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Attributing FX Option Valuation Differences to Volatility and Forward Inputs

Article Quant Q&A · Author: Ronnie
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Summary

The document considers how to explain a mark-to-market difference between two systems valuing the same AUD/JPY option under a Black–Scholes-style framework. The systems report different implied volatilities, forward rates, vegas, and USD deltas. The question is how much of the valuation gap comes from volatility and forward-rate inputs, and whether any remaining amount can also be explained.

The responses do not provide a numerical attribution procedure. They identify plausible sources of residual differences: gamma and other higher-order sensitivities, differing interest-rate assumptions, and inconsistent conventions for expressing USD delta on a cross-currency option. The suggested model is Garman–Kohlhagen, which extends Black–Scholes with separate rates for the two currencies. Comparing system outputs therefore requires aligning rate curves and clarifying the currency and hedge convention behind vega and delta; first-order Greek estimates alone may not account for the entire MtM gap.

Key ideas

  • A difference in implied volatility and forward rate can contribute to an FX option mark-to-market gap.
  • First-order Greek estimates may leave residual valuation differences because of gamma and other higher-order effects.
  • Different interest-rate assumptions can materially change a cross-currency option valuation.
  • Garman–Kohlhagen accounts for the two currencies' separate interest rates.
  • USD delta conventions can vary, so the meaning of each system's delta should be confirmed.

Tags

ForexOptionsDerivatives pricingVolatility
Full text
# How to break down an FX option P&L?


# How to break down an FX option P&L?












I am comparing the mark-to-market (MtM) valuations of two risk systems, with respect to FX Options.

My question is can I quantify the difference in MtM given the following:

System1

AUD/JPY, MTM = USD 461,000, Implied Vol. = 11.88%, Vega = USD 82,000, Forward Rate = 97.29 and USD Delta = -15,300,000

System2

AUD/JPY, MTM = USD 406,000, Implied Vol. = 12.14%, Vega = USD 77,000, Forward Rate = 97.81 and USD Delta = -13,600,000

Assuming both systems use Black Scholes, how can I quantify the difference in MtM (in USD) which is USD 55,000 by attributing it to:

- Difference in Implied Volatility and;

- Difference in Forward Rates?

I tried doing this and am still left with a small difference - is it possible to quantify that too?

## Answer by q.t.f. (score 1)

https://quant.stackexchange.com/a/16632

Some residual difference is expected due to gamma and other higher order greeks, and to rate assumptions as mentioned above. Also you should make sure about the meaning of delta expressed in USD for an AUD/JPY option; I am not sure that is standard across systems. It could be the dollar value of the JPY to hold as hedge taking AUD as the riskless currency, or dollar value of AUD to hold as hedge taking JPY as riskless currency, or either the value of AUD or JPY to hold for a combined hedge of both currencies' moves against USD. It could also be premium-included or forward versions of these.

## Answer by weismat (score 0)

https://quant.stackexchange.com/a/16614

Did you check that you use the same interest rate for AUD and JPY in both systems? The difference is quite large. The used model should be Garman–Kohlhagen which is Black Scholes with two interest rates. In which currency is the vega?

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