the stability of the numerical schemes used to solve the pricing equation. A more subtle observation, however, is that the coefficient in front of can in principle be of a very different magnitude than the other coefficients in the equation, depending on the definition of f(S). This may also signifi- cantly add to the numerical difficulty (Zvan et al., 1997–1998).
In practice, the additional convective term turns out not to be a significant difficulty if one considers the case of discrete sampling. Discrete sampling will be the main approach to path dependency in this book. The interested reader is referred to reference Zvan et al., 1997–1998, as a good example of work on the development of robust algorithms for the continuous case. An alternative approach to getting around the problem of absent diffusion in a particular coor- dinate is to introduce artificial diffusion in that direction and then obtain the limit as the artificial diffusion vanishes. This can be done numerically quite efficiently.
Discrete Sampling of Path Dependency
Discrete sampling of path dependency is of course a better approximation of what really happens, since the movement of the underlying processes can only be observed at discrete points in time. Discrete sampling, however, can also be viewed as a means of dealing with the problem introduced by the additional convective term and the absence of corresponding diffusion. The key observa- tion is that the value of the option immediately before the sampling time and immediately after the sampling time must be the same. This will be the case as long as sampling itself does not trigger cash flows. Denoting the sampling times by and the times immediately preceeding the sampling time by
this continuity condition is expressed as
(3.129) In Chapter 7 we will refer to this condition as displacement shock. The pricing equation between sampling times is obtained by applying Ito’s lemma to the relevant underlying processes. Since g(S,t) is constant between sam- pling times g(S,t) does not appear in the pricing equation. Between sampling times we must solve
(3.130) where the term f(S) does not enter. This equation must be solved within each sampling interval subject to initial conditions derived from the conti- nuity condition above.
The initial condition at the beginning of each sampling interval must be extracted from the solution at the end of the previous sampling interval such that the continuity conditions are satisfied. Notice that although the g dimension has
∂V
∂g ---
ti,i = 1, 2,…, ti–,i = 1, 2,…,
V S t( ( ),i g t( i, ti), ti) = VS t( ),i– ( g t( i–, ti–), ti–), i = 1 2, ,…
∂V∂t
--- rS∂V
∂S
--- 1
2---
+ + 2S2∂2V
∂S2
--- = rV
∂V
∂g ---
dropped from the PDE, the solution itself still has the g dimension. What has hap- pened is that the changes in the g dimension, which in the continuous sampling case occur through the differential equation, here occur through the initial con- ditions in each sampling interval. The application of the continuity condition given by Equation 3.129 has the effect of concentrating the convection at the boundaries of sampling intervals.
The practical implementation of the continuity condition or displace- ment shock will be discussed in greater detail in Chapter 7. The continuity condition is also referred to as a jump condition in the literature (Wilmott, DeWynne, and Howison, 1993).
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Scenario Generation
enerating scenarios of the underlying processes that determine the deriv- ative’s price is an essential and delicate task from an analytical perspective as well as from a system design viewpoint. This chapter sets up the nomen- clature we will use in the remaining chapters to refer to scenarios and describes the main issues and methods for generating scenarios for pricing.
The two main applications of scenarios are risk management and pric- ing. The objectives of using scenarios in risk management are to obtain dis- tributions of possible gains and losses as a function of future time and to determine the concentration of risk among various components of a portfo- lio of instruments. The main objective of scenarios in pricing is to compute the expectation that gives us the value of the financial instrument. Scenarios are used when this is done by simulation, the topic of the next two chapters.
These two applications have different implications in the methods used to generate scenarios. The most obvious difference is that if we are interested in determining probabilities of future gains and losses, we would expect our sce- narios to be based on real world probabilities. If we are using the scenarios for pricing, on the other hand, the scenarios must be based on the probabilities associated with the measure induced by the numeraire asset, as we discussed extensively in last two chapters. If we don’t do this, our expectation will give us the wrong price. When generating scenarios for pricing, using the right measure is essential. When generating scenarios for risk management, however, which probability measure we use to produce the scenarios is a much less significant issue. This is the case because the dispersion of values, especially over short time horizons, is primarily dominated by the volatility and correlation of the under- lying processes, rather than by their drifts. Since the difference in probability measure is determined by the drift of the underlying processes, as long as the time horizon is not large it may not matter whether we do value-at-risk analysis using scenarios meant for pricing. It must be clear, however, that doing pricing with scenarios meant for risk management is not possible in general.
This distinction between scenarios for pricing and scenarios for risk man- agement has a bearing on the design of pricing and risk management systems.
G
We can visualize a situation where value-at-risk analysis is conducted on a portfolio of instruments where some of the instruments are priced by simula- tion. In designing a system it is important to have an architecture that allows for proper separation between pricing and risk management scenarios.
In this chapter we will discuss issues pertaining to scenarios for pricing through simulation.