# Black Scholes Constant Implied Volatility

I hope someone can clarify my ideas about the constant implied volatility in the classical Black Scholes framework.

As well known, market practitioners quote the prices of vanilla call and put options in terms of implied volatilities. For inputs $K$, $S$, $r$, $T$ and the price of the option $V$, one can determine the implied volatility $σ$ such that

$V=BS(K,S,r,T,σ)$ (1)

When the market quoted implied volatilities are plotted against different strike prices for a fixed maturity $T$, the graph would tipically exhibit a 'smile' shape and hence the name volatility smile.

Theory says that this implies a deficiency in the Black Scholes model since it assumes a constant volatility parameter, not depending on $K$ nor $T$. Hence the volatility smile would be flat.

Here my ideas get confused. Assuming that $S$, $r$ and $T$ remain constant, for a fixed market price $V$ of a vanilla option the implied volatility will vary depending on the value of strike $K$ under the Black Scholes model (1). Hence, if the implied volatility is plotted against different strikes for a fixed $V$ it will indeed show a smile behaviour, which is in contrast to what theory states.

Furthermore, do the market quoted implied volatilities that form the volatility smile according to the theory correspond to a fixed vanilla option price $V$ and with varying $K$?

I think I am making a mistake in my reasoning but I do not understand where. I would be glad if someone can point me in the right thinking direction.

• What's the difference between constant and fixed? – BCLC Feb 7 '16 at 5:43

You seem somewhat lost between theory (the model) and practice (the market)

[Theory]

The Black-Scholes model postulates that the dynamics of 'the stock' follows a Geometric Brownian Motion with constant volatility, i.e. GBM$(r,\sigma)$.

Mathematically, this writes $$\frac {dS_t}{S_t} = r dt + \sigma dW_t^{\mathbb{Q}}$$

European option prices have a closed form expression under this modelling assumption, given by the celebrated Black-Scholes Formula.

If you believe the model, regardless of the option you will be pricing (in other words, whatever the strike $K$ or time to maturity $T$), you will therefore always plug the same volatility figure $\sigma$ in the BS formula. This is because all these options are written on the same underlying $S$, which has a unique dynamics, which was postulated to be a GBM with volatility $\sigma$ and nothing else.

[Practice]

Now, looking at real option quotes and assuming you have identified the relevant discount factors and forward curve, when you try to find the volatility values that need to be plugged in the BS formula to retrieve the observed market prices, you find that these numbers are not constant.

For a fixed maturity, this is what is known as the implied volatility smile. For a fixed strike, this is what is known as the implied volatility term structure.

Because the volatilities are not constant, the assumptions of the Black-Scholes modelling framework are violated.

Indeed, using different volatilities would effectively mean using different underlying dynamics (remember that one specific value of volatility = one specific dynamics in BS world) for each option you are trying to price, which does not make any sense since the underlying is unique.

In other words, contrary to what theory predicts, you cannot use a single volatility figure to retrieve all options' market price.

Quoting options in terms of BS volatility is strictly equivalent to quoting them in terms of price because, through the BS formula, there is a one-to-one relationship between volatility and price.

It is just more practical because: (1) IV varies less across strikes/maturities than prices would, which makes it easier to compare things on an equal footing (2) if you delta-hedge at the implied volatility your P&L will be proportional to the difference between realised and implied volatility. This is why people claim that buying options is like buying volatility (though this is not a pure volatility bet, because of the path-dependence of your P&L through the Gamma dollar).

To conclude I would say that it is not the Black-Scholes model which is used by market practitioners but rather the Black-Scholes pricing equation.

• Thank you for your very clear answer! This was exactly what I was looking for, it helps a lot! – Tinkerbell May 1 '16 at 10:56

Hence, if the implied volatility is plotted against different strikes for a fixed $V$ it will indeed show a smile behaviour, which is in contrast to what theory states.

The Black-Scholes theory says that price $V$ of a vanilla varies with strike according to the Black-Scholes formula. Where you go wrong is assuming $V$ constant as a function of strike.

• I still do not understand it fully. How can I deduce that the volatility smile is flat under the BS framework? If $K$ varies and $V$ does not remain constant, how can I deduce that $\sigma_{implied}$ has to be flat? Does it mean that given different prices $V$ for a specific option with as underlying the same asset, varying $K$ in the BS model would imply a constant implied volatility? Would this be in contrast to what the market shows because for different market option prices, varying $K$ would actually mean different implied volatilities? Could you maybe elaborate a bit in this direction? – Tinkerbell Feb 3 '16 at 14:46
• @Tinkerbell Volatility is assumed be constant and same for all strikes and all time horizons under the BS model. Period. Anything that makes this not true in practice is a smile behaviour. – SmallChess Feb 3 '16 at 14:58
• $\sigma_{implied}$ is defined as the volatility that, when put into the BS formula, gives back the target price $V$. For a BS world with constant vol $\sigma$, prices $V$ come from the BS formula with vol $\sigma$. So the vol to put back in to match that price is $\sigma_{implied}=\sigma.$ – q.t.f. Feb 3 '16 at 15:38
• I agree with q.t.f.'s comment. You are wrong to assume that BS should theoretically show constant implied volatility if you keep the option price fixed (along with the stock price, risk-free rate, and expiration). The volatility smile is "supposed" to be flat (but isn't) only if you just plot the implied volatilities of the different strikes for the same expiration date, stock price, and rfr. Of course the option prices should be different... why would anyone pay the same price for a near-the-money as an out-of-the-money? – Valtinho Mar 8 '16 at 23:03
• I understand that a smile can appears if we consider implied volatilities. What i personnaly don't understand is the sense of the vega given a constant $\sigma$. – were_cat Apr 7 '16 at 9:09

The black-scholes model requires that volatility is constant over time. The reason is because the theory assumes a random walk with a constant probability of each change in underlying price.

It makes no assumptions about volatility being the same for all strikes. You could argue that since volatility is supposed to be a measurement of the underlying asset, then it should be constant, but it doesn't invalidate the model. The real market determines prices of options, and since it tends to place a slightly higher value (all things being equal) for deep in-the-money options and deep out-of-the-money options, the implied volatility tends to be higher the further away from at-the-money you get.

• This is wrong. Black-Scholes assumes constant volatility. Period. Both over strike and over time. – Quantuple Apr 6 '16 at 23:38