Black Model

The Black formula

$c\; =\; e^\{-rT\}(FN(d\_1)\; -\; KN(d\_2))$

$r$ is the risk-free interest rate

$F$ is the current forward price of the underlying for the option maturity

$d\_1\; =\; \backslash frac\{log(\backslash frac\{F\}\{K\})\; +\; \backslash frac\{\backslash sigma^2t\}\{2\}\}\{\backslash sigma\backslash sqrt\; t\}$

$d\_2\; =\; d\_1\; -\; \backslash sigma\backslash sqrt\; t$

$\backslash sigma$ is the volatility of the forward price.

and $N(.)$ is the standard cumulative Normal distribution function.

$p\; =\; e^\{-rT\}(KN(-d\_2)\; -\; FN(-d\_1)).$

Derivation and assumptions

See also

• Financial mathematics
• Black-Scholes

External links

• Options on Futures: quantnotes.com or riskglossary.com
• Foreign exchange options: riskglossary.com
• Generalized Black-Scholes Calculator by Espen Gaarder Haug (himself)

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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