Estimating FX Volatility – Chap 12C, page 13.
So, let’s just cover a few points on how one might estimate volatility in the FX market.
To begin, the sigma we want to plug into the model is the sigma that will apply to the FX rate over the life of the option. There is no way to know the value of this forward-looking parameter. So like the future cash flows of a firm, volatility must be estimated.
There are basically two approaches – the historical approach and the implied volatility approach.
The historical approach uses past data over some period of time. A lot has to be decided if we’re going to use the historical approach.
What sample period do we use?
Do we use as much data as we can or only recent data? Using more data is better because it will increase the precision and accuracy of our estimate of sigma, but old could be from a different economic regime with different exchange rate behavior, different monetary and intervention policies and it could be less relevant, possibly irrelevant for predicting the future sigma.
Do we weight all of that data evenly or put more weight on recent data and less on older data?
What statistical method do we use for taking these numbers and producing a value of sigma?
The classical method, which is the formula embedded in Excel, and that we all learned in statistics is one method, but there are other ways to estimate sigma as well.
And if we use daily data, do we annualize it by multiplying by square root of 365 calendar days or 260 trading days, or somewhere in between?
So there are a lot of choices to be made in using the historical approach, and on the next slide I’ll show you a simple illustration of how it can make a difference.
The second approach, implied volatility, takes a completely different approach – you could call it an efficient market approach. How does it work? The Black-Scholes model says that an FX option price is a function of six variables – S, K, rd, rf, t, and . The first 5 are known, and only sigma is unknown. We can observe the price of the option in the market place (we’ll call this the market price). If the market price is equal to the model price, we have essentially one equation (the BS model) and one unknown () – So we can reverse the equation, putting on the left-hand-side, and solving for the value of that makes the market price and the model price identical. This value of sigma is given a special name – implied volatility.
As I’ve described it, implied volatility depends on two assumptions – (1) That the market is efficient so the market’s price embodies an unbiased expectation of the future volatility, and (2) that the BS model is the correct model. If either of those assumptions is false, the implied volatility approach will not return an accurate estimate of .
How would you actually implement the implied volatility approach? You need two things, an option price and a model.
But which model? Most likely, it would be the BS model, but there are others that could be used.
And which option price? For a single underlying asset (say £) there are many options – Many strike prices, many maturities, and both puts and calls. There could be 20, 30 or 40 £ options outstanding at any time. The BS model would predict that all of these options would be priced with the same , and so each would generate the same implied volatility. But in practice, this is not the case. Out of the money options have higher , which leads to the “volatility smile” as we will see in a moment.
This voice over terminates here. The final 3 slides of this section are on a short follow on piece.