Pricing And Hedging Over-The-Counter Options And Applications Of Backward Stochastic Differential Equations

A derivative contract is a contractual agreement to execute an exchange for some underlying assets at some future dates.The hedgers can use derivative contracts for risk management and the traders can use derivative contracts to reduce the transaction costs and overcome transaction restrictions.Therefore,derivatives market has devel-oped vigorously since its appearance.The valuation of financial derivatives,especially over-the-counter derivatives,has also attracted wide attentions.Since the Black-Scholes model has been published in 1973,the generalizations and applications of option pricing models have become one of the hot topics in financial mathematics.This thesis mainly studies the pricing and hedging of over-the-counter options and applications of backward stochastic differential equations.This thesis is organized as follows.In Section 1,we give the research background and a summary of the main results of this paper.Firstly,we introduce some overviews of the derivatives market to help understand the application scenarios of option pricing theory.Then we briefly describe the over-the-counter option pricing with the counterparty default risk.Finally,we give an overview of the main research content of this paper.In Section 2,we review the main scenarios of market makers applying the option pricing models.Firstly,we briefly describe some classical models of option pricing theory.We give the numerical solutions of Heston stochastic volatility model and its extended models.We review the Greeks introduced by the Black-Scholes model for risk hedging of option portfolios and give the analytical formulas for the risk factors of the CEV model and the numerical formulas for the risk factors of the stochastic volatility models.Finally,in Taiwan TXO option market,we demonstrate how to use market data to calibrate the option pricing model.In Section 3,we describe the phenomenon of "cointegration" in energy commodi-ty markets,and study the valuation of the commodity spread options with stochastic volatility and two-factor mean reversion model.We give the analytical pricing formula of futures spread and the numerical pricing formula of spread options.Firstly,we give closed-form expressions for the mean,variance and characteristic function of the spots spread process.Since the mean of the spots spread process has a closed-form expression,we obtain an analytical pricing formula for futures spread.Based on this analytical formula,we provide two parameters calibration methods.One method is based on the historical data of the spread of futures contracts with different expiration dates and the other one is based on the historical data of futures spread and spot spread.The nu?merical pricing formulas of European options with spots spread and European options with futures spread are given respectively.Furthermore,through numerical simulation,we show the influence of various parameters on option prices.Finally,one example of spreads in energy markets-the spread between Brent blend and WTI crude oil-is analyzed to illustrate the results.In Section 4,we study the valuation of the over-the-counter vulnerable options un-der the structural model.We use the double exponential jump model with stochastic volatility to describe the underlying asset and the counterparty firm's value.Firstly,we derive closed-form characteristic functions for this dynamic.Using COS method,we get numerical solutions for vulnerable European put options based on the characteristic functions.The inverse fast Fourier transform method provides a fast numerical algorith-m for the twice-exercisable vulnerable Bermuda put options.By virtue of the modified Geske-Johnson method,we obtain an approximate pricing formula of vulnerable Ameri-can put options.Numerical simulations are appropriate to provide for investigating the impact of stochastic volatility on vulnerable options.In Section 5,we study the valuation of the over-the-counter vulnerable options un-der the reduced-form model.In this model,we have considered the impact of funding costs and collateralization.It is found that,in the absence of arbitrage opportunities,the option price must lie within a no-arbitrage band.The boundaries of the no-arbitrage band are computed as solutions of backward stochastic differential equations of replicat-ing strategy and offsetting strategy.Under some conditions,we obtain the closed-form representations of the no-arbitrage band for local volatility models.In particular,the fully explicit expressions of the no-arbitrage band for the Black-Scholes model with time-dependent parameters are derived.Furthermore,we provide a strategy for the option holder by using the risky bond issued by the option writer to hedge the remaining po-tential losses.By virtue of numerical simulations,the impact of the default risk,funding costs and collateral can be observed visually.In Section 6,we construct a special class of g-expectations,by virtue of the adapted solutions of backward stochastic differential equations in a general filtered probability space satisfying the usual hypotheses and separability.Furthermore,under a domination assumption,we obtain a martingale representation theorem for the filtration consistent nonlinear expectations and demonstrate that these nonlinear expectations can be ex-pressed as special g-expectations.At last,we discuss the applications of this class of g-expectations.