16.1 Simulation techniques for option pricing

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.


16.1.1 Introduction to simulation techniques

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

$\displaystyle \int_{[0,1]^d} f(x)dx$ (16.1)

by evaluation of $ f(x)$ for independent uniform distributed random vectors $ X_1, X_2, \ldots, X_n$, $ X_i \in [0,1]^d$.

The arithmetic mean of the values $ f(X_i)$ can be seen as a guess for the expected value of the random variable $ f(X_i)$ 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 $ \epsilon$ is limited and decreases to zero with an increasing sample size if the variance of $ f$ is finite.


16.1.2 Pricing path independent European options on one underlying

For the case of a European option on one underlying we have to approximate the following integral via Monte Carlo simulation:


$\displaystyle e^{r(T-t)}\textrm{E}\left[C_T(S_T)\vert S_t\right]$ $\displaystyle =$ $\displaystyle \int_0^\infty
C_T(S_T)g(S_T\vert S_t,r,\sigma,T-t)dS_T$ (16.2)
  $\displaystyle =$ $\displaystyle \int_{[0,1)}C_T\{f(x,S_t,r,\sigma,T-t)\}dx$ (16.3)

Where

$\displaystyle g(S_T\vert S_t,r,\sigma,T-t)=\frac{\exp\left\{-\frac{(\log S_T-(\...
...r-0.5\sigma^2)(T-t)))^2}{2\sigma^2(T-t)}\right\}}{\sqrt{2\pi\sigma^2(T-t)}S_T} $

is the risk neutral density function of the Black Scholes model with parameters:

$ S_T$ : price of the underlying at maturity
$ S_t$ : price of the underlying at time $ t$
$ r$ : risk free interest rate
$ \sigma$ : volatility of log returns of the underlying
$ T-t$ : time to maturity
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$\displaystyle S_T =f(x,S_t,r,\sigma,T-t)=S_t\exp\{(r-\frac{1}{2}\sigma^2)(T-t)+\sigma\sqrt{T-t}F^{-1}(x)\} $

transforms the uniform distributed values $ x$ in $ g(S_T\vert S_t,r,\sigma,T-t) $ distributed underlying values $ S_T$. $ F^{-1}(x) $ is the inverse of the cumulative normal distribution function and $ C_T(y) $ is the payoff function of the option.

The Monte Carlo simulation calculates the value of the integral in the following way:

  1. $ n$ independent random underlying values $ S_T^1\ldots S_T^n $ are generated by computing $ f(x,S_t,r,\sigma,T-t) $ for a set of uniformly distributed pseudo random numbers $ X_1,\ldots,X_n $.
  2. the option payoff $ C_T(S_T^i)$ is calculated for each $ S_T^i $.
  3. the value of the integral in (16.3) is then approximated by the arithmetic mean of the option payoffs:

    $\displaystyle \bar{C}=\frac{1}{n}\sum_{i=1}^n C_T(S_T^i) $

We will now derive an estimate of the approximation error of the arithmetic mean. We assume that $ S_T^1\ldots S_T^n $ are independent random underlying samples of the $ g(S_T\vert S_t,r,\sigma,T-t) $ density. Using this assumption we can conclude that $ \bar{C} $ is a random variable with expected value

$\displaystyle \textrm{E}[\bar{C}]$ $\displaystyle =$ $\displaystyle e^{r(T-t)}C_t(S_t)$  

Additionally we have to assume that the variance of the option payoffs $ C_T(S_T) $ is given by:

$\displaystyle \textrm{Var}\left[ C_T(S_T)\right] = \int_{[0,\infty ]} C_T(S_T)^2 g(S_T\vert S_t,r,\sigma ,T-t)dS_T-\textrm{E}\left[ C_T(S_T)\right ]^2$ (16.4)

exists. Then we get:
$\displaystyle \textrm{Var}\left[ \bar{C}\right] =\frac{1}{n^2}\sum_{i=1}^n\textrm{Var}\left[ C_T(S_T^i) \right] =
\frac{1}{n}\textrm{Var}\left[ C_T(S_T) \right]$     (16.5)

because of the independence of $ S_T^1,\ldots ,S_T^n $.

The expected value of the random variable $ \bar{C} $ equals the value of the integral $ e^{r(T-t)}C_t(S_t) $ and its variance converges to zero with increasing $ n$. The probability that the approximation error is greater than a fixed positive value decreases to 0 with an increasing number $ n$. A first estimation of the error is given by the Chebychev inequality for $ \bar{C} $,

$\displaystyle \textrm{P}\left(\vert\bar{C}-e^{r(T-t)}C_t(S_t)\vert\ge a\right)\le\frac{\frac{1}{n}\textrm{Var}\left[C_T(S_T)\right]}{a^2}$    

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 $ \bar{C} $ converges to a normal distribution for $ n\rightarrow \infty $. It follows that the difference between the approximation and the integral, $ \bar{C}-e^{r(T-t)}C_t(S_t) $ is approximately normally distributed with mean 0 and standard deviation

$\displaystyle \sigma_{\bar{C}} = \sqrt{\frac{\textrm{Var}\left[C_T(S_T)\right]}{n}}$ (16.6)

for large $ n$. According to Boyle (1977) a value of $ n>1000 $ is sufficiently large in order to use the normal distribution for error estimation purposes.

We get the following equation if we assume that $ \bar{C}-e^{r(T-t)}C_t(S_t) $ is normal distributed:

$\displaystyle \textrm{P}\left(\left\vert \bar{C}-e^{r(T-t)}C_t(S_t)\right\vert\...
...nt_{-\epsilon}^{\epsilon} \exp\left\{-\frac{u^2} {2\sigma_{\bar{C}}}\right\} du$ (16.7)

If we choose $ k$ as a multiple of the standard deviation $ \sigma_{\bar{C}}$ of $ \bar{C} $, then we get:


$\displaystyle \textrm{P}\left(\left\vert \bar{C}-e^{r(T-t)}C_t(S_t)\right\vert\le k\sigma_{\bar{C}}\right)$ $\displaystyle =$ $\displaystyle \textrm{P}\left(\frac{\left\vert\bar{C}-e^{r(T-t)}C_t(S_t)\right\vert}{\sigma_{\bar{C}}}\le
k\right)$  
  $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{2\pi}}\int_{-k}^{k} \exp\left\{-\frac{u^2}{2}\right\} du$  
  $\displaystyle =$ $\displaystyle p$ (16.8)

Given a fixed probability level $ p$, the error converges to zero with $ {\mathcal{O}}(1/\sqrt{n}) $. The error interval holds for $ k =1,2,3$ with the respective probabilities $ p=0.682,0.955,0.997 $

The confidence intervals for a given probability level depend on the standard deviation of the payoff function $ C_T(S_T) $:

$\displaystyle \sigma_{C_T} = \sqrt{ \textrm{Var}\left[ C_T(S_T) \right]} \; .$ (16.9)

In general, this standard deviation cannot be calculated with analytical methods. Therefore one calculates the empirical standard deviation $ \bar{\sigma}$ and uses it as a proxy for the error bounds:

$\displaystyle \bar{\sigma}=\sqrt{\frac{1}{n-1}\sum_{k=1}^n \left\{C_T(S_T^i)-\bar{C}\right\}^2} \; .$ (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 $ {\mathcal{O}}\left(1/\sqrt{n}\right) $.

Figure 16.1: Absolute error of a European Call option price calculated by Monte Carlo simulations vs. $ n^{-1/2} $
\includegraphics[width=1.3\defpicwidth]{XplorePutKonvDiag.ps}

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 $ {\mathcal{O}}\left(1/\sqrt{n}\right) $ for a fixed probability level $ p$. Second, the probability that a fixed error bound holds converges to 1 with $ {\mathcal{O}}\left(1/\sqrt{n}\right) $, 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).


16.1.3 Pricing path dependent European options on one underlying

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 $ S_{t_1}$, $ \ldots$, $ S_{t_m} $ at several time points $ t_1\le\ldots\le t_m $ 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.


16.1.4 Pricing options on multiple underlyings

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.