Implied Vol vs. Calibrated Vol

Question

Consider the Black-Scholes model, in which the log stock return over a time period $\Delta t$ is given by

$$ \log(S_{i+1}/S_i) = (\mu - \sigma^2/2)\Delta t + \sigma \sqrt{\Delta t} Z_i, \qquad Z_i \sim \mathcal{N}(0,1). $$

The price of a call at time $T$ under this model (when we replace $\mu$ with $r$) is given by (emphasizing the dependence on $\sigma$)

$$ C(\sigma) = SN(d_1) - Ke^{-rT}N(d_2), $$

where

$$ d_1 = \frac{1}{\sigma{\sqrt{T}}}\left(\log(S/K) + (r + \sigma^2/2)T\right) = d_2 + \sigma \sqrt{T}. $$

Now, assuming $r$ is known, we have (at least) two methods of estimating $\sigma$, namely using a least-sqaures regression on the log returns, or calculating the implied vol.

Regression on log returns:

Note the log returns is a linear regression equation of the form

$$ Y_i = \beta_0 + \beta_1X_i + \sigma\sqrt{\Delta t} \epsilon_i $$

with $\beta_0 = (\mu - \sigma^2/2)\Delta t$, $\beta_1 = 0$ and $\epsilon_i \sim \mathcal{N}(0,1)$, independent. So, assuming we have a sample of $N$ log returns (denoted $Y_i$) and since $\beta_1 = 0$, we estimate $\beta_0$ in the usual regression way by

$$ \hat{\beta_0} = \frac{1}{N}\sum_{i=1}^N Y_i, $$ and then estimate $\sigma$ using the standard deviation of the residuals,

$$ \hat{\sigma} = \frac{std(Y_i - \hat{Y_i})}{\sqrt{\Delta t}}, $$

where the $\hat{Y_i}$ are the regression model-predicted log returns. This is one method to estimate $\sigma$ used in the pricing equation, and is in the least-sqaures sense our "best guess" at $\sigma$. This $\hat{\sigma}$ could then be used to compute all European call options for $S$ across all strikes and expirations.

Implied vol:

Given a market call price $C_{\text{observed}}$ for some strike and expiration, we can compute the $\sigma_{\text{implied}}$ such that $C(\sigma_{\text{implied}}) = C_{\text{observed}}$. We can compute such a $\sigma_{\text{implied}}$ for all call options we have prices for (again assuming $r$ is known). Then, when we would like to price a call using our pricing equation for some strike/expiration that is not observed, we can choose (or interpolate between a few) the $\sigma_{\text{implied}}$ that is closest to the strike/expiration we would like to price at and use this $\sigma_{\text{implied}}$ in our pricing equation.

So, we have two methods of deriving a suitable $\sigma$ to use in our pricing equation. It seems that much of the literature is devoted to implied vol, so I assume this is the preferred technique. My question is, is there any relation between the two, and when would you use one over the other?

Answer 1

The main difference is that one approach assumes that a certain dynamical structure properly describes the underlying instrument, while the other approach is really only a re-writing of the price in terms of an implied volatility.

Implied volatility

Implied volatility really only needs two things: the underlying stock price and the call option price (apart from the risk free rate and your choice of strike). Therefore all inputs are set by the market. At no point do you have any uncertainty about the different parameters that determine the implied volatility.

In the implied volatility approach you can therefore never detect miss-pricing of options, since the price is already set by the market. Why would that be useful? Well, maybe you hold some options on your book and you want to know what their fair value is. You are therefore interested in the markets view of the options price. But traders like to work with volatilities instead of prices, so you need not just the market-implied price of the option but rather the corresponding implied volatility. Again, this comes down to the fact that implied volatility is nothing more but a re-writing of the option price as set by the market.

Note that the BS model is not consistent with the implied volatility as set by the market. Why? Because the implied volatility is strike-dependent. If you extract the implied volatility you will find that typically at-the-money options have a lower implied volatility than options that are away from at-the-money. The implied volatility is said to form a smile, which refers to the shape of the graph formed by plotting implied volatility vs strike. This is not consistent with the BS model, since it can only work with one volatility.

Calibrated volatility

The regression approach calibrates everything to the underlying stock. Here you really try to infer what the fair value of the option price should be, given your model and the historical data. If you end up with an option price different from the market (which is pretty much guaranteed to happen), and you have a lot of faith in your calibration, then you would either go long or short an option and potentially make a profit. You would consider yourself "better informed" then the market.

This approach is therefore model-dependent. You made a choice in what model you use to describe the stock market. You assumed that a certain distribution describes your log-returns and it is completely fixed by only a handful statistical parameters. You could call this a type of model bias.

Second, you are dealing with statistical inference, and therefore statistical errors will creep in. Why is the error function you used the way to go? Why not the absolute norm or some other error function? Most likely your choice is made because you assumed your errors are distributed in a particular way, but this is then again an example of model bias. How much data will you use to calibrate your parameters? How sure are you of your statistical estimates? The implied volatility is just a price, so we will always agree on this. But for the calibrated volatility that is not the case and there will be some statistical error associated with it.

Finally, your model is fully calibrated to historical data. But options are all about future events. The historical volatility might not agree with the implied volatility because the market is expecting a more volatile future.

Now, your calibration approach might be too simplistic, but you could potentially improve on this. And this is indeed what some funds try to do. The calibration approach is therefore really about "trying to beat the markets view on option prices".

So is historical data never used as input in these pricing models? Not necessarily. For some markets there are no implied volatilities available (e.g. energy markets are very illiquid, or some institutes provide special options for their clients). In that case the historical data might be the only decent way of estimating your option prices.

Answer 2

It depends what you want volatility for. Theory will tell you that:

"Implied variance of short maturity ATM options is approximately equal to the expectation of the realised integrated variance of the underlying over the life of the option and under the risk neutral measure"

In math: $\sigma^2_{ATM}\approx E^Q\left(\frac{1}{T}\int_0^T\sigma^2_t dt\right)$

Now there are some interesting details here. For example, the relationship is model-free, but it holds for small horizons and for strikes around the current spot. Also, the expectation is under Q, not under the physical probability measure: even for short maturity ATM options there is a basis between implied and actual volatility. This is well documented in the literature and reflects the risk premium for trading volatility (see for example figure 6.5 in these notes).

Therefore, as I said, it depends what the purpose of the estimation is. If you want to price, stick to implied as they are under Q. For riskmanagement start with historical. There are valid reasons for using both, but one has to be careful and acknowledge the different measures.

Answer 3

This is pretty straight forward:

• The market prices vanilla options via implied volatility. You can like it or not like it but that is the way it is. So, the fair price of the option is the equivalent of the implied vol via BS.

• Now, if you believe the true price of an option should be different from the traded market price and you figure out that you have a better way of modeling future volatility then you should by all means do so and capitalize on such insight.

It is really that simple.