The Malliavin Derivative and Application to Pricing and Hedging a European Exchange Option

The exchange option was introduced by Margrabe in [1] and its price was explicitly computed therein, albeit with some small variations to the models considered here. After that important introduction of an option to exchange one commodity for another, a lot more work has been devoted to variations of exchange options with attention focusing mainly on pricing but not hedging. In this paper, we demonstrate the efficiency of the Malliavin derivative in computing both the price and hedging portfolio of an exchange option. For that to happen, we first give a preview of white noise analysis and theory of distributions.


Introduction
White noise analysis and theory of distributions is treated extensively in [2][3][4][5] and references therein.Applications in the form of the generalized Clark-Haussmann-Ocone (CHO) formula was studied in [6][7][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   1 X T being more than the seller's,   2 X T .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   F  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   F  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 F  may not be differen- tiable.Here represents the number of units of stock to be held at any time .
  F  is not differentiable everywhere.As a result, white noise theory justifies differentiability of F in distribution.The differential operator is the Malliavin derivative t .This operator is defined in the space of distributions S D  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  denote the action of  on  , then by the Bochner-Minlows theo- rem, there exists a probability measure P on S such that . In this case P is called the white noise probability measure and is the white noise probability space.

S B P 
As a result, we shall be considering the space S , as the sample space  , so that our asset prices will be defined on the probability space  , ,  where F 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 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.

B t
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
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 i 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 S   , the sample space, is rich enough to accommodate the concepts we require for our calculations.
The paper is organized as follows: The next section gi

W
prices defined on the filtered ves 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.

Complete Markets
and is a solution of the deterministic di fferential equation be the matrix of coefficients of volatility where for easy  .
Then if


where .If we let In all these cases we consider fin ve In this case we are considering .as the usual norm in .
for some ite time horizon T and throughou er, we are taking Tr to mean transposition.
An in stor who selects a portfolio consisting of the  1  assets will have to work out the proportions of alth that he has to invest in each of the his we Solving for V  we get , From now on, without loss of generality, we assume constant coefficients.Then Equation (2.5) becomes represents the investor's holdings at any time , where for each   is the n units of security number tha nvestor will hold.In future we shall refer to the ector of prices as the market and the vector as the portfolio.The holder of a portfoli y decide to liquidate his position at any time , and his wealth is the cumulative savings in the count 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 0 t  is given by 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) F D  and assume that the following conditions hold: s and app ing the generalized CHO formula to G , we ha ly ve where denotes the Malliavin derivative.By ness due to the Martingale Representation Theorem, we get 0 , d d and where as before and means transpose.Therefore

   
1 , , This gives the explicit number of units of stocks.The olding in the bank account can be found from h th   0 t  e self financing condition.The importance of these results is that in a complete market, every contingent claim with payoff   F  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 pay plifying (2.8) depends on the nature of the payoff.One may directly compute the expectation on condition that the distribution of   F off. Sim  is known.Sometimes it may be easier to determine the Black-Scholes partial differential equation sat d 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.

The Two Dimensional Market Model and
Transformation Theorems where as Then the payoff of the exchange option will be . ant to determine the price and the hedging portfolio of this optio eralized CHO formula.We assume in our case that the coefficients are constant.
The Girsanov chang We w n by using the gen e of measure for this setup can be Note that since we have assumed that t complete, then easily done by letting u ; and fficients, we can easily justif vikov conditions, so that th easure defined by Then with constant coe y that u satisfies the No e probability m Q With respe rtingale.In order to exploit the results from the previous discussions, we note here that the market is a special case e ered in the previous We assume that of th  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 is given by . In this case we note that from the CHO formula , for any contingent claim where

Transf ormation Theorems
In order to facilitate our computation and taking advantage of the distribution of the terminal values of the underlying securities In our case we have e e e e e X i t e 0 and variance 1 since e 1 To prove that 1 X   and 2 X are independent with respect to 2 e e 0.

X P
X be as given in Proposition 3.1 and let 1 2 1 , , y y  and 2  be real numbers.Then where and We have By using the notation in Proposition 3.1, then the previous expression can be re-written 1 1 We have shown in Proposition 3.1 that the random variables 2 X and 1 1 , so that with respect to the same probability measure, the random variable where

Price and Hedging Portfolio of an Exchange Option
Note that if, for a fixed time horizon , the random variables We are w ready ive the price and hedging portfolio of the European excha n.

Proposition 4.1 The price exchange option is given by
is the cumulative distribution function of the standard normal distribution.
Proof. had noted that with r of We espect to the equivalent martingale measure Q which we defined, the prices the two underlying assets 1 X and 2 X are given by fore the time zero European ge option is There price of the exchan If we define the probability measure equivalent to by By using the results of the previous proposition, we then conclude that the time zero price of the European exchange option is given by Note that this pr epend o he stocks nor on the market inte  , but just on the m rownian 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.

Hedging an Exchange Option
We now calculate the hedging portfolio .For this two dimensional case, ula, we get, from (2.9), that arket volatilities.This result is also similar to the one obtained in [1] but in that paper the author considers the case when the B , 1 , We thus have the following result Proposition 4. 2 The perfect hedge is given by where In order to calculate  Proof.

 1
, 1 ,2, we use the Markov property .We first calculate need to Therefore the previous expression becomes Note that, with respect to , we have Using the previous notations of and es to where Now, with respect to and for each j = 1, 2 we have is a normal distribution with zero mean and vari-  is a normally distributed random vector with mean zero (vector) and covariance matrix equal to the identity matrix.Moreover the previous expression becomes

Conclusion
ave 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 rice and hedging portfolio of European contingent of these results would be to get simiwith efully pleteness.Hedging an option is important in that the seller would know how much of each security to hol order to hedge his liability.In complete markets, this should al were as co related.In our case, we allowed the stock prices to each noise terms which are independent.Also in our paper, we have computed explicitly, the he-dging 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.

Acknowledgements
This work was supported by the University of Cape n Research Grant 461091 .REFER[1] W. Margrabe, "The Value of an Option to Exchange One Asset for Another," Journal of Finance, Vol.33, No. 1, 1978, pp.177-186.doi:10.1111/j.1540-6261.1978.tbWe h p claims.Extensions lar explicit results when modelling stock prices general Itô-Lévy processes, though one has to car consider the models of prices to avoid incom d in ways 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 given rdepend on the two Tow