International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 120253 | 22 pages | https://doi.org/10.5402/2011/120253

Some Sequential Boundary Crossing Results for Geometric Brownian Motion and Their Applications in Financial Engineering

Academic Editor: M. Skliar
Received21 Mar 2011
Accepted12 Apr 2011
Published11 Jul 2011

Abstract

This paper provides new explicit results for some boundary crossing distributions in a multidimensional geometric Brownian motion framework when the boundary is a piecewise constant function of time. Among their various possible applications, they enable accurate and efficient analytical valuation of a large number of option contracts traded in the financial markets belonging to the classes of barrier and look-back options.

1. Introduction

The joint law of the maximum (or the minimum) of a real-valued Brownian motion and its endpoint over a finite time interval is a central result in the study of Brownian motion, particularly with regard to the many applications of the theory in finance, medical imaging, robotics, and biology. It can be obtained as a consequence of the “reflection principle,” which derives from the strong Markov property of Brownian motion (Freedman [1]). Application of Girsanov’s theorem easily generalizes this seminal result to the case of a geometric Brownian motion (GBM), a frequently encountered diffusion process that is the building block of financial engineering. Alternatively, the law can be derived by a partial differential equation approach, using Kolmogorov’s equation for the transition density function of a diffusion process. The distribution of the first passage time by a one-dimensional GBM to a one-sided or a two-sided straight boundary then follows. A few cases where the boundary is curved have been handled (Barba Escribá [2]; Salminen [3]; Kunitomo and Ikeda [4]). For a comprehensive source of formulae, one may refer to Borodin and Salminen [5].

Various extensions to these results are often needed to solve practical engineering problems. In particular, one may look for joint distributions of the highest or lowest points hit over several time intervals, and one may deal with a multidimensional GBM. There are few known explicit results in these more general settings, partly because of the laborious analytical calculations involved, but also due to numerical obstacles: the rapidly increasing dimension of the boundary crossing problem leads to analytical solutions that are expressed in terms of functions that are hard to compute with accuracy. As far as one-sided boundaries are concerned, formulae have been published for the joint law of a sequence of maxima or minima of a one-dimensional GBM over several time intervals (Guillaume [6]), as well as for the joint law of the maxima or minima of a two-dimensional GBM over one time interval (Iyengar [7]; He et al. [8]). A formula for the joint law of the exit times of a one-dimensional GBM from two successive two-sided boundaries is also known (Guillaume [9]).

This paper focuses on a sequence of two one-sided straight boundaries conditional on two correlated GBMs, while the value of a third correlated GBM is taken into account at the endpoint of the time interval. The state space is thus three-dimensional. The choice of this particular distribution is motivated both by its usefulness in financial engineering applications and by the fact that it leads to tractable analytical solutions that can be computed with great accuracy and efficiency.

Section 2 presents the two main formulae of this paper. Section 3 deals with applications of these formulae and discusses their numerical implementation.

2. Main Formulae

This section contains the two main formulae of the paper. Let {𝑆1(𝑡),𝑡0} and {𝑆2(𝑡),𝑡0} be two geometric Brownian motions with constant drift coefficients, 𝛼1 and 𝛼2, respectively, under a probability measure , constant positive diffusion coefficients 𝜎1 and 𝜎2, respectively, and constant correlation coefficient 𝜌12. In other words, the processes 𝑆1(𝑡) and 𝑆2(𝑡) evolve in time according to the following dynamics: 𝑑𝑆1(𝑡)=𝛼1𝑆1(𝑡)𝑑𝑡+𝜎1𝑆1(𝑡)𝑑𝐵1(𝑡),𝑑𝑆2(𝑡)=𝛼2𝑆2(𝑡)𝑑𝑡+𝜎2𝑆2(𝑡)𝑑𝐵2(𝑡),(2.1) where 𝐵1(𝑡) and 𝐵2(𝑡) are standard real-valued Brownian motions and 𝑑[𝐵1,𝐵2](𝑡)=𝜌12𝑑𝑡. Let 𝐻1, 𝐻2, 𝐾1, 𝐾2, 𝐾3 be positive real numbers and 𝑃[](𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2) defined as one of the four following cumulative distribution functions, where 𝑡2𝑡1𝑡0=0: (1)𝑃[up-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2sup0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,sup𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,(2.2)(2)𝑃[down-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2inf0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,inf𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,(2.3)(3)𝑃[down-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2inf0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,sup𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,(2.4)(4)𝑃[up-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2sup0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,inf𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3.(2.5)

Let us introduce the following notations:𝑘1𝐾=ln1𝑆1𝑡0,𝑘2𝐾=ln2𝑆2𝑡0,𝑘3𝐾=ln3𝑆2𝑡0,1𝐻=ln1𝑆1𝑡0,2𝐻=ln2𝑆2𝑡0,𝜇1=𝛼1𝜎212,𝜇2=𝛼2𝜎222.(2.6) The next proposition provides an exact formula for the four above-mentioned cumulative distribution functions.

Proposition 2.1. Let 𝑁3[,,;𝜃12,𝜃13,𝜃23] denote the joint trivariate cumulative distribution function of three standard normal random variables 𝑋1, 𝑋2, 𝑋3, where 𝜃𝑎𝑏 is the correlation coefficient between 𝑋𝑎 and 𝑋𝑏, (𝑎,𝑏){1,2,3}.
Then, for the up-and-up and the down-and-down distributions, one has P[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2=𝑁3𝜆𝑘1𝜇1𝑡1𝜎1𝑡1𝑘,𝜆2𝜇2𝑡1𝜎2𝑡1𝑘,𝜆3𝜇2𝑡2𝜎2𝑡2;𝜌12,𝑡1𝑡2𝜌12,𝑡1𝑡2exp2𝜇22𝜎22𝑁3𝜆𝑘1𝜇1𝑡1𝜎1𝑡1+𝜌122𝜇2𝑡1𝜎2𝑘,𝜆2+𝜇2𝑡1𝜎2𝑡1𝑘,𝜆322𝜇2𝑡2𝜎2𝑡2;𝜌12,𝜌12𝑡1𝑡2,𝑡1𝑡2exp2𝜇11𝜎21×𝑁3𝜆𝑘121𝜇1𝑡1𝜎1𝑡1𝑘,𝜆2𝜇2𝑡1𝜎2𝑡1𝜌1221𝜎1𝑡1𝑘,𝜆3𝜇2𝑡2𝜎2𝑡2𝜌1221𝜎1𝑡2;𝜌12,𝜌12𝑡1𝑡2,𝑡1𝑡2+exp2𝜇1𝜎214𝜇2𝜌12𝜎1𝜎21+2𝜇22𝜎22×𝑁3𝜆𝑘121𝜇1𝑡1𝜎1𝑡1+𝜌122𝜇2𝑡1𝜎2𝑘,𝜆2+𝜇2𝑡1𝜎2𝑡1𝜌1221𝜎1𝑡1,𝜆𝑘322𝜇2𝑡2𝜎2𝑡2+𝜌1221𝜎1𝑡2;𝜌12,𝜌12𝑡1𝑡2,𝑡1𝑡2,(2.7) where 𝜆=1if𝑃[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2=𝑃[up-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2,𝜆=1if𝑃[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2=𝑃[down-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2.(2.8) And, for the up-and-down and the down-and-up distributions, one has 𝑃[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2=𝑁3𝜆𝑘1𝜇1𝑡1𝜎1𝑡1,𝜆𝑘2+𝜇2𝑡1𝜎2𝑡1,𝜆𝑘3+𝜇2𝑡2𝜎2𝑡2;𝜌12,𝑡1𝑡2𝜌12,𝑡1𝑡2exp2𝜇22𝜎22×𝑁3𝜆𝑘1𝜇1𝑡1𝜎1𝑡1+𝜌122𝜇2𝑡1𝜎2,𝜆𝑘2𝜇2𝑡1𝜎2𝑡1,𝜆𝑘3+22+𝜇2𝑡2𝜎2𝑡2;𝜌12,𝑡1𝑡2𝜌12,𝑡1𝑡2exp2𝜇11𝜎21×𝑁3𝜆𝑘121𝜇1𝑡1𝜎1𝑡1,𝜆𝑘2+𝜇2𝑡1𝜎2𝑡1+𝜌1221𝜎1𝑡1,𝜆𝑘3+𝜇2𝑡2𝜎2𝑡2+𝜌1221𝜎1𝑡2;𝜌12,𝑡1𝑡2𝜌12,𝑡1𝑡2+exp2𝜇1𝜎214𝜇2𝜌12𝜎1𝜎21+2𝜇22𝜎22×𝑁3𝜆𝑘121𝜇1𝑡1𝜎1𝑡1+𝜌122𝜇2𝑡1𝜎2,𝜆𝑘2𝜇2𝑡1𝜎2𝑡1+𝜌1221𝜎1𝑡1,𝜆𝑘3+22+𝜇2𝑡2𝜎2𝑡2𝜌1221𝜎1𝑡2;𝜌12,𝑡1𝑡2𝜌12,𝑡1𝑡2,(2.9) where 𝜆=1if𝑃[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2=𝑃[up-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2,𝜆=1if𝑃[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2=𝑃[down-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑡1,𝑡2.(2.10)

Corollary of Proposition 2.1
The four following joint cumulative distribution functions, that will be useful in Section 3, are deduced from Proposition 2.1:(1)sup0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,sup𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3=𝑃[up-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝑃[up-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,(2.11)(2)inf0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,inf𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3=𝑃[down-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝑃[down-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,(2.12)(3)inf0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,sup𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3=𝑃[down-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝑃[down-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,(2.13)(4)sup0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,inf𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3=𝑃[up-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝑃[up-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3.(2.14)

Proof of Proposition 2.1 is provided in the Appendices A, B, and C.

The numerical implementation of Proposition 2.1 and its corollary is easy using Genz’s algorithm for the computation of trivariate normal cumulative distribution functions (Genz [10]).

In the next Proposition, we introduce a third correlated geometric Brownian motion that will serve as the endpoint of the joint distribution, and we show that this can still be analytically valued. Let {𝑆3(𝑡),𝑡0} be a geometric Brownian motion driven by the following dynamics under the initial probability measure :𝑑𝑆3(𝑡)=𝛼3𝑆3(𝑡)𝑑𝑡+𝜎3𝑆3(𝑡)𝑑𝐵3(𝑡),(2.15) where 𝛼3 and 𝜎3 are real constants (𝜎3>0), 𝐵3(𝑡) is a real-valued standard Brownian motion and correlations are as follows:𝑑𝐵1,𝐵3(𝑡)=𝜌13𝐵𝑑𝑡,𝑑2,𝐵3(𝑡)=𝜌23𝑑𝑡.(2.16) Let 𝐾4 be a positive real number and 𝑃[](𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3) defined as one of the four following joint cumulative distribution functions, where 𝑡3𝑡2: (1)𝑃[up-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3sup0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,sup𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,𝑆3𝑡3𝐾4,(2.17)(2)𝑃[down-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3inf0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,inf𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,𝑆3𝑡3𝐾4,(2.18)(3)𝑃[down-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3inf0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,sup𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,𝑆3𝑡3𝐾4,(2.19)(4)𝑃[up-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3sup0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,inf𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,𝑆3𝑡3𝐾4.(2.20)

Let us introduce the following new notations:𝑘4𝐾=ln4𝑆3𝑡0,𝜇3=𝛼3𝜎232.(2.21) The next proposition provides an exact formula for the four above-mentioned joint cumulative distribution functions.

Proposition 2.2. Let the real function Φ[𝑏1,𝑏2,𝑏3,𝑏4;𝜃12,𝜃13,𝜃14,𝜃23,𝜃34], where {𝑏1,𝑏2,𝑏3,𝑏4}4 and each of the real numbers 𝜃12,𝜃13,𝜃14,𝜃23,𝜃34 is included in ]1,1[, be defined as follows: Φ𝑏1,𝑏2,𝑏3,𝑏4;𝜃12,𝜃13,𝜃14,𝜃23,𝜃34=𝑏1𝑥1=𝑏2𝑥2=𝑏3𝑥3=1𝜙21𝜙328𝜋3𝑥exp222𝑥1𝜃12𝑥222𝜙221𝑥3𝜃23𝑥222𝜙232𝑏×𝑁4𝜃14𝑥1𝜃341𝑥3𝜃13𝑥1/1𝜃213𝜙413𝑑𝑥3𝑑𝑥2𝑑𝑥1,(2.22) where 𝑁[] is the standard normal cumulative distribution function and the following definitions apply: 𝜙21=1𝜃212,𝜙32=1𝜃223,𝜃341=𝜃34𝜃13𝜃14𝜙31,𝜙413=1𝜃214𝜃2341.(2.23) Then, for the up-and-up and the down-and-down distributions, one has P[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3𝜆𝑘=Φ1𝜇1𝑡1𝜎1𝑡1𝑘,𝜆2𝜇2𝑡1𝜎2𝑡1𝑘,𝜆3𝜇2𝑡2𝜎2𝑡2𝑘,𝜆4𝜇3𝑡3𝜎3𝑡3;𝜃12=𝜌12,𝜃13=𝑡1𝑡2𝜌12,𝜃14=𝑡1𝑡3𝜌13,𝜃23=𝑡1𝑡2,𝜃34=𝑡2𝑡3𝜌23exp2𝜇22𝜎22𝜆𝑘×Φ1𝜇1𝑡1𝜎1𝑡1+𝜌122𝜇2𝑡1𝜎2𝑡1𝑘,𝜆2+𝜇2𝑡1𝜎2𝑡1𝑘,𝜆322𝜇2𝑡2𝜎2𝑡2,𝜆𝑘4𝜇3𝑡3𝜎3𝑡3𝜃34𝜃13𝜃141𝜃21322𝜎2𝑡2+𝜃13𝜌122𝜇2𝑡1𝜎2𝑡1+𝜃14𝜌122𝜇2𝑡1𝜎2𝑡1;𝜃12=𝜌12,𝜃13=𝑡1𝑡2𝜌12,𝜃14=𝑡1𝑡3𝜌13,𝜃23=𝑡1𝑡2,𝜃34=𝑡2𝑡3𝜌23exp2𝜇11𝜎21𝜆𝑘×Φ121𝜇1𝑡1𝜎1𝑡1𝑘,𝜆2𝜇2𝑡1𝜎2𝑡1𝜌1221𝜎1𝑡1𝑘,𝜆3𝜇2𝑡2𝜎2𝑡2𝜌1221𝜎1𝑡2,𝜆𝑘4𝜇3𝑡3𝜎3𝑡3𝜃1421𝜎1𝑡1𝜃34𝜃13𝜃141𝜃213𝜌1221𝜎1𝑡2𝜃1321𝜎1𝑡1;𝜃12=𝜌12,𝜃13=𝑡1𝑡2𝜌12,𝜃14=𝑡1𝑡3𝜌13,𝜃23=𝑡1𝑡2,𝜃34=𝑡2𝑡3𝜌23+exp2𝜇1𝜎214𝜇2𝜌12𝜎1𝜎21+2𝜇22𝜎22𝜆𝑘×Φ121𝜇1𝑡1𝜎1𝑡1+𝜌122𝜇2𝑡1𝜎2𝑡1𝑘,𝜆2+𝜇2𝑡1𝜎2𝑡1𝜌1221𝜎1𝑡1,𝜆𝑘322𝜇2𝑡2𝜎2𝑡2+𝜌1221𝜎1𝑡2,𝜆𝑘4𝜇3𝑡3𝜎3𝑡3+𝜃14𝜌122𝜇2𝑡1𝜎221𝜎1𝑡1𝜃34𝜃13𝜃141𝜃21322𝜎2𝑡2𝜌1221𝜎1𝑡2+𝜃13𝜌122𝜇2𝑡1𝜎221𝜎1𝑡1;𝜃12=𝜌12,𝜃13=𝑡1𝑡2𝜌12,𝜃14=𝑡1𝑡3𝜌13,𝜃23=𝑡1𝑡2,𝜃34=𝑡2𝑡3𝜌23,(2.24) where 𝜆=1if𝑃[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3=𝑃[up-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3,𝜆=1if𝑃[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3=𝑃[down-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3.(2.25) And, for the up-and-down and the down-and-up distributions, one has 𝑃[.]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3𝜆=Φ𝑘1+𝜇1𝑡1𝜎1𝑡1𝑘,𝜆2𝜇2𝑡1𝜎2𝑡1𝑘,𝜆3𝜇2𝑡2𝜎2𝑡2𝑘,𝜆4𝜇3𝑡3𝜎3𝑡3;𝜃12=𝜌12,𝜃13=𝑡1𝑡2𝜌12,𝜃14=𝑡1𝑡3𝜌13,𝜃23=𝑡1𝑡2,𝜃34=𝑡2𝑡3𝜌23exp2𝜇22𝜎22𝜆×Φ𝑘1+𝜇1𝑡1𝜎1𝑡1+𝜌122𝜇2𝑡1𝜎2𝑡1𝑘,𝜆2+𝜇2𝑡1𝜎2𝑡1𝑘,𝜆322𝜇2𝑡2𝜎2𝑡2,𝜆𝑘4𝜇3𝑡3𝜎3𝑡3𝜃34𝜃13𝜃141𝜃21322𝜎2𝑡2+𝜃13𝜌122𝜇2𝑡1𝜎2+𝜃14𝜌122𝜇2𝑡1𝜎2;𝜃12=𝜌12,𝜃13=𝑡1𝑡2𝜌12,𝜃14=𝑡1𝑡3𝜌13,𝜃23=𝑡1𝑡2,𝜃34=𝑡2𝑡3𝜌23exp2𝜇11𝜎21𝜆×Φ𝑘1+21+𝜇1𝑡1𝜎1𝑡1𝑘,𝜆2𝜇2𝑡1𝜎2𝑡1𝜌1221𝜎1𝑡1𝑘,𝜆3𝜇2𝑡2𝜎2𝑡2𝜌1221𝜎1𝑡2,𝜆𝑘4𝜇3𝑡3𝜎3𝑡3𝜃1421𝜎1𝑡1𝜃34𝜃13𝜃141𝜃213𝜌1221𝜎1𝑡2𝜃1321𝜎1𝑡1;𝜃12=𝜌12,𝜃13=𝑡1𝑡2𝜌12,𝜃14=𝑡1𝑡3𝜌13,𝜃23=𝑡1𝑡2,𝜃34=𝑡2𝑡3𝜌23+exp2𝜇1𝜎214𝜇2𝜌12𝜎1𝜎21+2𝜇22𝜎22𝜆×Φ𝑘1+21+𝜇1𝑡1𝜎1𝑡1+𝜌122𝜇2𝑡1𝜎2𝑘,𝜆2+𝜇2𝑡1𝜎2𝑡1𝜌1221𝜎1𝑡1,𝜆𝑘322𝜇2𝑡2𝜎2𝑡2+𝜌1221𝜎1𝑡2,𝜆𝑘4𝜇3𝑡3𝜎3𝑡3+𝜃14𝜌122𝜇2𝑡1𝜎221𝜎1𝑡1𝜃34𝜃13𝜃141𝜃21322𝜎2𝑡2𝜌1221𝜎1𝑡2+𝜃13𝜌122𝜇2𝑡1𝜎221𝜎1𝑡1;𝜃12=𝜌12,𝜃13=𝑡1𝑡2𝜌12,𝜃14=𝑡1𝑡3𝜌13,𝜃23=𝑡1𝑡2,𝜃34=𝑡2𝑡3𝜌23,(2.26) where 𝜆=1if𝑃[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3=𝑃[down-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3,𝜆=1if𝑃[]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3=𝑃[up-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,𝑡1,𝑡2,𝑡3.(2.27)

Corollary of Proposition 2.2
The four following cumulative distribution functions, that will be useful in Section 3, are deduced from Proposition 2.2:(1)sup0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,sup𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,𝑆3𝑡3𝐾4=𝑃[up-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑃[up-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,(2.28)(2)inf0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,inf𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,𝑆3𝑡3𝐾4=𝑃[down-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑃[down-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,(2.29)(3)inf0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,sup𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,𝑆3𝑡3𝐾4=𝑃[down-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑃[down-and-up]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4,(2.30)(4)sup0𝑡𝑡1𝑆1(𝑡)𝐻1,𝑆1𝑡1𝐾1,𝑆2𝑡1𝐾2,inf𝑡1𝑡𝑡2𝑆2(𝑡)𝐻2,𝑆2𝑡2𝐾3,𝑆3𝑡3𝐾4=𝑃[up-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝑃[up-and-down]𝐻1,𝐻2,𝐾1,𝐾2,𝐾3,𝐾4.(2.31)

Proof of Proposition 2.2 is provided in the Appendices A, B, and C.

Practical use of Proposition 2.2 and its corollary in engineering applications depend on the accuracy and the efficiency with which the function Φ can be numerically implemented. This question is dealt with in Section 3.

3. Applications and Numerical Implementation

This Section deals with applications of the results provided in Section 1 in financial engineering. Indeed, Propositions 2.1 and 2.2 can be used as the building blocks for the valuation and the risk management of a large number of option contracts. The main class of instruments under consideration will be barrier options. In their standard form, the latter are contracts whereby the holder is entitled (but not obligated) to buy (call option) or sell (put option) an asset at a prespecified future date (the option expiry) at a prespecified price (the strike price), on condition that the asset price has not (knock-out type) or has (knock-in type) crossed a specific upward level (called the up-barrier) or downward level (called the down-barrier) at any given time from valuation date to expiry. In standard option pricing theory, asset prices are modelled as geometric Brownian motions, so that boundary crossing distributions such as those provided in Section 1 are of immediate use in financial engineering.

It must be emphasized that the market for barrier options is huge. They are the most actively traded class of nonstandard options (usually referred to as exotic options). Many variations on the standard barrier option payoff have been designed to match investors’ demand more closely. One of them is the partial-time barrier option, which specifies that the barrier level is monitored during only a fraction of the option lifespan. The basic contracts were studied by Heynen and Kat [11], and more general contracts were valued by Armstrong [12] and Guillaume [13]. Another variation is the step barrier option (Guillaume [6, 9]), whereby the barrier evolves as a step function of time. It is also common to encounter outside barrier options, whose payoff is a function of two asset prices: one of them is compared with the strike price at expiry to determine the moneyness of the option, while the other one is monitored up until expiry to check whether the barrier level has been crossed. Outside barrier options were originally valued by Heynen and Kat [14], and they were further studied by Kwok et al. [15] and Wong and Kwok [16].

These different features: partial time barrier, step barrier, and outside barrier, often combine to allow for increased flexibility. But then, performing analytical valuation becomes more and more involved and practitioners have to turn to numerical methods that are slow and relatively inaccurate. This is when the results provided in Section 1 become valuable. Indeed, they enable to price a large number of barrier option contracts sharing the three above-mentioned innovative features (partial time, step, and outside barrier), that is, options based on two or even three assets that may knock out depending on whether the underlying assets move within a sequence of prespecified ranges of prices over all or part of the option lifespan. More precisely, Proposition 2.1 and its corollary are the building blocks for the valuation of sequential two-asset knock-out calls and puts whose four pay-off structures are defined as follows:(1)(𝐾𝑆𝑡(2)2)+𝕀{sup10𝑡𝑡𝑆𝑡(1)<𝐻1,sup𝑡12𝑡𝑡𝑆𝑡(2)<𝐻2} for a two-asset up-and-up knock-out put (substitute (𝑆𝑡(2)2𝐾)+ for (𝐾𝑆𝑡(2)2)+ for an up-and-up knock-out call),(2)±(𝑆𝑡(2)2𝐾)+𝕀{inf10𝑡𝑡𝑆𝑡(1)>𝐻1,inf𝑡12𝑡𝑡𝑆𝑡(2)>𝐻2} for a two-asset down-and-down knock-out call or put, (3)±(𝑆𝑡(2)2𝐾)+𝕀{sup10𝑡𝑡𝑆𝑡(1)<𝐻1,inf𝑡12𝑡𝑡𝑆𝑡(2)>𝐻2} for a two-asset up-and-down knock-out call or put,(4)±(𝐾𝑆𝑡(2)2)+𝕀{inf10𝑡𝑡𝑆𝑡(1)>𝐻1,sup𝑡12𝑡𝑡𝑆𝑡(2)<𝐻2} for a two-asset down-and-up knock-out put or call,

where:(i)𝕀{} is the indicator function, (ii){𝑆1(𝑡),𝑡0} and {𝑆2(𝑡),𝑡0} are the price processes of two risky assets,(iii)𝐾 is the strike price and 𝑡2 is the option expiry, (iv)𝐻1 is a knock-out barrier monitored with respect to {𝑆1(𝑡),𝑡0} over the time interval [𝑡0,𝑡1],(v)𝐻2 is a knock-out barrier monitored with respect to {𝑆2(𝑡),𝑡0} over the time interval [𝑡1,𝑡2].

The no-arbitrage price of these eight different option contracts is now provided by the following proposition.

Proposition 3.1. Let two geometric Brownian motions {𝑆1(𝑡),𝑡0} and {𝑆2(𝑡),𝑡0} be the price processes of two risky assets with constant volatilities 𝜎1 and 𝜎2, respectively, and constant correlation coefficient 𝜌12. Let 𝛿1 and 𝛿2 be two constant pay-out rates on assets 𝑆1 and 𝑆2, respectively, and 𝑟 be the constant risk-free interest rate. Let 𝜐1=𝑟𝛿1+𝜎1𝜎2𝜌12 and 𝜐2=𝑟𝛿2+𝜎22.
The value, at time 𝑡0=0, of a two-asset sequential knock-out option with strike price 𝐾, with knock-out barriers 𝐻1 and 𝐻2 monitored with respect to 𝑆1 over the time interval [𝑡0,𝑡1] and 𝑆2 over the time interval [𝑡1,𝑡2], respectively, and with maturity 𝑡2, 𝑡2𝑡1𝑡0, is given by 𝑉𝑆1𝑡0,𝑆2𝑡0,𝑡1,𝑡2,𝐾,𝐻1,𝐻2=±𝐾exp𝑟𝑡2𝑃1𝑆2𝑡0exp𝛿2𝑡2𝑃2,(3.1) where the following specifications hold. (1) If the option is an up-and-up put, ±=1, and 𝑃1 is obtained by substituting 𝛼1=𝑟𝛿1 and 𝛼2=𝑟𝛿2 in 𝑃[up-and-up](𝐻1,𝐻2,𝐻1,𝐻2,𝐾,𝑡1,𝑡2) in Proposition 2.1, and 𝑃2 is obtained by substituting 𝛼1=𝜐1 and 𝛼2=𝜐2 in 𝑃[up-and-up](𝐻1,𝐻2,𝐻1,𝐻2,𝐾,𝑡1,𝑡2) in Proposition 2.1. (2) If the option is a down-and-down call,±=1, and 𝑃1 is obtained by substituting 𝛼1=𝑟𝛿1 and 𝛼2=𝑟𝛿2 in 𝑃[down-and-down](𝐻1,𝐻2,𝐻1,𝐻2,𝐾,𝑡1,𝑡2) in Proposition 2.1, and 𝑃2 is obtained by substituting 𝛼1=𝜐1 and 𝛼2=𝜐2 in 𝑃[down-and-down](𝐻1,𝐻2,𝐻1,𝐻2,𝐾,𝑡1,𝑡2) in Proposition 2.1. (3) If the option is an up-and-down call,±=1, and 𝑃1 is obtained by substituting 𝛼1=𝑟𝛿1 and 𝛼2=𝑟𝛿2 in 𝑃[up-and-down](𝐻1,𝐻2,𝐻1,𝐻2,𝐾,𝑡1,𝑡2) in Proposition 2.1 and 𝑃2 is obtained by substituting 𝛼1=𝜐1 and 𝛼2=𝜐2 in 𝑃[up-and-down](𝐻1,𝐻2,𝐻1,𝐻2,𝐾,𝑡1,𝑡2) in Proposition 2.1. (4) If the option is a down-and-up put, ±=1, and 𝑃1 is obtained by substituting 𝛼1=𝑟𝛿1 and 𝛼2=𝑟𝛿2 in 𝑃[down-and-up](𝐻1,𝐻2,𝐻1,𝐻2,𝐾,𝑡1,𝑡2) in Proposition 2.1 and 𝑃2 is obtained by substituting 𝛼1=𝜐1 and 𝛼2=𝜐2 in 𝑃[down-and-up](𝐻1,𝐻2,𝐻1,𝐻2,𝐾,𝑡1,𝑡2) in Proposition 2.1. By using Corollary of Proposition 2.1, one can value up-and-up and down-and-up call options, as well as down-and-down and up-and-down put options.

A sketch of proof of Proposition 3.1 is provided in the Appendices A, B, and C.

Just as Proposition 2.1 enables to value two-asset knock-out calls and puts, Proposition 2.2 can be used to value three-asset knock-out calls and puts. More specifically, if {𝑆3(𝑡),𝑡0} is the price process of a third risky asset and 𝑡3 is the new option expiry, with 𝑡3𝑡2𝑡1𝑡0, then the four following pay-off structures can be analytically tackled:(1)±(𝐾𝑆𝑡(3)3)+𝕀{sup10𝑡𝑡𝑆𝑡(1)<𝐻1,sup𝑡12𝑡𝑡𝑆𝑡(2)<𝐻2} for a three-asset up-and-up knock-out put or call,(2)±(𝑆𝑡(3)3𝐾)+𝕀{inf10𝑡𝑡𝑆𝑡(1)>𝐻1,inf𝑡12𝑡𝑡𝑆𝑡(2)>𝐻2} for a three-asset down-and-down knock-out call or put, (3)±(𝑆𝑡(3)3𝐾)+𝕀{sup10𝑡𝑡𝑆𝑡(1)<𝐻1,inf𝑡12𝑡𝑡𝑆𝑡(2)>𝐻2} for a three-asset up-and-down knock-out call or put,(4)±(𝐾𝑆𝑡(3)3)+𝕀{inf10𝑡𝑡𝑆𝑡(1)>𝐻1,sup𝑡12𝑡𝑡𝑆𝑡(2)<𝐻2} for a three-asset down-and-up knock-out put or call.

The no-arbitrage prices of the eight types of option under consideration are now provided by the following proposition.

Proposition 3.2. Let three geometric Brownian motions {𝑆1(𝑡),𝑡0}, {𝑆2(𝑡),𝑡0}, and {𝑆3(𝑡),𝑡0} be the price processes of three risky assets with constant volatilities 𝜎1, 𝜎2, and 𝜎3, respectively, and constant pairwise correlation coefficients 𝜌𝑎𝑏, (𝑎,𝑏){1,2,3}, at any fixed time 𝑡0. Let 𝛿1, 𝛿2, and 𝛿3 be three constant pay-out rates on assets 𝑆1, 𝑆2, and 𝑆3, respectively, and 𝑟 be the constant risk-free interest rate. Let 𝜐1=𝑟𝛿1+𝜎1𝜎3𝜌13, 𝜐2=𝑟𝛿2+𝜎2𝜎3𝜌23, and 𝜐3=𝑟𝛿3+𝜎23.
The value, at time 𝑡0=0, of a three-asset sequential knock-out option with strike price 𝐾, with knock-out barriers 𝐻1 and 𝐻2 monitored with respect to 𝑆1 over the time interval [𝑡0,𝑡1] and 𝑆2 over the time interval [𝑡1,𝑡2], respectively, with maturity 𝑡3, 𝑡3𝑡2𝑡1𝑡0, is given by 𝑉𝑆1𝑡0,𝑆2𝑡0,𝑆3𝑡0,𝑡1,𝑡2,𝑡3,𝐾,𝐻1,𝐻2=±𝐾exp𝑟𝑡3𝑃1𝑆3𝑡0exp𝛿3𝑡3𝑃2,(3.2) where the following specifications hold. (1) If the option is an up-and-up put, ±=1, and 𝑃1 is obtained by substituting 𝛼1=𝑟𝛿1, 𝛼2=𝑟𝛿2 and 𝛼3=𝑟𝛿3 in 𝑃[up-and-up](𝐻1,𝐻2,𝐻1,𝐻2,𝐻2,𝐾,𝑡1,𝑡2,𝑡3) in Proposition 2.2, and 𝑃2 is obtained by substituting 𝛼1=𝜐1, 𝛼2=𝜐2 and 𝛼3=𝜐3 in 𝑃[up-and-up](𝐻1,𝐻2,𝐻1,𝐻2,𝐻2,𝐾,𝑡1,𝑡2,𝑡3) in Proposition 2.2. (2) If the option is a down-and-down call,±=1, and 𝑃1 is obtained by substituting 𝛼1=𝑟𝛿1, 𝛼2=𝑟𝛿2 and 𝛼3=𝑟𝛿3 in 𝑃[down-and-down](𝐻1,𝐻2,𝐻1,𝐻2,𝐻2,𝐾,𝑡1,𝑡2,𝑡3) in Proposition 2.2, and 𝑃2 is obtained by substituting 𝛼1=𝜐1, 𝛼2=𝜐2 and 𝛼3=𝜐3 in 𝑃[down-and-down](𝐻1,𝐻2,𝐻1,𝐻2,𝐻2,𝐾,𝑡1,𝑡2,𝑡3) in Proposition 2.2. (3) If the option is an up-and-down call, ±=1, and 𝑃1 is obtained by substituting 𝛼1=𝑟𝛿1, 𝛼2=𝑟𝛿2 and 𝛼3=𝑟𝛿3 in 𝑃[up-and-down](𝐻1,𝐻2,𝐻1,𝐻2,𝐻2,𝐾,𝑡1,𝑡2,𝑡3) in Proposition 2.2 and 𝑃2 is obtained by substituting 𝛼1=𝜐1, 𝛼2=𝜐2 and 𝛼3=𝜐3 in 𝑃[up-and-down](𝐻1,𝐻2,𝐻1,𝐻2,𝐻2,𝐾,𝑡1,𝑡2,𝑡3) in Proposition 2.2. (4) If the option is a down-and-up put,±=1, and 𝑃1 is obtained by substituting 𝛼1=𝑟𝛿1, 𝛼2=𝑟𝛿2 and 𝛼3=𝑟𝛿3 in 𝑃[down-and-up](𝐻1,𝐻2,𝐻1,𝐻2,