The Hidden Risk in Simplicity

Finance Published: October 10, 2007

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Hidden Costs of Volatility Drag

The European call option with an exercise price of $22 is a relatively simple example, but its valuation is not as straightforward as one might expect.

That said, in the context of basic Monte Carlo methods simulation, we wish to price a European call option with an exercise price of $22 and payoﬀfunction V (ST) = (ST −22)+. Assume for the present that the interest rate is 0%.

In this case, since the distribution is very simple, we can price the call option explicitly; EQV (ST ) = EQ(ST −22)+ = (23 −22) 4 16 + (24 −22) 1 16 = 3.8

However, the ability to value an option explicitly is a rare luxury. An alternative would be to generate a large number (say n = 1000) independent simulations of the stock price ST under the measure Q and average the returns from the option. Say the simulations yielded values for ST of 22, 20, 23, 21, 22, 23, 20, 24.

CHAPTER 3. BASIC MONTE CARLO METHODS

The law of large numbers assures us for a large number of simulations n, the average V (ST ) will approximate the true expectation EQV (ST). In fact, while it would be foolish to use simulation in a simple problem like this, there are many models in which it is much easier to randomly generate values of the process ST than it is to establish its exact distribution.

Randomly generating a value of ST for the discrete distribution above is easy, provided that we can produce independent random uniform random numbers on a computer. For example, if we were able to generate a random number Yi which has a uniform distribution on the integers {0, 1, 2, 15} then we could deﬁne ST for the i0th simulation as follows: If Yi is in set {0}{1, 2, 3, 4}{5, 6, 7, 8, 9, 10}{11, 12, 13, 14}{15} define ST = 20

Of course, to get a reasonably accurate estimate of the price of a complex derivative may well require a large number of simulations, but this is decreasingly a problem with increasingly fast computer processors. The first ingredient in a simulation is a stream of uniform random numbers Yi used above. In practice all other distributions are generated by processing discrete uniform random numbers. Their generation is discussed in the next section.

Uniform Random Number Generation

The ﬁrst requirement of a stochastic model is the ability to generate “random” variables or something resembling them. Early such generators attached to computers exploited physical phenomena such as the least significant digits in UNIFORM RANDOM NUMBER GENERATION 99 an accurate measure of time, or the amount of background cosmic radiation as the basis for such a generator, but these suffer from a number of disadvan- tages.

They may well be “random” in some more general sense than are the pseudo-random number generators that are presently used but their properties are diﬃcult to establish, and the sequences are impossible to reproduce. The ability to reproduce a sequence of random numbers is important for debugging a simulation program and for reducing its variance. It is quite remarkable that some very simple recursion formulae deﬁne se- quences that behave like sequences of independent random numbers and appear to more or less obey the major laws of probability such as the law of large num- bers, the central limit theorem, the Glivenko-Cantelli theorem, etc. Although computer generated pseudo random numbers have become more and more like independent random variables as the knowledge of these generators grows, the main limit theorems in probability such as the law of large numbers and the central limit theorem still do not have versions which directly apply to depen- dent sequences such as those output by a random number generator.

The fact that certain pseudo-random sequences appear to share the properties of inde- pendent sequences is still a matter of observation rather than proof, indicating that many results in probability hold under much more general circumstances than the relatively restrictive conditions under which these theorems have so far been proven. One would intuitively expect an enormous diﬀerence between the behaviour of independent random variables Xn and a deterministic (i.e. non- random) sequence satisfying a recursion of the form xn = g(xn−1) for a simple function g. Surprisingly, for many carefully selected such functions g it is quite diﬃcult to determine the diﬀerence between such a sequence and an indepen- ent sequence. Of course, numbers generated from a simple recursion such as this are neither random, nor are xn−1 and xn independent.

Numbers generated from a recursive formula typically exhibit statistical properties that are typical of sequences with high levels of dependence. The distribution of the differences between consecutive values may be influenced by the underlying function g, leading to an apparent lack of independence between the variables. While it is true that many random number generators produce seemingly unpredictable output, the relationship between the generated numbers and their predecessors can often be predicted using statistical methods.

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SECTION HEADERS

The Hidden Cost of Volatility Drag

The European call option with an exercise price of $22 is a relatively simple example, but its valuation is not as straightforward as one might expect.

In this case, since the distribution is very simple, we can price the call option explicitly; EQV (ST ) = EQ(ST −22)+ = (23 −22) 4 16 + (24 −22) 1 16 = 3.8

However, the ability to value an option explicitly is a rare luxury.

On the flip side..., it's essential to recognize that while explicit valuation may not be feasible for every option, simulation can still provide valuable insights into the underlying dynamics of these securities.

In this case, since the distribution is very simple, we can price the call option explicitly; EQV (ST ) = EQ(ST −22)+ = (23 −22) 4 16 + (24 −22) 1 16 = 3.8

However, the ability to value an option explicitly is a rare luxury.

A large number of simulations is required to obtain an accurate estimate of the call option's price. This can be challenging because the payoff function V (ST) and the underlying distribution are not well-defined for all values of ST. Moreover, the random numbers used in the simulation may not accurately reflect the real-world behavior of the stock.

That said, on the flip side..., it is essential to recognize that while explicit valuation may not be feasible for every option, simulation can still provide valuable insights into the underlying dynamics of these securities.

Why Most Investors Miss This Pattern

Most investors do not take advantage of Monte Carlo methods because they are not familiar with the underlying mathematical concepts and simulations. They often rely on intuition or anecdotal evidence to make investment decisions, which may lead them astray.

In this case, since the distribution is very simple, we can price the call option explicitly; EQV (ST ) = EQ(ST −22)+ = (23 −22) 4 16 + (24 −22) 1 16 = 3.8

However, the ability to value an option explicitly is a rare luxury.

Real-World Applications of Monte Carlo Methods

Monte Carlo methods have numerous real-world applications in finance and other fields. For example, they can be used to model complex financial instruments such as credit default swaps (CDS) or options on foreign currencies.

That said, in the context of basic Monte Carlo methods simulation,

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