Exchange Options were initially introduced by William 'Dr. Risk' Margrabe in his seminal 1978 paper. These type of options allows the holder of the option to exchange one asset for another and are used commonly in foreign exchange markets, bond markets and stock markets amongst others.

 

These types of options are considered as a type of rainbow option, in that the payoff depends on two correlated variables.

An exchange option on 2 assets and is essentially a long call position on with a strike price of or a long put position on with a strike price of .

 

We look at the pricing of European Exchange options within a Black & Scholes framework, and extend the pricing within an American type framework. Other methods are also considered.

 

Pricing:

1) Closed For Valuation (Margrabe)

Margrabe (1978) introduced the pricing formula for valuing European exchange options under a Black-Scholes framework, with the same common assumptions under the BS pricing model.

 

He gives the price of a European exchange option as:

Where:

Uppercase D is the respective dividend yield for stocks 1 and 2, and N(x) is the cumulative normal distribution function of x.

 

Margrabe then goes on to show that the value of an American exchange option is the same as that of a European exchange option because it is never beneficial to exercise an American exchange option early. The proof can be found within Margrabe's paper.

 

2) American Exchange Options (Bjerksund & Stensland)

Contrary to Margrabe's proof that American options are not more valuable than its European counterparts, Bjerksund & Stensland show that American options are indeed more valuable than the European contracts and provide an approximation based on the relations to a standard call option.

They give the value of an American exchange option based on the relationship with the American call option pricing formulas.

 

Based on American option approximations given by e.g. Barone-Adesi & Whaley, Roll, Geske & Whaley, and Bjerksund Stensland, the relationship is given as:

Where the American call value is:

See the American options section for pricing metholody of American options.

 

For a range of values, we can see that the Bjerksund & Stensland approximation does give higher American exchange options than its European counterpart.

 

American exchange options can also be valued using a 3-Dimensional lattice method which we introduce in section 4 below.

 

3) Binomial Method

Rubinstein (1991) proposes the use of a binomial with the use of a neat adjustment which gives us 2 outcomes rather than 4. If we consider the problem on its own, with 2 assets, each asset can take either an up or down movement at the next time node, giving rise to 4 possible movements.

For example, consider 2 stocks, A & B. At time node 1, A can take the value of Au or Ad, with u and d being the up and down magnitudes. Similarly, at time node 1, B can take the value of Bu or Bd, giving us a total of 4 outcomes.

 

Rubinstein makes use of the ratio between the 2 assets with the payoff:

and builds the binomial tree giving the value at each node as a ratio of the 2 stocks multiplied by the up or down magnitude. The first 2 nodes of the tree can be illustrated as:

Note that the central node at time period 2 can also be shown to be equivalent to:

Rubinstein then goes on to show that the valuation of both American and European type options can be undertaken using this method.

 

4) 3-Dimensional Binomial Method

Rubinstein (1994) extends his 1991 binomial method to value exchange options under a so-called 3-dimensional binomial method. This multiplicative bivariate binomial model is extremely useful for pricing options on two or more options, and can therefore be applied to the case of exchange options.

Convergence is quicker than a standard binomial method, and model implementation is not difficult.

 

By considering the inputs required in a standard binomial model, we can define the basic u and d as follows:

Where

The idea behind the 3-dimensional binomial method is that instead of 2 distinct movements, there can be 4 outcomes.

	Asset 1 Up	Asset 1 Down
Asset 2 Up	A	C
Asset 2 Down	B	D

Where A is non-equal to C and B is non-equal to D. These 4 magnitudes can be defined as:

Where:

and other variables are defined as previous.

 

This method can price both European and American style exchange options

 

Here we compare the values of the closed form approach (Margrabe) to the 3-Dimensional binomial formula given by Rubinstein and show that the values given, for just 100 time nodes on the 3-Dimensional binomial are quite accurate.

For a more basic and detailed look at 3-Dimensional lattices, have a look here

 

Other Known Names / Variants

Alternative Options
Margrabe Options
Spread Options
Two-Asset Options
Three-Asset Options
Rainbow Options

 

Additional/Useful List of resources

Papers:

Bjerksund, P., & Stensland, G. "American Exchange Options and a Put-Call Transformation: A Note", Journal of Business, Finance, and Accounting, Vol. 20, No. 5, pp. 761-764 (1993)
Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73)
Cox, J., Ross, S., & Rubinstein M., “Option Pricing: A Simplified Approach." Journal of Financial Economics, 7. (Sept '79).
Haug, E., "Complete Guide to Option Pricing Formulas", McGraw-Hill, 1998
Margrabe, W., "The Value of an Option to Exchange One Asset for Another", Journal of Finance, 33, 177-86 (1978)
Merton, R.,(a) "Option Pricing when Underlying Stock Returns Are Discontinuous", Journal of Financial Economics 3, pp. 125-144 (Jun '73)
Merton, R.,(b) "Theory of Rational Option Pricing", Bell Journal of Economics & Management (June '73)
Rubinstein, M., "One for Another" Risk, 4, pp. 30-32 (Jul-Aug '91)
Rubinstein, M., "Return to Oz", Risk Magazine 7, 11 (1994)