Improving Monte Carlo: Control Variates

I’ve already discussed quite a lot about Monte Carlo in quantitative finance. MC can be used to value products for which an analytical price is not available in a given model, which includes most exotic derivatives in most models. However, two big problems are the time that it takes to run and the ‘Monte Carlo error’ in results.

One technique for improving MC is to use a ‘control variate’. The idea is to find a product whose price is strongly correlated to the product that we’re trying to price, but which is more easy to calculate or which we already know the price of. When we simulate a path in MC, it will almost surely give either an under-estimate or an over-estimate of the true price, but we don’t know which, and averaging all of these errors is what leads to the Monte Carlo error in the final result. The insight in the control variate technique is to use the knowledge given to us by the control variate to reduce this error. If the two prices are strongly correlated and a path produces an over-estimate of product price, it most likely also produces an over-estimate of the control variate and visa versa, which will allow us to improve our estimate of the product we’re trying to price.

The textbook example is the Asian Option. Although the arithmetic version of the asian option discussed in previous posts has no analytic expression in BS, a similar Geometric asian option does have an analytic price. So, for a given set of model parameters, we can calculate the price of the option. As a reminder, the payoff of an arithmetic asian option at expiry is

    \[C_{\rm arit}(T) = \Bigl({1\over N}\sum_{i=0}^{N-1} S(t_i) - K \Bigr)^+\]

and the payoff of the related geometric averaging asian is

    \[C_{\rm geo}(T) = \Bigl( \bigl(\prod_{i=0}^{N-1} S(t_i)\bigr)^{1\over N} - K \Bigr)^+\]

Denoting the price of the arithmetic option as X and the geometric option as Y, the traditional monte carlo approach is to generate N paths, and for each one calculate X_i (the realisation of the payoff along the path) and take the average over all paths, so that

    \[{\mathbb E}[X] = {1 \over N} \sum_{i=0}^{N-1} X_i\]

which will get closer to the true price as N \to \infty.

Using Y as a control variate, we instead calculate

    \[{\mathbb E}[X] = {1\over N} \sum_{i=0}^{N-1}\bigl( X_i - \lambda( Y_i - {\mathbb E}[Y] ) \bigr)\]

where {\mathbb E}[Y] is the price of the geometric option known from the analytical expression, and \lambda is a constant (in this case we will set it to 1).

What do we gain from this? Well, consider the variance of X_i - \lambda (Y_i - {\mathbb E } [Y])

    \[{\rm Var}\bigl( X_i - ( Y_i - {\mathbb E}[Y] ) \bigr) = {\rm Var}( X_i ) + \lambda^2 {\rm Var} ( Y_i ) - 2 \lambda {\rm Cov}( X_i Y_i )\]

(since {\mathbb E } [Y] is known, so has zero variance) which is minimal for \lambda = \sqrt{ {\rm Var}(X_i) \over {\rm Var}(Y_i) }\cdot \rho(X_i,Y_i) in which case

    \[{ {\rm Var}\bigl( X_i - ( Y_i - {\rm E}[Y] ) \bigr) \over {\rm Var}( X_i )} = 1 - \rho(X_i,Y_i)^2\]

that is, if the two prices are strongly correlated, the variance of the price calculated using the control variate will be a significantly smaller. I’ve plotted a sketch of the prices of the two types of average for 100 paths – the correlation is about 99.98%. Consequently, we expect to see a reduction in variance of about 2000 times for a given number of paths (although we now have to do a little more work on each path, as we need to calculate the geometric average as well as the arithmetic average of spots). This is roughly 45 times smaller standard error on pricing – well over an extra decimal place, which isn’t bad – and this is certainly much easier than running 2000 times as many paths to achieve the same result.

The relationship between payout for a geometric and an arithmetic asian option, which here demonstrate a 99.98% sample correlation
The relationship between payout for a geometric and an arithmetic asian option, which here demonstrate a 99.98% sample correlation. Parameters used were: r: 3%; vol: 10%; K: 105; S(0): 100; averaging dates: monthly intervals for a year

Asian options III: the Geometric Asian

I’ve introduced the Asian Option before, it is similar to a vanilla option but its payoff is based on the average price over a period of time, rather than solely the price at expiry. We saw that it was exotic – it is not possible to give it a unique price using only the information available from the market.

Today I’m going to introduce another type of option – the geometric asian option. This option is similar, but its payoff is now based on the geometric average, rather than the arithmetic average, of the spot over the averaging dates

    \[C(T) = \Bigl( \bigl( \prod_{i=1}^{N} S_i \bigr)^{1 \over N} - K \Bigr)^+\]

This option is exotic, just like the regular (arithmetic-average) asian option. At first sight it seems even more complicated; and what’s more, it is almost never encountered in practice, so why bother with it? Well, it’s a very useful as an exercise because in many models where the arithmetic asian’s price has no closed form, the geometric asian happens to have one! Further, since an arithmetic average is ALWAYS higher than a geometric average for a set of numbers, the price of the geometric asian will give us a strict lower bound on the price of the arithmetic asian.

Considering the Black-Scholes model in its simplest form for the moment (although the pricing formula can be extended to more general models), let’s consider what the spot will look like at each of the averaging times, and as we did in the earlier post, considering a simple geometric asian averaging over only two times t_1 and t_2 so that the payoff is C(T) = \Bigl( \sqrt{ S_1 \cdot S_2 } - K \Bigr)^+

At t_0S(t_0) = S_0. At t_1,

    \[S(t_1) = S_1 = S_0 \exp{ \Bigl\{ (r - {1\over 2} \sigma^2 )t_1 + \sigma \sqrt{t_1} x_1 \Bigr\} }\]

where x_1 \sim {\mathbb N}(0,1). At t_2,

    \[S(t_2) = S_2 = S_1 \exp{ \Bigl\{ (r - {1\over 2} \sigma^2 )(t_2 - t_1) + \sigma \sqrt{t_2-t_1} x_2 \Bigr\} }\]

where x_2 \sim {\mathbb N}(0,1) also, and importantly x_2 is uncorrelated with x_1 due to the independent increments of the Brownian motion.

Now the reason we couldn’t make a closed form solution for the arithmetic asian was that S_1 + S_2 is the sum of two lognormal distributions, which itself is NOT lognormally distributed. However, as discussed in my post on distributions, the product S_1 \cdot S_2 of two lognormal distributions IS lognormal, so valuing an asian that depends on the product of these two is similar to pricing a vanilla, with a slight change to the parameters that we need

    \[\begin{matrix} \sqrt{ S_1 \cdot S_2 } & = & S_0 \exp \Bigl\{ {1\over 2}(r - {1\over 2} \sigma^2 )(t_1 + t_2) + \sigma ( \sqrt{t_1} x_1 + {1\over 2} \sqrt{t_2 - t_1} x_2 ) \Bigr\} \\ & = & S_0 \exp \Bigl\{ {1\over 2}(r - {1\over 2} \sigma^2 )(t_1 + t_2) + {1 \over 2} \sigma \sqrt{ 3 t_1 + t_2 } x_3 \Bigr\} \end{matrix}\]

where x_3 is another normal variable (we’ve used the result about the sum of two normally distributed variables here).

If we re-write this as

    \[\begin{matrix} \sqrt{ S_1 \cdot S_2 } & = & S_0 \exp \Big\{ {1 \over 2} r ( t_1 + t_2 ) \Bigr\} \exp \Big\{ -{1 \over 2} \sigma^2 (t_1 + t_2) + {1 \over 2} \sigma \sqrt{ 3 t_1 + t_2 } x_3 \Big\} \\ & = & \sqrt{ F_1 \cdot F_2 } \exp \Big\{ {1 \over 2} \sigma^2 ( \tilde{t} - \tilde{\tilde{t}} ) \Big\} \exp \Big\{ -{1 \over 2} \sigma^2 \tilde{t} + \sigma \tilde{t} x_3 \Big\} \end{matrix}\]

where F_1 and F_2 are the forwards at the respective times and \tilde{t} and \tilde{\tilde{t}} are defined below. This is just the same as the vanilla pricing problem solved here. So, we can use a vanilla pricer to price a geometric asian with two averaging dates, but we need to enter transformed parameters

\begin{matrix} \tilde{t} & = & {1 \over 2} \sqrt{ 3t_1 + t_3 } \\ \tilde{\tilde{t}} & = & {1 \over 2} (t_1 + t_2) \\ F & \to & \sqrt{ F_1 \cdot F_2 } \cdot e^{{1 \over 2} \sigma^2 (\tilde{t} - \tilde{\tilde{t}}) } \\ \sigma^2 t & \to & \sigma^2 \tilde{\tilde{t}} \end{matrix}

In fact this result is quite general, we can price a geometric asian with any number of averaging dates, using the general transformations below (have a go at demonstrating this following the logic above)

 \begin{matrix} \tilde{t} & = & {1 \over n} \sqrt{  \sum_{i , j=0}^n {\rm min}(t_i, t_j) } \\ \tilde{\tilde{t}} & = & {1 \over n} \sum_{i =1}^n t_i \\ F & \to & \Big( \prod_{i=1}^n F_i \Big)^{1 \over n} \cdot e^{{1 \over 2} \sigma^2 (\tilde{t} - \tilde{\tilde{t}}) } \\ \sigma^2 t & \to & \sigma^2 \tilde{\tilde{t}} \end{matrix}

Asian Options II: Monte Carlo

In a recent post, I introduced Asian options, an exotic option which pay the average of the underlying over a sequence of fixing dates.

The payoff of an Asian call is

C(T) = \Big({1 \over N}\sum^N_{i=0} S_{t_i} - K \Big)^+

and to price the option at an earlier time via risk-neutral valuation requires us to solve the integral

C(t) = P_{t,T}\ \big( \iint\cdots\int \big) \Big({1 \over N}\sum^N_{i=0} S_{t_i} - K \Big)^+ \phi(S_{t_1},S_{t_1},\cdots,S_{t_N}) dS_{t_1}dS_{t_2}\cdots dS_{t_N}

A simple way to do this is using Monte Carlo – we simulate a spot path drawn from the distribution \inline \phi(S_{t_1},S_{t_1},\cdots,S_{t_N}) and evaluate the payoff at expiry of that path, then average over a large number of paths to get an estimate of C(t).

It’s particularly easy to do this in Black-Scholes, all we have to do is simulate N gaussian variables, and evolve the underlying according to a geometric brownian motion.

I’ve updated the C++ and Excel Monte Carlo pricer on the blog to be able to price asian puts and calls by Monte Carlo – have a go and see how they behave relative to vanilla options. One subtlety is that we can no longer input a single expiry date, we now need an array of dates as our function input. If you have a look in the code, you’ll see that I’ve adjusted the function optionPricer to take a datatype MyArray, which is the default array used by XLW (it won’t be happy if you tell it to take std::vector). This array can be entered as a row or column of excel cells, and should be the dates, expressed as years from the present date, to average over.

Asian Options I

Today I’m going to introduce the first exotic option that I’ve looked at in my blog, the Asian option. There is quite a lot to be said about them so I’ll do it over a few posts. I’ll also be updating the pricers to price these options in a variety of ways.

The simplest form of Asian option has a payoff similar to a vanilla option, but instead of depending on the value of the underlying spot price at expiry, it instead depends on the average the underlying spot price over a series of dates specified in the option contract. That is, the value of the option at expiry C(T) is the payoff

    \[C(T) = \Big( {1\over N}\sum^{N}_{i=0} S_{t_i} - K\Big)^+\]

where K is the strike, N is the number of averaging dates, and S_t is the value of the spot that is realised at each of these dates.

First of all, what does it mean that this is an exotic option? From the market, we will be able to see the prices of vanilla options at or near each of the dates that the contract depends on. As I discussed in my post on Risk-Neutral Valuation, from these prices we can calculate the expected distribution of possible future spot prices at each date – shouldn’t we be able to use this to calculate a unique price of the Asian option consistent with the vanilla prices?

The answer, unfortunately, is no. Although we know the distribution at each time, this doesn’t tell us anything about the correlations between the price at time t_1 and time t_2, which will be critical for the Asian option. To illustrate this, let’s consider a simple Asian that only averages over two dates, t_1 and t_2, and we’ll take t_2 to be the payoff date for ease, and try to calculate its price via risk-neutral valuation. The price at expiry will be

    \[C(t_2) = \Big( {1\over 2}\big(S_{t_1} + S_{t_2}\big) - K\Big)^+\]

To calculate the price at earlier times, we use the martingale property of the discounted price in the risk-neutral measure

    \[C(0) = N(0)\cdot{\mathbb E}^{RN} \Bigl[ \ {C(t_2) \over N(t_2)}\ |\ {\cal F}_0\ \Bigr]\]

and expanding this out as an integral, we have

    \[C(0) = \delta_{0,t_2} \int \int \Big( {1 \over 2} \big( S_{t_1} + S_{t_2} \big) - K \Big)^+\ p(S_{t_1},S_{t_2})\ dS_{t_1} dS_{t_2}\]

where \delta denotes the discount factor to avoid confusion with the pdfs inside the integral. From the market, we know both p(S_{t_1}) and p(S_{t_2}), which are respectively the market-implied probability distributions of the spot price at times t_1 and t_2 calculated from vanilla call prices using the expression we derived before

    \[p(S_{t}) = {1 \over \delta_{0,t}}{\partial^2 C(t) \over \partial K^2}\]

But, we don’t know the JOINT distribution p(S_{t_1},S_{t_2}) which expresses the effect of the dependence of the two distributions. We know from basic statistics that

    \[p(S_{t_2},S_{t_1}) = p(S_{t_2} | S_{t_1}) \cdot p(S_{t_1})\]

and since the spot at the later time depends on the realised value of the spot at the earlier time p(S_{t_2} | S_{t_1}) \neq p(S_{t_2}), so we don’t have enough information to solve the problem from only the market information available from vanilla options.

What is this telling us? It is saying that the market-available information isn’t enough to uniquely determine a model for underlying price movements with time. There are a variety of models that will all re-create the same vanilla prices, but will disagree on the price of anything more exotic, including our simple Asian option. However, observed vanilla prices do allow us to put some bounds on the price of the option. For example, a careful application of the triangle inequality tells us that

    \[\Big( {1\over 2}\big(S_{t_1} + S_{t_2}\big) - K \Big)^+ \leq {1\over 2} \Big( S_{t_1} - K_1 \Big)^+ + {1\over 2} \Big( S_{t_2} - K_2 \Big)^+\]

for any K_1 + K_2 = 2K; the expressions on the right are the payoff at expiry of call options on S_{t_1} and S_{t_2} respectively, and we can see the prices of these on the market, which allow an upper bound on the asian price. This means that, while we can’t uniquely price the option using the market information alone, we CAN be sure that any model consistent with market vanilla prices will produce an asian price that is lower than the limit implied by this inequality.

In order to go further, we need to make a choice of model. Our choice will determine the price that we get for our option, and changing model may well lead to a change in price – analysing this ‘model risk’ is one of a quant’s main jobs. A very simple choice such as Black Scholes [but note that BS might not be able to re-create an arbitrary vanilla option surface, for example if there is a vol smile at some time slices] will allow us to solve the integral above by numerical integration or by Monte Carlo (this option form still doesn’t have a closed form solution even in BS).

In the coming posts I’m going to develop the Monte Carlo pricer on the blog to price asian options in BS, and look at a number of simplifying assumptions that can be made which will allow us to find closed form approximations to the price of Asian options.

 

Finally, a brief note on the naming and usage of Asian options. Opinion seems divided – some people believe they are called Asian options because they’re neither European nor American, while some people say it’s because they were first traded in Japan. They are often traded so that the days on which the averaging is done are the business days of the last month of the option’s life. As the look up period is spread out over a period, they are less susceptible to market manipulation on any particular date, which can be important in less liquid markets, although the theoretical prices of Asian options like this are usually very similar in price to vanilla options at any of the look up dates. More exotic Asians might average over the last day of a month for a year. This average is much less volatile than the underlying over the same period, so Asian options are much cheaper than a portfolio of vanillas, but it’s easy to imagine that they might well match the risk exposure of a company very well, since companies trading in two or more currency areas will probably be happy to hedge their average exchange rate over a period of time rather than the exchange rate at any one particular time.