The Malliavin Derivative and Application to Pricing and Hedging a European Exchange Option ()
1. Introduction
White noise analysis and theory of distributions is treated extensively in [2-5] and references therein. Applications in the form of the generalized Clark-Haussmann-Ocone (CHO) formula was studied in [6-8] and references therein. The theorem takes advantage of the martingale representation theorem which expresses every square integrable martingale as a sum of a previsible process and an Itô integral. The power of the generalized CHO is that one can take advantage of the Malliavin derivative for computing the hedging portfolio. The Malliavin derivative is a better mathematical operation as opposed to the delta hedging approach whose limitations are a failure to explain differentiation of some payoffs which are not differentiable everywhere or if the underlying security is not Markovian. Most of the attention in contingent claim analysis is directed at pricing because of its importance to market practitioners. It is in this regard that explicit results of hedging portfolios for different options are not always readily available. In this paper, we present both explicit results of the price and hedging portfolio of an exchange option, written on two underlying securities with independent Brownian motions. The ground-breaking work was done in [1]. The market setup is a complete market setup to escape the problem of not finding a perfect hedge.
Hedging portfolios are just as important as prices of options in that they give us an understanding of how sellers or writers can managed dynamically to replicate the payoff of a contingent claim. The price at any time of the contingent claim equals the intrinsic value of the hedging portfolio at that point.In the case of a European exchange option, the payoff 
is the difference in terminal value of the underlying securities, conditional on the buyer’s terminal asset price 
being more than the seller’s,
. A more interesting problem will be to look at an American exchange option where the buyer would exercise on or before maturity. Such an exercise time will be a stopping time and the price for such an option will be the essential supremum, over all stopping times, of the payoff above. Our attention in this paper is on the European exchange option.
The price of the exchange option will be determined from the CHO formula as the discounted expectation of the payoff 
while the hedging portfolio will be obtained from the integrant in the martingale representation theorem setup of the the payoff. This integrant involves the Malliavin derivative of the payoff and its market price of risk and in the case that the latter is time-dependent, it reduces to the discounted expectation of the Malliavin derivative of 
conditioned with respect to the filtration.
Preliminaries
The following is a summary of important results from [6] and [7]. One of the weaknesses of the delta hedging approach is its failure to justify fully the delta 
because 
may not be differentiable. Here 
represents the number of units of stock to be held at any time
.
In this setup, if for example 
, then 
is not differentiable everywhere. As a result, white noise theory justifies differentiability of 
in distribution. The differential operator is the Malliavin derivative
. This operator is defined in the space of distributions 
discussed fully in [6] and summarized below.
Let 
be the Schwartz space of rapidly decreasing smooth functions and 
be its dual, which is the space of tempered distributions.Now, for 
and
, let 
denote the action of 
on
, then by the Bochner-Minlows theorem, there exists a probability measure P on 
such that
(1.1)
where
. In this case P is called the white noise probability measure and 
is the white noise probability space.
As a result, we shall be considering the space
, as the sample space
, so that our asset prices will be defined on the probability space 
where 
is the family of all Borel subsets of
. The construction of a version of the Brownian motion then is a direct consequence of the Bochner-Minlows theorem in that if
then clearly 
and thus
so that immediately we conclude that 
where 
is normal with mean 0 and variance
. One can easily prove that 
is really a standard Brownian motion described in [7] as a continuous modification of the white noise process constructed above.
The Brownian motion constructed this way is a distribution and thus special operations like the Malliavin derivative, defined below, are possible. Note that the Brownian motion is not differentiable in the classical sense but is differentiable in the Malliavin sense. The Malliavin derivative is a stochastic version of the directional derivative in classical calculus, with the direction carefully chosen. The following definition is from [7].
Definition 1.1 Assume that 
has a directional derivative in all directions 
of the form 
where 
for fixed
, in the strong sense that
exists in 
and assume further that there exists 
such that
, then we say that 
is differentiable and we call
the Malliavin derivative of
.
Just like any operation where using “first principles” is not usually easy operationally, one can use a series of characterizations to the above definition, which includes the chain rule, to compute the Malliavin derivative of any random variable which is differentiable. The set of all differentiable square integrable random variables was denoted by 
in [7]. As an illustration, we see that 
and the chain rule yield that,
. Here and elsewhere in this paper,
.
Therefore classically, one sees that the Malliavin derivative, in some sense, mimics differentiation in deterministic calculus. This is a big departure from Itô derivation which does not in any way make sense as a derivative in classical sense. Thus the space
, the sample space, is rich enough to accommodate the concepts we require for our calculations.
The paper is organized as follows: The next section gives the general pricing and hedging formulae for general contingent claims. The next section defines our market model and the final section gives our pricing and hedging results for the European exchange option.
2. General Pricing and Hedging Models in Complete Markets
We now consider the asset prices defined on the filtered probability space 
where 
is the standard filtration generated by the Brownian motions, and which is rich enough to represent the information available to traders about all assets on the market at any time
.
The first security is a risk-free asset, e.g. bank account where the balance in the bank is 
and is a solution of the deterministic differential equation
(2.1)
under the assumption of existence of a unique solution
. In this case 
is the interest rate which we shall later assume is constant for computational advantages.
The other securities are risky securities, e.g. stocks where for each
, the price 
of stock number 
is given by the Ito diffusion
(2.2)
where 
is the appreciation rate of security number 
and 
is the volatility coefficient of the Brownian motion 
in security
.
Let 
be the vector of appreciation rates for the stocks and the matrix
be the matrix of coefficients of volatility where for easy writing we suppressed dependence on time and noise and the risk factors are modeled by the 
dimensional Brownian motion
.
Then if
, we have
(2.3)
In all these cases we consider 
for some finite time horizon T and throughout this paper, we are taking 
to mean transposition.
An investor who selects a portfolio consisting of the 
assets will have to work out the proportions of his wealth that he has to invest in each of the 
securities. The vector 
represents the investor’s holdings at any time
, where for each
, 
is the number of units of security number 
that the investor will hold.
In future we shall refer to the vector of prices 
as the market and the vector 
as the portfolio. The holder of a portfolio 
may decide to liquidate his position at any time
, and his wealth is the cumulative savings in the bank account plus the trading gains up to and including the date of liquidation. We assume that the portfolio is self financing, so that, the value of this portfolio at time 
is given by
The portfolio 
is called admissible if it is self financing and the value process 
is bounded below.
We note that by writing the value of the portfolio 
as 
and assuming that the portfolio is self financing and admissible, then, if 
is invertible, we have
where
. If we let
, where 
is the vector of stock prices and if we further assume that 
satisfies the Novikov conditions, then, by the Girsanov theorem, 
given by 
is a Brownian vector with respect to the probability measure 
given by
.
In this case we are considering 
as the usual norm in
.
Therefore
(2.4)
Solving for 
we get
(2.5)
From now on, without loss of generality, we assume constant coefficients. Then equation (2.5) becomes
(2.6)
This is a particular version of the Martingale Representation Theorem which can be found for example, in [9] applied to a particular square integrable martingale 
. It is this Martingale Representation theorem which the CHO formula relies on. We state here the theorem without proof and refer the reader to [6] for more details.
Theorem 2.1 (The generalized Clark-Ocone-Haussmann formula)
Suppose that 
and assume that the following conditions hold:
1) 
2) 
3) 
then
where 
is the Girsanov kernel, 
is the equivalent martingale measure and 
is a Brownian motion with respect to
.
By letting 
and applying the generalized CHO formula to
, we have
(2.7)
where 
denotes the Malliavin derivative.
By uniqueness due to the Martingale Representation Theorem, we get
(2.8)
and
(2.9)
where as before 
and 
means transpose.
Therefore
(2.10)
This gives the explicit number of units of stocks. The holding 
in the bank account can be found from the self financing condition.
The importance of these results is that in a complete market, every contingent claim with payoff 
is attainable by a portfolio of stocks and bonds. Therefore
, the initial value of a self financing portfolio, equals the price of such a derivative, since 
. It then shows that the time zero price of such a contingent claim is the discounted expectation of the payoff. Simplifying (2.8) depends on the nature of the payoff. One may directly compute the expectation on condition that the distribution of 
is known. Sometimes it may be easier to determine the BlackScholes partial differential equation satisfied by the value function with corresponding boundary conditions. If such a boundary value problem can be simplified explicitly, or through numerical techniques, then the price can be determined either explicitly or as a good approximation respectively. Other direct numerical methods of solution like the Monte Carlo simulations involve simulations of the underlying security itself and approximations of the expected values give estimate of (2.8). In this paper, we will find explicit results using some important change of measure transformations which we prove first.
3. The Two Dimensional Market Model and Transformation Theorems
Suppose that traders will agree to trade an option which gives the holder the right, but not obligation to exchange a predetermined risky security with another predetermined risky security, then we call such an option an exchange option. We shall consider the case of a European exchange option. Suppose that the underlying securities in question have time 
prices 
and 
and to simplify our computations, we assume that the market consists of a bank account and these two stocks. The price of the bond is given by (2.1) albeit with constant force of interest
. The risky securities are given by
(3.1)
where as before 
is a standard Brownian motion.
Assume that these stochastic processes are defined on a filtered probability space 
where 
is a filtration for the assets such that the stochastic processes 
are adapted. Suppose that at terminal time
, then 
. Then the payoff of the exchange option will be
.
We want to determine the price and the hedging portfolio of this option by using the generalized CHO formula. We assume in our case that the coefficients are constant.
The Girsanov change of measure for this setup can be easily done by letting 
where
;
; 
and
.
Then with constant coefficients, we can easily justify that 
satisfies the Novikov conditions, so that the probability measure 
defined by
(3.2)
is equivalent to 
and 
is a martingale with respect to
. In this case
is a 
martingale.
Moreover 
is a Brownian motion with respect to
. Let
.
With respect to 
price 
is
so that
is a Q-martingale.
In order to exploit the results from the previous discussions, we note here that the market 
is a special case of the 
-dimensional market considered in the previous section with n = 2 in this case. We assume that 
is invertible so that the market is complete. Therefore if we choose a self financing portfolio 
which is also admissible, then the discounted value of the portfolio at any time 
is given by
where
.
In this case we note that from the CHO formula , for any contingent claim
, we get
(3.3)
and
(3.4)
where 
and
.
Note that since we have assumed that the market is complete, then
.
Transformation Theorems
In order to facilitate our computation and taking advantage of the distribution of the terminal values of the underlying securities 
and
, we provide some important transformation results. Their usefulness will be evident in simplifying both (3.3) and (3.4).
Proposition 3.1 Let 
and 
be two independent standard normal random variables and let
. Define a probability measure 
equivalent to 
with density
.
Then the random Gaussian variable 
and 
are independent standard normal variables with respect to
.
Proof.
We have to prove first that 
and 
are independent normally distributed random variables with respect to the probability measure
.
Recall that a random variable 
with mean
and variance 
is normally distributed if its characteristic function is
In our case we have
Therefore 
is normal with mean 
and variance 1, with respect to
. Therefore 
is normal with mean zero and variance 1 with respect to
.
In the same way we can show that 
is normal with mean
and variance 1 since
.
To prove that 
and 
are independent with respect to 
it suffices to prove that 
and 
are uncorrelated , that is,
.
Now
Corollary 3.1
Let 
and 
be as given in Proposition 3.1 and let 
and 
be real numbers. Then
where 
and 
Proof.
We have
By using the notation in Proposition 3.1, then the previous expression can be re-written
(6)
We have shown in Proposition 3.1 that the random variables 
and 
are standard normal distritions with respect to
, so that with respect to the same probability measure, the random variable 
has a normal distribution with mean zero and variance
. In the same way, with respect to
, the random variables 
and 
are standard normal distributions so that 
is a normal distribution with mean zero and variance
.
Therefore (6) becomes
□
Remark 3.1
Proposition 3.1 still holds in an m-dimensional case where both 
and 
are independent multivariate standard normal random vectors. In that case, the following results will be extensions of the preceding proposition and corollary respectively.
Proposition 3.2
Let 
and 
be two independent m-dimensional normal random vectors each with mean equal to the zero vector and covariance matrix equal to the identity matrix and let 
be any non-zero vector. Define a probability measure
, equivalent to 
with density
, where 
is the usual norm in
.
Then 
and 
are independent Gaussian vectors with zero mean (vector) and covariance matrix equal to the identity.
Consequently Corollary 3.1 can be extended as follows:
Corollary 3.2 Let 
and 
be as in Proposition 3.2 and let 
and 
be real numbers. If 
and 
are m-dimensional vectors, then
where 
and 
and 
denotes the usual norm in
.
4. Price and Hedging Portfolio of an Exchange Option
Note that if, for a fixed time horizon
, the random variables 
and 
in the previous proposition are Brownian vectors 
and 
respectively, then the equivalent probability measure 
will be given by the density 
and we would also insist that the vector 
satisfy the Novikov conditions.
We are now ready give the price and hedging portfolio of the European exchange option.
4.1. Price of a European Exchange Option
Proposition 4.1 The price of the European exchange option is given by
where 
is the cumulative distribution function of the standard normal distribution.
Proof.
We had noted that with respect to the equivalent martingale measure 
which we defined, the prices of the two underlying assets 
and 
are given by
where 
and
.
Therefore the time zero price of the European exchange option is
If we define the probability measure 
equivalent to 
by
, then
By using the results of the previous proposition, we then conclude that the time zero price of the European exchange option is given by 
where
and
Note that
.
Moreover, since 
and
, then 
and ss
so that the time zero price of the European exchange option becomes
Note that this price does not depend on the appreciation rates of the stocks nor on the market interest rate
, but just on the market volatilities. This result is also similar to the one obtained in [1] but in that paper the author considers the case when the Brownian motions are correlated and also with a special assumption that the noise terms for each stock are different. We have allowed that stock prices to depend on the two Brownian motions.
4.2. Hedging an Exchange Option
We now calculate the hedging portfolio 
. For this two dimensional case, thanks to the CHO formula, we get, from (2.9), that
, where, as before
, with 
and
.
Now
where
. Therefore
We thus have the following result Proposition 4.2 The perfect hedge 
is given by
and
where
,
and
Proof.
In order to calculate
, we need to use the Markov property . We first calculate 
as follows:
where
.
Therefore the previous expression becomes
.
Note that, with respect to
, we have
.
Therefore since 
is independent of 
then
where
Using the previous notations of 
and
, then the previous expression reduces to
where
.
Now, with respect to 
and for each j = 1, 2 we have 
is a normal distribution with zero mean and variance 
then 
is a normally distributed random vector with mean zero (vector) and covariance matrix equal to the identity matrix. Moreover the previous expression becomes
where
Therefore
with
where
where
In the same way we can also calculate 
to get the result as 
where
In summary we have
.
Therefore the equation 
gives
.
Solving for 
and 
we get:
and
where
.
The number of units in the bond, at any time 
will be computed from the expression
□
To the best of our knowledge, this result has not been obtained before in its explicit form. However, since the market is Markovian, we could still check this result of the hedging portfolio by differentiating the value function with respect to the stock prices 
and
.
5. Conclusion
We have shown that white noise analysis is of vital importance to Finance in that the generalized CHO formula becomes important in finding explicit expressions for the price and hedging portfolio of European contingent claims. Extensions of these results would be to get similar explicit results when modelling stock prices with general Itô-Lévy processes, though one has to carefully consider the models of prices to avoid incompleteness. Hedging an option is important in that the seller would know how much of each security to hold in order to hedge his liability. In complete markets, this should always be possible and thus the results in this paper can be applied to any European contingent claim. The [1] opened the door for pricing the exchange options though in that paper, the stock prices were influenced each by one Brownian motion and the two were given as correlated. In our case, we allowed the stock prices to each depend on the two noise terms which are independent. Also in our paper, we have computed explicitly, the hedging portfolio, something which was not done in [1]. As a result, our results are extensions of that paper with the strength of using white noise analysis.
6. Acknowledgements
This work was supported by the University of Cape Town Research Grant 461091 .