Formula Magrabe

In financial mathematics , the Magrabe formula is one of the option pricing formulas. It is applied to the option to exchange (the Magrabe option) of one risky asset to another, at the time of redemption. The formula was independently proposed by William Magrabe and Stanley Fisher in 1978.

Content

Definition

Let be $S_{1}(t)$ ${\ displaystyle S_ {1} (t)}$ and $S_{2}(t)$ ${\ displaystyle S_ {2} (t)}$ - the price of two risky assets at the time $t$ ${\ displaystyle t}$ each has a fixed continuous dividend equal to $q_{i}$ ${\ displaystyle q_ {i}}$ . Option $C$ ${\ displaystyle C}$ which we want to evaluate gives the buyer the right (but not the obligation) to exchange the second asset for the first at the time of maturity $T$ ${\ displaystyle T}$ . In other words his winnings $C(T)$ ${\ displaystyle C (T)}$ will make $\max(0,S_{1}(T)-S_{2}(T))$ ${\ displaystyle \ max (0, S_ {1} (T) -S_ {2} (T))}$ .

The Magrabe model of the market assumes only the existence of two risky assets, whose prices follow the geometric Brownian motion . The volatility of these Brownian movements is not constant, but it is important that the volatility $\sigma$ ${\ displaystyle \ sigma}$ their relationship $S_{1}/S_{2}$ ${\ displaystyle S_ {1} / S_ {2}}$ is a constant. In particular, the model does not assume the existence of a risk-free asset (such as a zero- coupon bond ) or any interest rate norm.

If volatility $S_{i}$ ${\ displaystyle S_ {i}}$ are equal $\sigma _{i}$ ${\ displaystyle \ sigma _ {i}}$ then $\textstyle \sigma ={\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}-2\sigma _{1}\sigma _{2}\rho }}$ ${\ displaystyle \ textstyle \ sigma = {\ sqrt {\ sigma _ {1} ^ {2} + \ sigma _ {2} ^ {2} -2 \ sigma _ {1} \ sigma _ {2} \ rho} }}$ then $\rho$ ${\ displaystyle \ rho}$ - correlation coefficient of Brownian movements $S_{i}$ ${\ displaystyle S_ {i}}$ .

The Magrabe formula establishes a fair price for an option at the initial moment of time as:

e^{-q_{1}T}S_{1}(0)N(d_{1})-e^{-q_{2}T}S_{2}(0)N(d_{2})

{\ displaystyle e ^ {- q_ {1} T} S_ {1} (0) N (d_ {1}) - e ^ {- q_ {2} T} S_ {2} (0) N (d_ {2 })}

where through $N$ ${\ displaystyle N}$ denotes the cumulative standard normal distribution ,

$d_{1}={\frac {\ln(S_{1}(0)/S_{2}(0))+(q_{2}-q_{1}+\sigma ^{2}/2)T}{\sigma {\sqrt {T}}}}$ ${\ displaystyle d_ {1} = {\ frac {\ ln (S_ {1} (0) / S_ {2} (0)) + (q_ {2} -q_ {1} + \ sigma ^ {2} / 2) T} {\ sigma {\ sqrt {T}}}}}$ ,

$d_{2}=d_{1}-\sigma {\sqrt {T}}$ ${\ displaystyle d_ {2} = d_ {1} - \ sigma {\ sqrt {T}}}$ .

Proof

The formula is proved by reducing to the Black - Scholes formula :

First, consider both assets valued in units. $S_{2}$ ${\ displaystyle S_ {2}}$ (in such cases it is said that $S_{2}$ ${\ displaystyle S_ {2}}$ used as counting money ), this means that the unit of the first asset is now worth $S_{1}/S_{2}$ ${\ displaystyle S_ {1} / S_ {2}}$ units of the second asset, and the second asset is exactly 1.
With this choice of counting money, the second asset becomes risk-free and its dividend rate $q_{2}$ ${\ displaystyle q_ {2}}$ coincides with the rate of interest. The income of the option recalculated in accordance with the change in the counting money is equal to $\max(0,S_{1}(T)/S_{2}(T)-1)$ ${\ displaystyle \ max (0, S_ {1} (T) / S_ {2} (T) -1)}$ .
Thus, the initial option becomes a call option on the first underlying asset (with its counting price) with a strike price equal to 1 unit of the risk-free asset. Note that the dividend rate $q_{1}$ ${\ displaystyle q_ {1}}$ the first asset remains the same even after recalculation.
By applying the Black - Scholes formula to these values as corresponding input data, for example, the value of the original asset $S_{1}(0)/S_{2}(0)$ ${\ displaystyle S_ {1} (0) / S_ {2} (0)}$ , interest rate $q_{2}$ ${\ displaystyle q_ {2}}$ volatility $\sigma$ ${\ displaystyle \ sigma}$ and so on, we get the option price, expressed in the counting money.
Since the final price of an option is expressed in units $S_{2}$ ${\ displaystyle S_ {2}}$ , then multiplying by $S_{2}(0)$ ${\ displaystyle S_ {2} (0)}$ will translate the answer into the original units, that is, the usual currency in which we obtain the Magrabe formula.

Links

The Margrabe Formula, Rolf Poulsen, Center for Finance, University of Gothenburg

Literature

William Margrabe. The One Of The Asset For Another. Journal of Finance , 33: 177-186, 1978
Stanley Fischer. Call Option Pricing When the Exercise Price is Uncertain, and the Valuation of Index Bonds. Journal of Finance , 33: 169-176, 1978