Sembra tutto facile e comprensibile anche per uno come me digiuno di opzioni che conosce solo la B & S.
Siamo all'ABC, o poco piu'. Bastera' leggere con attenzione e credo che i dubbi verranno facilmente dissipati.
Rate Caps ***8211; Black Model - John C. Hull, Options, Futures and other Derivative Securities, 2nd ed., 1993,
The basic model for rate caps or floors is as described in Hull.
Let us first consider the case of a rate cap. We lose no generality in considering one caplet at a time since the total cap can be treated as a sum of the individual caplets.
Let K be the cap rate, let T be the time from the valuation date until the start of the caplet, let t be the length of the caplet period (from T to T+ t) and let L be the (unknown) reset rate for the caplet at time T.
As previously defined the cash flow for the caplet period is t P max (L - K, 0), at time T+ t, or a value of t P.max(L - K, 0) / (1+ t L) at time T. Thus, the pay-out for the caplet looks much like the pay-out from a European option terminating at time T and a commonly used valuation model treats it as such.
Clearly, at the valuation date, one does not know the reset rate L. One only has the implied forward rate F. In the basic valuation a simplification is made. Letting D=1/(1+ t F) be the discount factor from time T to T+ t, one assumes that the value at time T is D t P max (F -K,0). Now F is treated as a stochastic term while the rest of the terms are held fixed, including D (though it depends on F). Assuming F has a lognormal distribution (with a constant volatility s), one may model the caplet as a European call option on the forward rate F using the Black model.
Consider the Black model with a forward rate of F, a strike of K, a volatility s, an option period of T and a risk-free interest rate of r (compounded continuously). The fair value of a call option is
C (vedi formula sotto)
and N( . ) is the cumulative standard normal distribution . Note that e(-rT) is simply the discount factor that applies on the option expiry date. This can be obtained directly from the zero curve and e(-rT) is replaced with Z. Hence, for the caplet, one applies the Black model with a forward rate of F, a strike of K, a volatility s , an option period of T and a discount factor of Z.
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