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European Pricing with Simulation
s we saw in Chapter 3, the price of a derivative security can be ex- pressed as an expectation of discounted payoffs. In the case of European derivatives, where the times when future cash flows occur are known, Monte Carlo simulation is a robust and well-established way to get an esti- mate of this expectation. In the case of derivatives with early exercise fea- tures, such as American or Bermudan options, pricing with simulation is a much more challenging problem. It is only due to very recent advances that simulation can be viewed as a practical approach for pricing early exercise derivatives.
In this chapter we will discuss some of the standard methods and issues associated with pricing European derivatives. The discussion on methods and issues is by no means exhaustive, but it consists of a selection made on the basis of practicality and usefulness.
Monte Carlo in Pricing
In this case we are interested in computing (or estimating) an expectation. In the case of a European derivative, we are interested in the expectation
(5.1) where B(.) is the normalizing asset and V(T) is the known payoff at matu- rity.
In the case of an option with early exercise, we are interested in (5.2) where is a stopping time and F() is the payoff at t . What is impor- tant about these two problems is that we are trying to compute expecta- tions and we assume that the distribution properties of the function whose expectation we want are determined by underlying processes (Ito or Poisson processes, for example). These processes are given to us as part of the problem formulation, and we accept the distributions that result for . Typically, we are not looking for properties other than the expectation. In applying Monte Carlo to pricing a European derivative, we face two challenges.
■ How do we construct the function from the underlying processes?
■ How do we estimate the expectation efficiently and accurately?
The first item is important, because we may not have an analytical for- mula for as a function of the underlying processes. In fact, in many cases of practical interest, we don’t. The second item is important because we must get the answer with known error bounds (if at all possible) and within time constraints.
In computing the distribution of the discounted payoffs, we must work with processes that are specified in an appropriate measure. In its simplest form, Monte Carlo pricing is carried out in the pricing measure used to derive the derivative price. However, since we are only interested in an expectation, we do not necessarily have to carry out our simula- tion in the pricing measure. We may be able to carry out our simulation in a measure other than the pricing measure, but one that is more suitable for speed and accuracy. Or, we may be able to work with a function that is not the discounted payoff, but one with the same expectation as the discounted payoff. These aspects are quite different in risk management applications of simulation.
V( )0 B( )E0 0B V T( ) B T( ) ---
=
V( )0 B( )0 sup
∈{0,T}E0B F( ) B( ) ---
=
=
V T( ) B T( ) ---
V T( ) B T( ) ---
V T( ) B T( ) ---
In summary, here are the main challenges in simulation applied to pricing.
■ Speed: The computation of the expectation should be fast enough to be satisfactory for trading. At the time of this writing, this amounts to the order of a second or less on high-end desk computers.
■ Accuracy: Accuracy should be good enough for trading and hedging.
■ Early exercise: Although significant advances have occurred in this area, at the time of this writing the use of simulation in early exercise still has room for significant improvements. We will discuss this topic extensively in the next chapter.
Monte Carlo in Risk Management
The applications of Monte Carlo simulations to risk management have to do with estimating losses that can occur with a given probability over a given time horizon. The area where this is relevant is known as value-at- risk computations (Jorion (2000) is a comprehensive source on this sub- ject). In value-at-risk-computations we want to arrive at statements such as,
“We are 90 percent confident that over the next 24 hours there is less than a 1 percent chance that losses will exceed 20 million dollars.” This state- ment tells us that in risk management we are typically interested in resolv- ing the tails of distributions. This poses different challenges than estimating the expectation, which is what matters in pricing.
The appropriate measure for risk management applications of simula- tion is the market or real world measure. This may or may not be critical, depending on the time horizon and the nature of the portfolio. Furthermore, the fact that we are interested in events that occur in the tails of distributions also means that we are interested in the dependency structure of extreme events. This dependency is not properly described by correlations and brings in considerations quite different from what is relevant to pricing. A good reference in this area is in Embrechts, Resnik, and Samorodnitsky (1999).
Furthermore, unlike pricing applications, the demand on accuracy in risk management applications refers to the aggregation of a large number of financial contracts, not to the individual contracts. The same can be said of computational speed. We will not elaborate on risk management applications of simulation in this book.
To summarize, the main challenges of simulation in risk management are presented below.
■ Speed: The entire portfolio of the institution must be completed over a period of a few hours. This may mean that the methodology of pricing used for the purpose of risk management could be different than the one used for trading.
■ Reliability: The results obtained must pass backtesting standards. This is a modeling issue, highly influenced by statistical considerations.
■ Relevance: This refers to the question of whether the results of simula- tion can be used for decision making or purely for compliance with reg- ulatory requirements.