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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)?

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    $\begingroup$ On #2: volatility surface, derived from the prices of vanilla (simple) options, is useful in pricing other (exotic) options, such as barrier options. $\endgroup$
    – Alex C
    Aug 17, 2019 at 23:30
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    $\begingroup$ The vol is only constant for calls and puts at any chosen strike.It is NOT constant across different strikes. That is how the vol surface embeds expectations of skewed and kurtotic returns into options premia. The vol surface is essentially the market's correction for the fact that returns are not perfectly lognormally distributed. The vol surface essentially embeds the market's best guess how the market would then behave at different strikes. $\endgroup$
    – demully
    Aug 17, 2019 at 23:42
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    $\begingroup$ @demully: I guess you stated the truth about vol surface. what I'd wlike to know is why, assuming BS model is completely correct, the vol surface shuld be flat, a statement I saw at investopedia.com/articles/stock-analysis/081916/…. IV in pratice is calculated using iteratiion. there is not a formula for IV. $\endgroup$
    – techie11
    Aug 18, 2019 at 0:12
  • $\begingroup$ #1: Under the Black Scholes theory the Implied Volatility should be equal to the market's expectation of the volatility for the underlying stock from now until maturity. But if two options on the same stock (with same maturity but different strikes) have different implied vols, then this cannot be right. $\endgroup$
    – Alex C
    Aug 18, 2019 at 0:16
  • $\begingroup$ from your URL: "The volatility surface is far from flat and often varies over time because the assumptions of the Black-Scholes model are not always true." ;-) $\endgroup$
    – demully
    Aug 18, 2019 at 0:20

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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.

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  • $\begingroup$ It seems to me the solution to G-Browian process is a set of PDF whose standard deviations are a function of time. $\endgroup$
    – techie11
    Aug 18, 2019 at 0:29
  • $\begingroup$ hmm, I don't follow you? " .... No matter what time to maturity or strike is then the volatility of the asset is the same ... " $\endgroup$
    – Sanjay
    Aug 18, 2019 at 0:36
  • $\begingroup$ I was referring to the first of "Properties of a one-dimensional Wiener process" at $${\displaystyle f_{W_{t}}(x)={\frac {1}{\sqrt {2\pi t}}}e^{-x^{2}/(2t)}}$$ at en.wikipedia.org/wiki/Wiener_process $\endgroup$
    – techie11
    Aug 18, 2019 at 0:44
  • $\begingroup$ If the $$ \sigma $$ is a constant in $$ {\frac {dS}{S}}=\mu \,dt+\sigma \,dW\ $$, then the answer is self-evident. this $$ \sigma $$ is not the same as the standard deviation in PDFs of the Wiener process. $\endgroup$
    – techie11
    Aug 18, 2019 at 0:54
  • $\begingroup$ But the the phrase implied volatility is exactly referring to this $\sigma$. That’s it. $\endgroup$
    – Sanjay
    Aug 18, 2019 at 1:11

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