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.


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.

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  • $\begingroup$ Thanks for your answer. But \tilde{x}_1 = x_1 - z \frac{z_2^2}{z} + \frac{z_2^2}{2} = \log U_F - \frac{z^2}{2}\\ this equation is problematic.... $\endgroup$ – yita17 May 12 '13 at 17:07
  • $\begingroup$ What's wrong with it? $\log U_F = x_1 + \frac{z_1^2}{2}$ $\endgroup$ – achille hui May 12 '13 at 17:23
  • $\begingroup$ The formula for $\tilde{x}_1$ is correct. There is a implicit change of variable which generate the extra factor you failed to see. I have modified the answer to make that change of variable from $u$ to $\tilde{u} = u + (z_2^2/z)$ explicit. Can you see that now? $\endgroup$ – achille hui May 13 '13 at 7:58
  • $\begingroup$ oh, it's all my fault...thanks so much...by the way, does PAR FORWARD have realistic financial meaning? $\endgroup$ – yita17 May 13 '13 at 9:08
  • $\begingroup$ "Par Forward" originates in the FX (foreign exchange) market. It is an agreement to exchange a series of cash-flows over time from one currency to another currency with a pre-agreed rate. We also use "par forward" to describe the currency rate which will be fair for both parties. Its usage has been extended beyond currencies market. Sometimes we simply call "the fair rate to use" for any non-contingent exchange between any tradeable assets in future as "par forward". $\endgroup$ – achille hui May 13 '13 at 9:58

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