Exchange, Basket and other multiasset options
Igor Hlivka
MUFG Securities International, LONDON
This application demonstration reviews several classes of multiasset options and as such extends the concept on multivariability in Finance presented / discussed in other applications. Multivariability is important concept in financial engineering as many nonstandard structured products in the market are exposed to multiple source of randomness. Multivariability is not trivial in terms of handling multiple dependencies, however suitable change of martingale measure and dimensionreduction techniques can help simplifying the multivariable process into more manageable routines.
Although multidependency in many instances requires numerical processing, we will show that with Maple we can do better. Our aim is to devise an analytical solution to this problem and will show how Maple's symbolic engine can efficiently cope with this task
Exchange Options
Exchange Options are special class of bivariate options that give the buyer the right to exchange at option maturity the asset for . This means that the option holder will exercise his (her) right to exchange the asset if one has greater value than the other. In this respect, we can define the option's payoff as:
A key to valuation of such options is to assess the behavior of assets under two sources of randomness: and
Both and in isolation are independent Wiener processes with with following properties:
In case of two source of randomness, we define the SDE for two stochastic processes as:
From here we can establish:
where is the correlation between
To value this option analytically, we first change the option's payoff by performing the "change of measure"
The standard way of valuing option is to apply the riskneutral measure, such that the asset processes when normalized by appropriate numeraire become martingale:
where B is moneymarket account
We can therefore express the Exchange option payoff with new measure as follows:
To see why the change is measured is required, we examine two stock process in detail. Under standard riskneutral measure P we get:
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(1.1) 
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(1.2) 
We observe that the volatility of the ratio of two assets is equal to = so for the ratio process to be martingale, the following condition has to to satisfied:
However, under the riskneutral measure P, this is not the case as:
≠ ( because in bivariate setting
Let's build the new volatility and transformed measure vectors under new measure Q
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(1.3) 
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(1.4) 
As we can see, under this measure transformation, the ratio can be expressed as: .
What does this imply?
 Under the transformed measure Q, the bivariate process `(W[1], W[2])" align="center" border="0">can be reduced to the univariate counterpart with :
We use this result for the option valuation:
 We will first define the new martingale process X
 Decide the limits of integration for which the option payoff holds
 Integrate the payoff similar to BlackScholes with
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(1.5) 
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(1.6) 
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(1.7) 
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(1.8) 
Maple calculates the sensitivities with ease. Since is a denominator in the exchange option formula, the Delta (first derivative w.r.t ) for this factor is negative.
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As the chart above shows, the Exchange Option is quite sensitive to correlation and volatility. Negative correlation between two assets will make the the ratio volatility higher and hence lead to a higher premium.
Best / worst of two options
Results for the Exchange options easily extends into the valuation of option contract that pays the "best" or "worst" of two assets:
Best of two assets
The value of the Best of two option is then:
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(2.1.1) 
Worst of two assets
The value of the Best of two option is then:
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(2.2.1) 
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It is worth mentioning that "Best of" / "Worst of" options are relatively expensive as their premium is close to the underlying asset price.
Basket options
Basket options are special case of multiasset options where several underlying assets are bundled together and this basket performance then determines the option's payoff. By virtue of construction, baskets options are essentially a particular case of a broader family of portfolio options where the stochastic variable is the baskets (portfolio) of assets. As such, basket options can be utilized to monetize the expectation of the future behavior of combined assets pool.
Similar to the univariate case, basket options can be structured as Puts or Calls. Puts provide a positive payoff if the weightedaverage of basket assets drops below a strike level. Calls give rise to a positive payoff if the weightedaverarge at maturity is greater than the strike.
Formally:
= where is ith asset and is its weight in the basket with = 1
A key to valuation of basket options analytically is to determine the parameters of joint distribution. We observe that each asset return in the basket individually (under the appropriate measure) follows a normal distribute: and return dependency structure is characterized by the correlation matrix Σ. This allows us deriving basket's variance as a product of individual weights and variancecovariance matrix. Once basket variance has been determined, we can proceed with valuation as in univariate case  just replace the asset with the weightedaverage of basket assets and volatility with basket's volatility.
Let's examine this is practice and valued 5 stocks Call Basket Option:
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Here we have derived the basket's variancecovariance matrix. We will obtain the basket's volatility and weighted average of assets through simple factor/matrix manipulation:
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(3.3) 
The valuation is now straightforward we have reduced the multivariate dependency to a univariate case which we know how to value. Assuming joint lognormality of each assets we set:
where and =
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This is the "fullform" valuation, and despite the long string of functional parameters Maple handles it well. We can now compute the option premium with parameters substitution:

(3.5) 
On the other hand, we could have taken a "shortcut"  computing the premium symbolically for the set of theoretical parameters and then making the substitution for volatility and mean.
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(3.8) 
As the above analysis indicates, the basket options can be quite complex, particularly when the number of underlying assets becomes large. In this respect, basket options are also more difficult to manage as multidependency and multivariability creates complex web of sensitivities  including multiple deltas, gammas, vegas etc.
Conclusion
This application example shows that multiasset options can become quite complex when number of underlying assets increases. Computational efficiency greatly increases when CAS is employed to simplify processes symbolically. Maple, as this example again demonstrates, handles these tasks very well.
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