We introduce Monte Carlo techniques and Quasi Monte Carlo techniques for option pricing. First, we give an idea how to use simulation techniques to determine option prices, then - using the developed basic methods - we give examples how to price more complex i.e. exotic options even on more than one underlying. Finally we present a short guideline how to price exotic options with the proposed techniques.
First, we take a look at a European put on one underlying stock, a pricing problem which can be solved analytically e.g. by using the Black-Scholes formula. We start with this problem not only because it has become a kind of "standard problem" but also to have the possibility to compare the results of our approximation with an analytical solution. At the same time we look at the time-complexity of the used simulation technique. Next, we show how to price path dependent options with Monte Carlo methods. Afterwards, we show how to price a stock option on several underlyings. This implies that we have to solve a multi-dimensional simulation problem.
The idea behind randomized algorithms is that a random sample from a population (of input variables) is representative for the whole population. As a consequence, a randomized algorithm can be interpreted as a probability distribution on a set of deterministic algorithms.
We will see that there are three main advantages to randomized algorithms: 1. Performance: For many problems, it can be shown that randomized algorithms run faster than the best known deterministic algorithm. 2. Simplicity: Randomized algorithms are easier to describe and implement than comparable deterministic algorithms. 3. Flexibility: Randomized algorithms can be easily adapted.
In general one distinguishes two types of randomized algorithms. Las Vegas algorithms are randomized algorithms that always give correct results with only the variation from one run to another being its running time. Monte Carlo algorithms are randomized algorithms that may produce an incorrect solution for which one can bound the probability of occurrence. The quality of the solution can be seen as a random variable.
Within this chapter, we focus on Monte Carlo algorithms calculating the value of the following integral
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(16.1) |
The arithmetic mean of the values can be seen as a guess for
the expected value of the random variable
and therefore
can be interpreted as an approximation for the value of the integral.
According to the strong law of large numbers the estimator for the
expected value (the arithmetic mean of the random function values)
is converging to the expected value
(the value of the integral) with an increasing sample size.
The probability that the absolute error of the approximation result
exceeds a fixed positive value
is limited and decreases to
zero with an increasing sample size if the variance of
is finite.
For the case of a European option on one underlying we have to approximate the following integral via Monte Carlo simulation:
>
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: | price of the underlying at maturity |
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: | price of the underlying at
time ![]() |
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: | risk free interest rate |
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: | volatility of log returns of the underlying |
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: | time to maturity |
The Monte Carlo simulation calculates the value of the integral in the following way:
We will now derive an estimate of the approximation error of the arithmetic
mean.
We assume that
are independent random
underlying samples of the
density. Using this
assumption we can conclude that
is a random variable with expected value
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Additionally we have to assume that the variance of the option payoffs
is given by:
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(16.4) |
The expected value of the random variable equals the value
of the integral
and its variance converges to
zero with increasing
. The probability
that the approximation error is greater than a fixed positive value decreases
to 0 with an increasing number
. A first estimation of the error is given by
the Chebychev inequality for
,
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The bound given by this equation is rather imprecise since we do not make any
assumptions on the distribution of the random variable. Only the expected value
and the variance are used in the previous equation. According to the central limit theorem
the distribution of converges to a normal distribution for
. It follows that the difference between the approximation and the integral,
is approximately normally distributed with mean 0 and standard
deviation
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(16.6) |
We get the following equation if we assume that
is normal distributed:
If we choose as a multiple of the standard deviation
of
, then we get:
Given a fixed probability level , the error converges to zero
with
.
The error interval holds for
with the respective
probabilities
The confidence intervals for a given probability level depend on
the standard deviation of the payoff function :
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(16.9) |
In general, this standard deviation cannot be calculated with analytical methods.
Therefore one calculates the empirical standard
deviation
and uses it as a proxy for the error bounds:
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(16.10) |
Figure 16.1 shows the evolution of the absolute error of the price for
a European call option calculated by Monte Carlo methods compared with
the analytic solution. One can observe that the error tends to zero with
.
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We would like to give some of the main properties of algorithms using Monte Carlo techniques.
First from (16.9) it follows that the error bound tends to zero with
for a fixed probability level
. Second,
the probability that a fixed error bound holds converges to 1 with
, Mavin H. Kalos (1986). Since these results hold
independent of the dimension of the problem, which affects only the variance
of the payoff function with respect to the Black-Scholes risk neutral density,
the Monte Carlo method is especially well suited for the evaluation of option
prices in multidimensional settings. Competing pricing methods e.g finite
differences have exponential growing computational costs in the dimension of
the problem. Another advantage of the Monte Carlo
pricing method is the error estimate given by the empirical standard deviation
which can be computed with a small additional effort.
The two most important drawbacks of the Monte Carlo simulation, mentioned in literature are its small convergence speed compared to other techniques for options on few underlyings and the difficulties occurring for options with early exercise possibilities. For example, American options giving the investor the possibility to exercise the option at any time before and at maturity, are difficult to price. To evaluate an American option means to find an optimal exercise strategy which leads - using only basic Monte Carlo techniques - to a recursive algorithm with exponential time-complexity. But more advanced techniques using importance sampling methods show that Monte Carlo simulations can be applied to evaluate American contracts, Broadie (2000).
There are two categories of options.
Path-independent options are options whose payoff depends only on the
underlying prices at maturity. Path-dependent options are options whose payoff depends on underlying
price outcomes
,
,
at several time points
within the lifetime of the respective
option.
Within the group of path-dependent options one can distinguish options with a payoff function depending on a continuously defined path variable and options with a payoff function depending on a fixed number of underlying values. The price of an option with many - usually equally spaced - exercise dates is often approximated by the price of an option with a continuously defined path variable and vice versa.
Examples for path-dependent options are barrier options, lookback options, and Asian options. The latter have a payoff function which is linked to the average value of the underlying on a specific set of dates during the life of the option. One distinguishes two basic forms of Asian options: options on the geometric mean (for which the price can be calculated with standard techniques) and options on the arithmetic mean (for which the price can not be determined using standard approaches). Asian options are frequently used in commodity markets. The volatility of the underlying prices of the commodities is usually very high so that prices for vanilla options are more expensive than for comparable Asian-style options.
In this section we show how to extend the Monte Carlo simulation technique to higher dimensions. The problem is not only that one has to deal with higher dimensional integrals, but also that one has to incorporate the underlying correlation structure between the considered securities. In our framework we need the covariance matrix of the log returns on an annual basis.
In general, a basket option is an option on several underlyings (a basket of underlyings). Basket options can be European-, American or even Asian-style options. Normally, the average of the underlying prices is taken to calculate the price of the basket option, but sometimes other functions are used.
The advantage of the usage of basket options instead of a series of one dimensional options is that the correlation between securities is taken into account. This may lead to better portfolio hedges. We will look at a basket option on five underlyings where the underlying price of the best security in the basket is taken to calculate the option price.