using bid ask prices to imply bid ask volatilities

Question

Let's say i have bid / ask feed of an option prices (across strikes and expiries, calls and puts), what is the accurate way of implying out vols from these bid / asks

For eg; to get the bid vol, should i be using the bid prices on calls and puts. To get the ask vol, should i be using the ask prices on calls and puts?

First of all, i will need to use put call parity (for index options) F = (c-p)/DF + K to get the forwards ( i may get slight differences depending on K) but how do i correctly apply the bid and ask option prices here? Should i use bid prices to get a bid? forward which i used to imply the vol (likewise use ask prices to get the ask forward to imply ask vol) or is this not accurate?

Assume i have access to BS model (black76) for implying vols, given price, forwards, rate, etc As you can imagine, im not a quant, just an average joe trying to improve my reporting for the traders and management.

Answer 1

@Stéphane answer is quite good. There are a few more details to consider:

• the mids may not be arbitrage-free. If you need arbitrage-free prices, you will need to find the closest arbitrage-free quotes, for example following Appendix B of An arbitrage-free interpolation of class C2 for option prices
• you may incorporate bid-ask spread information using a weight in the minimization, for example weight=1/spread.

Finally, as @will mentioned in the comments, you can also do the minimization directly in terms of bids and asks. The drawback is that it adds complexity: which forward do you use (the same for both or the implied forward for each)? I presume it will be the latter, even though the concept of implied forward is not so clean in that case since the bid (respectively ask) prices are not arbitrage-free.

For this reason, the mid approach using weights on the spread is more common and missing bid/ask points are just removed. But yes, if the market is skewed towards bids (or asks), then it may be more practical to minimize directly on the whole set.

Answer 2

As regards option prices, most people would use the midpoint of the spread as the fair price. Then, risk free rates would be computed by using the yield on very safe government fixed income instruments like a US Treasury Bond. You can pick two bonds with times to maturity straddling that of your option contract and you interpolate them linearly to get a risk-free rate. You could do it differently, but that's an easy one. Now, you MUST make some adjustement on the index for dividends. One way would involve using discounted realized dividends. Another way would be to impose put-call parity -- since you are only missing the dividend yield.

Once you have your cleaned up data, you can invert Black-Scholes to get implied volatility -- but you need a numerical algorithm to do it. There are many ways to get an approximate value in closed form, so you could use the approximate value to initialize a Newton-Raphson algorithm to invert the Black-Scholes formula numerically.