Research Article | Open Access
Some Sequential Boundary Crossing Results for Geometric Brownian Motion and Their Applications in Financial Engineering
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.
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 ). 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 ; Salminen ; Kunitomo and Ikeda ). For a comprehensive source of formulae, one may refer to Borodin and Salminen .
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 ), as well as for the joint law of the maxima or minima of a two-dimensional GBM over one time interval (Iyengar ; He et al. ). 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 ).
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.
2. Main Formulae
This section contains the two main formulae of the paper. Let and be two geometric Brownian motions with constant drift coefficients, and , respectively, under a probability measure , constant positive diffusion coefficients and , respectively, and constant correlation coefficient . In other words, the processes and evolve in time according to the following dynamics: where and are standard real-valued Brownian motions and . Let , , , , be positive real numbers and defined as one of the four following cumulative distribution functions, where : (1)(2)(3)(4)
Let us introduce the following notations: The next proposition provides an exact formula for the four above-mentioned cumulative distribution functions.
Proposition 2.1. Let denote the joint trivariate cumulative distribution function of three standard normal random variables , , , where is the correlation coefficient between and , .
Then, for the up-and-up and the down-and-down distributions, one has where And, for the up-and-down and the down-and-up distributions, one has where
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 be a geometric Brownian motion driven by the following dynamics under the initial probability measure : where and are real constants (), is a real-valued standard Brownian motion and correlations are as follows: Let be a positive real number and defined as one of the four following joint cumulative distribution functions, where : (1)(2)(3)(4)
Let us introduce the following new notations: The next proposition provides an exact formula for the four above-mentioned joint cumulative distribution functions.
Proposition 2.2. Let the real function , where and each of the real numbers is included in , be defined as follows: where is the standard normal cumulative distribution function and the following definitions apply: Then, for the up-and-up and the down-and-down distributions, one has where And, for the up-and-down and the down-and-up distributions, one has where
Practical use of Proposition 2.2 and its corollary in engineering applications depend on the accuracy and the efficiency with which the function 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 , and more general contracts were valued by Armstrong  and Guillaume . 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 , and they were further studied by Kwok et al.  and Wong and Kwok .
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) for a two-asset up-and-up knock-out put (substitute for for an up-and-up knock-out call),(2) for a two-asset down-and-down knock-out call or put, (3) for a two-asset up-and-down knock-out call or put,(4) for a two-asset down-and-up knock-out put or call,
where:(i) is the indicator function, (ii) and are the price processes of two risky assets,(iii) is the strike price and is the option expiry, (iv) is a knock-out barrier monitored with respect to over the time interval ,(v) is a knock-out barrier monitored with respect to over the time interval .
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 and be the price processes of two risky assets with constant volatilities and , respectively, and constant correlation coefficient . Let and be two constant pay-out rates on assets and , respectively, and be the constant risk-free interest rate. Let and .
The value, at time , of a two-asset sequential knock-out option with strike price , with knock-out barriers and monitored with respect to over the time interval and over the time interval , respectively, and with maturity , , is given by where the following specifications hold. (1) If the option is an up-and-up put, , and is obtained by substituting and in in Proposition 2.1, and is obtained by substituting and in in Proposition 2.1. (2) If the option is a down-and-down call,, and is obtained by substituting and in in Proposition 2.1, and is obtained by substituting and in in Proposition 2.1. (3) If the option is an up-and-down call,, and is obtained by substituting and in in Proposition 2.1 and is obtained by substituting and in in Proposition 2.1. (4) If the option is a down-and-up put, , and is obtained by substituting and in in Proposition 2.1 and is obtained by substituting and in 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.
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 is the price process of a third risky asset and is the new option expiry, with , then the four following pay-off structures can be analytically tackled:(1) for a three-asset up-and-up knock-out put or call,(2) for a three-asset down-and-down knock-out call or put, (3) for a three-asset up-and-down knock-out call or put,(4) 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 , , and be the price processes of three risky assets with constant volatilities , , and , respectively, and constant pairwise correlation coefficients , , at any fixed time . Let , , and be three constant pay-out rates on assets , , and , respectively, and be the constant risk-free interest rate. Let , , and .
The value, at time , of a three-asset sequential knock-out option with strike price , with knock-out barriers and monitored with respect to over the time interval and over the time interval , respectively, with maturity , , is given by where the following specifications hold. (1) If the option is an up-and-up put, , and is obtained by substituting , and in in Proposition 2.2, and is obtained by substituting , and in in Proposition 2.2. (2) If the option is a down-and-down call,, and is obtained by substituting , and in in Proposition 2.2, and is obtained by substituting , and in in Proposition 2.2. (3) If the option is an up-and-down call, , and is obtained by substituting , and in in Proposition 2.2 and is obtained by substituting , and in in Proposition 2.2. (4) If the option is a down-and-up put,, and is obtained by substituting , and in