Friday, December 13, 2013

BS-F -> Black model for IR-sensitive underliers

Generalized BS-Formula assumes a constant interest rate Rgrow :

Call (at valuation time 0) := C0 = Z0 * ( F0 N(d1) – K N(d2) ), where

Z0 := P(0,T) := observed time-0 price of a T-maturity zero coupon bond. There’s no uncertainty in this price as it’s already revealed and observed. We don’t need to assume constant interest rate.
F0 := S0 exp(Rgrow * T), which also appears inside d1 and d2

Q: How does this model tally with the Black model?
A: Simply redefine

F0 := S0 / Z0 , which is the time-0 “fwd price” of the asset S

Now the pricing formula becomes the Black formula for interest-rate-sensitive options. Luckily this applies even if S is the price of 5Y junk bond (or muni bond), and we know 5Y interest rate is stochastic and changes every day.

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