Black-76 Pricing for an Option on a Forward Contract

# Question about Black76 derivation


# Question about Black76 derivation












Below is my attempted derivation of the Black76 formula to price a $K$-strike forward option where upon option expiry $T_1$, the holder has the right to enter into a $K$-delivery $T_2$-maturity forward contract. The option's payoff is $(F(T_1,T_2)-K)_+$ at $T_1.$ Note that $Z(t,T)$ is the price at $t$ of a $T$-maturity ZCB, $C_K(t,T_1)$ is the price of the $K$-strike $T_1$-maturity call, and $F(t,T_2)$ is the forward price at $t$ for the $T_2$-delivery forward.

By the fundamental theorem, $\frac{C_K(t, T_1)}{Z(t,T_2)}=\mathbb{E}_{\mathbb{Q}^{Z(\cdot, T_2)}}\bigg[\frac{C_K(T_1,T_2)}{Z(T_1,T_2)}\big|\mathcal{F}_t\bigg]=\mathbb{E}_{\mathbb{Q}^{Z(\cdot, T_2)}}\bigg[\frac{F(T_1,T_2)-K)_+}{Z(T_1,T_2)}\big|\mathcal{F}_t\bigg].$

Thus, \begin{align*} C_K(t,T_1)&=Z(t,T_2)\mathbb{E}_{\mathbb{Q}^{Z(\cdot, T_2)}}\bigg[\frac{F(T_1,T_2)-K)_+}{Z(T_1,T_2)}\big|\mathcal{F}_t\bigg]\\&=\mathbb{E}_{\mathbb{Q}^{Z(\cdot, T_2)}}\bigg[Z(t,T_1)(F(T_1,T_2)-K)_+\big|\mathcal{F}_t\bigg]\tag{since $Z(t,T_1)Z(T_1,T_2)=Z(t,T_2)$}\\ &=Z(t,T_1)\mathbb{E}_{\mathbb{Q}^{Z(\cdot, T_2)}}\bigg[(F(T_1,T_2)-K)_+\big|\mathcal{F}_t\bigg]. \end{align*} Note that $F(t,T_2)=\frac{S_t}{Z(t,T_2)}$ is the ratio of tradable assets to the $Z(\cdot, T_2)$ numeraire so $F(t,T_2)$ is a martingale under the $Z(\cdot, T_2)$ measure. This means if we choose lognormal $dF(t,T_2)=F(t,T_2)\sigma dW_t$, then $$\ln F(T_1, T_2)|\mathcal{F}_t\sim \mathcal {N}\left(\ln F(t, T_2)-\frac{1}{2}\sigma^2(T_1-t),\sigma^2(T_1-t)\right).$$ Thus, the price of the option on the forward is \begin{align*}C_K(t,T_1)&=Z(t,T_1)\int_{\ln K}^\infty e^y\frac{1}{\sigma\sqrt{T_1-t}\sqrt{2\pi}}e^{-\frac{\left(y-\ln F(t,T_2)+\frac{1}{2}\sigma^2(T_1-t)\right)^2}{2\sigma^2(T_1-t)}}dy\\&=\boxed{Z(t,T_1)[F(t,T_2)\Phi(d_1)-K\Phi(d_2)].}\end{align*}

Is this correct? I'm not sure I'm allowed to substitute $Z(t,T_1)Z(T_1,T_2)=Z(t,T_2)$, since $E[A/B]$ isn't necessarily $E[A]/E[B]$, which is kind of what I'm doing in order to get rid of Z(T_1,T_2) from the denominator.

## Answer by THATS MY QUANT MY QUANTITATIVE (score 3)

https://quant.stackexchange.com/a/81541

In the black76 model, the discounted parameter is the interest-rate. In the black-scholes, it is the interest rate - dividend rate. ($r-d$). In both models, the discount parameter is constant. Therefore, you can split up the discount parameter as follows; $e^{r t_2} = e^{r{t_1}}e^{r (t_2 - t_1)}$, which is what you want.

Since we are pricing a European option, the fact that the option expires at $T_1$, does not mean it's random. $T$ is not a stopping time by the definition you may have seen, since it always stops at the expiry. Therefore, you can split the exponential up like I showed above.

I suspect you may have over-complicated it for yourself as you seem more than capable with what you have written. When doing the proof, we state that the $r$ is a constant value from $t_0$ to $T$, therefore we can split it up and remove it outside the expectation without having to do any proof.