Black model
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The Black model (sometimes known as the Black-76 model) is a variant of the Black-Scholes option pricing model. It is widely used in the futures market and interest rate market for pricing bond options. 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.
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[edit] The Black formula
The Black formula is similar to the Black-Scholes formula for valuating stock options except that the spot price of the underlying is replaced by the forward price.
The Black formula for a call option on an underlying struck at K, expiring T years in the future is
- c = e − rT(FN(d1) − KN(d2))
where
- r is the risk-free interest rate
- F is the current forward price of the underlying for the option maturity
- σ is the volatility of the forward price.
- and N(.) is the standard cumulative Normal distribution function.
The put price is
- p = e − rT(KN( − d2) − FN( − d1)).
[edit] Derivation and assumptions
The derivation of the 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.
[edit] See also
[edit] External links
- Options on Futures: quantnotes.com or riskglossary.com
- Foreign exchange options: riskglossary.com
[edit] 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.
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