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Another look at the integral of exponential Brownian motion and the pricing of Asian options

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Abstract

It is shown that Marc Yor’s formula (Adv. Appl. Probab. 24:509–531, 1992) for the density of the integral of exponential Brownian motion taken over a finite time interval is an extremal member of a family of previously unknown integral formulae for the same density. The derivation is independent from the one by Yor and obtained from a simple time-reversibility feature, in conjunction with a Fokker–Planck type argument. Similar arguments lead to an independent derivation of Dufresne’s result (Scand. Actuar. J. 90:39–79, 1990) for the law of the integral taken over an infinite time interval. The numerical aspects of the new formulae are developed, with concrete applications to Asian options.

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Notes

  1. The most common use of (1.3) has been to identify the integral component(s) on the right side as diffusion(s) (in \(t\)) and rely on diffusion theory for the study of the distribution of the associated component(s) on the left side. In contrast, the time dynamics play no role in the present study, except for the fact that an infinitesimal generator for either side exists.

  2. In Dufresne [11, Proposition 4.4.4], this property is established as a consequence from the \(\mathrm{arcsine}\) law. Alternatively, it can be seen as a direct application of the law of the iterated logarithm, since \(\int_{0}^{\infty }e^{\sigma ((1+\varepsilon ) \sqrt{2s\log (\log s)}+\nu s)}\,\mathrm{d}s<\infty \) for every choice of \(\sigma ,\varepsilon >0\) and \(\nu <0\).

  3. Dufresne [11] also discretizes the finite time integral and relies on the discrete time result from Vervaat [31] (effectively, on the result of Chamayou and Letac [5]), before passing to the limit. Yor [34] connects (3.3) with Getoor’s result [16] on escape times of Brownian motion.

  4. Strictly speaking, although identical in terms of dynamics, the SDE that one can write for the process \(Z\) from (3.1) is different for the SDE that one can write for the integral taken over finite time (see [3, 19, 20, 21, 22, 25]) in the sense that the filtration of \(Z\) is larger, and this is what allows us to “fit” the limiting distribution into the initial state \(Z_{0}\), which then makes passing to the limit redundant. In fact, the coincidence between the two SDEs is an accident of a sort, because the process \(Z\) originates from perturbing in a particular way the integral over infinite time, and the result \(Z\) cannot depend on the history of the Brownian motion until time \(t\) alone.

  5. The connections between the distributions of \(X_{t}\), \(M_{t}\) and \(L_{t}\), the local time of \(W\) in 0, have been studied extensively by several authors—see the survey papers by Matsumoto and Yor [20, 21].

  6. Processes of this form have played an important role in other studies—see the survey papers by Matsumoto and Yor [20, 21].

  7. See http://dlmf.nist.gov/10.32.E9 regarding the identity involving \(K_{1}\).

  8. Yor’s formula (1.1) is essentially a consequence from the contour integral representation of the density of the Hartman–Watson law.

  9. Equation (5.3) is the only condition that we want the function \(g_{t,p}(\,\cdot \,)\) to satisfy. Because of the magic of the Cauchy formula and the calculus of residues, the left side of (5.3) is nothing but a differentiation operation of order \(p-2\) applied to \(g_{t,p}(\,\cdot \,)\), while the right side is given. It should be clear that (5.3) can be satisfied with infinitely many choices for \(g_{t,p}(\,\cdot \,)\).

  10. Apparently, there are infinitely many choices for the contour \(\zeta \) that can be used for constructing integral representations that are of interest to us. The study of which choices would lead to the most computationally efficient representations is beyond the scope of the present paper. Our choices here appear to be “natural”, but we are also trying to mimic the contour integral representations from [32] simply because we insist on developing integral representation of which (1.1) is a special case.

  11. Generally, the GSL C-library is much faster, but in all experiments described here, both methods return the results practically in an instant.

  12. With \(p=3\) and \(f(x)=x\) or \(f(x)=(x-K)^{+}\), the integration with respect to \(x\) can be closed, but the integration with respect to the remaining variables \(u\) and \(y\) can only be carried out as an iterated double integral (in this order).

  13. The first function uses the GSL C-library, while the second one uses the QUADPACK package.

  14. One possible explanation for this phenomenon is that the closed-form expression for \({\mathfrak{R}}(F _{t,\sigma ,\mu ,k}(u,w))\) is quite long and requires a substantially larger number of evaluations.

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Acknowledgements

The author is greatly indebted to two exceptionally conscientious and constructive anonymous referees, and also to Catherine Donati-Martin, Helyette Geman, Yannis Karatzas, Alex Novikov, Hiroyuki Matsumoto, and especially to Marc Yor, for their interest in the present work and for many helpful comments.

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  1. Questrom School of Business, Boston University, Boston, MA, United States

    Andrew Lyasoff

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Lyasoff, A. Another look at the integral of exponential Brownian motion and the pricing of Asian options. Finance Stoch 20, 1061–1096 (2016). https://doi.org/10.1007/s00780-016-0307-1

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