汇率衍生品多因子定价模型

Multi⁃Factor Pricing Model of Exchange Rate Derivatives

汇率是非常重要的宏观变量之一,在如今中国经济加速与世界融合的大背景下,汇率金融衍生品是兼顾投资、保值和规避风险等较为全面功能的交易品种。本文紧密联系当前国际上关于汇率随机模型研究的前沿理论,创新性地在仿射模型框架下提出将汇率和利率共同建模并引入汇率的随机波动率与跳跃,推导了本国零息债券价格的解析解、外国零息债券价格的解析解和基于傅里叶变换的汇率欧式期权价格的半解析解,这对于中国汇率衍生品定价的协调一致性与风险管理有着非常重要的意义。同时,使用最小化损失函数的方法进行参数校准,并利用蒙特卡洛模拟验证了方法的有效性,进而作为例子将其应用到人民币美元汇率期权的实证数据中,得到了参数校准的结果。

Exchange rate,as the representation of the external price of a country's currency,is a crucial part of the macroeconomic operation in China.In the macro field,exchange rate functions to connect the currency market,the foreign exchange market,and the macroconomy,exerting sig-nificant influence on individuals'and enterprises'savings,credit,and investment behaviors.In the micro aspect,exchange rate serves as an important benchmark and target for asset pricing,deriva-tive product design,risk management,arbitrage,and speculation.With the comprehensive opening of China's financial market and the gradual progress of RMB's free convertibility,the links among the currency market,the capital market,and the foreign exchange market are becoming closer and closer,resulting in a complex dynamic feedback system where exchange rate and other economic variables interact and influence each other.In the context of the current economic globalization,foreign exchange derivatives,as a common derivative instrument,are a comprehensive trading vari-ety that combines investment,value preservation,and risk hedging functions.Compared with ma-ture foreign derivative markets and their operational mechanisms,China is still in its infancy in this regard.For example,the current types of exchange rate financial derivatives in China are relatively limited and far from meeting market demand.Given this situation,it is imperative to develop China's derivatives market,and the core of this task is to conduct in-depth research on the pricing principles of derivative products and implement them.Secifically,in line with the objcctives of this paper,proper pricing of currency derivatives can not only hedge the profit risks of multination-al enterprises,but also generate investment retums in the foreign exchange market,providing an important foundation and powerful tool for Chinese enterprises and institutions to participate in in-temational financial market competition.Additionally,it can offer a scientifically rational reference for investors'investment decisions and risk management,thus having significant practical implica-tions in the flourishing development of China's financial market.Therefore,this paper aims to closcly connect with cutting edge theoretical research on exchange rate stochastic models interma-tionally,with the goal of constructing excellent and practical currency derivative pricing models and methods.The focus of exchange rate modeling lies In improving the empirical fitting effect of the model to the implied volatility surface,so as to better apply it to the development of China's foreign ex-change market.This paper aims to jointly model exchange rates and interest rates,derive analytical formulas for European option prices,and test their empirical fiting eleet.In our model,the ex-change rate and its volatility process are extended from the Heston model by introducing a jump term in the exchange rate process,while boch domestic and foreign interest rate processes use the two-factor stochastic interest rate model(CIR).Our model incorporates three impotant empirical features:the crrelation between exchange rates and their volatility,the presence of jumps in the exchange rate process,and stochastie interest rate processes.These features are beneficial for the development of complex exchange rate derivatiss with longer maturities and belp to expand the variety of Chin's exchange rate derivative products.One important cousideration is the examination of the correlation between exchange rales and their vlaility with domestie and foreign interest rates.The model involves four processes,and if all of them are intercomected,the model becomes owerly complex.Therefore,it is necessary to reasonably assume the corelation coeffcient matrix.We assume that there is no correlation between domestic and foreign interest rates and exchange rates and their volatility.This modeling approach is selected based on empirical evidence,and it lows for the derivation of analyical fomulas for various related derivatve prices.These formulas can then be fficiently implemented using Four'er transtform methods,thereby enabling efcient calibration of model parameters.The focus of this work is to ensure accuracy,fficiency,and sta-bility in the implementation of the formulas,which can be tested by compering the resuls wih those generated by Monte Carlo simulations.There is extenstive discussion on what constitutes a god stochastie exchange rate model.Somer gue that a sccessful exchange rate stochastic model should be able to generate appropriate implied volatility surfaces that are qualitatively consistent with market data.Others argue that a sccessfl exchange rate stochastie model should be able to generate model prices for derivaives that are close to observed market prices.Oten,models are selected through numerical optimization and goodnes of-ft or diference related measures,as well a information criteria.This paper takes the latter view and uses a loss function constructed from exchange rate option prices,implied volatili-ties of exchange rate options,and zer0-coupon bond yelds to calibrate exchange rate model paran-eters via numerical optinization metbods.The findings of this paper have significant theoretical and practical implications.The four factor moodel obtained satisfactory calibration results when using Monte Carlo simulation data for parame-ter calibration,demonstrating the ffectiveness and robustness of the ealibration method.Further-more,when fiting real RMB/USD exchange ralte option data,the model exhibited good perfom-ance.Additionally,besides exchange rate option data,the four-factor model also achieved a good ft with Chinese and Us govermment bond data,whbch suggests is potential ppicblity in joinly modeling interest rates and exchange rates and pricing relevant derivatives,thereby contrnbuting to the development of China's foeign exchange market.