- The Stochastic Process Followed by the Volatility
- The Merton-Garman Equation
- The ``Straightforward'' Solution when
- A Quantum Mechanical Formulation of the Problem
- Merton's Theorem
- An Extension to Merton's Theorem
- Risk-Neutrality
- Brownian Motion on a Riemmanian Manifold

Since the appearance of the analysis by Black and Scholes, several people have tried to extend it and relax the assumptions on which the theory has been based. Merton[9] dropped the assumption of constant interest rates and showed that in this case, an option can be priced in terms of a bond price. In the same paper, Merton also showed how the Black-Scholes formula can be extended to cover the situation in which the volatility is a deterministic function of time. This is more realistic as the strong assumption of constant volatility is known not to be true[10]. Research has also been done assuming different processes for the evolution of stock prices by Merton[11], Cox and Ross[12] and Jones[13]. Cox and Ross[12] and Rubinstein[14] have solved the problem for the case when the volatility is a function of the underlying security price.

Empirical evidence investigating the distribution of stock returns has shown mixed results. Kon[15] finds that the observed distributions are consistent with stochastic volatility while Scott[16] shows that the hypothesis that stock returns are distributed independently over time can be rejected. Bodurtha and Courtadon[17] and Hull and White[18] also support the hypothesis of stochastic volatility. Considering these results, it seems reasonable to model volatility as another stochastic variable.

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where and are constants representing the mean reversion strength and the mean value of the volatility respectively. All the processes above except for (4.3)

Hence, to keep our discussion as general as possible, we have chosen this stochastic process for the volatility. Note that this process has one more free parameter as compared to the others, namely . This allows for greater flexibility in the model. Most of the processes considered by researchers have been mean-reverting as there is some empirical evidence to show this

The above, however, does not completely define the process followed by
the volatility as there is still a possibility of a correlation
between the white noise in the stock price process and , the
white noise in the volatility process. Again, to keep the discussion
as general as possible, we will assume that the correlation is ^{}.

The process we are considering is

where
and are constants,
and and are white noise processes with correlation
. Using Ito's lemma, we obtain the following expression for
the process followed by a derivative dependent on the underlying
security and the volatility of that security

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We now consider two different options, and on the same
underlying security with strike prices and maturities given by
and respectively. We form a portfolio

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to obtain

Since the portfolio is now risk-less, it must increase at the risk-free
interest rate by the principle of no arbitrage. In other words, we
must have

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is difficult to estimate empirically and there is some
evidence that it is non-zero[27]. To estimate this
quantity, we consider the Cox, Ingersoll and Ross model where the
consumption growth has constant correlation with the spot-asset
return. This gives rise to a risk premium which is proportional to the
volatility. We assume this model for simplicity as it has only the
effect of redefining in the above equation. Henceforth, we shall
assume that the market price of risk has been included in the
Merton-Garman equation by redefining . Therefore, the
Merton-Garman equation for the process we are considering is

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where is the probability distribution function for the mean of the volatility (which is a delta function for a deterministic process) and and are the same variables as defined in the previous chapter.

We will give two simple examples to illustrate this. First, let us
consider a deterministic process. We will choose the process

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We will now consider a stochastic volatility process. We
choose^{}

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However, it is usually extremely difficult to find the distribution of the mean of the volatility and hence, we usually perform a Monte-Carlo simulation of the stochastic process, finding the average volatility and the Black-Scholes price using that average volatility at each step. The prices can then be averaged over to give an estimate of the final solution. For most processes, a few tens of thousands of configurations suffice to give a solution accurate to about 4 significant figures which is more than sufficient for our purpose. Such a simulation generally takes only a couple of minutes on a Pentium 200.

We will derive Merton's theorem once we have laid the foundation of the theory for stochastic volatility. The theorem follows naturally and elegantly from the quantum mechanical formulation of the theory.

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While this looks deceptively simple, no analytic solution has been obtained for this equation. The special case was solved using a series method by Hull and White[19] and using elementary probability techniques by Heston[20]. A solution for was obtained by Baaquie[8] using the path integral formalism of quantum mechanics.

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To determine a Lagrangian for the problem, we will have to discretise
it. We discretise the time so that there are time steps.The time
step is then
. The continuous variables
and are then discretised to and where . The operator
can
then be decomposed to

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We see that if we can find , we can find the propagator and hence the option price. Therefore, let us look at this quantity more closely. Before we consider this quantity for the stochastic volatility case, let us consider the Black-Scholes (constant volatility) case as it is simpler and retains the essential features.

In the Black-Scholes case, we only have one variable (as is
just a constant). We write

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where .

For the stochastic volatility case, we have

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which can be simplified to

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It should be emphasized that the above Lagrangian is only exactly correct in the limit and the complete Lagrangian may include terms of order and greater apart from the above expression.

where we define

(again and ). We note that the action is quadratic in . This enables us to integrate over the stock price.

We also define

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We will now find . The -dependent term in the
Lagrangian is

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The integration over is easily performed. We obtain

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The above integration can be repeatedly performed over all the
variables to obtain

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which, on taking the limit, , becomes

(for the term arises from the fact that and it will be easily seen that when , that term is replaced by ) where

when follows the random process (4.4). Hence, if we can find the joint probability density functions for and with following the given stochastic process, we can get an analytic solution for the problem which will be given by

where is the joint probability distribution function. However, we were unable find the joint probability distribution function for the above quantities.

Hence, we retain the discrete solution which finally gives us

We are now in a position to derive a Monte-Carlo algorithm to calculate option prices with the volatility performing the stochastic process (4.4) with any correlation with almost the same efficiency as the straightforward solution when . However, the method has a disadvantage in that it cannot handle lumpy dividends (for a continuous dividend yield , we can just replace by ).

The last expression is particularly interesting as which finally determines the option price is the same as that for the Black-Scholes case with replacing . In other words, we just have to replace the constant volatility in the Black-Scholes equation by the average volatility during the time period under consideration.

This is precisely the substance of Merton's theorem. While we have assumed a specific process for the volatility, the astute reader will notice that the final result does not depend on that process as long as that process is independent of .

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(since the payoff of the option is given by ) with the average taken over the paths for . Since the propagator is in the form of a Gaussian, we can perform this integration to obtain

( denotes the cumulative normal distribution as in the previous chapter) where and are the initial and final volatilities of the path respectively and and are given by

The reader should be easily able to verify that equation (4.65) is the same as the Black-Scholes equation for any single volatility path when .

When
, several simplifications occur. We see
that
are known and . In that
case, (4.65) reduces to

Hence, we see that we have a straightforward solution for even when the correlation is not zero.

When
and , we obtain a similar
simplification since
and . In this case,
we obtain the following expression for the option price

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For the case considered in Baaquie[8], we have and . In this case, we have
which gives us

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which is somewhat more complicated since two functionals, and of the volatility path are involved. In this case, however, a perturbation analysis can be used to derive an approximate form for the probability distribution functions of the functionals. Due to this fortunate occurence, a series solution to this problem can be obtained.

The probability density function for the functionals is a very
difficult quantity to obtain. The probability density function for
obtained for the special case in
Stein and Stein[21]^{}. Stein and Stein[21] have used this
probability distribution function and the ``straightforward'' solution
for to get an analytic form of the solution for this
case. We now see that the result can be extended to non-zero
if we can find the joint probability density function of this
functional and . While the individual probability
distribution functions can be obtained (the pdf for is obtained
in Stein[21] and the pdf for is trivial), they are
not independent.

We first change variables from to in (4.5) (changing
to in accordance with risk-neutral valuation) to obtain

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While the result is simple, it has important consequences. We note that risk-neutrality alone cannot determine any constraints for the volatility process. Any volatility process whatsoever satisfies risk-neutrality.

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Using this interpretation, we can derive the discrete form of the
Lagrangian (4.37). Further, we can also derive a continuous
time limit for the Lagrangian which is given by

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However, this Lagrangian is very difficult to deal with directly and hence, we use the discrete action to derive an efficient algorithm.