Price vs. Implied Vol

Something people often comment on when they start out in quantitative finance is that it’s odd that prices tend to be quoted in terms of implied vol instead of… well, price! This seems a bit strange, surely price is both more useful and more meaningful, given that implied vol is based on a model which isn’t really correct?

Briefly, when a vanilla option is priced in the Black-Scholes, its price is given by the following formula

 C(F,K,\sigma,\tau,\delta) = \delta(\tau) \phi \Bigl( F \Phi(\phi \cdot d_1) - K \Phi(\phi \cdot d_2) \Bigr)

d_1 = {\ln{F \over K} + {1 \over 2}\sigma^2 \tau \over \sigma \sqrt{\tau}}

d_2 = d_1 - \sigma \sqrt{\tau}

with \inline \tau the time to expiry, \inline \Phi(x) the standard normal cumulative density of x, \inline \delta(\tau) the discount factor to expiry, \inline \phi +1 for a put and -1 for a call, F the forward to expiry, K the strike, and\inline \sigma the Black-Scholes volatility (there are a few different ways of expressing this formula, I’ll come back to it another time).

Importantly, for both puts and calls there is a 1-to-1 correspondence between price and vol – in both cases, increased vol means increased price, since more vol means a higher chance of greater returns, while our losses are capped at zero. However, the BS price is derived by assuming that vol is a constant parameter (or at least that it only varies with time), but we know that in reality it also varies with strike (this is called the vol smile, and it is a VERY important phenomenon which I’ll talk about LOTS in these posts!). What vol should we put into the equation to get a sensible price?

Actually, we usually think about this in reverse – prices are quoted on the market, and we can invert the BS price to give us instead an implied vol. In fact, usually even the quotes that we receive will be given in terms of implied vol!

There are a few reasons for this. Firstly, a price varys depending on the notional of an option – in physics we’d call it an extrinsic variable, while imp vol is an intrinsic one. But it’s more that price doesn’t really give us as much information about where the option is as vol does. Have a look at the graphs below:

Two graphs of price variation with strike,for options with a flat BS vol (grey) and from a vol smile(red). On the left, their comparative prices are plotted with the forward price for reference. On the right, the corresponding BS implied vols are plotted. In all cases, the forward price is 100, time to expiry is 1 year, the flat vol is a constant 0.1 while the SABR parameters are instant vol = 0.1, vol of vol = 0.5 and rho = -0.2.

These graphs show the price variation with strike for two vol surfaces. Although they come from very different vol surfaces, we really can’t see that from the price graphs. Because the scale is so large, the relatively small price differences are overwhelmed. On one end they look like essentially forwards, while on the other end they are effectively zero.

But when we look at the implied vols instead, we see that they’re in fact very different options. One set has an (unrealistic) constant vol of 10%, while the other set shows higher vols away from the money (ie. at high and low strikes), which is what we typically see in the market. If we didn’t take these into account and priced them using the same vols, we’d be exposing ourselves to significant arbitrage opportunities (incidentally, this vol smile comes from a model commonly used to model and interpolate vol smiles called SABR – we’ll be seeing a lot more of this in the future).

Finally, implied vols give us a feeling for what is happening – since vol is annualised, this is the same order as the percentage change that we would expect in the underlying in a typical year. This gives us an important intuition check on our results that could easily be forgotten in the decimal points or trailing zeroes of a price given in dollars or euros.

As an aside, I’m in the process of upgrading the vanilla pricer to do implied vol calculations as well – so you will be able to either enter a vol and calculate the price of the option, or else enter a price and work out the corresponding vol. Have fun!

[This requires some root-finding (once again, no closed form for the normal cdfs…), and once again I’m taking the path of least resistance for the moment and coding a bisection solver. Since this involves many, many calls to the normal cdf code I used before, I should probably use a quicker method eventually, so I’ll be coding a brent solver soon, which will probably be a post in itself]

-QuantoDrifter

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