RY819 - Black-Scholes pricer: negative spot rate at opt. with & days to expiratn

SAP Error Message - Details

• Message type: E = Error

• Message class: RY - Messages for RM Basis

• Message number: 819

• Message text: Black-Scholes pricer: negative spot rate at opt. with & days to expiratn

What causes this issue?

Negative spot values might occur in a <ZH>regular</> way if some of the
following conditions are combined:
Options on interest rate instruments (such as caplets, floorlets,
swaptions or implicit options arising from bounded variable interests)
You have chosen the <LS>Swaption as Interest Rate Option (Black-Scholes
Model) </>valuation model (as opposed to the <LS>Standard</><ZK>
</>valuation model which values swaptions as bond options).
Yields are quoted less than zero
When an absolute but sufficiently large negative shift is applied,
either a market data shift or a shift from historical VaR simulation.
The following <ZH>irregular</> circumstance can also lead to negative
spots:
Forward calculation results in negative yields when the curve of
discount factors does not decrease monotonically.

System Response

The basic assumption of the Black-Scholes model that the spot price S(t)
is a positive number for which possible future values S(T) are
log-normally distributed is not applicable. An alternative assumption on
the distribution is not available.
The option price can still be calculated on the basis of the following
assumption:
Consider S(t)<0 as an extreme case with virtually zero probability that
S will increase beyond strike value K by expiration date T. In other
words: The probability of S(T)>K is so minute as to almost disappear.
The domain of the original Black-Scholes model can subsequently be
extended. In this way, the price function and its first derivatives
steadily continue beyond the original positive spot domain. Therefore,
sensitivity calculation, as well as delta and gamma calculation in VaR
runs will not suffer from indefinite numbers that would occur without
steady continuation.
Price formulas for the domain extension area S(t)<0:

The system issues an error message and will not allow you to continue with this transaction until the error is resolved.

Price value of a put:,,P = B(t,h,T) * [ K - S(t,T) ]
Price value of a call: ,,C = 0
where:
K: Strike price
S(t,T): expectation spot value on expiration date T
B(t,h,T): discount factor taking into account any horizon date h

How to fix this error?

Check yield curves and market data yields for inconsistencies.
If there are no irregular reasons for negative spots, you may consider
either keeping the Black-Scholes model and its domain extension or
changing the valuation model.
In the case of swaptions, for example, you can switch from the
<LS>Swaption as Interest Rate Option (Black-Scholes Model)</> to the
<LS>Standard</> valuation model. Or, for swaptions, caps, and floors,
you can use the <LS>Hull-White</> pricing method.

Error message extract from SAP system. Copyright SAP SE.