The Study Of The Expected Shortfall Of Quadratic Portfolio Under The Multivariate Laplace Distribution

The People’s Bank of China has been constantly promoting market-orientedprocess of interest rate in recent years, thus the financial risk faced by financialinstitution would be more complex, and risk management would be paid more and moreattention to by financial institution. In step with the maturing of the country’s financialmarket, a variety of financial products such as the RMB foreign exchange optionintroduced by State Administration of Foreign Exchange are constantly emerging. Theemergence of these financial products is a double-edged sword for financial institution.On one hand, these financial can be used for hedging, pricing and risk management. Onthe other hand, they could bring new financial risk since their high leverage and lowtransaction cost. So it is a practical problem to work with a rational and valid riskmeasurement for these quadratic portfolios.The Value-at-Risk method is a widely used approach. While it does not satisfy thesubadditivity axiom and would deny the conclusion that diversified investment wouldlower the risk. The expected shortfall overcomes such a weakness though it is not thatwidely used. Besides, the previous models always assume the yield rate of financialmarket variables follow the normal distribution. But such an assumption does not reflectthe fat-tail character adequately. What’s more, for working with VaR or ES, the historysimulation or Monte Carlo simulation method are widely used. And the analyticalsolution cannot be obtained. To address the above issues, this thesis devotes to work outan analytical solution of the expected shortfall of the quadratic portfolio under themultivariate Laplace distribution. To be specific, it contains the following.First of all, using the maximum likelihood estimate method based on the definitionof Laplace distribution for the yield rate distribution to get a consistent asymptoticallynormal effective estimation. The results show that the Laplace distribution describesbetter than the normal distribution for the exchange rate yield curve because of its fat-tail character.Secondly, it explains the reasons to choose the expected shortfall as the riskmeasure rather than the Value-at-Risk. It introduces the four axioms of the coherent riskmeasures and gives the economic explanation for each axiom. By comparing thestrengths and weaknesses of VaR and ES, it chooses the expected shortfall as the riskmeasure rather than the Value-at-Risk. Thirdly, based on the Delta-Gamma-Theta model it decomposes the risk exposureof the portfolio into several basic risk factors. Then it obtains the mathematicalexpression. In order to gain the analytical solution to the expected shortfall, it convertsto work with the moment generating function and the characteristic function of Q bythe Delta-Gamma-Theta transformation. While the characteristic function of Q isintractable, prompting for an indirect approach. Finally it obtains the analytical solutionto expected shortfall via the result of Geary (1944).Finally, it collects three foreign exchange option data to calculate the expectedshortfall and does a test of model validity. According to the expected shortfall model ofquadratic portfolio under the multivariate Laplace distribution, we get the expectedshortfall of the next trading day which is close to the result by Monte Carlo simulation.We also make contrast to the actual daily expected loss and use the likelihood ratio to doback testing. The results pass the test, indicating the model is valid.