Options and Volatility Smile

An equity option represents the right to buy (“call” option) or sell (“put” option) a unit of underlying stock at a pre-specified price (strike) at a predetermined maturity date (European option) or at any time up to the predetermined date (American option).

Option writer sells an option and option holder buys an option.

For a European call option on an index with strike 8,000 and index level of 8200 at maturity, the option holder receives the difference of $200 from option writer. This is called the instrinsic value or payoff of the option from the holder’s point of view.

The payoff function for a call option is $$ h_{T}(S,K) = max[S_{T}-K, 0] \tag{Eq. 1}$$

where T = maturity date, $\ S_T $ is the index level at maturity and K is the strike price.

In-the-money: a call (put) is in-the-money when S > K (S < K)
At-the-money: call or put is at-the-money when $\ S \approx K $
Out-of-the-money: a call is out-of-the-money when S < K (S > K)

A fair present value (is different than payoff) of a European call option is given by Black-Scholes formula:

$\ C_{0}^{*} = C^{BSM}(S_{0},K,T,r,\sigma) \tag{Eq. 2}$

$\S_{0} $ current index level (spot)
K strike price of the option
T time-to-maturity of the option
r risk-free short rate
$\ \sigma $ volatility or the std dev of the index returns

$\ C^{BSM} = S_{t} . N(d_{1}) – e^{r(T-t)} . K. N(d_{2})\tag{Eq. 3} $


$\displaystyle N(d) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{d} e^{-\frac{1}{2}x^{2}} dx$

$\displaystyle d1 = \frac{\log\frac{S_{t}}{K} + (r + \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$

$\displaystyle d2 = \frac{\log\frac{S_{t}}{K} + (r – \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$


The present value of the option is always higher than the undiscounted payoff, the difference being the time value. In other words, the option’s present value is composed of payoff plus the time value. Time value indicates that there is always a chance of option going in-the-money or more in-the-money during that time.

Simulating Returns

The Geometric Brownian motion model of the BS equation is given by

$$\displaystyle dS_{t} = rS_{t}dt + \sigma S_{t} dt dZ_{t}\tag{Eq.4}$$

The discretized version is

$$\displaystyle S_{t} = S_{t – \Delta t} e^{(r – \frac{1}{2}\sigma^2) \Delta t + \sigma \sqrt{\Delta t} z_{t}}\tag{Eq.5}$$

where t $\in {(\Delta t, 2\Delta t,…..,T)}$

Using the above discretized version, we will simulate the spot prices with $S_{0}$=100, T=10, r = 0.05 and $\sigma$=0.2

Implied Volatility is the value of $\sigma$ that solves Eq. 2 given the option market quote $C_{0}^{*}$

Volatility surface is the plot of the implied volatilities for different option strikes and different option maturities on the same underlying (an option chain).

Vol Surfaces exhibit :
Smiles: option implied volatilities exhibit a smile form, i.e. for calls the OTM implied volatilities are higher than the ATM ones; sometimes they rise again for ITM options
term structure:smiles are more pronounced for short-term options than for longer-term options; a phenomenon sometimes called volatility term structure

To demonstrate Vol Surface, I will use an option chain on AAPL stock as of 5/11/2017. I have downloaded this data from a reputable vendor, you can find this file here AAPL_BBG_vols

Once we have the implied volatilities, we will generate a grid of strikes and maturities and use Cubic interpolation to derive the missing implied volatilities needed for a smooth surface.

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