Pricing Best-of Options with Margrabe’s Exchange-Option Formula

# Pricing Equation for Best of Options


# Pricing Equation for Best of Options












I am trying to derive a martingale pricing equation (closed form solution) for a best-of option. But I am getting stuck at a point.

There are 2 stocks $U(t)$ and $V(t)$ they both follow GBM with a correlation of $\rho$.

I want to price an European option whose final payoff is $\max \left\{ U(t), V(t) \right\}$. For simplicity I am assuming that interest rates are 0.

$$ V(0) = \mathbb{E} \left[ \max \left\{ U(t), V(t) \right\} \right] $$

By theory of total expectation

$$ V(0) = \mathbb{E} \left[ U(t) \right] \mathbb{P} \left\{ U(t) > V(t) \right\} + \mathbb{E} \left[ V(t) \right] \left( 1 - \mathbb{P} \left\{ U(t) > V(t) \right\} \right) $$

But I guess this is not the correct approach.

Any help is highly appreciated.

## Answer by JejeBelfort (score 2, accepted)

https://quant.stackexchange.com/a/35309

For best-of options, the usual approach is to use Magrabe's formula.

Starting from the following relationship:

$$\max\{U(t),V(t)\} = U(t) + \max\{V(t) - U(t), 0\} ,$$

you end up with the sum of one of the underlying asset, $U(t)$, and an exchange option paying $\max\{V(t) - U(t), 0\}$.

While the present value of the first term is easy to find, the second requires Magrabe's results stating that the price of the exchange option is given by:

$$V(t) N(d_U) e^{-q_V \tau} - U(t) N(d_V) e^{-q_U \tau}$$

where

$$d_V = \frac{\ln\left(V(t) / U(t)\right) + \left( q_U - q_V - \sigma^2 / 2\right)\tau}{\sigma \sqrt{\tau}}$$

$$d_U = d_V + \sigma \sqrt{\tau}$$

and

$$\sigma^2 = \sigma^2_U + \sigma^2_V - 2 \rho \sigma_U \sigma_V$$

with $\tau$ being the time-to-expiration, $q_x$ and $\sigma_x$ being respectively the dividend yield and the volatility of asset $x$, $x \in \{U,V \}$

Therefore, the price of your best-of option becomes:

$$U(t) e^{-q_U \tau} + V(t) N(d_U) e^{-q_V \tau} - U(t) N(d_V) e^{-q_U \tau}$$

which simplifies to

$$V(t) N(d_U) e^{-q_V \tau} + U(t) N(-d_V) e^{-q_U \tau}$$

## Answer by Tom Gladd (score -1)

https://quant.stackexchange.com/a/35324

Assuming GBM dynamics, I work through the (tedious) details of a worst-of call option in the more general case of a nonzero strike.

ntgladd.com

tab = Finance, subsection = Option models, file = Worst Of Call Option - Three Ways

In a real world application, you would use Monte Carlo with some sort of market calibrated local vol or stochastic vol model.