Stochastic processes and point processes of excursions

Electrical Engineering, Mathematics and Computer Scienc

Indonesian options

Jakarta Stock Exchange Indonesia has started to trade Indonesian options at September 9th, 2004. An Indonesian option can be considered as an American style barrier option with immediate (forced) exercise if the price hits or crosses the barrier before maturity. The payoff of the option is based on a moving average of the price of the underlying stock. The barrier is fixed at the strike price plus or minus a 10 percent. The option is automatically exercised when the underlying stock hits or crosses the barrier and the difference between strike and barrier is paid immediately. We will refer to type of this option as Indonesian option. In this paper we study the pricing of the Indonesian option in a Black-Scholes economy. We will derive analytic approximations for the option price. We will discuss volatility and it turns out that expression we cannot calculate the implied volatilities.Electrical Engineering, Mathematics and Computer Scienc

Stochastic renewal process models for estimation of damage cost over the life-cycle of a structure

In the life-cycle cost analysis of a structure, the total cost of damage caused by external hazards like earthquakes, wind storms and flood is an important but highly uncertain component. In the literature, the expected damage cost is typically analyzed under the assumption of either the homogeneous Poisson process or the renewal process in an infinite time horizon (i.e., asymptotic solution). The paper reformulates the damage cost estimation problem as a compound renewal process and derives general solutions for the mean and variance of total cost, with and without discounting, over the life cycle of the structure. The paper highlights a fundamental property of the renewal process, referred to as renewal decomposition, which is a key to solving a wide range of life cycle analysis problems. The proposed formulation generalizes the results given in the literature, and it can be used to optimize the design and life cycle performance of structures.Applied Probabilit

Efficient pricing of Asian options under Lévy processes based on Fourier cosine expansions. Part II. Early-exercise features and GPU implementation

In this article, we propose an efficient pricing method for Asian options with early–exercise features. It is based on a two–dimensional integration and a backward recursion of the Fourier coefficients, in which several numerical techniques, like Fourier cosine expansions, Clenshaw–Curtis quadrature and the Fast Fourier transform (FFT) are employed. Rapid convergence of the pricing method is illustrated by an error analysis. Its performance is further demonstrated by various numerical examples, where we also show the power of an implementation on the Graphics Processing Unit (GPU).Electrical Engineering, Mathematics and Computer Scienc

A Benchmark Approach of Counterparty Credit Exposure of Bermudan Option under Lévy Process: The Monte Carlo-COS Method

An advanced method, which we call Monte Carlo-COS method, is proposed for computing the counterparty credit exposure profile of Bermudan options under Lévy process. The different exposure profiles and exercise intensity under different mea- sures, P and Q, are discussed. Since the COS method [1] delivers accurate Bermudan prices, and no change of measure [2] needed to get the P-probability distribution, the exposure profile produced by the Monte Carlo-COS algorithm can be used as a benchmark result, E.g., to analyse the reliability of the popular American Monte Carlo method [3], [4] and [5]. The efficient calculation of expected exposure (EE) [6] can be further applied to the computation of credit value adjustment (CVA) [6].Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc

Higher order saddlepoint approximations in the Vasicek portfolio credit loss model

This paper utilizes the saddlepoint approximation as an efficient tool to estimate the portfolio credit loss distribution in the Vasicek model. Value at Risk (VaR), the risk measure chosen in the Basel II Accord for the evaluation of capital requirement, can then be found by inverting the loss distribution. VaR Contribution (VaRC), Expected Shortfall (ES) and ES Contribution (ESC) can all be calculated accurately. Saddlepoint approximation is well known to provide good approximations to very small tail probabilities, which makes it a very suitable technique in the context of portfolio credit loss. The portfolio credit model we employ is the Vasicek one factor model, which has an analytical solution if the portfolio is well diversified. The Vasicek asymptotic formula however fails when the portfolio is dominated by a few loans. We show that saddlepoint approximation is able to handle such exposure concentration. We also point out that the saddlepoint approximation technique can be readily applied to more general Bernoulli mixture models (possibly multi-factor). It can further handle portfolios with random LGD.Electrical Engineering, Mathematics and Computer Scienc