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The * Black options valuation model* is a modified version of the BSM model used for options on underlying securities that are costless to carry, including options on futures and forward contracts.

Similar to the BSM model, the Black model assumes that future prices follow geometric Brownian motion. The Black option model values for European call and put options are expressed as:

$$ \begin{align*} \text{European call}: c_0 &= e^{–rT} \left[F_0(T)N(d_1) – KN(d_2) \right] \\ \text{European put}: p_0 &=e^{-rT} \left[KN\left({-d}_2\right)-F_0\left(T\right)N\left(-d_1\right) \right] \end{align*} $$

Where:

$$ d_1=\frac{\ln{\left(\frac{F_0\left(T\right)}{K}\right)}+\frac{\sigma^2}{2}T}{\sigma\sqrt T} $$

and

$$ d_2=d_1-\sigma\sqrt T $$

\(F_0\left(T\right)\) = Futures price at time 0 that expires at time T.

\(\sigma\) = Volatility of returns on the futures price.

The NASDAQ index currently stands at $12,900. A three-month futures contract on the index trades at $12,800. The exercise price is $12,750, the continuously compounded risk-free rate is 1%, and volatility is 15%. Finally, the index has a dividend yield of 2%.

The above information has been used to obtain the following results for both call and put options on the futures contract:

$$ \begin{array}{cc|cc} \textbf{Calls} & & \textbf{Puts} & \\ \hline N(d_1) & 0.5636 & N(–d_1) & 0.4364 \\ \hline N(d_2) & 0.5339 & N(–d_2) & 0.4661 \\ \hline c_0 & 405.84 & p_0 & 355.83 \end{array} $$

The values of a European call option and the put option on the futures contract are *closest* to:

$$ \begin{align*} \text{European call}: c_0& = e^{–rT} \left[F_0(T)N(d_1) – KN(d_2) \right] \\ c_0 &= e^{–0.01×0.25} \left[12,800×0.5636– 12,750×0.5339 \right] \\ &=$405.84 \\ \text{European put}: p_0 &=e^{-rT} \left[KN\left({-d}_2\right)-F_0\left(T\right)N\left(-d_1\right) \right] \\ p_0 &=e^{-0.01\times0.25}\left[12,750\times0.4661-12,800\times0.4364\right]\\ & =$355.96 \end{align*} $$

## Question

The US 30 index is at $30,605. A futures contract on it trades at $30,400. The exercise price is $30,000, the continuously compounded risk-free rate is 1.75%, the time to a futures contract and options expiration is two months, and the volatility is 15%. The US 30 dividend yield is 1.8%. The following results have been determined using the above information:

$$ \begin{array}{cc|cc} \textbf{Calls} & & \textbf{Puts} & \\ \hline N(d_1) & 0.6388 & N(–d_1) & 0.3612 \\ \hline N(d_2) & 0.6156 & N(–d_2) & 0.3844 \\ \hline c_0 & 948.75 & p_0 & 549.91 \end{array} $$

Which of the following options

bestdescribes how the black model is used to value a European call option on the futures contract?The call value is the present value of the difference between:

- The current futures price times 0.6388 and the exercise price times 0.6156.
- The exercise price times 0.6156, and the current futures price times 0.6388.
- The current spot price times 0.6388, and the exercise price times 0.6156.
## Solution

The correct answer is A.The value of a European call option on a futures contract is obtained using the formula:

$$ c_0= e^{–rT}\left[F_0(T)N(d_1)– KN(d_2) \right] $$

The above formula implies that the valuation of a European call option based on the black model involves calculating the present value of the difference between the futures price and the exercise price.

Reading 34: Valuation of Contingent Claims

*LOS 34 (i) Describe how the Black model is used to value European options on futures.*