Pricing a Heat Rate Option on Two Independent Lognormal Assets

# pricing of heat rate-linked derivative


# pricing of heat rate-linked derivative












It's a simplified model. Suppose $U_t$ is a random variables subject to Lognormal($x_1$, $z_1^2$)distribution. $V_t$ is a random variables subject to Lognormal($x_2$, $z_2^2$)distribution. Suppose they are independent here. The payoff of the heat rate-linked derivatives is $\max(U_T - V_T, 0)$. How to price this option? It's a integration stuff.

## Answer by achille hui (score 5, accepted)

https://quant.stackexchange.com/a/7982

This might be a surprise to you, you can evaluate the option using Black Scholes.

The key concept is change your numéraire from dollar to the asset associated with $V$. The $V$ in your payout $\max(U_t-V_t,0)$ will effectively get replaced by a constant, the par forward of asset $V$ at maturity $t$.

Since $U_t$ and $V_t$ are independent, you can parametrize them by two standard normal random variables $\eta_1$, $\eta_2$ with mean $0$ and standard derivation $1$:

$$U_t = e^{x_1 + z_1 \eta_1}\quad\text{ and }\quad V_t = e^{x_2 + z_2 \eta_2}$$

Let $(\cdots)^{+}$ stands for the function $\max(\cdots,0)$, the future value of the option is given by the integral:

$$\begin{align}\text{F.V.} = & \int ( U_t - V_t )^{+} \exp( -\frac{\eta_1^2 + \eta_2^2}{2}) \frac{d\eta_1 d\eta_2}{2\pi}\\ = &\int ( e^{x_1 + z_1 \eta_1} - e^{x_2 + z_2 \eta_2} )^{+} \exp( -\frac{\eta_1^2 + \eta_2^2}{2}) \frac{d\eta_1 d\eta_2}{2\pi}\\ = &\int ( e^{x_1 + ( z_1 \eta_1 - z_2 \eta_2 ) } - e^{x_2} )^{+} \exp( z_2\eta_2 -\frac{\eta_1^2 + \eta_2^2}{2}) \frac{d\eta_1 d\eta_2}{2\pi}\tag{*1} \end{align}$$

Let $$z = \sqrt{z_1^2+z_2^2}\quad\text{ and }\quad\begin{cases}u = \frac{z_1\eta_1 - z_2\eta_2}{z}\\ \\ v = \frac{z_2\eta_1 + z_1\eta_2}{z}\end{cases} \Longleftrightarrow \begin{cases}\eta_1 = \frac{z_1 u + z_2 v}{z}\\ \\ \eta_2 = \frac{z_1 v - z_2 u }{z}\end{cases} $$

It is easy to check:

$$ u^2 + v^2 = \eta_1^2 + \eta_2^2 \quad\text{ and }\quad du dv = d\eta_1 d\eta_2$$

Let $U_F = e^{x_1 + \frac{z_1^2}{2}}$ and $V_F = e^{x_2 + \frac{z_2^2}{2}}$ be the par forward of asset $U$ and $V$ at maturity $t$. We can rewrite $(*1)$ as: $$ \begin{align} &\int ( e^{x_1 + z u } - e^{x_2} )^{+} \exp\left( \frac{z_2(z_1 v - z_2 u)}{z} -\frac{u^2 + v^2}{2}\right) \frac{du dv}{2\pi}\\ = & \int ( e^{x_1 + z u } - e^{x_2} )^{+} \exp\left( \frac{z_2^2}{2}-\frac{(u + (z_2^2/z))^2 + ( v - (z_1z_2/z))^2}{2}\right) \frac{du dv}{2\pi}\\ = & \int ( e^{\tilde{x}_1 + z \tilde{u}} - e^{\tilde{x}_2} )^{+} e^{-\frac{\tilde{u}^2}{2}} \frac{d\tilde{u}}{\sqrt{2\pi}} \quad\text{ where } \begin{cases} \tilde{u}\; = u + (z_2^2/z)\\ \tilde{x}_1 = x_1 - z \frac{z_2^2}{z} + \frac{z_2^2}{2} = \log U_F - \frac{z^2}{2}\\ \tilde{x}_2 = x_2 + \frac{z_2^2}{2} = \log V_F \end{cases}\end{align}$$ As a result, we have: $$\text{F.V.} = \int ( U_F\,e^{z\tilde{u} - \frac{z^2}{2}} - V_F )^{+} e^{-\frac{\tilde{u}^2}{2}} \frac{d\tilde{u}}{\sqrt{2\pi}}\tag{*2}$$

This is nothing but the future value of a call option with strike $V_F$ on an asset with par forward $U_F$ and standard derivation $z$ at maturity. You can finish the integral using Black Scholes.

If $U_t$ and $V_t$ are not independent to each other, you can still transform F.V. to an integral of the form $(*2)$. The only difference is $U_F$ and $z$ there will be adjusted by some factors.

If you want to learn how to deal with the case with correlation, pickup any standard textbook on option pricing and look for the pricing of quanto option. The issues you encountered in pricing a quanto option is similar to the one you need to price your heat-linked option under the log normal model.