/i(*) = A ( t ) { l - p ( t ) } . (2) Since germinated seeds are observable, \x can be estimated by using kernel
smoothing fi(t) = ^2t. k({t - ti}/h), where k is a kernel and h the band- width. Replacing /j,(t) in equation (2) by £(£), we can obtain a smoothed estimate of X(t) by two methods. The first is to replace p(t) by its estimate obtained from the re-construction of the sample path by using the estimate v. The second is to substitute p(t) = exp{— JQ u>dVd(t - s)dX(s)ds} into equation (2) to get an integral equation that can be solved numerically, where u>d is the volume of a unit ball in Rd.
Numerous simulation studies for various density functions suggested that the integral equation method and the reconstruction method produce very similar results. Figure 3 shows the estimates of A given in equation (1) with A = 6, 7 = 1, k = 2, v = 0.2, d = 1 and L = 50 from ten indepen- dent simulated realisations by using the integral equation method. Further simulation results can be found in Molchanov and Chiu19.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 3. Left: Estimates of A from ten independent simulated realisations by using the integral equation method. Right: The mean and pointwise 95% confidence bounds from these ten estimates; the true A is shown as a solid line.
3.4. Maximum likelihood estimation of the parameters
The parameter estimation mentioned in Section 3.2 can also be improved because in the estimation of the speed v we already have the explicit form of
the likelihood of a given realisation if A is of known analytical form. Thus, if there are only a finite number of unknown parameters, the maximum likelihood estimates can be obtained.
In Section 3.1 we have the maximum likelihood estimator for v if we know the germination times and locations. An important feature of the approach in the current section is that if only the germination times, but not the locations, are known, we are still able to write down the likelihood as a function of v and the parameters of A. Chiu et al.5 compared the two different cases and the results are given in Tables 4 and 5.
A difficulty of the maximum likelihood estimation in this context is that unless A is a very simple function, we do not have a closed form for the optimal solution and so have to rely on numerical approximation.
Table 4. Means and standard deviations of the maximum likelihood estimates obtained from 50 replicates of 100 independent realisations for d = 1.
times and locations times only
S 7 k v (a = 6) (7 = 1) (fc = 2) (u = 0.2)
6.069 (0.598) 1.004 (0.174) 2.008 (0.175) 0.202 (0.002) 5.912 (1.048) 1.080 (0.273) 2.036 (0.199) 0.192 (0.031)
Table 5. Means and standard deviations of the maximum likelihood estimates obtained from 50 replicates of 100 independent realisations for d = 2.
times and locations times only
5 7 k v ( a = 6) (7 = 1) (k = 2) (v = 0.2)
5.988 (0.400) 1.054 (0.115) 2.060 (0.118) 0.201 (0.001) 6.026 (0.654) 1.020 (0.170) 2.012 (0.162) 0.193 (0.022)
Acknowledgments
This paper results from several joint work with various collaborators. I thank all of them and especially thank M. P. Quine (Sydney), from whom I obtained Tables 1 and 2 and Figure 2, and I. S. Molchanov (Berne), from whom I obtained Figure 3. This research is partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKBU/2075/98P).
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THE VALUATION OF BASKET DEFAULT S W A P S
ZHIFENG ZHANG, KIN PANG, PETER COTTON, CHAK WONG AND SHIKHAR RANJAN
Morgan Stanley Dean Witter, 1585 Broadway, New York, NY 10036, USA E-mail: eZhifeng. Zhang@morganstanley. com
Basket default swaps are complex credit derivatives that are difficult to price an- alytically and are typically priced by using Monte Carlo simulations. The pricing and risk management of basket default swaps present challenging computational problems. We present a method for efficiently generating correlated default times whose marginal distributions are consistent with a reduced form stochastic hazard rate model.
1. Introduction
The global credit market has been growing rapidly. According to the annual surveys carried out by the International Swaps and Derivatives Association (ISDA), the global notional outstanding volume of credit derivatives trans- actions has grown steadily from USD 631.5 billion during the first half of 2001 to about USD 1.6 trillion by the middle of 2002. With the recent slow down in the global financial markets and marked increase in corporate and sovereign defaults, these products have an obvious appeal to market participants in managing risk in times of volatility and uncertainty. While conventional credit default swaps continue to dominate the market, more complicated derivatives such as basket default swaps are quickly gaining popularity. As trading volumes increase, pricing and risk management of these more complicated credit derivatives are becoming increasingly impor- tant.
In this paper we describe an algorithm for efficient simulation of corre- lated defaults and how this algorithm is employed in a Monte Carlo pricing methodology for the valuation of basket default swaps. After a brief descrip- tion of two of the most popular credit derivative products in the market, we lay down a general framework for treating similar pricing and hedging problems.
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