The options we have priced so far are often referred to as “plain vanilla options”.
They are straightforward in that they have only one underlying and their value at maturity only depends on the value of their underlying at maturity. However, there is an abundance of other options nowadays that do not satisfy this assumption. Let us list some examples of theseexotic options:
8.4 Exotic Options and the Monte Carlo Method 299
8.4.1 Barrier Option
There are different variants of barrier options. A down-and-out call, e.g., has the same payoff as a call,providedthe price of the underlying never falls below a certain
“barrier” level. Otherwise the option pays back zero at maturity.
8.4.2 Asian Option
This is essentially a call option where the strike is given by theaverageof the price over time.
8.4.3 Fixed-Strike Average
This is a call option where instead of the price of the underlying, its average over times is used to compute the final payoff.
8.4.4 Variance Swap
The payoff of this option is determined by the difference between observed variance (i.e., the square of the volatility) and a predefined value.
8.4.5 Rainbow Option
Here not only one, but several underlying assets are used. The payoff of this option is determined by the average of these underlyings. However, as in the case of the barrier option, there is no payoff if a predefined barrier is hit. In the case of a rainbow option, however, it is sufficient ifoneof the underlyings falls below this barrier level at some point.
This (by no means complete) list of exotic options demonstrates not only the creativity of issuers, but also the need for more advanced methods for pricing of such options. It is important to notice that these exotic options are not all rare. In fact, some of them (in particular barrier options and rainbow options) are used to construct structured financial products for the retail market which are enjoying a huge popularity in recent years, particularly in Europe and East Asia.6 A typical structured product is the following “reverse convertible”: at maturity (after one year) this product yields the average return of three selected stocks, but not more than 10%. There is a capital protection, i.e., if this average is below the starting price at maturity, the starting price is paid instead, but this capital protection is only valid if
6Restrictive laws seem to hinder their success in the US.
none of the three stocks fell below of70% of its starting price during the one year period. We see immediately that we need several options to hedge such a structured product, in particular a rainbow option.
In fact, variants of this product are among the most popular structured products on the market. They are particularly popular with retail customers. For reasons for this popularity and a theoretical analysis of structured products see [Rie10,Rie12, HR14]. Fundamental for the understanding of these products is the study of the probability that a barrier is reached. For theoretical work on this, see [SR09].
How can we price exotic options? For many of these options there are by now sophisticated methods to obtain pricing formulas using similar methods as for the Black-Scholes formula. Some examples for this can be found, e.g., in [KK01, Chap. 4.1]. There are, however, cases where it is either not possible to follow this route or options are so new that there are simply no mathematical results available yet. In such situations numerical approximation methods are used extensively.
Moreover, they have the additional advantage that exchanging the underlying stochastic process with a more sophisticated and realistic variant than the geometric Brownian motion is usually much simpler.7In the following we will sketch just one of the many methods to price options numerically, theMonte Carlo method.
The key idea for this method is to use risk-free probabilities, also called equivalent martingale measure (see Sect.4.2.2) and to take the discounted expected value over the payoff of the option. This expected value can be computed by simulating n independent random paths for the underlying(s) and computing the value of the option in each of these cases.
How can we implement this idea? The first difficulty we encounter is that the price of the underlying follows a continuous stochastic process, but we can compute only finitely many values. Thus we need to discretize the process. In the case of a geometric Brownian motion this means that we use N discrete time steps t D 0;T=N; 2T=N; : : :, and generate for each time step an independent random number y.t/that follows a standard normal distribution. The approximation for a Brownian motionB.t/is then fornD1; : : : ;Ngiven by
B.nT=N/DB..n1/T=N/Cp
T=Ny.nT=N/:
Between these points, we can interpolateB.t/, e.g., by piecewise affine functions.
Having constructed a realization of the Brownian processB.t/, we can compute the priceS.t/and – given its path – the payoff of the option at maturity.
If we have several underlyings, the path of each of them can be constructed in the same way. However, if they are not independent (which will usually be the case for stocks or indices as underlyings) we need to construct a process that respects the correlations between them, which adds more difficulties.
The method relies heavily on a good source of random numbers. This is occasionally a crucial issue, since computers do not actually provide real random
7The need for studying other processes will be discussed in Sect.8.8.