The black option 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.<\/p>\n
Similar to the BSM model, the black model assumes that future prices follow geometric Brownian motion.<\/p>\n
The black option model values for European call and put options are expressed as:<\/p>\n
European call: \\(c_{0}=e^{-rT}[F_{0}(T)N(d_{1})-KN(d_{2})]\\)<\/p>\n
European put: \\(p_{0}=e^{-rT}[KN(-d_{2})-F_{0}(T)N(-d_{1})]\\)<\/p>\n
Where:\u00a0<\/p>\n
$$d_{1}=\\frac{ln\\bigg(\\frac{F_{0}(T)}{K}\\bigg)+\\frac{\\sigma^{2}}{2}T}{\\sigma\\sqrt{T}}$$<\/p>\n
and\u00a0<\/p>\n
$$d_{2}=d_{1}-\\sigma\\sqrt{T}$$<\/p>\n
\\(F_{0}(T)\\) = Futures price at time 0 that expires at time T<\/p>\n
\\(\\sigma\\)= volatility of returns on the futures price<\/p>\n
The NASDAQ index is currently 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%, volatility is 15%, and the index has a dividend yield of 2%.<\/p>\n
The above information has been used to obtain the following results for both call and put options on the futures contract:<\/p>\n
$$\\small{\\begin{array}{|cc|cc|}\\hline\\textbf{Calls}&{}&\\textbf{Puts}&{}\\\\\\hline\\text{N}(\\text{d}_{1})&0.5636&\\text{N}(-\\text{d}_{1})&0.4364\\\\ \\hline\\text{N}(\\text{d}_{2})&0.5339&\\text{N}(-\\text{d}_{2})&0.4661\\\\ \\hline\\text{C}_{0}&405.84&\\text{p}_{0}&355.83\\\\ \\hline\\end{array}}$$<\/p>\n
The values of a European call option and the put option on the futures contract are closest to:<\/p>\n
European call: \\(c_{0}=e^{-rT}[F_{0}(T)N(d_{1})-KN(d_{2})]\\)<\/p>\n
$$c_{0}=e^{-0.01\\times0.25}[12,800\\times0.5636-12,750\\times0.5339]=$405.84$$<\/p>\n
European put: \\(p_{0}=e^{-rT}[KN(-d_{2})-F_{0}(T)N(-d_{1})]\\)<\/p>\n
$$p_{0}=e^{-0.01\\times0.25}[12,750\\times0.4661-12,800\\times0.4364]=$355.96$$<\/p>\n
\nQuestion<\/h2>\n
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:<\/p>\n
$$\\small{\\begin{array}{|cc|cc|}\\hline\\textbf{Calls}&{}&\\textbf{Puts}&{}\\\\\\hline\\text{N}(\\text{d}_{1})&0.6388&\\text{N}(-\\text{d}_{1})&0.3612\\\\ \\hline\\text{N}(\\text{d}_{2})&0.6156&\\text{N}(-\\text{d}_{2})&0.3844\\\\ \\hline\\text{C}_{0}&948.75&\\text{p}_{0}&549.91\\\\ \\hline\\end{array}}$$<\/p>\n
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:<\/p>\n
\n
- The current futures price times 0.6388 and the exercise price times 0.6156.<\/li>\n
- The exercise price times 6156 and the current futures price times 0.6388.<\/li>\n
- The current spot price times 0.6388, and the exercise price times 0.6156.<\/li>\n<\/ol>\n
Solution<\/h3>\n
The correct answer is A:<\/strong>\u00a0<\/strong><\/p>\n
The value of a European call option on a futures contract is obtained using the formula:<\/p>\n
$$c_{0}=e^{-rT}[F_{0}(T)N(d_{1})-KN(d_{2})]$$<\/p>\n
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.<\/p>\n<\/blockquote>\n
Reading 38: Valuation of Contingent Claims<\/em><\/p>\n
LOS 38 (i) describe how the Black model is used to value European options on futures;<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"
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