Black Scholes ( Black 76 ) Formula in Endur with Delayed Payment (from Exercise Date)

Black Scholes ( Black 76 ) Formula in Endur with Delayed Payment (from Exercise Date)

by Israr Ahmed

 

In this Article we look at the academic definition of how Black Scholes is used to value options on forwards/futures – in the context of delayed payment from exercise date. We then look at how this is implemented in Endur. We also have a version of the Endur BS Formula in an online Spreadsheet format.

 

Academic Model

Black 76

Black Scholes is used to price Spot prices. To model Future/Forward prices we assumes Futures are like a stock paying dividend q equal to r. The standard Black-Scholes with dividend collapses into:

 

 

 

 

Black 76 extension: Delayed Payment Date

 

In reality the actual Payment Date generally occurs later than the Expiry Date ie .

 

We can think of the derivative becoming a cash position (equal to it’s value) at expiration. Such a cash position does not accrue interest until payment date. Therefore this type of contract must be further discounted to take into account the delay in payment.

 

 

 

=      (A)

 

where is the discount from the Payment Date back to the Exercise Date ie

 

 

 

now substituting this into (A) we can also re-write

 

 

        (B)

 

 

(A) and (B) are equivalent

 

where

 

is the zero rate at Expiry Date

 

is the zero rate at Payment Date

 

 

 

OpenLink Model

 

OpenLinks B-S ( Black 76 ) Model with Delayed Payment Date

 

There are a number of modifications to consider:

  • OpenLink sets q (the dividend) to = r (risk free rate) so that the B-S Model essentially becomes the Black 76 Model
  • OpenLink actually calculates the B-S (ie Black 76) value as at the Payment Date using the modified Black 76 extension shown above.
  • = time to Expiration (with CC Zero Rate r )
  • = time to Payment (with CC Zero Rate )

 

Given that


then

 

The formula can then be written in two different way – essentially exactly the same – but Formula 2 is more explicit.

 

 

Formula 1 ( as implemented by OpenLink )

 

 

 

Formula 2 (more explicit)

 

 

 

 

where we have the usual , time t (exercise date ) calculated;

 

 

 

 

Note: OpenLink gets the value for {ie ‘R’ in model input screen} by recalculating the CC Zero Rate using Act/365 rather than what is on the LIBOR curve setting

 


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