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The '''Black model''' (sometimes known as the '''Black-76 model''') is a variant of the [[Black–Scholes]] option pricing model.  Its primary applications are for pricing [[bond option]]s, interest rate caps / floors, and [[swaption]]s.  It was first presented in a paper written by [[Fischer Black]] in 1976.
 
Black's model can be generalized into a class of models known as log-normal forward models, also referred to as [[LIBOR market model]].
 
==The Black formula==
The Black formula is similar to the [[Black–Scholes formula]] for valuing [[stock option]]s except that the [[spot price]] of the underlying is replaced by a discounted [[futures price]] F.
 
Suppose there is constant [[risk-free interest rate]] ''r'' and the futures price ''F(t)'' of a particular underlying is log-normal with constant volatility ''&sigma;''. Then the Black formula states the price for a [[European call option]] of maturity ''T'' on a [[futures contract]] with strike price ''K'' and delivery date ''T''' (with <math>T' \geq T</math>) is
 
:<math> c = e^{-r T} [FN(d_1) - KN(d_2)]</math>
 
The corresponding put price is
 
:<math> p = e^{-r T} [KN(-d_2) - FN(-d_1)]</math>
 
where
 
:<math> d_1 = \frac{\ln(F/K) + (\sigma^2/2)T}{\sigma\sqrt{T}} </math>
 
:<math> d_2 = \frac{\ln(F/K) - (\sigma^2/2)T}{\sigma\sqrt{T}} = d_1 - \sigma\sqrt{T}, </math>
 
and N(.) is the [[Normal distribution#Cumulative distribution function|cumulative normal distribution function]].
 
Note that ''T' '' doesn't appear in the formulae even though it could be greater than ''T''. This is because futures contracts are marked to market and so the payoff is realized when the option is exercised. If we consider an option on a [[forward contract]] expiring at time ''T' > T'', the payoff doesn't occur until ''T' ''. Thus the discount factor <math>e^{-rT}</math> is replaced by <math>e^{-rT '}</math> since one must take into account the [[time value of money]]. The difference in the two cases is clear from the derivation below.
 
==Derivation and assumptions==
The Black formula is easily derived from use of [[Margrabe's formula]], which in turn is a simple, but clever, application of the [[Black–Scholes formula]]. 
 
The payoff of the call option on the futures contract is ''max (0, F(T) - K)''. We can consider this an exchange (Margrabe) option by considering the first asset to be <math>e^{-r(T-t)}F(t)</math> and the second asset to be the riskless bond paying off $1 at time ''T''. Then the call option is exercised at time ''T'' when the first asset is worth more than ''K'' riskless bonds.  The assumptions of Margrabe's formula are satisfied with these assets. 
 
The only remaining thing to check is that the first asset is indeed an asset.  This can be seen by considering a portfolio formed at time 0 by going long a ''forward'' contract with delivery date ''T'' and short ''F(0)'' riskless bonds (note that under the deterministic interest rate, the forward and futures prices are equal so there is no ambiguity here). Then at any time ''t'' you can unwind your obligation for the forward contract by shorting another forward with the same delivery date to get the difference in forward prices, but discounted to present value: <math>e^{-r(T-t)}[F(t) - F(0)]</math>. Liquidating the ''F(0)'' riskless bonds, each of which is worth <math>e^{-r(T-t)}</math>,  results in a net payoff of <math>e^{-r(T-t)}F(t)</math>.
 
==See also==
*[[Financial mathematics]]
*[[Black–Scholes]]
 
==External links==
'''Discussion'''
*[https://www.ma.utexas.edu/users/mcudina/Lecture24_3.pdf Bond Options, Caps and the Black Model] Dr. Milica Cudina, [[University of Texas at Austin]]
'''Online tools'''
*[http://lombok.demon.co.uk/financialTap/options/bond/shortterm Caplet And Floorlet Calculator] Dr. Shing Hing Man, Thomson-Reuters' Risk Management
*[http://www.cba.ua.edu/~rpascala/greeks2/GFOPMForm.php 'Greeks' Calculator using the Black model], Razvan Pascalau, Univ. of Alabama
 
==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.
*Miltersen, K., Sandmann, K. et Sondermann, D., (1997): "Closed Form Solutions for Term Structure Derivates with Log-Normal Interest Rates", Journal of Finance, 52(1), 409-430.
 
{{Derivatives market}}
 
[[Category:Mathematical finance]]
[[Category:Options (finance)]]
[[Category:Finance theories]]

Latest revision as of 10:00, 11 November 2014

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