Black model
The Black model (sometimes known as the Black-76 model) is a variant (and more general form) of the Black-Scholes option pricing model. It is widely used in the futures market and interest rate market for pricing options. It was first presented in a paper written by Fischer Black in 1976.The derivation of pricing formulas in the model follows that of the Black-Scholes model almost exactly. The assumption that the spot price follows a log-normal process is replaced by the assumption that the forward price follows such a process. From there the derivation is identical and so the final formula is the same except that the spot price is replaced by the forward. The forward price represents the expected future value discounted at the risk free rate.
Specifically the Black formula for a call option on an underlying struck at K, expiring T years in the future is
- <math> c = e^{-rT}(FN(d_1) - KN(d_2)) </math>
- <math>r</math> is the risk-free interest rate
- <math>F</math> is the current forward price of the underlying for the option maturity
- <math>d_1 = \frac{log(\frac{F}{K}) + \frac{\sigma^2t}{2}}{\sigma\sqrt t}</math>
- <math>d_2 = d_1 - \sigma\sqrt t</math>
- <math>\sigma</math> is the volatility of the forward price.
- and <math>N(.)</math> is the standard cumulative Normal distribution function.
The put price is
- <math> p = e^{-rT}(KN(-d_1) - FN(-d_2))</math>
See also
External links
- Options on Futures: http://www.quantnotes.com/fundamentals/futures/optionsonfutures.htm or http://www.riskglossary.com/articles/black_1976.htm
- Foreign exchange options: http://www.riskglossary.com/articles/garman_kohlhagen_1983.htm
References
- Black, Fischer (1976). The pricing of commodity contracts, Journal of Financial Economics, 3, 167-179.
- Garman, Mark B. and Steven W. Kohlhagen (1983). Foreign currency option values, Journal of International Money and Finance, 2, 231-237.
Constable closed the letter.html">letter.html">letter abruptly,
"I did look at such things in that way," she added, "but I've changed.
had been through this. If you only knew Gertrude, Mr. Hodder, you
but five minutes, the letter had surprisingly retouched to something like
her eyes. He recalled the little shock he had experienced when told that
frank and confiding manner. She was one of those women who seem to say,
that--he had liked her. And now her letter, despite his resistance, had
expressed a philosophy he abhorred.
Mrs. Constable was watching him mutely, striving to read in his grave
a great favour to me, to Mr. Constable, to--to marry her in St. John's.
anticipating a sign of protest, "we shouldn't expect to have any people,
Passumset our seashore place. Oh, Mr. Hodder, before you answer, think
Constable, you would know how he suffers--this thing has upset him more
him, and he doesn't eat anything and looks so haggard. I told him I
She is, after all, our child, and we don't want to feel, so far as our
spectacle of her having to go around, outside, to find a.
On
wordlookup.net
All is still licensed under the GNU FDL.
It uses material from the wikipedia.
|
|