# FX Options price vs implied vol

From the screenshot below, what is the difference between the option price by strike in the table versus the implied volatilities by delta in the chart at the bottom?

https://www.investing.com/currencies/forex-options

No difference really if you look closely.

Looking at the table, you see delta of a put is negative; delta of a call is positive. To the left of the middle of the chart (50 Delta) you have out the money puts, to he right, out the money calls. Now, delta is associated with a strike. For the same strike, you have ITM calls, if OTM puts and vice versa.

In the case of CNY (and many others), there is a caveat that is ignored here. The delta is generally delta premium included. Now what this means exactly is somewhat involved and not very important and also not incorporated in the chart you look at. Ignoring this for now, you see that the 6m option with K=6.32 has a put delta of -0.14 and a call delta of 0.86. The chart is somewhat imprecise but you can tell that around -0.1 it has a vol of about 5% which corresponds to the vol in the chart (the displayed region in the table is kind of 5% throughout but the point is that calls and puts with same maturity and strike have the same implied vol.

You can can read P.409 chapter 19 “OPTIONS, FUTURES, AND OTHER DERIVATIVES - John C. Hull: 8th edition” Side remark, I switched the notation to match FX (Hull uses Black Scholes for equity).

Put-call parity states $$𝑝 + 𝑆 * e^{-r_{ccy1}*t} = 𝑐 + 𝐾*e^{-r_{ccy2}*t}$$ holds for market prices (pmkt and cmkt) and for Garman Kohlhagen prices (pbs and cbs) As a result, pmkt− pbs =cmkt− cbs.

When pbs = pmkt, it must be true that cbs = cmkt.

It follows that the implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity. Now the same strike means one is ITM, the other is OTM - thus the volatility smile for European call options should be exactly the same as that for European put options.

Now, to understand FX option pricing, the best starting point is to read FX Volatility smile construction by Dimitri Reiswich and Uwe Wystup. I have a some useful comments, here and also there.

In a nutshell, FX options are quoted in At-the-money Delta neutral straddles (ATM DNS), as well as Risk Reversals (RR) and Butterflies (BF) for varying delta levels. ATM determines the level, RR the skew (how its tilted, here towards OTM calls) and BF the kurtosis (how pronounced the general wings are).

The above mentioned paper also explains the simplified Malz formula. Using this, one can quickly demonstrate this with a few lines of code in Julia.

If you only have ATM quotes, you are kind of in the "Black Scholes" world where vol is known and constant.

RR determines the skew.

BF the kurtosis.

and combined you get the full vol surface.

Notation wise, these charts are a but sloppy. I just used delta in terms of put, which is why it is from 0 to 1. However, as explained above, 10 delta put (10DP) is equal to 90 delta call (90DC). 10DP = 90DC. It's customary to use OTM quotes only as these are the main options of interest. Hence, why the chart you provided uses 0.5 in the middle, and 0.1 on the sides (with - being the put). Frequently tools just display this as 10DP and 10DC though.

I had a quick look at the website. You can also display premium in pips as opposed to percent. Look at the link if you are interested in this.

• thanks for the explain, very useful. I am new to options in particular currency options. From the screenshot above I see for OTM calls there is an inverse vol\price relationship. In FX, what usually drives the relationship between price and vol? – Student Apr 29 at 12:09
• All else equal, higher IVOL will ALWAYS be higher price for a vanilla option. The issue with looking at different moneyness levels is that the further OTM, the less likely it will be profitable. USDCNH (I just realized I wrote CNY which is onshore but you have CNH - offshore but logic is exactly the same) call on USD (that is not clear but explained here with K = 6.68 is ~0.78 with vol 5.78%. With 5% it would be about 0.71%. Think of an extreme example. Call on USD with K = 200. Right to buy USD at 200. Is obviously worthless. – AKdemy Apr 29 at 12:28
• Think of it intuitively. The hokey stick payoff means the maximum loss is the premium you pay for the option (like insurance). The more time, or the more wildly the underlying moves, the better for you. Yes, it could move against you but you do not care much as the right to exercise means you simple do not use the option if you ae not ITM. You have all the upside potential and (very) limited downside. Therefore, the higher IVOL or the more time to maturity the more expensive your option. By the way, that holds for all options, not just FX. – AKdemy Apr 29 at 20:36