Monte Carlo. As the asset returns correlation increases, the effectiveness of the LHS decreases, but so does the effectiveness of the standard Monte Carlo. The standard deviation ratios remain the same. The batching size is the same as in Table 5.3.
EFFECT OF DISCRETIZATION ON ACCURACY AND
scheme. The Euler scheme, for example, has an order of weak convergence of 0.5, meaning that the discretization error in computing paths is of order , where t is the time integration step. The discretization error on moments arises from the strong convergence properties of the numerical scheme. Using the Euler scheme we can compute moments with a discreti- zation error of order t.
The only way to generate trajectories of general underlying processes that are not subject to error is to solve the stochastic differential equation analytically. In many cases of practical importance this is not possible. This is particularly the case when we work with implied local volatility functions In such cases we must construct tra- jectories by numerical solution of the stochastic differential equation.
If we use a numerical scheme (such as the Euler scheme) to build approximate trajectories of the underlying process and then use Monte Carlo simulation to price an option that depends on that underlying pro- cess, we confront a complication in defining the convergence of our Monte Carlo simulation. Our simulation will converge at the rate , where n is the number of Monte Carlo replications, to a value that is consistent with the approximate trajectories calculated with the numerical scheme. Clearly, this value, to which the Monte Carlo converges, is not the true value we seek. That true value will be consistent with exact solutions of the SDE governing the underlying process, not with the approximate solutions given by the numerical integration scheme.
Two questions arise.
■ How far should the Monte Carlo be pushed, given that the trajectories it uses are approximate?
■ How does the computational effort of the Monte Carlo scale with the true accuracy of the solution?
As far as the first question goes, it does not make sense to try to extract more accuracy out of the simulation than is embedded in the approximate nature of the trajectories. An elaboration of this idea gives us the answer to the second question. We will see that when there is numerical discretization error of order t, the scaling law between accuracy and computational effort jumps from to . This is extremely important and means that when using a numerical scheme such as Euler, eventually we have to use eight times more effort to double the accuracy of our calcula- tion. Of course, this transition from the second to the third power of the accuracy may occur after we have attained the accuracy we want. However, if the accuracy we want, given the discretization parameters we are cur- rently using, places our computation in the scaling, we confront
∆t
(the (S t, ) in dS---S = dt+(S t, )dW).
1 ---n
1 accuracy2
--- 1
accuracy3 ---
1 accuracy3 ---
what we call a computational barrier, where further increases in accuracy come at much higher computational expense.
We now quantify these ideas. We are solving a pricing problem by sim- ulation and we will assume we build trajectories of the underlying process, S(t), by numerically solving
(5.106) using the Euler scheme, with an order of weak convergence t. The error in computing functionals of S(t) will be
t t (5.107)
where is a constant. The error due to the Monte Carlo convergence is (5.108) whereC is the standard deviation of our option value estimator. Since it makes sense that these two errors should be of the same order, we set MC tand get the relationship
(5.109) where T is the maturity over which we solve the underlying stochastic dif- ferential equation numerically. Since there are evaluations of the under- lying process in each Monte Carlo replication, the computational effort, WC, in the simulation scales as follows:
(5.110) Since the error in the computation, , is assumed to be of the same order as the Monte Carlo and discretization errors, with Equations 5.109 and 5.110 we get
(5.111) If there were no discretization of the underlying process involved, the computational effort would scale like regardless of the level of accuracy we seek.
In the next susbsections we will determine the number of Monte Carlo replications beyond which no further improvement in accuracy is possible
dS t( ) = a S t( , )dt+b S t( , )dW
=
MC C ---n
=
=
∆t
---T 1
T ---C
---n
=
T t ---
WC T t---n
∝
WC TC2
3 ---
∝
C2
2 ---
without a much larger computational effort. Doing this analysis for general cases of nonlinear stochastic differential equations is difficult and is not done in practical applications. The purpose of the analysis that follows is to gain an understanding of what may happen when discretization error and the standard Monte Carlo error are equally dominant.
We will do the analysis on the standard log-normal process integrated with the Euler scheme. Of course, in practice we don’t have to integrate the log-normal process approximately with a numerical scheme because the solution to the stochastic differential equation is known analytically. But our objective is to illustrate the fundamentals. To this end, the simple log- normal process is ideal.
We will consider two cases in detail. First, we look at the discretization error of the mean and variance of the log-normal process when the SDE is integrated with the Euler scheme. Then, we consider the error induced by the discretization when trajectories of a price process are used to price a European call option.
Discretization Error for the Log-Normal Process Consider the log-normal process described by
(5.112) where and are constant and W(t) is a Wiener process.
Consider now the process defined by the following difference (not dif- ferential) stochastic equation, which results from applying the Euler scheme to Equation 5.112:
(5.113) where t is finite and constant. The solution is an approximation to the exact solution S(t). We will determine how the expected value and vari- ance of process differ from the expected value and variance of S(t) using a simple first order perturbation technique.1
The exact process S(t) and the approximate process are related as follows:
(5.114)
1Actually, in the simple case of the log-normal process, the expectation and variance of the approximate process can be computed exactly. We will use the perturbation technique, however, because it is a method of general validity.
dS
---S = dt+dW t( )
Sˆ(t+∆t)–Sˆ( )t = Sˆ( )t ∆t+Sˆ( )t ∆W Sˆ( )t Sˆ( )t
Sˆ( )t Sˆ(n t∆ ) = S n t( ∆ )+(n t∆ ), n= 1 2, ,…
where n is the integration step number and (.) is an error term, itself a sto- chastic process (this is not the same as the one in the previous section). We will interpret the functions and as continuous functions of time that agree with the expectation and variance of the the process at ,
Discretization Error of the Mean Taking expectation of both sides of Equation 5.113, the mean of the approximate log-normal process satisfies
(5.115) where we remember to assume is a constant. The mean of the exact pro- cess follows the differential form of the last equation:
(5.116) Expanding the left-hand side of Equation 5.115 to second order in t we have
(5.117) Replacing
(5.118) in Equation 5.117 we get
(5.119) where we drop the arguments in S(.) and (.) for convenience. Notice that the fourth term on the left is of higher order. Since we are doing a first- order analysis, we can neglect this term. Replacing Equation 5.116 into Equation 5.119 we get
(5.120) Introducing Equation 5.116 into Equation 5.120, we get the following first-order ordinary differential equation for the error of the mean:
E[Sˆ( )t ] var[Sˆ( )t ]
Sˆ( )t t = n t∆ n = 1 2, ,…
E[Sˆ(t+∆t)]–E[Sˆ( )t ] = E[Sˆ( )t ]∆t
dE[S t( )]
---dt = E[S t( )]
dE[Sˆ( )t ] ---dt ∆t
---2 d2E[Sˆ( )t ] dt2 ---
+ = E[Sˆ( )t ]
E[Sˆ( )t ] = E[S t( )]+E[( )t ]
dE[ ]S
---dt dE[ ] ---dt ∆t
---2 d2E[ ]S dt2 --- ∆t
---2 d2E[ ] dt2 ---
+ + + = E[ ]S +E[ ]
dE[ ]
---dt E[ ] ∆t
---2 d2E[ ]S dt2 --- –
=
(5.121) The mean of the exact log-normal process is
(5.122) where S(0) is the initial process value. Since the initial conditions of the approximate process, , and the exact process, S(t), are the same, Equa- tion 5.121 can be written as follows:
(5.123) Since, by definition, the initial error is zero, the solution of Equation 5.123 is
(5.124) Remember that is what we need to add to the exact solution to get the approximate solution. Since is negative, this equation tells us that the first-order effect of the Euler discretization on the log-normal process is to depress the mean. To the first order, this effect is only due to the drift (the volatility does not enter in Equation 5.124). Effectively, the discretization induces a downward drift. If the drift of the log-normal process is negative, this distortion will grow at the beginning but will eventually die out as the exponential term begins to dominate. If the drift is positive, the first-order error will grow exponentially.
This simple analysis reveals that the behavior of the discretized process is by no means intuitive. Of course, this result does not carry over to other processes and only serves to illustrate what can be expected.
Discretization Error of the Variance A little algebra shows that the vari- ance of the approximate process is given by
(5.125)
while the ordinary differential equation for the variance of the exact log- normal process is
dE[ ]
---dt = E[ ] –∆---2tdE---dt[ ]S
E[S t( )] = S( )0 exp ( )t
Sˆ( )t
dE[ ]
---dt = E[ ] –∆---S2t ( )0 exp( )t
E[( )t ] = –∆---S2t ( )0 texp( )t
var[Sˆ(t+∆t)]–var[Sˆ( )t ]
∆t
--- = (2 + 2)var[Sˆ( )t ]+E2[Sˆ( )t ]2 t2var[Sˆ( )t ]
∆ +
(5.126) Expanding the left-hand side of Equation 5.125 and retaining up to second- order terms int, we get
(5.127)
We define the error of the variance as follows:
(5.128) Replacing Equations 5.128 and 5.114 in Equation 5.127 we get
(5.129) Replacing Equation 5.126 into Equation 5.129 and neglecting higher-order terms, we get the following first-order ordinary differential equation for the error of the variance:
(5.130) The variance for the exact process can be obtained by integrating Equation 5.126:
(5.131) Replacing this expression for the variance as well as E[S(t)] and E[] in Equation 5.130, we can rewrite the equation for the error of the variance as follows:
(5.132) where
(5.133) dvar [S t( )]
---dt = (2 + 2)var [S t( )]+E2[S t( )]2
dvar [Sˆ( )t ] ---dt ∆t
---2 d2var[Sˆ( )t ] dt2 ---
+ = (2 + 2)var [Sˆ( )t ]+E2[Sˆ( )t ]2 t2var[Sˆ( )t ]
∆ +
var[Sˆ( )t ] = var[S t( )]+( )t
dvar [ ]S ---dt d
---dt ∆t
---2 d2var [ ]S dt2 --- ∆t
---2 d2 dt2
--- = (2 + 2)var [ ]S +(2 + 2)
+ + +
E[ ]S +E[ ]
( )22 ∆t2(var[S t( )] )
+ + +
d
---dt (2 + 2) ∆---2td2var[ ]S
dt2
--- E2[ ]S 2+2E[ ]ES [ ] 2+∆t2var[ ]S
–
=
var [ ]S = S( )0 2exp 2( t)(exp(2t)–1)
d
---dt = (2 + 2)–aexp 2(( + 2)t)+exp 2( t)(b–ct)
a tS( )0 2 2 22 4 ---2
+ +
∆
=
(5.134) (5.135) Solving Equation 5.132 subject to the initial condition 0, we obtain the following expression for the error of the variance, valid to first order in t:
(5.136) The quantity in the square bracket in the right-hand side of Equation 5.136 is negative. This tells us that the first-order discretization effect on the trajectories is a lower variance in the approximate process. As we saw in the previous section, the discretization effect is also a lower mean in the approximate process. We must emphasize that these conclusions are only applicable to the log-normal process. The main purpose of the analysis is to illustrate how nonintuitive these effects can be.
Of course, instead of working with the equation for the variance directly, we could have worked with the equations for the moments; the result would have been the same.
The next two figures show how significant the effect of discretization error can be. Figures 5.15 and 5.16 show a case where the log-normal stochastic differential equation was integrated with the Euler scheme using quarterly time steps. Especially for the variance, the numerical dis- tortion can be very significant when the drift is large and positive.
So far we have discussed the effect of discretization on the properties of the trajectories themselves. How does this translate into pricing errors?
How much accuracy can we get with the Monte Carlo method before we hit a computational barrier? We discuss this in the next section for a Euro- pean call.
Discretization Error and Computational Barriers for a European Call2
We use the log-normal process to gain insight into the effect of discretiza- tion on the derivative price. We analyze the case of a simple European call.
A European call on a process with deterministic volatility would not be
2 This analysis was done jointly with Dr. Ervin Zhao.
b = ∆tS( )0 22 c = ∆tS( )0 22
(t=0) =
tS( )0 2 exp 2(( + 2)t) 2t exp(–2t) 2 22 4 ---2
+ +
t
∆ –
=
FIGURE 5.15 Discretization error effect on the mean. Log-normal process inte- grated with Euler scheme with time step 0.25, 0.4. Normalized with S(0) 1.
FIGURE 5.16 Discretization error effect on the variance. Log-normal process inte- grated with Euler scheme with time step 0.25, 0.4. Normalized with S(0) 1.
0 1.5 0.75 3 2.25
3.75 –0.2 –0.08
0.04 0.16 –0.05
–0.04 –0.03 –0.02 –0.01 0
Time (years) Drift
Induced error in the mean
= =
0 0.75 1.5 3 2.25
3.75 –0.2
–0.08
0.04 0.16 –1
–0.8 –0.6 –0.4 –0.2 0
Induced error in the variance
Drift Time (years)
= =
priced by simulation, but this insight can be useful in cases where we need to use a numerical scheme to construct scenarios.
This is the strategy for analysis.
■ We compute the errors induced by discretization in the moments of the log-normal process.
■ We construct an approximate probability density function for the sim- ulated log-normal process using the Edgeworth expansion. This allows us to introduce the discretization error in the representation of the approximate pdf.
■ The discretization error of the call price is approximately the difference between the price obtained with the approximate probability density repre- sented by the Edgeworth expansion and the log-normal probability density.
As in the previous section, we start out with the approximate process, (5.137) We define the error in the moments of the underlying process as follows:
(5.138) We will not repeat the algebraic steps, but it is straightforward to see that this error is governed by the following ordinary differential equation:
(5.139) where
(5.140)
Solving Equation 5.139 subject to the initial condition (0) 0, we get the following expression for the discretization error of the moments:
Sˆ(t+∆t)–Sˆ( )t = Sˆ( )t ∆t+Sˆ( )t ∆W
E[ ]Sˆn = E[ ]Sn +n+ᏻ(∆t2)
dn
---dt –Ann = ∆---B2t nE[ ]Sn
A1=
An= n+12---n n( –1)2,n>1 B1 = 0
B2 = 22
B3 = 62+62
Bn = n n( –1)2+n n( –1)(n–2)2 1
4---n n( –1)(n–2)(n–3)4,n>3 +
=
(5.141) To obtain an expression for the discretization error of the option price, we express the probability density function (pdf) of the simulated price pro- cess as an Edgeworth asymptotic expansion (Abramovitz and Stegun, 1964) about a log-normal pdf whose first two moments are matched to the first two moments of the discretized process. Denoting the pdf of the simu- lated process at the option maturity by and the log-normal pdf with the first two moments matched by f(.), we have
(5.142)
with , where the summation on the right-hand side starts with i 3 because the first two moments are matched. Equation 5.142 is an asymp- totic expansion and will not converge in general. Here we will limit this expansion to the first two terms, which means that we match the first four moments. We determine the coefficients ai by requiring that the moments of
should equal the moments of its Edgeworth expansion.
With the definitions,
(5.143)
(5.144) we get the following recursive relationship for the coefficients in Equation 5.142:
(5.145)
In this equation, and n are the nth moments of and S, respectively, evaluated at t T, where T is the maturity of the call option.
We are now ready to get the effect of the numerically induced error on the option price. The values of the exact and simulated calls are
(5.146) n ∆t
---S2 ( )0 nBnt exp(Ant)
=
fˆ( ).
fˆ( ) f( ) ai ----i!∂if
∂i --- i=3 i=
∑
+
=
∈ =
fˆ
ˆn nfˆ( ) d
∫
0=
n nf( ) d
∫
0=
an ( )–1 n(ˆn –n) ( )–1 n i– n
i
ain i– i=1
i n= –1
∑
+
=
ˆn Sˆ
=
C exp(–rT) (S–K)f S( )dS
∫
K=
(5.147) where r is the risk-free rate and K is the strike. Using Equation 5.142, we find
(5.148)
The error in the price is then
(5.149)
To determine the number of Monte Carlo replications beyond which we cannot increase the accuracy of the simulation for a given integration time step (remember we assume we are using the Euler scheme to solve the stochastic differential equation), we equate C to the standard error of the option price estimator. The following table (5.5) shows exact call option prices for the same underlying process discussed in the previous
TABLE 5.5 Discretization error effect on a European call.
Strike T 1 T 1.25 T 1.5 T 1.75 T 2
30 21.434 21.789 22.143 22.496 22.847
35 15.731 17.179 17.627 18.071 18.510
40 12.253 12.834 13.401 13.951 14.486
45 8.311 9.021 9.693 10.333 10.946
50 5.193 5.957 6.673 7.352 8.000
55 2.997 3.709 4.389 5.043 5.674
60 1.608 2.190 2.772 3.350 3.921
65 0.811 1.236 1.691 2.165 2.650
70 0.387 0.671 1.002 1.368 1.759
Note: The bolded digits represent the accuracy bound consistent with a quarterly time step.
Log-normal process integrated with Euler Scheme with time step 0.25, 0.4. Spot 50, risk-free rate 0.1. Maturities in years.
Cˆ = exp(–rT) (S–K)fˆ( )S dS
∫
KCˆ C exp(–rT) ai ----i!∂if
∂i ---( )K i=3
i=
∑
–
=
C exp(–rT) ai ----i!∂if
∂i ---( )K i=3
i=
∑
–
=
= = = = =
= =
=
section. The integration step is quarterly. The bolded figures in the table indicate the digits that cannot be improved upon, given the fact that we are solving the trajectories by quarterly integration. Given that increasing the accuracy can only be achieved by reducing the time step, we see that moving the order of accuracy by one order of magnitude would cause the computational work to increase by a magnitude that scales like 103. This marks the onset of a computational barrier that becomes increasingly costly to overcome.
6
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