The Stochastic Volatility Model Of Market Maker With Inventory Risk And Optimal Quoting Strategy

The inventory models for market maker's optimal quoting explain the bid-ask spreads by inventory holding cost, and make a strategic decision to maximize the dealer's expected utility of terminal wealth. In light of the current models which commonly set a fixed volatility, this thesis takes the time-varying volatility of underlying asset into consideration and studies the optimal quoting derived from an inventory model with stochastic volatility which may evaluate the risk of position more appropriately. Then a method is studied to solve this complex model and elicit optimal quoting to control the exposed risk.The study is started with the volatility of underlying asset, setting it as a stochastic process, and building an inventory model with explicit stochastic volatility. Then the volatility effect on quoting strategy is revealed by solving this model. The volatility mode including Geometric Brownian Motion and Mean-Reverting Process is studied and the analysis formulas of optimal quoting are induces respectively. A set of numerical simulations are showed at last. The numerical simulations make a comparison between these two volatility modes, showing that under a drastic volatility, dealer takes more risks under the mode of Geometric Brownian Motion, and quoting is more sensitive to position and a larger spread is required, while two modes have no significantly difference to the quoting under a mild volatility. The simulation for successive trading days shows the return and risk characteristics of market making. The model allows dealer to make a trade-off between return and risk and to limit the risk within an acceptable range by adjusting risk aversion coefficient.The market making quoting strategy for derivative asset is studied further more. The volatility implied in the market price of relative option is treated as a stochastic process and an inventory model with implied stochastic volatility is built. The quantitative relation between implied volatility and actual volatility is analyzed at first and the volatility effect on quoting strategy is disclosed by solving this model. Under the guidance of an elaborately designed sampling algorithm which is used to estimate the expectation of stochastic item, a simulation of sampling is made to estimate the expectation of stochastic item. Then the simulation of intraday trading shows the return and risk characteristic of market making for option.A comparison between characteristics of stock and option deriving from simulations of intraday trading respectively reveals that they have a similar return and risk characteristic. Consequently, a deduction is formed that after trading for several consecutive days, dealer would obtains a relatively large probability to make a profit and a positive expected return by the accumulation of position expected return and positive skewness from intraday trading. The comparison also shows that the Sharp Ratio of the latter is smaller while the skewness is larger, which means that the market making for option takes a relatively low risk.This thesis introduces a stochastic volatility, which makes it possible to have a more accurate inventory risk assessment for underlying asset market maker and to hedge the Delta risk meanwhile control the Vega risk which can't be hedged for option market maker.