Stochastic methods in risk management

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Stochastic methods, such as stochastic modeling and simulation, risk neutral valuation, derivative pricing, etc., are widely used in the finance industry. Under Solvency II framework, in order to protect the benefit of shareholder and policyholder, the insurance company should be adequately capitalized to fulfill the capital requirement for solvency. Therefore, two main quantities are taken into account, i.e. the available capital (or basic own funds) and the required capital. In general, these two quantities are calculated by means of stochastic simulation and hence an Economic Scenario Generator (ESG) is used to simulate the potential evolution of risk factors of the economies and financial markets over time. For the calculation of available capital (defined as the difference between the market value of assets and liabilities), the stochastic cash flow projection model is used to perform the market consistent valuation of assets and liabilities given the risk neutral scenarios. For the calculation of required capital, the probability distribution of available capital over a one-year time horizon and a risk measure based on such distribution is taken into account. For instance, the Solvency Capital Requirement (SCR) is measured by the Value-at-Risk at confidence level of 99.5%. We began by reviewing the existing literature and found that most authors used stochastic methods in risk management under Solvency II framework on one of the three components of the partial internal model, i.e. the input model, the valuation model or the risk capital model. In this thesis, we aimed to build a partial internal model including all components and show how we can use stochastic methods to do market consistent valuation and calculate the required capital. For the input model, instead of using academic preferred simple ESG models, e.g. one factor short rate interest rate model along with geometric Brownian motion equity model, we developed advanced models that are more suitable in practice. For the modeling of interest rate, we used the extended three-factor Cox-Ingersoll-Ross model, which is able to capture the three main principle components of yield curve. We derived the pricing of zero coupon options by Fourier transformation of the characteristic function of the linear combination of state variables and subsequently the pricing of swaption using stochastic duration approximation. For the modeling of equity, we used the stochastic volatility model (Heston model) along with above-mentioned stochastic interest rate. Similarly, we first showed the closed-form of discounted characteristic function of log equity price by solving a system of Ordinary Differential Equations (ODEs) resulting from an affine Partial Differential Equation (PDE). We then derived the price of European options by Fourier techniques as well. In addition, we formulated the method of generating economic scenarios by using Monte Carlo simulation with Euler discretization scheme and variance reduction technique of antithetic variates. For the valuation model, we built a stochastic cash flow projection model to capture the development of balance sheet as well as the asset portfolio consisting of coupon bonds and stocks and the liability portfolio consisting of German traditional participating life insurance contracts. We then derived market consistent valuation of assets and liabilities based on the cash flows projected by the stochastic model along with the input of risk neutral economic scenarios. Furthermore, we modeled the management rules. For instance, we developed a constant asset allocation strategy to rebalance the asset portfolio. We considered the unrealized gain and loss by modeling the book value and market value of assets. Additionally, we modeled the MUST-case for the investment surplus distribution between shareholders and policyholders. For the risk capital model, we first implemented the nested stochastic simulation to determine the required risk capital. Since nested simulation requires high computational time, we also investigated the proxy methods of least squared Monte Carlo, replicating portfolio and curve fitting. In particular, we developed a general strategy to construct a good replicating portfolio. First, we described the construction of asset pool. Second, we illustrated the construction of sensitivity sets through recalibration or reweighting techniques. Third, we proposed a calibration procedure, by using the least square optimization and subset selection with certain criteria, to select the optimal replicating portfolio and calculate the required capital. Finally, we performed an empirical application to illustrate the full process, including the calibration of ESG models to real market data, economic scenario generation and validation, market consistent valuation and determination of SCR by nested simulation and replicating portfolio.