To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Margrabe's formula

From Wikipedia, the free encyclopedia

In mathematical finance, Margrabe's formula[1] is an option pricing formula applicable to an option to exchange one risky asset for another risky asset at maturity. It was derived by William Margrabe (PhD Chicago) in 1978. Margrabe's paper has been cited by over 2000 subsequent articles.[2]

YouTube Encyclopedic

  • 1/1
    Views:
    3 604
  • Margrabe model (option formula). Change of numeraire

Transcription

Formula

Suppose S1(t) and S2(t) are the prices of two risky assets at time t, and that each has a constant continuous dividend yield qi. The option, C, that we wish to price gives the buyer the right, but not the obligation, to exchange the second asset for the first at the time of maturity T. In other words, its payoff, C(T), is max(0, S1(T) - S2(T)).

If the volatilities of Si's are σi, then , where ρ is the Pearson's correlation coefficient of the Brownian motions of the Si 's.

Margrabe's formula states that the fair price for the option at time 0 is:

where:
are the expected dividend rates of the prices under the appropriate risk-neutral measure,
denotes the cumulative distribution function for a standard normal,
,
.

Derivation

Margrabe's model of the market assumes only the existence of the two risky assets, whose prices, as usual, are assumed to follow a geometric Brownian motion. The volatilities of these Brownian motions do not need to be constant, but it is important that the volatility of S1/S2, σ, is constant. In particular, the model does not assume the existence of a riskless asset (such as a zero-coupon bond) or any kind of interest rate. The model does not require an equivalent risk-neutral probability measure, but an equivalent measure under S2.

The formula is quickly proven by reducing the situation to one where we can apply the Black-Scholes formula.

  • First, consider both assets as priced in units of S2 (this is called 'using S2 as numeraire'); this means that a unit of the first asset now is worth S1/S2 units of the second asset, and a unit of the second asset is worth 1.
  • Under this change of numeraire pricing, the second asset is now a riskless asset and its dividend rate q2 is the interest rate. The payoff of the option, repriced under this change of numeraire, is max(0, S1(T)/S2(T) - 1).
  • So the original option has become a call option on the first asset (with its numeraire pricing) with a strike of 1 unit of the riskless asset. Note the dividend rate q1 of the first asset remains the same even with change of pricing.
  • Applying the Black-Scholes formula with these values as the appropriate inputs, e.g. initial asset value S1(0)/S2(0), interest rate q2, volatility σ, etc., gives us the price of the option under numeraire pricing.
  • Since the resulting option price is in units of S2, multiplying through by S2(0) will undo our change of numeraire, and give us the price in our original currency, which is the formula above. Alternatively, one can show it by the Girsanov theorem.

External links and references

Notes

Primary reference

Discussion

This page was last edited on 28 May 2023, at 04:07
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.