Why I use Black formula rather than Black-Scholes

When I need to price a European option, I use Black formula rather than Black-Scholes. Although both formulas give the same result when applied correctly, I think that Black formula is a bit more general. Let me show why.

The Black-Scholes formula gives the value of a European option. For example, at time t, the price of a call option maturing at time T on the underlying asset with spot price S and given the risk-free interest rate r is

where

σ is the volatility of the underlying. The formula as written above assumes that the underlying pays no dividends.

The Black formula looks similar, but instead of spot price of the underlying it uses its forward price F:

where

is the price of a zero-coupon bond paying 1 at time T,

20090924-d2-Black-Scholes

σ is the volatility of the underlying.

Firstly, the Black formula does not assume a constant risk-free interest rate. Instead, we have to know the price of a zero-coupon bond maturing at T, which is usually easy to get.

Secondly, if the underlying asset pays dividends, the Black formula still works, because the dividends (whether discrete or continuous) are already taken into account by the forward price F. Finding the forward price is no more difficult than the spot price; moreover, in some cases forwards are actually more liquid than the stock itself.

Thirdly, the same formula can be applied to foreign exchange options and interest rate derivatives (caps, floors and swaptions).

If we assume that the interest rate is constant and no dividends are paid, the Black formula is equivalent to the Black-Scholes formula. In that case,

20090924-FisSerT

20090924-PtTiserT

20090924-d1-equivalence

20090924-Black-equivalence

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