Why Inverse FX Options Can Have Different Delta Conventions
Summary
The document examines whether a volatility surface for an inverse FX quote can be obtained by reversing call and put conventions around the at-the-money level. Its central explanation is that option delta depends on the currency used to measure the hedge. A single option can therefore have different numerical deltas when expressed in the two currencies, even though the underlying economic position corresponds to a call in one quote and a put in the inverse quote.
The answers also discuss reciprocal strikes and at-the-money definitions, then give formulas for mapping call and put deltas to strikes using the forward, implied volatility, time, interest rate, and inverse normal distribution. These are conceptual explanations and illustrative calculations rather than market data or empirical evidence. The precise mapping depends on quote, numeraire, delta convention, and at-the-money definition, so simply matching delta labels may not produce reciprocal strikes under every convention.
Key ideas
- FX option delta changes numerically with the currency used to express the hedge.
- A call in one currency quote corresponds economically to a put in the inverse quote.
- Reciprocal strikes and at-the-money conventions matter when translating an FX volatility surface.
- Delta-to-strike formulas depend on forward price, implied volatility, time, rates, and the selected convention.
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Full text
# Volatility Surface for Inverse FX pair / Indirect Quote
# Volatility Surface for Inverse FX pair / Indirect Quote
I understand that, based on market convention, you can construct the volatility surface of an indirect quote FX pair by flipping the volatility surface of the direct quote around the ATM level. For example, the 25 Delta Call volatility of USD/JPY is the same as the 25 Delta Put volatility of JPY/USD.
I am wondering if this is an approximation, though, because I have tried deriving the strikes of (1) USD/JPY call option that gives you 25% delta and (2) JPY/USD put option that gives you -25% delta, but they are not exactly inverse of each other.
Any insight on this matter is greatly appreciated!
## Answer by dm63 (score 1)
https://quant.stackexchange.com/a/31205
An example will illustrate things. Let's say usdjpy forward is 100 and you are considering a usd1mm worth of a 120 call on usdjpy, which entitles you to buy 1mm usd using 120mm yen. Let's say the correct delta hedge is 0.2mm usd versus 20mm yen. Then viewed as an option on the usd, the delta is 0.20/1=20pct. But viewed as an option on jpy, the delta is 20/120= 16.66pct. Thus the same option has different deltas when measured 'the other way round. There's no theoretical problem with this since we are using different numeraires.
## Answer by Randor (score 0)
https://quant.stackexchange.com/a/30307
remember that atm strike of usdjpy is defined as fwd× exp(-0.5xvolsq.T) - ie it is the strike at which delta call = delta put for a jpyusd deal and atm strike of jpyusd = 1/ atm strike of usd jpy
also remember that a usd call = jpy put so if you have the usdjpy vol at strike k, then by this identity that vol also applies to jpyusd at strike 1/k
## Answer by Will Gu (score 0)
https://quant.stackexchange.com/a/31678
Basically the strike of USDJPY call corresponding to the 25% delta is
$K_c = Fe^{\frac {\sigma_K^2}{2}T - \sigma_K \sqrt T N^{-1}(\Delta_c e^{qT})}$
where $F$ is the forward price, $\sigma_K$ is the implied vol for a given delta, $N^{-1}(x)$ is the inverse normal CDF, $q$ is the asset currency interest rates, and $\Delta_c \in [0, 0.5]$.
Similarly for put, $K_p$ is the strike for a put with delta $-\Delta_p$, and $\Delta_p \in [0, 0.5]$
$K_p = Fe^{\frac {\sigma_K^2}{2}T + \sigma_K \sqrt T N^{-1}(\Delta_p e^{qT})}$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.