Journal of Industrial and Management Optimization
April 2006 , Volume 2 , Issue 2
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This special issue is based on, but not limited to, contributions from invited speakers of the International Workshop in Financial Mathematics and Statistics held at the Hong Kong Polytechnic University, on December 16, 2004. The workshop was well attended by experts in the field all over the world.
The issue aims to look at leading-edge research on the interface between derivatives, insurance, securities and quantitative finance. As financial mathematics and statistics are two essential components in these four areas, the issue, as we hope, will give the readers a survey of the important tools of mathematics and statistics being used in the modern financial institutions.
In this special issue, 7 papers are included. The papers cover mathematical finance topics, such as option pricing, interest models and stochastic volatility; topics in risk management, such as Value at Risk, liquidity risk management; and actuarial science topics, such as ruin theory.
The papers in the issue were selected with a view towards readers coming from finance, actuarial science, mathematics or statistics. Hopefully this is a first step to provide a platform for people who are interested in the interplay among theory and practice of these disciplines.
Some widely used short interest rate term structure models are discussed. By establishing certain estimates on the solutions of stochastic differential inequalities, we found that the interest rate processes obtained from these models do not have enough integrability which leads to some defects in several applications.
Liquidity risks arise from the presence of time lags on execution of market orders in trading securities and ''quantity'' effect (liquidation discount) on security price. In this paper, we consider an investor who is holding a portfolio of stock and cash (in the form of market money account) with the objective to unwind his position on the risky asset so that the expected value of cash at the end of a fixed time horizon is maximized. Assuming that the executive time lags and liquidation discount are deterministic, we construct the numerical algorithms for computing the optimal trading strategy that maximizes the expected terminal value of cash position in the portfolio. We also investigate the probability of meeting the target cash level under different liquidation discount functions.
This paper proposes a method of estimating Value at Risk (VaR) based on the assumption that the financial returns follow a switching regime ARCH model. We use the simple switching-regime model, the traditional GARCH(1,1) model and the switching-regime ARCH model to do some empirical analysis and to calculate the VaR values under different confidence levels for Shanghai and Shenzhen Stock Index. The calculated VaR values are compared. The results of back-testing and the Proportion of Failure test show the VaR values calculated by the switching-regime ARCH model are preferred to other methods.
In this paper, ruin probabilities are examined in a discrete time risk model in which the interest rates follow a Markov chain with a denumerable state space and the net losses(the claim amount minus the premium income) are assumed to have a dependent AR(1) structure. An upper bound for ultimate ruin probability is obtained by martingale approach. Recursive equations for both finite time ruin probabilities and ultimate ruin probability are derived. By integrating the inductive method and the recursive equation, an upper bound is given for both finite time ruin probabilities and ultimate ruin probability.
This paper develops a valuation model for options under the class of self-exciting threshold autoregressive (SETAR) models and their variants for the price dynamics of the underlying asset using the self-exciting threshold autoregressive Esscher transform (SETARET). In particular, we focus on the first generation SETAR models first proposed by Tong (1977, 1978) and later developed in Tong (1980, 1983) and Tong and Lim (1980), and the second generation models, including the SETAR-GARCH model proposed in Tong (1990) and the double-threshold autoregressive heteroskedastic time series model (DTARCH) proposed by Li and Li (1996). The class of SETAR-GARCH models has the advantage of modelling the non-linearity of the conditional first moment and the varying conditional second moment of the financial time series. We adopt the SETARET to identify an equivalent martingale measure for option valuation in the incomplete market described by the discrete-time SETAR models. We are able to justify our choice of probability measure by the SETARET by considering the self-exciting threshold dynamic utility maximization. Simulation studies will be conducted to investigate the impacts of the threshold effect in the conditional mean described by the first generation model and that in the conditional variance described by the second generation model on the qualitative behaviors of the option prices as the strike price varies.
In this paper we extend the standard LIBOR market model to accommodate the pronounced phenomenon of implied volatility smiles/skews. We adopt a multiplicative stochastic factor to the volatility functions of all relevant forward rates. The stochastic factor follows a square-root diffusion process, and it can be correlated to the forward rates. For any swap rate, we derive an approximate process under its corresponding forward swap measure. By utilizing the analytical tractability of the approximate process, we develop a closed-form formula for swaptions in term of Fourier transforms. Extensive numerical tests are carried out to support the swaptions formula. The extended model captures the downward volatility skews by taking negative correlations between forward rates and their volatilities, which is consistent with empirical findings.
In this paper, we consider a renewal risk process with negative risk sums. We derive integral equations and integro-differential equations for the survival and ruin probabilities for the proposed model. Exact expression and upper and lower bounds for the ruin probability are obtained. We also present some closed form expressions for the survival and ruin probabilities under some certain choices of the claim amount distribution and the distribution of the inter-occurrence time of the claims.
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