This is typically a far more powerful way of reducing the variance in pric- ing than using antithetic variates. When properly designed, computations can be accelerated by one or even two orders of magnitude.
In financial applications of the control variate method, the emphasis has traditionally been on using a closely related financial instrument whose value is known analytically to compute the value of another instrument by simulation. The combination of the two instruments allows us to construct an estimator with much less variance if both instruments are closely related.
The best-known example in finance is the arithmetic average Asian option, priced by simulation using the geometric average Asian option as control variate. While there is a closed form solution of the geometric average option, there is no analytical solution in the arithmetic average case.
The fact that we are pricing a financial instrument does not mean that we have to use another financial instrument as control variate. In fact, any other function of the same underlying processes whose expectation is known, and which is highly correlated with the instrument we are interested in will work. Also, we don’t have to know the control variate’s expectation analyti- cally. Even if we know the control variate expectation numerically, the tech- nique will work as long as we have a reasonably good assessment of the control variate expectation.
We need some nomenclature and definitions.
■ is the uncontrolled estimator of the value of the instrument we are interested in.
■ is the controlled estimator. This estimator has an expectation equal to or very close to the expectation we are seeking, but its variance is much smaller than that of .
V Vc
V
■ is the estimator of the value of another instrument closely related to the one we are interested in. We call this instrument the control variate instru- ment. In general, the control variate does not have to be a valid financial instrument at all, although in most cases it is. For simplicity, we will refer to the control variate as an instrument, or simply as the control variate.
■ Va is the value of the control variate instrument. It is important that we should know this value very accurately.
■ is a constant to be chosen optimally.
■ is the variance of the control variate.
■ is the variance of the instrument we are interested in.
The controlled estimator is constructed as follows:
(5.32) In terms of samples, this can be written as
(5.33) We can select an optimal by minimizing the variance of the con- trolled estimator:
(5.34) The minimum variance corresponds to
(5.35) This gives
(5.36) The optimal value of that minimizes the variance of the controlled estimator is
(5.37)
With this value of , the minimum variance of the controlled estimator is (5.38) Va
V2a
V2
Vc = V+(Va–Va)
Vc 1 n--- Vj
j=1 j n=
∑
Va 1n--- Vja j=1 j n=∑
–
+
=
var[Vc] V2 2V V, aVVa 2Va
+ 2 –
=
∂var[Vc] --- = 0
2V,VaVVa
– 2Va
2 =0 +
opt
V,VaVVa
Va
---2
=
var[Vc]min var[ ]V 1
V V, a
– 2
( )
=
If the control variate estimator and the uncontrolled estimator were perfectly correlated, we would be able to compute the value of the instru- ment we are interested in with one Monte Carlo cycle. This makes sense, because if both the instrument we want to price and the control variate instrument were perfectly correlated, then the control variate would simply be a rescaling of the instrument we are pricing. This implies that pricing the control variate instrument was essentially the same as pricing the instru- ment we want, save for a constant. This situation would not materialize in practice, but we can get fairly close to it.
Efficiency of Control Variates
The acceleration factor, Equation 5.21, becomes
(5.39) In order to benefit from the control variate, the correlation coefficient between the instrument and the control instrument must be arranged such that
(5.40) Equivalently,
(5.41) If it took as much time to sample the control instrument as the original instrument, , and the minimum correlation coefficient for which the control variate technique will work is
(5.42) This is a fairly severe limitation. The reason is that the correlation coef- ficient between control and instrument enters as 2. We need to have very tight correlations in order to gain significantly in speed. The minimum value of increases rapidly as the incremental effort to compute the control variate increases.
Case Study: Application of Control Variates to Discretely Sampled Step-Up Barrier Options
This simple example shows some of the practical issues to be considered in applying control variates. The traditional approach to control variates has been to select a highly correlated financial instrument as a control variate, for
y var[ ]V var[Vc]
---
1–2 ---
= =
1–2 --->1
> 1–
= 12---
1–0.5, min = 0.707
which the price is known, preferably analytically. This approach is typically limited to academic examples (the best-known one is the arithmetic Asian option computed with the geometric Asian as control variate).
A much more practical approach is to select a control variate that can be solved numerically very accurately with a simple and reliable implemen- tation, by simulation or other methods, such as finite differences or trees.
This numerically “priced” control variate can then be used repeatedly to price variations of the instrument of interest. We put priced in quotes because, as we said earlier, the control variate does not need to be a mean- ingful financial instrument, although in most cases it will be. We now use a discretely sampled barrier option to illustrate this case.
Consider the pricing of a weekly monitored European knockout call with maturity one year, such that in the first half of the year the barrier, H, is set to $125, and in the second half of the year it is set to $127. Assume the underlying process is log-normal with risk-free rate r 0.07, dividend yield d 0.02, and volatility 0.2. This problem is very simple to solve by simulation. If the spot price S(0) 100, the price of the option is about $3, and the variance of the uncontrolled estimator is approximately 35. This means that the number of cycles needed to estimate the price within $0.01 is There are several ways to select an effective control variate. Intuitively, we can think that a discretely sampled barrier option with a constant barrier could be a reasonable control variate. This is very attractive, because the case of a discretely sampled constant barrier can be solved almost analyti- cally by applying a correction to the continuously sampled case (Broadie, Glasserman, and Kou, 1996). However, the approximate nature of this solu- tion will contaminate the expectation of the controlled estimator. A more practical and robust approach is to use a discretely sampled constant barrier option that has been priced using finite differences. If the finite difference technique works well for the constant barrier option, why not use it also for the case we are interested in? We can certainly do this. However, there are three reasons why we may still want to use simulation for our case and finite differences for the control variate. One reason is that applying finite differ- ences to a constant barrier option is extremely simple; another is that the price obtained with finite differences is reliable and very accurate; and yet another reason is that this control variate can be reused for different config- urations of the instrument we are interested in. The main requirement of a good control variate is that the price must be known very accurately. A fast computation of the control’s price is welcome, but it is not a critical require- ment if the control can be used more than once.
The fact that our problem contains a barrier that steps up in the second half of the year means that the second half of the barrier will be less likely to be hit when compared with the second half of the barrier of the control, if
=
= =
=
35 0.012
---≈350,000.
everything else stays the same. If we can somehow change something about the control to decrease the likelihood that the barrier will be hit in the sec- ond half of the year, it is possible that we will get a better control variate.
One way we can accomplish this is to use a different underlying process for the control. For example, if we use a process with a lower volatility for the control instrument, the likelihood of the barrier being hit in the second half of the year will decrease, and we would expect that a slightly less volatile underlying will lead to a more efficient control. Another way to accomplish a similar effect is to increase the dividend yield of the control underlying. If we do this, the underlying price of the control variate will grow less on the average in the risk neutral world, and will therefore be less likely to hit the barrier compared with the case of lower dividend yield.
For the parameters we are discussing here, this is a very effective way of producing a control variate. Figure 5.1 shows the correlation coefficient of the uncontrolled estimator and the control variate estimator for varying divi- dend yield of the price process used in the control variate. We see that if the control variate is constructed with the same process as that of the problem we are trying to solve, the correlation coefficient between the uncontrolled esti- mator and the control variate is approximately 0.87. Assuming that the con- trol variate doubles the computational effort (very much the case here), this translates into an acceleration factor approximately equal to 2. If we solve the
FIGURE 5.1 Effect of yield of control variate process on correlation between uncontrolled estimator of discretely sampled barrier and control variate. European knockout call with H 125 for t< 0.5 and H 127 for t> 0.5, S(0) 100, K 100, r 0.07, d 0.02, 0.2, T 1. Control variate parameters are identical except for the barrier: H 125 for and the dividend yield varies as shown.
0.86 0.88 0.90 0.92 0.94 0.96 0.98
0.02 0.025 0.03 0.035 0.04 0.045 0.05 Dividend yield of control variate process
Correlation with control variate
= = =
= = = = =
= 0≤ ≤t T
control variate with an underlying price process with a dividend yield of 0.035, the correlation coefficient goes up to approximately 0.97. This means that the acceleration factor is now approximately 8.4. This is a very signifi- cant gain. We can now get the desired accuracy almost ten times faster.
In this case, the variance of the control variate is approximately 23, and the covariance of the instrument with the control variate is approxi- mately 28. This gives The controlled estimator is then,
(5.43) The value of the step-up barrier option is 2.962. The expectation of the control variate with d 0.035 is 2.452.