# 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!

• Could you show us more details about what you're doing here and give us an example? Are you factoring in rollover rates? – barrycarter Sep 16 '16 at 16:50

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.

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

• Hi, thanks for replying! Yeah, I think your explanation holds for delta-neutral ATM strike, i.e. atm strike of jpyusd is 1/atm strike of usdjpy and both the call and put have the same delta. But what about non-ATM strikes? I tried deriving and I found that the strike that gives usdjpy call 25% delta is not exactly the inverse of strike that gives jpyusd put 25% delta. – psiblade Sep 28 '16 at 6:19
• yeah fx is tricky... note delta also depends on premium currency – Randor Sep 28 '16 at 13:41

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 $\Delta_p \in [0, 0.5]$

$K_p = Fe^{\frac {\sigma_K^2}{2}T + \sigma_K \sqrt T N^{-1}(\Delta_p e^{qT})}$