Kirk’s approximation - A numerical experiment
Margrabe’s formula
Let \(S_1(t)\) and \(S_2(t)\) denote the prices of two risky assets which have dynamics:
\[ \begin{align*} dS_1(t)/ S_1(t) &= r dt + \sigma_1 dW_1^{\mathbb{Q}}(t) \\ dS_2(t)/ S_2(t) &= r dt + \sigma_2 dW_2^{\mathbb{Q}}(t) \end{align*} \]
where \(r\) is the constant risk-free rate, \(W_1^{\mathbb{Q}}(t)\) and \(W_2^{\mathbb{Q}}(t)\) are brownian motions with instantaneous correlation \(\rho\).
We are interested to price the payoff
\[ V_T = (S_1(T) - S_2(T))^+ \]
By the risk-neutral pricing formula,
\[ \begin{align*} V_0 &= M(0)\mathbb{E}^{\mathbb{Q}}\left[\frac{V(T)}{M(T)}\right]\\ &= S_2(0)\mathbb{E}^{\mathbb{Q}^{S_2}}\left[\frac{V(T)}{S_2(T)}\right]\\ & \quad \{\text{Switching from }\mathbb{Q}\text{ to }\mathbb{Q}^{S_2}\text{-measure.}\}\\ &= S_2(0)\mathbb{E}^{\mathbb{Q}^{S_2}}\left[\frac{1}{S_2(T)}S_1(T) - S_2(T) 1_{S_1(T) > S_2(T)}\right]\\ &= S_2(0)\mathbb{E}^{\mathbb{Q}^{S_2}}\left[\left(\frac{S_1(T)}{S_2(T)} - 1 \right) 1_{S_1(T) > S_2(T)}\right]\\ \end{align*} \]
Define the asset price process \(Y(t)\) as:
\[ Y(t) := \frac{S_1(t)}{S_2(t)} \]
So, we want to compute the expectation
\[ V_0 = S_2(0) \mathbb{E}^{\mathbb{Q}^{S_2}} \left[(Y_T - 1)^+\right] \]
Dynamics of \((Y_t)\)
We know that \((Y_t,t\geq 0)\) is a \(\mathbb{Q}^{S_2}\) martingale. The \(\mathbb{Q}\)-dynamics of \((Y_t)\) is:
\[ \begin{aligned} dY_{t} & =d\left(\frac{S_{1}( t)}{S_{2}( t)}\right)\\ & \left\{\text{Applying Ito's product rule }\right\}\\ & =S_{1}( t) d\left(\frac{1}{S_{2}( t)}\right) +\frac{1}{S_{2}( t)} dS_{1}( t) +dS_{1}( t) d\left(\frac{1}{S_{2}( t)}\right)\\ & =-S_{1}( t)\left[\frac{1}{S_{2}( t)^{2}} dS_{2}( t) +\frac{1}{2}\left(\frac{2}{S_{2}( t)^{3}}\right) dS_{2}( t) \cdot dS_{2}( t)\right] +\frac{1}{S_{2}( t)}\left( rS_{1}( t) dt+\sigma _{1} S_{1}( t) dW_{1}^{\mathbb{Q}}( t)\right)\\ & +S_{1}\left( rdt+\sigma _{1} dW_{1}^{\mathbb{Q}}( t)\right)\left[ -\frac{1}{S_{2}( t)^{2}} dS_{2}( t) +\frac{1}{2}\left(\frac{2}{S_{2}( t)^{3}}\right) dS_{2}( t) \cdot dS_{2}( t)\right]\\ & =-\frac{S_{1}( t)}{S_{2}( t)}\left(\cancel{rdt} +\sigma _{2} dW_{2}^{\mathbb{Q}}( t)\right) -\frac{S_{1}( t)}{S_{2}( t)} \sigma _{2}^{2} dt+\frac{S_{1}( t)}{S_{2}( t)}\left(\cancel{rdt} +\sigma _{1} dW_{1}^{\mathbb{Q}}( t)\right)\\ & +\frac{S_{1}}{S_{2}}\left( rdt+\sigma _{1} dW_{1}^{\mathbb{Q}}( t)\right)\left[ -\left( rdt+\sigma _{2} dW_{2}^{\mathbb{Q}}( t)\right) +\sigma _{2}^{2} dt\right]\\ & =\frac{S_{1}}{S_{2}}\left[ -\sigma _{2} dW_{2}^{\mathbb{Q}}( t) +\sigma _{1} dW_{1}^{\mathbb{Q}}( t) -\rho \sigma _{1} \sigma _{2} dt-\sigma _{2}^{2} dt\right]\\ &=Y_t\left[ -\sigma _{2} dW_{2}^{\mathbb{Q}}( t) +\sigma _{1} dW_{1}^{\mathbb{Q}}( t) -\rho \sigma _{1} \sigma _{2} dt-\sigma _{2}^{2} dt\right] \end{aligned} \]
Since we know, the \(Y_t\) is the price of \(S_1(t)\) expressed in units of \(S_2(t)\), it is a \(\mathbb{Q}^{S_2}\)-martingale. So, we can just drop the \((...)dt\) terms and write:
\[ dY_t = Y_t \left[ -\sigma_{2} dW_{2}^{\mathbb{Q}^{S_2}}( t) +\sigma_{1} dW_{1}^{\mathbb{Q}^{S_2}}(t) \right] \]
We can perform an orthogonal decomposition of the correlated brownian motions \(W_1^{\mathbb{Q}^{S_2}}(t)\) and \(W_2^{\mathbb{Q}^{S_2}}(t)\) and write:
\[ \begin{align*} dY_t = Y_t \left[ -\sigma_{2} (\rho dB_1^{\mathbb{Q}^{S_2}} (t) + \sqrt{1 - \rho^2} dB_2^{\mathbb{Q}^{S_2}}(t)) +\sigma_{1} dB_{1}^{\mathbb{Q}^{S_2}}(t) \right]\\ dY_t = Y_t \left[(\sigma_1 - \rho \sigma_2) dB_1^{\mathbb{Q}^{S_2}} (t) - \sigma_2 \sqrt{1 - \rho^2}dB_2^{\mathbb{Q}^{S_2}}(t)\right] \end{align*} \]
Define the process \((X_t,t\geq 0)\) as:
\[ dX_t = \frac{1}{\sigma} \left[(\sigma_1 - \rho \sigma_2) dB_1^{\mathbb{Q}^{S_2}} (t) - \sigma_2 \sqrt{1 - \rho^2}dB_2^{\mathbb{Q}^{S_2}}(t)\right] \]
where \(\sigma = \sqrt{\sigma_1^2 + \sigma_2^2 - 2\rho \sigma_1 \sigma_2}\).
It follows that \((X_t,t\geq 0)\) is a martingale and
\[ \begin{align*} dX_t \cdot dX_t &=\frac{1}{\sigma^2}\left[ \sigma_2^2(\rho dB_1^{\mathbb{Q}^{S_2}} (t) + \sqrt{1 - \rho^2} dB_2^{\mathbb{Q}^{S_2}}(t))^2 + \sigma_1^2 dt - 2\rho \sigma_1 \sigma_2 dt\right]\\ &=\frac{1}{\sigma^2}(\sigma_1^2 + \sigma_2^2 - 2\rho \sigma_1 \sigma_2)dt \\ &= dt \end{align*} \]
By Levy’s characterization theorem, \((X_t,t\geq 0)\) is a standard brownian motion. Hence, \((Y_t)\) given by the SDE:
\[ dY_t = \sigma Y_t dX_t \]
follows lognormal dynamics.
Analytical formula
We can thus price the claim \(\mathbb{E}^{\mathbb{Q}^{S_2}}\left[(Y_T - 1)^+\right]\) using the Black formula for a european call option with the asset price given by \(Y_t = S_1(t)/S_2(t)\), strike \(K = 1\), the volatility parameter \(\sigma = \sqrt{\sigma_1^2 + \sigma_2^2 - 2\rho \sigma_1 \sigma_2}\) and riskfree rate \(r=0\). Subbing these quantities in the Black formula, we have:
\[ \begin{align*} V(0) &= S_2(0) (F\Phi(d_{+}) - K\Phi(d_{-})) \\ &= S_2(0)\left(\frac{S_1(0)}{S_2(0)}\Phi(d_{+}) - \Phi(d_{-})\right)\\ &=S_1(0)\Phi(d_{+}) - S_2(0)\Phi(d_{-}) \end{align*} \]
where
\[ d_{\pm} = \frac{\ln\left(\frac{S_1(0)}{S_2(0)}\right) \pm \frac{\sigma^2}{2}T}{\sigma\sqrt{T}} \]
References
- Margrabe’s formula, Wikipedia.