Research Article | Open Access
On the Computation of the Survival Probability of Brownian Motion with Drift in a Closed Time Interval When the Absorbing Boundary Is a Step Function
This paper provides explicit formulae for the probability that an arithmetic or a geometric Brownian motion will not cross an absorbing boundary defined as a step function during a finite time interval. Various combinations of downward and upward steps are handled. Numerical computation of the survival probability is done quasi-instantaneously and with utmost precision. The sensitivity of the survival probability to the number and the ordering of the steps in the boundary is analyzed.
The question of the first passage time of a diffusion process to an absorbing boundary is of central importance in many mathematical sciences. As mentioned in Wang and Pötzelberger , it arises in biology, economics, engineering reliability, epidemiology, finance, genetics, seismology, and sequential statistical analysis. Typically, one needs to compute the probability that some random dynamics modelled as a diffusion process will remain under or above some critical threshold over a given time interval. This is often referred to as a survival probability. Some papers focus on specific forms of the boundary, for example, a square root one (Breiman  and Sato ) or a curved one (Groeneboom  and Daniels ). Other contributions focus on a specific diffusion process, mostly Brownian motion (Park and Schuurmann , Jennen and Lerche , Salminen , Scheike , Novikov et al. , and Fu and Wu , 2010) or Ornstein-Uhlenbeck process (Alili et al.  and Lo and Hui ). Alternatively, some papers provide results for general classes of diffusion processes or boundaries (Durbin , Pötzelberger and Wang , Wang and Pötzelberger , and Downes and Borovkov ).
The contributions mentioned in this short noncomprehensive survey either focus on numerical algorithms usually involving recursive multidimensional quadrature or they seek to obtain approximate solutions by substituting the initial boundary with another one for which computations are easier and then deriving a bound for the error entailed by using the approximating boundary. This is because no closed form solution is known except in very few special cases when the boundary is linear and the underlying process is a Brownian motion. The main contribution of this paper is to provide a closed form solution to the problem of the computation of the survival probability of an arithmetic or geometric Brownian when the boundary is a step function of time. In general, this problem is numerically approximated by Monte Carlo simulation. The reason for this is twofold: from an analytical standpoint, the calculations involved are cumbersome and relatively complicated; from a numerical standpoint, the dimension of the integrals involved increases linearly with the number of steps in the boundary. In this paper, explicit formulae are provided up to five steps. An extension to higher dimensions is analytically straightforward. These formulae can be numerically computed quasi instantaneously with utmost precision in contrast to the slow and inaccurate approximations yielded by a Monte Carlo simulation. Nonmonotonic sequences of upper and lower steps are handled. The sensitivity of the survival probability with regard to the number and the ordering of the steps in the boundary is examined essentially as a function of the volatility of the underlying process. The problem of the location of the maximum or the minimum of Brownian motion with drift over several time intervals is also tackled. Although the purpose of this paper is not to study a specific engineering problem, one can mention that an immediate application of the results derived in this paper is the pricing and the hedging of many kinds of popular financial contracts known as barrier options and lookback options.
Section 2 provides a formula for the survival probability of an arithmetic Brownian motion when the absorbing boundary includes three steps. Section 3 is dedicated to numerical experiments whose purpose is to gain insight into the sensitivity of the survival probability to the number and the ordering of the steps. Section 4 extends the results of Section 2 to higher dimensions of the absorbing boundary. Section 5 provides a generalization to two-sided absorbing boundaries.
2. Formula for the Survival Probability of an Arithmetic Brownian Motion through a Sequence of Three Absorbing Steps
Let be a real constant and let be a positive real constant. Let be a standard Brownian motion. Under a given probability measure , let be the process driven byA finite time interval is considered and divided into subintervals , . Let us define a piecewise constant boundary made up of horizontal line segments matching time subintervals, in other words, a step function. These line segments are called the steps of the boundary. They take on constant values in each time subinterval , . The boundary may or may not be monotonically increasing or decreasing in time. When , the step is said to be upward, and when , it is said to be downward. When all the steps of the boundary are upward, the boundary itself is said to be upward and may also be called an upper boundary, while a downward boundary (also referred to as a lower boundary) is one whose steps are all downward.
The process is said to have survived at maturity when none of the steps has been hit at any moment in continuous time from to .
The tractability of the calculation of the survival probability depends on the number of steps in the boundary. Indeed, each additional step increases the dimension of the integration problem under consideration. We first come up with a general closed form solution for boundaries made up of three steps. Higher dimensions are considered in Section 3.
Let be defined as one of the six following cumulative distribution functions, where and , , are real constants: Then, we have the following.
Proposition 1. where if and zero otherwise; if and zero otherwise; if and zero otherwise; if and zero otherwise; if and zero otherwise, if and zero otherwise.And the function is the cumulative distribution function of three correlated standard normal random variables , , and with upper bounds and correlation coefficients between and , between and , and between and ; the numerical evaluation of this function can be carried out with double precision and computational time of approximately 0.01 second using the algorithm by Genz .
Proof of Proposition 1. , , and are absolutely continuous random variables. Moreover, is a Markov process. Hence, by applying the law of total probability and by conditioning, we havewhere , , , , , and and , and are jointly normal random variables and each pairwise correlation coefficient is equal to , , . Besides, the distribution of the maximum (or the minimum) of a Brownian motion with drift on a closed time interval, conditional on the two endpoints, is given byThus, expanding the trivariate normal probability density function, using (5) and substituting into (4), we obtainwhereIt is possible to solve this triple integral in closed form as a linear combination of sixteen trivariate standard normal cumulative distribution functions and this yields Proposition 1. The details are quite cumbersome, so they are omitted. However, it is easy to verify that a Monte Carlo simulation will produce numbers that slowly converge to the exact values provided by Proposition 1 whatever the parameters considered. It is highly recommended to use a conditional Monte Carlo approach inasmuch as the latter enables approximating the value of the integral in (6) by randomly drawing three and only three values of the process at each run, without having to discretize the time interval; for some background on the conditional Monte Carlo approach, also known as Brownian Bridge Monte Carlo, the reader is referred to Dagpunar . After implementing conditional Monte Carlo and drawing random numbers from the Mersenne Twister generator, we observed that at least 20,000,000 simulations were required to be sure to attain a minimum level of convergence with the values provided by Proposition 1. These empirical findings were obtained out of a sample of 1,500 lists of randomly drawn parameters used as inputs to Proposition 1. They are reported in more detail in Table 1.
Before turning to Section 3, one may notice that the following combinations of steps are not handled by Proposition 1: This is not because (8) and (9) are more difficult to obtain using the method outlined in the proof of Proposition 1 but simply because the resulting formulae involve twice as many multivariate distribution functions as in the statement of Proposition 1, which would make the latter very bulky. Closed form formulae for (8) and (9) are available from the author upon request.
|The average divergence is the average of the absolute values of the differences between Proposition 1 and CMC divided by Proposition 1 out of a sample of 1,500 lists of randomly drawn parameters.|
The maximum divergence is the maximum of the absolute values of the differences between Proposition 1 and CMC divided by Proposition 1 out of a sample of 1,500 lists of randomly drawn parameters.
Computational time is measured on an i3 2.5 Ghz processor.
3. Numerical Experiments
Let us consider a geometric Brownian motion driven bywhere is a constant, is a positive constant, and is a standard Brownian motion.
Without loss of generality, this process is assumed to start at level at time . We first deal with sequences of maxima or minima, that is, -type and -cumulative distribution functions. We are interested in the following survival probabilities:An elementary application of Ito’s lemma to shows that (9) and (10) can be obtained by computingwhere and is driven byThus, Proposition 1 can be used to compute exact survival probabilities of the geometric Brownian motion .
The boundaries under consideration are said to “widen” on a given time interval when the current step is located farther from the initial process value than the previous step. Likewise, the boundaries are said to “shrink” on a given time interval when the current step is nearer to the initial process value than the previous step. Our objective is threefold:(i)To measure the effect of widening the boundary at one or two steps, compared with keeping a fixed boundary until maturity.(ii)To measure the effect of ordering the steps of the boundary in different ways; in particular, we compare survival probabilities when the farthest step is located in the last time interval and the nearest step is located in the first time interval.(iii)To compare, conversely, survival probabilities when the boundary is upward and when it is downward.For the latter comparison to be unbiased or neutral, we select upward and downward steps whose distances to the origin are the same. Additionally, the probability of not hitting a boundary is a function of the deterministic part of the stochastic differential equation driving the process: a positive drift coefficient increases the chances of hitting an upward boundary, while a negative drift coefficient increases the chances of hitting a downward boundary. That is why we set symmetric values of of 3% and % when examining sequences of maxima and sequences of minima, respectively.
Three levels of the dispersion coefficient are selected: 18% (defining a “low volatility regime”), 36% (defining an “intermediate volatility regime”), and 64% (defining a “high volatility regime”).
|All the values reported in this table are obtained by using Proposition 1 with , , , , , , , and .|
|All the values reported in this table are obtained by using Proposition 1 with , , , , , , , and .|
First, it can be noticed that the increase in the survival probability entailed by a widening of the boundary is not symmetric whether one deals with an upward or a downward boundary. In a low volatility regime, the probability of remaining above a downward boundary is higher than the probability of remaining below an upward boundary. The reverse holds in a high volatility regime. In the intermediate volatility regime, the result depends on the order of the steps of the boundary.
It must be stressed that not all these observations are intuitive. Recall that the drift and the dispersion coefficients of a geometric Brownian motion such as are proportional to at any given , so that the drift coefficient of is equal to . When , we have if as in Table 2 and if