FX Option Curvature and Interest Rate Cross Gamma Under FRTB-SA

Summary

The document raises a question about whether cross-gamma effects belong in curvature calculations under the Fundamental Review of the Trading Book standardised approach. It describes an FX option whose risk is represented in the two currency legs relative to the reporting currency. FX shocks change option delta and therefore alter the associated currency positions; interest-rate shocks also change option value and produce GIRR curvature risk. The author says the translation effects from those rate shocks have been modelled, then asks whether the resulting rate-driven delta change and consequent FX-position change should also be captured as cross gamma.

The text provides a problem setup rather than a resolution. It distinguishes direct FX shock effects from interest-rate shock effects that flow through option delta into FX exposure, making the interaction relevant to risk aggregation and model design. No regulatory interpretation, calculation method, or worked example resolves the question. Any implementation conclusion therefore requires further guidance on the applicable FRTB-SA rules and conventions.

Key ideas

• An FX option can create risk positions in both currency legs when measured against a reporting currency.
• FX shocks can change option delta and alter the resulting currency positions.
• Interest-rate shocks can change option value and produce GIRR curvature risk.
• A rate shock may also change delta, creating a secondary change in FX exposure described as cross gamma.
• The document poses whether that interaction belongs in curvature calculations but does not provide an answer or regulatory method.

Tags

# FRTB - SA Curvature cross gamma with FX options


# FRTB - SA Curvature cross gamma with FX options












Under the FRTB we do an downwards and upwards shocks reating to the each of the risk factors.

FX risk is also calculated as the change in respect to the reporting currency to each of legs of the trade.

For example a USDJPY FX option with say $10m Notional, when the reporting currency is EUR, we need to calculate the FX risk based on EUR to USD and EUR to JPY. This would be;

Notional x delta = USD FX position

Notional x strike x delta = JPY FX position

These positions are used to calculate the FX risk.

When calculating the curvature - the FX shocks changes the delta, and we get new positons in USD and JPY. Quite streight forward.

However if I shock the USD and JPY interest rates, I get a change in the OV giving the GIRR CRV risk impact. These GIRR risk give rise to translation risk as well in that we have a risk position in USD and JPY. This I have modelled.

However the shock in interest rates leads to a change in the delta, which in tern leads to a change in the FX position which is cross gamma. Should this cross gamma impact be picked up in the CRV calculations as well?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.