Why Implied Volatility Varies Across Strikes and Maturities
Summary
The document explains implied volatility as the volatility input that makes a Black–Scholes option price match an observed market price. Under Black–Scholes assumptions, the underlying follows geometric Brownian motion with constant volatility, so the model implies the same volatility across strikes and maturities. Market implied volatilities can nevertheless vary, producing a smile or surface that signals prices differ from the model’s constant-volatility assumptions.
It describes one practical use of the surface: calibrating a different asset-pricing model. Compute model prices under a proposed risk-neutral asset process, compare them with observed option prices across strikes, and choose model parameters to reduce pricing errors, for example by least squares. A calibrated model can then be used to price other derivatives on the underlying. The explanation is introductory: it does not specify a particular alternative model, address calibration stability or arbitrage constraints, or give empirical results. Its fitting equation is schematic, and model calibration does not by itself establish that the assumed dynamics will forecast future prices accurately.
Key ideas
- Black–Scholes assumes constant asset volatility, implying no strike or maturity variation in model implied volatility.
- Market implied volatility is the input that reproduces an observed option price in Black–Scholes.
- A volatility smile indicates that market prices depart from the model’s assumptions.
- Option prices across strikes can be used to fit parameters of an alternative risk-neutral asset model.
- A calibrated model may price related derivatives, but its usefulness depends on the model assumptions and fit.
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# Question about volatility surfaces
# Question about volatility surfaces
As a learner, I'm curious to know the answer to 2 questions regarding volatility surfaces
1) It's stated that volatility surface should be flat accdording to Black-Scholes model. Why is that? Time (in relative to to maturity) is a variable in the BS equation:
$$ {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0 $$
2) How volatility surface is useful in pratice (simulation, modeling, trading)?
## Answer by Sanjay (score 5, accepted)
https://quant.stackexchange.com/a/47167
Let's take a step back to look at what implied volatility (IV) really is. If we know the price of a call option, the interest rate (we can use the spot rate corresponding the option maturity) then Implied volatility is that level of volatility that will result in the option price when putting into the Black-Scholes formula for a call option value. If we express the price of a call option as a function of volatility ($c_t^{BS}(\sigma;....)$) and we observe a market price for an option $c^{observed}$ then implied volatility is defined according to following equation
$$c^{observed} = c_t^{BS}(\sigma^{implied};...)$$
Now to your question: The rest is trivial. BS model is based on Geometric Brownian Motion (GBM) processes for assets. No matter what time to maturity or strike is then the volatility of the asset is the same.
Question 2 I do not have professional experience so there might be more to it but here is the basic idea of Alex C's comment.
When a (non-linear) smile is observed then we know that the market is NOT priced according to BS. Please note that implied volatilities can easily be transformed to prices.
If instead of BS/GBM model we will make use of another model for the asset $$dS_t = \text{some brownian motion (maybe multiple) driven dynamics} $$ The expression for $dS_t$ will of course include some parameters.
According our new model for $S_t$ we can compute the price of an option according to $$c_t(k,S_t,T) = D(t,T)E_t^Q[(S_T-k)^+]$$ where D is the discount factor. In order to estimate the parameters of the model for the asset ($dS_t=...$), we can match the observed prices with the prices generated by our model. Hence the model parameters must satisfy $$ D(t,T)E_t^Q[(S_T-k^*)^+]=c_t^{BS}(\sigma^{implied},k^*)=\text{observed prices} $$
In practice more strikes and implied volatilities are being used and the equality will not necessarily hold strictly. One way to estimate the parameters is then least squares error method. The idea is to use more points and determine parameters such that the sum of squared differences are minimized
$$ \min_{\text{model parameters}} \sum_i^n \left(D(t,T)E_t^Q[(S_T-k_i)^+]=c_t^{BS}(\sigma^{implied},k_i)\right)^2 $$
So we fit the model to market prices. When the parameters are estimated we know the distribution of $S_t$ which we can use to price other derivatives with $S$ as the underlying.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.