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The exponentially-distributed random timestepping algorithm with boundary test is implemented to evaluate the prices of some variety of single one-sided barrier option contracts within the framework of Black-Scholes model, giving efficient estimation of their hitting times. It is numerically shown that this algorithm, as for the Brownian bridge technique, can improve the rate of weak convergence from order one-half for the standard Monte Carlo to order 1. The exponential timestepping algorithm, however, displays better results, for a given amount of CPU time, than the Brownian bridge technique as the step size becomes larger or the volatility grows up. This is due to the features of the exponential distribution which is more strongly peaked near the origin and has a higher kurtosis compared to the normal distribution, giving more stability of the exponential timestepping algorithm at large time steps and high levels of volatility.

Barrier options are path-dependent exotic options with payoff depending on the price of the underlying asset at expiration and whether or not the asset price reaches a pre-specified barrier during the option’s life [

Some barrier options specify that a fixed cash rebate is to be given to the option-holder if the knock-out and knock-in options become worthless. This can make the barrier options more attractive to the potential purchasers by compensating them for the loss of the option when the knock-out option ceases to exist or when knock-in option comes into existence [

However, the rebate options are not necessarily be combined with the standard barrier options and in this case the rebate options are usually called binary or digital barrier options [

Since early nineties, more complicated structures of barrier options have been innovated according to clients and investors needs. For controlling starting and ending time of the monitoring period, one can use partial-time barrier option, where the underlying price is monitored during a fraction of the option’s lifetime [

The analytical expressions for the barrier options, however, are available only under these particular frameworks and in fact, many other cases such as options with multiple assets and some path-dependent options have not explicit formulae yet. Therefore, accurate numerical and Monte Carlo simulation procedures play crucial role in this situation.

Barrier options are considered as exit time problems and therefore large errors can occur when direct Monte Carlo simulations are used. Specifically, Monte Carlo algorithm for pricing the continuously monitored barrier options has slow convergence and produces high statistical and hitting time errors, due to the knockout feature of such options [

A clever idea to reduce this kind of error efficiently is to apply a simple hitting test after each time step using the distribution of the Brownian bridge pinned between the discrete computational nodes, in order to check if the barrier is crossed during the time step or not [

In this paper, we present an analogous method called the exponential timestepping algorithm introduced by Jansons and Lythe [

time step δ t then has the expectation E [ δ t ] = 1 λ as equivalent to the fixed

time step Δ t . Analogously to the Brownian bridge technique, the probability that the barrier has been hit during the time step can also be taken into account using an efficient boundary hitting test at the end of each time step. Numerical experiments for the double-well potential physical problem [

but its mean value is given as 1 λ = Δ t . The total elapsed time after N time steps is thus a random variable with mean N λ [

The contribution of this work is to present an efficient method for simulating the exit time or functional of the exit time of one dimensional diffusion models such as barrier options in environment of Black-Scholes models. The remainder of this paper is structured as follows. Section 2 discuses the analytical expre- ssions for valuating some variety of one-sided barrier options contracts in Black- Scholes environment. In Section 3, we describe the implementation of the exponential timestepping algorithm with boundary test for pricing the three types of one-sided single barrier options outlined in the previous section. Section 4 includes our numerical experiments concerning these barrier options, in order to compare such an algorithm for efficiency and accuracy with the well-known algorithm called the Brownian bridge technique. The hitting time errors are also discussed and analyzed. Section 5 presents our concluding remarks and some ideas for future work.

We here consider some of different structures of one-sided barrier options within the framework of Black-Scholes model. Thus, under a risk-neutral measure Q the asset price S t is represented by a geometric Brownian model:

d S t = μ S t d t + σ S t d W t , S 0 = S , (1)

where W t is a standard Brownian motion and σ is the constant volatility of the asset price. μ S t = ( r − q ) S t , is the drift under the risk-neutral probability, where r and q represent the risk-free interest rate and the continuous dividend payable, respectively. The first hitting time of the asset price S t with barrier H , where 0 < t < T and T is expiry time, is thus defined by

τ H = { i n f { t ≥ 0 : S t ≤ H } for down-option , i n f { t ≥ 0 : S t ≥ H } for up-option . (2)

Based on above, we will discuss the valuation of some variety of one-sided barrier option contracts in the Black-Scholes environment as follows.

As a case study, we consider up-and-out put vanilla barrier option with cash rebate and other forms can be dealt with, in the same manner. The discounted payoff of such an option at risk-free interest rate r is given by [

f vanilla = { e − r T max ( K − S T , 0 ) if S t < H for all 0 < t < T e − r τ H R if S t ≥ H for some 0 < t < T (3)

The value of the option price at time t = 0 can be formally given as [

V u o p r = E Q [ f vanilla ] , (4)

where the expectation here is taken with respect to the risk-neutral probability measure Q , R is rebate payment and K is strike price of the barrier option.

The theoretical value of such an option can be calculated as the sum of the values of the up-and-out put option with zero rebate V u o p and a pure rebate

option V r o p . To calculate V u o p , we first consider d = ln ( H / S 0 ) σ T , c = ln ( K / S 0 ) σ T , θ 0 = ( r − q + σ 2 2 ) T σ and θ 1 = ( ( r − q + σ 2 2 ) + σ 2 ) T σ , where σ and q represent the volatility and dividend payable respectively. The value of the up-and-out put option with zero rebate is thus given by [

V u o p = K e − r T F ( min ( c , d ) , d , θ 0 ) − S 0 e − q T F ( min ( c , d ) , d , θ 1 ) , (5)

where

F ( a , b , θ ) = N ( a − θ ) − e 2 b θ N ( a − 2 b − θ ) .

N denotes the standard normal distribution function defined as

N ( x ) = 1 2 π ∫ − ∞ x e − y 2 2 d y .

The pure rebate option V r o p can be calculated as [

V r o p = R e − r T F ( d , d , θ 0 ) . (6)

Consequently, the up-and-out put option with cash rebate can be given as

V u o p r = V u o p + V r o p . (7)

Here, we consider the down-and-out cash-or-nothing option as an example of binary barrier options. The holder of this contract will receive a fixed cash amount R only if the underlying asset price never hits the barrier H from above before the expiry date T . Otherwise, the option will expire without value. The discounted payoff of such an option with S 0 > H at risk-free interest rate r is given by [

f b i n = { e − r T R if S t > H for all 0 < t < T 0 if S t ≤ H for some 0 < t < T (8)

The value of the down-and-out cash-or-nothing option price at time t = 0 can be formally given as [

V b d o = E Q [ f b i n ] , (9)

where the expectation here is taken with respect to the risk-neutral probability measure Q and R is pre-specified cash amount.

The theoretical value of the down-and-out cash-or-nothing option V b d o at time t = 0 with barrier H , S 0 > H can be calculated as [

V b d o = R e ( θ 0 − θ 2 ) d ( 1 − F ( − d , − d , − θ 2 ) ) , (10)

where θ 2 = θ 0 2 + 2 r T . d , θ 0 and F are defined as above.

The type A (early-ending) partial-time barrier option is considered here as an example of partial-time single-asset barrier options. This type of option is defined such that the barrier starts at time t = 0 and ends at some time t 1 < T . Since the barrier will end before the expiration time T , we do not need to distinguish whether K > H or K < H . Therefore, we have a total of eight varieties of type A partial-time barrier option. As an example, we discuss a down-and-out call partial-time barrier option, where the option is knocked out during the interval [ 0, t 1 ] as soon as the underlying price is below the barrier H . The discounted expected payoff for such an option at risk-free interest rate r can be thus written as: [

f A = { e − r T max ( S T − K , 0 ) if S t > H for all 0 < t < t 1 0 if S t ≤ H for some 0 < t < t 1 (11)

The value of the down-and-out call partial-time barrier option at time t = 0 can be formally given as [

V A d o = E Q [ f A ] , (12)

where the expectation here is taken with respect to the risk-neutral probability measure Q .

The closed form formula for pricing this type of options was originally derived by Heynen and Kat as [

V A d o = S 0 e − q T [ M ( d 1 , e 1 ; ρ ) − ( H S 0 ) 2 ( μ + 1 ) M ( d 3 , e 3 , ρ ) ] − K e − r T [ M ( d 2 , e 2 ; ρ ) − ( H S 0 ) 2 μ M ( d 4 , e 4 , ρ ) ] , (13)

where d 1 = l n K S 0 + ( r − q + σ 2 2 ) T σ T , d 2 = d 1 − σ T , d 3 = d 1 + 2 ln H S 0 σ T , d 4 = d 3 − σ T , e 1 = l n S 0 H + ( r − q + σ 2 2 ) t 1 σ t 1 , e 2 = e 1 − σ t 1 , e 3 = e 1 + 2 ln H S 0 σ t 1 , e 4 = e 3 − σ t 1 , μ = r − q + σ 2 2 σ 2 , ρ = t 1 T and M ( .,. ; . ) is bivariate normal distribution.

The strategy of this algorithm is based on approximating the asset price S t of the underlying barrier option at each time by a Brownian motion with a constant drift Y t = μ t + σ W t , Y 0 = 0 , where μ and σ are constants, with parameters determined at the current position and in this case, the increment W δ t has a symmetric exponential distribution [

p ( S ) = ∫ 0 ∞ λ exp ( − λ t ) 1 ( 2 π σ 2 t ) exp ( − ( S − t μ ) 2 2 σ 2 t ) d t = λ σ 2 G − 1 { exp ( − S ( − G − F ) ) if S < 0 , exp ( − S ( G − F ) ) otherwise , (14)

where F = μ σ 2 and G = ( F ) 2 + 2 λ σ 2 . Integrating over the density of Y δ t defined in Equation (14) yields

ℙ ( Y δ t > S ) = G + F 2 G e x p ( − ( G − F ) S ) , (15)

and thus Y δ t has exponential distribution and can be easily sampled. To produce updates for S t using these calculations, we first consider a uniformly distributed random variable u n in ( 0,1 ) and an exponentially distributed random variable p n that can be generated as p n = − ln v n , where v n ~ U ( 0,1 ) and is independent of u n . Then with given the value S n ≈ S ( t n ) , we generate the value of S n + 1 ≈ S ( t n + 1 ) for t n + 1 = t n + δ t , where δ t is exponential time step

with E [ δ t ] = 1 λ , as [

S n + 1 = S n + [ g ( S n ) G ( S n ) − g ( S n ) F ( S n ) ] p n , (16)

where F ( S n ) = ( r − q ) S n σ 2 S n 2 and G ( S n ) = ( F ( S n ) ) 2 + 2 λ σ 2 S n 2 . g ( S n ) = s i g n ( [ G ( S n ) + F ( S n ) 2 G ( S n ) ] − u n ) where the quantity G ( S n ) + F ( S n ) 2 G ( S n ) comes from Equation (15) by setting S = 0 .

Next, a simple posteriori test is performed after each time step in order to calculate the conditional probability of a given barrier H being hit during the time step [

P Up = ℙ ( τ H < δ t | S n , S n + 1 ) = e − 2 G ( S n ) [ H − max ( S n , S n + 1 ) ] if S n , S n + 1 < H , for up options (17)

and

P Down = ℙ ( τ H < δ t | S n , S n + 1 ) = e − 2 G ( S n ) [ min ( S n , S n + 1 ) − H ] if S n , S n + 1 > H , for down options . (18)

Then, an excursion is deduced in [ t n , t n + 1 ] if

S n + 1 > H or z n < P Up ,

for the case of up barrier options and

S n + 1 < H or z n < P Down ,

for down barrier options, where z n is a uniformly distributed random variable. The time stepping will be repeated until this hitting event is detected or the maximum number of exponential time steps is reached. The output is hitting

time approximated as n λ , where n is the number of taken time steps. Based

on this, the discounted payoffs of the three cases of the underlying barrier options discussed in Section 2 are calculated using (3), (8) and (11), respectively. The prices of the underlying barrier options are then computed as the expectations of such discounted payoffs under the risk-neutral measure. A monte Carlo procedure is therefore used to estimate these expectations by a sample average of M independent simulations; see Appendix A for full algorithm.

In our numerical experiments, the mean value 1 λ of the random time step δ t

is used in the exponential timestepping algorithm as equivalent to the fixed time step Δ t in the Brownian bridge technique. We employ these simulation algorithms for the three types of one-sided single barrier options outlined in Section 2, and compare the efficiency of such techniques. The hitting time errors will be discussed and analyzed.

For the first type, we consider up-and-out put vanilla barrier option with rebate payment, and its computational results are displayed in

simulation techniques against the discretization parameter Δ t = 1 λ are shown

in log-log scale at the top of

Volatility parameter | Hitting Time Error using Brownian Bridge Technique | Standard Error using Brownian Bridge Technique | Hitting Time Error using Exponential Timestepping algorithm | Standard Error using Exponential Timestepping algorithm |
---|---|---|---|---|

0.20 | 0.0031 | 0.0026 | 0.0046 | 0.0026 |

0.30 | 0.0040 | 0.0039 | 0.0056 | 0.0039 |

0.40 | 0.0098 | 0.0050 | 0.0051 | 0.0050 |

0.50 | 0.0126 | 0.0060 | 0.0021 | 0.0060 |

0.60 | 0.0173 | 0.0068 | 0.0043 | 0.0068 |

S 0 = 100 , the strike price K = 100 , the barrier H = 130 , the cash rebate R = 1.5 and the expiration time T = 1 . Using (7), we get the theoretical value of 15.5550 and use it to check the efficiency of the two algorithms. We choose a discretization of N = 10 , 25 , 40 , 100 , 250 time steps per year and, the averages are taken over M = 10 7 paths in order to avoid the effects of statistical errors. For a reference, a line of slope one is included. For both simulation techniques,

the first hitting time error was found to be proportional to O ( Δ t ) = 1 λ , achieving

a first order weak convergence. However, in this case, the exponential time stepping algorithm is more accurate for a given amount of CPU time, particularly for high frequency monitoring or large time steps.

For the second type, we consider down and out cash-or-nothing barrier option, and display the behavior of its hitting time errors obtained by the two methods in

the up-and-out put vanilla barrier option with rebate payment considered above. Thus, for the present example, as we increase the volatility more, the random timestepping algorithm shows greater accuracy than the Brownian bridge technique.

For the third type, we consider the type A partial time (early-ending) call barrier option.

Finally, we present some prices of these three different types of barrier options using the underlying two algorithms with Δ t = E [ δ t ] = 0.02 , σ = 0.50 and M = 10 6 . The other parameters for the considered barrier options are chosen as in Figures 1-3, respectively. The results associated with the resultant standard errors and the corresponding analytical values are shown in

Type of barrier option | Pricing using Brownian Bridge Technique ± standard Error | Pricing using Exponential Timestepping algorithm ± standard Error | Analytical result |
---|---|---|---|

up-and-out put vanilla barrier option with cash rebate | 14.0218 | ||

down-and-out cash-or-nothing option | 1.5048 | ||

type A (early-ending) partial-time barrier option | 17.3410 |

choice of volatility, we see that the approximations obtained using the expo- nential timestepping algorithm are more accurate than those obtained using the Brownian bridge technique, coinciding with our observations discussed above. For an instance, using the random timestepping algorithm, the pricing value of type A (early-ending) partial-time barrier option is 17.3413, which is very close to the analytical value (17.3410), compared to the value obtained using the Brownian bridge technique (17.3005).

Barrier options have become increasingly popular in financial markets, parti- cularly in over-the-counter market, since they are cheaper than the plain vanilla options and they can offer a protection for the investor when are used as hedges. We have discussed four various types of single one-sided barrier options within the framework of Black-Scholes environment, including up-and-out put vanilla barrier option with cash rebate, down-and-out cash-or-nothing barrier option and early-ending partial-time barrier option. The barrier options are the most popular class of path-dependent options, where their closed-form pricing formulas are available only under particular frameworks. Therefore, accurate numerical techniques and Monte Carlo simulations play a crucial role in such situation. However, for pricing barrier options, a standard Monte Carlo algorithm yields an over-estimation of hitting time since there is a possibility that the barrier may be hit between the discrete computational nodes, causing large hitting time errors and slow convergence of weak order O ( Δ t ) .

In order to reduce this kind of errors efficiently, we have implemented a method called exponential timestepping algorithm with boundary test introduced by Jansons and Lythe [

chose its mean duration ( 1 λ ) as equivalent to the fixed time step Δ t used for

the Brownian bridge technique. As observed from our numerical experiments, both methods significantly improved the weak order of convergence from one- half order to one order with the same level of standards errors. However, in spite of similarity between their respective rates of convergence, the random time- stepping algorithm displayed better results, for a given amount of CPU time, than the Brownian bridge technique as the time step grows up or the volatility becomes high, due to the features of the exponential distribution. To be specified, the random time step takes samples of exponential distribution and this distri- bution is more strongly peaked near the origin than that of the normal distri- bution. Thus, the symmetric exponential distribution has a higher kurtosis compared to the normal distribution and this gives more stability of exponential timestepping algorithm at large time steps and high levels of volatility [

For the present work, the exponential timestepping algorithm is implemented for only one-asset barrier options, giving efficient estimation for their hitting times. The challenge is how to develop this technique to deal with the barrier options with multiple assets efficiently. Jansons and Lythe [

The author would like to thank the anonymous referees for their valuable comments that greatly improved the manuscript. The author also would like to thank the members of AJMC for their professional performance.

Alzubaidi, H. (2017) Random Time-Stepping Algorithm with Exponential Distribution for Pricing Various Structures of One-Sided Barrier Options. American Journal of Computational Mathematics, 7, 228-242. https://doi.org/10.4236/ajcm.2017.73020

Algorithm of exponential timestepping with boundary test for the barrier options.