximated by asymptotic expansion about the exponential parameter. Furthermore, we give error estimation of the approximation formula.The empirical results show that SSCEV model can fit quite well the market data of European option whose expirations are greater than90days. Also the dynamics of the market smile predicted by SSCEV is consistent with the observed market behavior. So hedging based on this model will perform more stably.In addition, based on Jim Gatheralâ€™s work, we improve Pierre HenryLabordereâ€™s approximation result and give a new approximation formula of implied volatility for general local volatility model. Based on the Taylor series expansion, the implied volatility can be approximated by a simple polynomial series. The numerical results show that, our formula has high approximation accuracy under the homogeneous and nonhomogeneous CEV model, and also has high computational efficiency. Therefore, it could be a good alternative for CEV modelâ€™s exact solution. We also compare our approximation formula with Pierre HenryLabordereâ€™s in nonhomogeneous CEV model. Our formula shows not only higher accuracy but also better stability, which illustrates that our improvements are very good and necessary.  Keywords/Search Tags:  option pricing, implied volatility, volatility smile, local volatility, stochastic volatility, singular perturbation, asymptotic expansion, CEV model, SSCEV model  PDF Full Text Request  Related items 
 
