Using Put-Call Parity to Align Call and Put Implied Volatility
Summary
The document addresses why separately fitted call and put prices at the same strike might imply different volatilities or vegas. For European options with matching terms, put-call parity implies equal call and put vegas because the parity relationship’s forward and discounting component does not depend on volatility. If the market-implied forward is used consistently, calls and puts at a given strike should therefore fit with the same implied volatility.
The suggested workflow is to infer the forward from synchronous call and put prices and the discount factor, then fit volatility using that forward. In practice, the result can be noisy: bid and ask quotes may be sparse, prices may not be observed at the same time, and different strikes may imply slightly different forwards. Choosing representative quotes and mids also matters. The parity argument is a useful diagnostic under its assumptions, but inconsistent inputs or noisy market data can produce apparent differences.
Key ideas
- European calls and puts with matching terms have equal vega under put-call parity.
- The forward implied by call and put prices should be used when fitting implied volatility.
- Synchronous prices and a consistent discount factor are prerequisites for a meaningful comparison.
- Sparse quotes, timing differences, and strike-specific forward estimates can create noisy fitted results.
Tags
Full text
# Can Call and Put Vega be different (for the same strike)
# Can Call and Put Vega be different (for the same strike)
I'm calculating the volatility of an options market (description of market below) by fitting 2 functions: 1. fitting the on book call prices 2. fitting the on book put prices
And I'm getting a strange result: the volatility of each function is different i.e. the volatility for the calls isn't correlated to the volatility of the puts which in turn means the call and put (on the same strike) have different vega values. My question is, is this some kind of mistake on my part ?
- Side note the mean of the functions do move together.
Market description: Index options market in a market that has no futures. the options series expires in 22 days.
## Answer by siou0107 (score 1)
https://quant.stackexchange.com/a/53867
By put-call parity, put and call must have the same vega : \begin{align} & c - p = PV\left(F_T - K\right) \\ \Rightarrow & \partial_\sigma c - \partial_\sigma p = \partial_\sigma PV\left(F_T - K\right) = 0 \\ \Rightarrow & \partial_\sigma c \equiv \partial_\sigma p \end{align}
## Answer by Ivan (score 1)
https://quant.stackexchange.com/a/53870
Assuming the options are European (they should be since the underlying is an index) and assuming the prices you have are synchronous so that the whole exercise makes sense in the first place, then provided the forward you are using is that which the market implies you should find the same implied vols for calls and puts.
So you need to start by finding the implied forward which is $\frac{C_K- P_K}{df}+ K$ where $df$ is your discount factor to maturity.
Once you have this then by construction you will fit the prices of these two options with one vol.
Of course this process is noisy in practice, and if all you have are sparse bid offers you need to decide what points to retain (not all $K$ may point to the exact same forward), what constitutes a mid price for calls and puts etc. but the above should guide your thinking. You have probably decided on a forward before trying to fit the vols but that is inconsistent.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.