Mathematical and Statistical Methods for Actuarial Sciences and Finance

314 G. Villani

[0. 90 ,1] and so the sample size is about 440 000. We can observe that this stratified
sample can account for an improvement in efficiency of about 23.

6 Conclusions

In this paper we have shown a generalisation of the antithetic method and ana-
sampling procedure to value exchange options improving on the Monte Carlo simu-
lation. Using the delivery assetDas numeraire, we have reduced the bi-dimensionality
of evaluation to one stochastic variablePthat is the ratio between assetsVandD.
But the particular evolution of assetPrequires a new sampling procedure to concen-
trate the simulations in the range in whichPis more sensitive in order to reduce the
variance. The paper can be improved choosinga∗in order to minimise the variance of
simulation through an endogenic process. To realise this objective, a short simulation,
to estimate some optimala∗, and then thea∗-stratification, may be used.

Acknowledgement.Many thanks to the anonymous reviewers for their constructive comments.

References

1. Armada, M.R., Kryzanowsky, L., Pereira, P.J.: A modified finite-lived American exchange
   option methodology applied to real options valuation. Global Finan. J. 17, 419–438 (2007)


2. Barraquand, J., Martineau. D.: Numerical valuation of high dimensional multivariate amer-
   ican securities. J. Finan. Quant. Anal. 30,383–405 (1995)


3. Boyle, P.: Options: a Monte Carlo approach. J. Finan. Econ. 4, 323–338 (1977)


4. Broadie, M., Glasserman, P.: Pricing American-style securities using simulation. J. Econ.
   Dyn. Control 21, 1323–1352 (1997)


5. Carr, P.: The valuation of sequential exchange opportunities. J. Finan. 43,1235–1256
   (1988)


6. Carr, P.: The valuation of American exchange options with application to real options. In:
   Trigeorgis, L. (ed.) Real Options in Capital Investment: Models, Strategies and Applica-
   tions. Praeger, Westport, CT, London (1995)


7. Margrabe, W.: The value of an exchange option to exchange one asset for another. J. Finan.
   33, 177–186 (1978)


8. McDonald, R.L., Siegel, D.R.: Investment and the valuation of firms when there is an
   option to shut down. Int. Econ. Rev. 28, 331–349 (1985)


9. Raymar, S., Zwecher, M.: A Monte Carlo valuation of american call options on the maxi-
   mum of several stocks. J. Derivatives 5, 7–23 (1997)


10. Tilley, J.: Valuing american options in a path simulation model. Trans. Soc. Actuaries 45,
    83–104 (1993)