Multiasset derivatives and exchange options

Definition Two positive asset prices follow correlated geometric Brownian motions under a pricing measure $\mathbb{Q}$ with money-market account $B_t = e^{rt}$ when

for $i = 1, 2$, with a constant rate $r$, constant continuous dividend yields $q_i$, constant volatilities $\sigma_i \ge 0$, and $\mathbb{Q}$-Brownian motions with $d[W^{1,\mathbb{Q}}, W^{2,\mathbb{Q}}]_t = \rho \, dt$ for a constant $-1 \le \rho \le 1$. One construction takes $W^{2,\mathbb{Q}} = \rho W^{1,\mathbb{Q}} + \sqrt{1 - \rho^2}\, Z$ with $Z$ a $\mathbb{Q}$-Brownian motion independent of $W^{1,\mathbb{Q}}$. The covariation of the prices is

and for $\sigma_1, \sigma_2 > 0$ the pair $(\ln S^1_T, \ln S^2_T)$ is bivariate normal with correlation $\rho$. With $d$ assets the matrix $(\rho_{jk})$ of driver correlations must be symmetric with unit diagonal and positive semidefinite, and pairwise values between $-1$ and $1$ do not guarantee this. The law of each price on its own does not depend on $\rho$, but the joint law does, and so does the price of a payoff such as $(S^1_T - S^2_T)^{+}$.

Definition In the correlated model the ratio $R_t = S^1_t/S^2_t$ satisfies

To derive it, Itô’s lemma gives $d(1/S^2_t) = (1/S^2_t)\bigl[(q_2 - r + \sigma_2^2)\, dt - \sigma_2 \, dW^{2,\mathbb{Q}}_t\bigr]$, where the second-order term contributes $\sigma_2^2$, and the product rule for $S^1 \cdot (1/S^2)$ adds the covariation $d[S^1, 1/S^2]_t = -\rho \sigma_1 \sigma_2 R_t \, dt$. The Brownian part has quadratic variation $\sigma_{\text{eff}}^2 \, dt$, where the effective volatility $\sigma_{\text{eff}} \ge 0$ is given by

so $d[\ln R]_t = \sigma_{\text{eff}}^2 \, dt$. With $\sigma_1, \sigma_2 > 0$, it is zero exactly when $\rho = 1$ and $\sigma_1 = \sigma_2$, and it equals $\sqrt{\sigma_1^2 + \sigma_2^2}$ only when $\rho = 0$. The effective volatility is the same under every equivalent measure, since a change of measure alters drifts and not quadratic variation.

Definition Let $N_t = e^{q_2 t} S^2_t$ be the value of one unit of asset 2 bought at time $0$ with its dividends reinvested in the asset, a strictly positive traded price. In the correlated model, $e^{-rt} N_t = S^2_0 \exp\bigl(\sigma_2 W^{2,\mathbb{Q}}_t - \tfrac12 \sigma_2^2 t\bigr)$ is a positive $\mathbb{Q}$-martingale, and the measure $\mathbb{Q}^{N}$ on $\mathcal{F}_T$ is defined by

The exponent is a constant times a Brownian motion, so Novikov’s condition holds, and Girsanov’s theorem with this positive sign gives the $\mathbb{Q}^{N}$-Brownian motions

for $0 \le t \le T$, still with correlation $\rho$. For a claim $X$ paid at $T$ with $\mathbb{E}^{\mathbb{Q}}[\lvert X \rvert] < \infty$, Bayes’ formula with the density process $Z_t = e^{-rt} N_t/N_0$ gives

Substituting $dW^{1,\mathbb{Q}}_t = dW^{1,N}_t + \rho \sigma_2 \, dt$ and $dW^{2,\mathbb{Q}}_t = dW^{2,N}_t + \sigma_2 \, dt$ into the ratio dynamics adds $\rho \sigma_1 \sigma_2 - \sigma_2^2$ to the drift, so

under $\mathbb{Q}^{N}$, with the same effective volatility. The general change-of-numeraire theorem is developed later, in Numeraires and forward measures, and this card uses only the explicit Gaussian density above.

Definition An exchange option gives its holder the right at $T$ to receive one unit of asset 1 by delivering one unit of asset 2, so it pays $(S^1_T - S^2_T)^{+}$, with no cash strike. In the correlated model with $\tau = T - t > 0$ and $\sigma_{\text{eff}} > 0$, its value at $t$ is

with the carry term $b = q_2 - q_1 + \sigma_{\text{eff}}^2/2$. This is Margrabe’s formula. To derive it, note that $N_T = e^{q_2 T} S^2_T$, so the payoff in units of the numeraire is $(S^1_T - S^2_T)^{+}/N_T = e^{-q_2 T}(R_T - 1)^{+}$. Writing $\mathbb{E}^{N}_t$ for the conditional expectation under $\mathbb{Q}^{N}$ given $\mathcal{F}_t$, the pricing identity gives

Given $\mathcal{F}_t$, the variable $\ln R_T$ is normal under $\mathbb{Q}^{N}$ with mean $\ln R_t + (q_2 - q_1 - \sigma_{\text{eff}}^2/2)\tau$ and variance $\sigma_{\text{eff}}^2 \tau$, and the lognormal call computation with unit strike gives the formula. The rate $r$ does not appear, because both legs are traded assets and no cash is paid. Here $\Phi(d_2) = \mathbb{Q}^{N}(S^1_T > S^2_T \mid \mathcal{F}_t)$. If $\sigma_{\text{eff}} = 0$, the ratio is deterministic and $V_t = \bigl(S^1_t e^{-q_1 \tau} - S^2_t e^{-q_2 \tau}\bigr)^{+}$, and at $\tau = 0$ the value is the payoff. Since $(x)^{+} - (-x)^{+} = x$, the option to receive asset 1 for asset 2 and the option to receive asset 2 for asset 1 differ in value by $S^1_t e^{-q_1 \tau} - S^2_t e^{-q_2 \tau}$.

Definition For assets $S^1, \dots, S^d$ in the correlated model with correlation matrix $(\rho_{jk})$ and fixed quantities $w_j$, the basket is $B_t = \sum_j w_j S^j_t$, and a basket call with strike $K > 0$ pays $(B_T - K)^{+}$ at $T$. When $B_t \ne 0$, the value shares are $a_j(t) = w_j S^j_t / B_t$ and the instantaneous proportional variance of the basket is

so that $d[B]_t = \sigma_B^2(t) B_t^2 \, dt$, with every covariance term included. This number is exact at the current composition, but the shares $a_j(t)$ change as prices move, so $\sigma_B^2(t)$ is random and $B$ is in general not a geometric Brownian motion. A sum of lognormal prices is in general not lognormal, so $B_T$ has no lognormal law to integrate against. Dividing by one asset does not help either, since for two assets $(w_1 S^1_T + w_2 S^2_T - K)^{+}/S^2_T = (w_1 R_T + w_2 - K/S^2_T)^{+}$ still depends on $S^2_T$.

Definition Lognormal moment matching approximates $B_T$ by a lognormal variable $L = \exp(\mu_L + s_L Z)$ with $Z \sim N(0, 1)$ that has the same first two $\mathbb{Q}$-moments. With forward prices $F^j = S^j_0 e^{(r - q_j)T}$ these moments are exact,

The matching needs $M_1 > 0$ and $M_2 > M_1^2$, and then $s_L^2 = \ln(M_2/M_1^2)$ and $\mu_L = \ln M_1 - s_L^2/2$. The approximate call price is

This is an approximation, not a statement about the law of $B_T$. Two moments do not fix the third moment or the tails, the method carries no general error bound, and its accuracy has to be checked numerically, for example against Monte Carlo with its standard error. With negative weights, as in a spread, the basket can be zero or negative with positive probability, which no lognormal variable can match, and the mean $M_1$ can be zero or negative.