Pricing Discrete Monitoring Asian Options With Multi-dimensional Stochastic Volatility

As one of the core contents of financial mathematics research,option pricing has always been an research hotspot at home and abroad.With the rapid development of financial markets in recent years,many new options and exotic options have appeared.Asian options,one of the most active exotic options,is a strongly path-dependent option.Its profits depend on the average price of the underlying assets throughout the entire period of validity,which can effectively circumvent the risk of artificially manipulating and reduce the risks of both parties.Compared with standard options,Asian options are more sensitive to volatility and cheaper.Investors who hold Asian options as they seek hedging in the market can hedge average price risk,so Asian options is a good tool for risk management.Volatility is an intrinsic feature of financial markets.It can reflect the uncertainty of assets.The classical Black-Scholes option pricing model assumes that the volatility is constant.And there is a large deviation in the practice application of Black-Scholes model,which is called “volatility smile”.In order to better explain this phenomenon,Heston proposed stochastic volatility model.But financial market is complex and changeable in reality,and there's a lot of uncertain factors in it.For example,the volatility is sometimes fast and sometimes slow,sometimes high and sometimes low.Obviously only one process of volatility is insufficient to describe these phenomena.Therefore,It is necessary to introduce multi-dimensional stochastic volatility model.Furthermore,there will be a certain dynamic correlation between different markets.So for questions like “Is the correlation between the markets same,or does it vary over time?”,the multi-dimensional stochastic volatility model can help us analyze well.In this paper,the Wishart matrix process is used to measure the dynamic changes of the multi-dimensional stochastic volatility and the stochastic correlation of the underlying asset and the volatility.The multi-dimensional stochastic volatility model is actually a multi-dimensional improvement of Heston model.The multi-dimensional stochastic volatility model is actually a multi-dimensional improvement of Heston model.Under this model,we use methods such as Feynman-Kac theorem,multidimensional random variable,Inverse Fourier transform,linearized Riccati matrix equation,to derive the numerical solution of European-style geometric average Asian options with fixed and floating strike price of discrete time.And by means of the control variable Monte Carlo simulation method,we obtain the numerical approximate calculation of European-style arithmetic average Asian options with fixed and floating strike price.In addition,through numerical examples,we also analyze the influence on price that results from volatility matrix,correlation coefficient matrix,the matrix of mean reversion speeds,volatility of the volatility matrix in Asian option.The result shows that many market factors have significant influence on price.When the market is at highly positive correlation,the transfer effect between different markets is strengthened,thus option price fluctuates greatly.At the same time,it can be seen that the positive impact on the price is obvious when the elements with slow mean reversion speed change.And short-term volatility has a great impact on the Asian options price.Secondly,we extend the theoretical method to the power-Asian options with discrete monitoring,to derive the pricing formula of geometric average power-Asian call options with fixed and floating strike price.Through numerical examples analysis based on the different values of ?,it is found that the smaller the value of ?,the better the conditions for investors to hedge.By selecting the appropriate exponent,investors can reduce the impact of the parameters on option prices,lower investment risk.In order to better explain the complex and changeable financial market,we use Wishart process to measure multi-dimensional stochastic volatility.Then we study the Asian option with discrete monitoring and derive a analytical formula based on it.Not only can provide a reasonable reference for investors to estimate risk,but also the research of multidimensional stochastic volatility model has practical significance for promoting the application of relevant financial fields.