Limited Time Offer: Save 10% on all 2021 and 2022 Premium Study Packages with promo code: BLOG10    Select your Premium Package »

Black Option Valuation Model

Black Option Valuation Model

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.

Example: Black Option Valuation

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:

Solution

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

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 best describes 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:

  1. The current futures price times 0.6388 and the exercise price times 0.6156.
  2. The exercise price times 0.6156, and the current futures price times 0.6388.
  3. 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.

Featured Study with Us
CFA® Exam and FRM® Exam Prep Platform offered by AnalystPrep

Study Platform

Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Online Tutoring
    Our videos feature professional educators presenting in-depth explanations of all topics introduced in the curriculum.

    Video Lessons



    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.
    Nyka Smith
    Nyka Smith
    2021-02-18
    Every concept is very well explained by Nilay Arun. kudos to you man!
    Badr Moubile
    Badr Moubile
    2021-02-13
    Very helpfull!
    Agustin Olcese
    Agustin Olcese
    2021-01-27
    Excellent explantions, very clear!
    Jaak Jay
    Jaak Jay
    2021-01-14
    Awesome content, kudos to Prof.James Frojan
    sindhushree reddy
    sindhushree reddy
    2021-01-07
    Crisp and short ppt of Frm chapters and great explanation with examples.