Confusion with volatility smiles implied by different models

Question

I am reading a book "The concepts and practice of mathematical finance" by Mark Joshi. In Chapter 18 he discusses the shapes and dynamics of smiles under different models. I do not understand what is meant by "smile implied by a model".

As I understand it, given a vanilla option with a fixed maturity $T$ and strike $K$, implied volatility is defined as the value of the parameter $\sigma$ we need to input into the Black-Scholes formula in order to get the price observed in the market. Repeating this for different strikes, we obtain the smile as a function of $K$.

Correct me if I am wrong, but implied volatilities are always meant to be as above and relate to the Black-Scholes model. In this case, can anyone please give me a definition of "smile implied by a model" for a more general model?

For example, Joshi discusses the smiles of the stochastic volatility model:

$$\frac{dS}{S} = \mu dt + V^{1/2} dW^{(1)},$$ $$ dV = \lambda (V_r - V) dt + \sigma_V V^\alpha dW^{(2)}.$$

What totally confuses me is when he talks about the smile implied by this model. We are allowed to calibrate any of $\alpha$, $\lambda$, $V_r$ and $\sigma_V$ so that the model matches market prices, but volatility of the stock is a stochastic process so there is no way we can introduce "implied volatility".

I have a similar confusion in the case of jump-diffusions or variance gamma. There could be even more parameters and the solution set that gives the market price would be multidimensional. Let alone the fact that volatility of the stock may not be a parameter.

Thank you for your help.

Answer 1

In the context of option pricing, "implied volatility" always refers to the equivalent diffusion coefficient in the geometric Brownian motion (GBM) dynamics that is necessary to match an observed European plain vanilla price for a given strike and maturity.

When talking about "model implied volatility smile", what is meant is that:

1. You choose some pricing model such as the ones mentioned in your question.
2. You fix the model's parameters.
3. Under the given model dynamics and parameters you value European plain vanilla option prices for all strikes of the maturity of interest.
4. Finally, you then treat these model prices as the observed inputs to your implied volatility computation. I.e. you compute the equivalent GBM volatility such that the previously computed model prices are matched.

Answer 2

Just wanted to point out a few small issues in your statement and maybe help with the conceptual model of these formulas.

implied volatility is defined as the value of the parameter σ we need to input into the Black-Scholes formula in order to get the price observed in the market.

That is actually backward. Implied Volatility is actually better thought of as the primary output and we later reverse the formula for modeling purposes.

The price is a fundamental fact. The price is what it is and a formula doesn't change that. It is better to rephrase your statement above to something like this;

Given a vanilla option with a fixed maturity T and strike K, implied volatility is derived from the option price as the risk-neutral probability distribution described by the parameter σ using the Black-Scholes formula (or other). Repeating this for different strikes, we obtain the implied volatility of each strike based on its own pricing which leads to the formation of the volatility Smile.

We do often use the IV as the input, but that is purely for the purpose of modeling what might happen to option prices as factors change.

There are numerous reasons for Smile effects. The primary reason would be that each options pricing model that is capable of describing Implied Volatility relies on some form of implicit or explicit probability distribution as a reference to determine the volatility Implied by the specific options price.

Smile effects happen when the market pricing does not agree with the fundamental assumption of the probability distribution. If market participants feel that tail risks are higher than the model implies and price that into the options then your calculation of IV will give you excess IV in the options further from ATM. This will give you a smile effect. If the risk is priced in as lower in the outer strikes it will appear as a frown. Skew effects can form in the same way.