FX Options price vs implied vol

Question

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

Answer 1

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.