option pricing - Ways of treating time in the BS formula

The Black-scholes formula typically has time as $\sqrt{T-t}$ or some such. My questions:

• What is the granularity of this? If we treat $t$ as the number of days, then logically on the day of expiry, d₁ would involve division by zero.


• Given that $t$ is the number of days in the year, what value should we divide it by? 252 or 365? It is that important? It it worth using quantlib to take account of various calendars?

Answer

Two quick points:

1. 
   

   Recall that the derivation involves continuous time and $(t, t+\Delta t)$ arguments---so the granularity is (at the margin) infinite. And hence time zero does not really get reached until we actually are at expiry.

   
   
   


2. 
   

   Generally speaking want the number of business days, not calendar days, and holidays do matter. So one generally uses the 'number of trading days til expiry / 252'.

   
   

Here is a quick example moving maturity from 1 year to 0.9975 years

R> library('RQuantLib')
R> EuropeanOption("call", 100, 100, 0.01, 0.03, 1, 0.4)
Concise summary of valuation for EuropeanOption 
   value    delta    gamma     vega    theta      rho   divRho 

 16.5382   0.5927   0.0096  38.2821  -8.3458  42.7367 -59.2749 
R> EuropeanOption("call", 100, 100, 0.01, 0.03, 0.9975, 0.4)
Concise summary of valuation for EuropeanOption 
   value    delta    gamma     vega    theta      rho   divRho 
 16.5150   0.5926   0.0096  38.2332  -8.3578  42.6295 -59.0986 
R>