Option Volatility and Binomial Model

In my previous post Options and Volatility Smile , we used Black-Scholes formula to derive Implied Volatility from given option strike and tenor. Most of the options traded on exchanges are American (with a few index options being European) and can be exercised at any time prior to expiration. Whether it is optimal to exercise an option early depends on whether the stock pays dividend or the level of interest rates and is a very complex subject. What I want to focus on is using Binomial model to price an American option. I summarize below Binomial model theory from the excellent book Derivative Markets 3rd Ed. by Robert McDonald

There are many flavors of the Binomial model but they all have following steps in common:

  • Simulate future prices of underlying stock at various points in time until expiry
  • Calculate the payoff of the option at expiry
  • Discount the payoff back to today to get the option price

1. Simulate future prices of the underlying

Assume the price of a stock is $S_0$ and over a small time period $\Delta t$, the price could either be $S_0u$ or $S_0d$ where u is a factor by which price rises and d is a factor by which price drops. The stock is assumed to follow a random walk, also assume that $p$ is the probability of the stock price rising and $(1-p)$ is the probability of it falling. There are many ways to approach the values of $u$,$d$ and $p$ and the various Binomial models differ in the ways these three parameters are calculated.

In Cox-Ross-Rubinstein (CRR) model, $u = \frac{1}{d}$ is assumed. Since we have 3 unknowns, 2 more equations are needed and those come from risk neutral pricing assumption. Over a small $\Delta t$ the expected return of the stock is

$$pu + (1-p)d = e^{r \Delta t}$$

and the expected variance of the returns is

$$pu^2 + (1-p)d^2 – (e^{r \Delta t})^2 = \sigma ^2 \Delta t$$

Solving for $u$, $d$ and $p$, we get

$$p = \frac{e^{r\Delta t} – d}{u-d}$$

$$u = e^{\sigma \sqrt{\Delta t}}$$

$$d = e^{-\sigma\sqrt{\Delta t}}$$

The CRR model generates a following tree as we simulate multi step stock price movements, this a recombining tree centered around $S_0$.


2. Calculating payoffs at expiry

In this step, we calculate the payoff at each node that corresponds to expiry.

For a put option, $payoff = max(K – S_N, 0)$

For a call option, $payoff = max(S_N – K, 0)$

where $N$ is node at expiry with a stock price $S_N$ and $K$ is the strike.

3. Discounting the payoffs

In this step, we discount the payoffs at expiry back to today using backward induction where we start at expiry node and step backwards through time calculating option value at each node of the tree.

For American put, $V_n = max(K – S_n, e^{-r \Delta t} (p V_u + (1-p) V_d))$

For American call, $V_n = max(S_n – K, e^{-r \Delta t} (p V_u + (1-p) V_d))$

$V_n$ is the option value at node n

$S_n$ is the stock price at node n

$r$ is risk free interest rate

$\Delta t$ is time step

$V_u$ is the option value from the upper node at n+1

$V_d$ is the option value from the lower node at n+1

All the variants of Binomial model, including CRR, converge to Black-Scholes in the limit $\Delta t \to 0$ but the rate of convergence is different. The variant of Binomial model that I would like to use is called Leisen-Reimer Model which converges much faster. Please see the original paper for formulas and a C++ implementation at Volopta.com  which I have ported to Python in the next section.

The python code is going to look very similar to Options and Volatility Smile post except that we will swap out Black-Scholes framework with Leisen-Reimer model. We will also use the same option chain data AAPL_BBG_vols.csv

A note on Python code

I usually do not write code like below, I am purposely avoiding using any classes so that the focus remains on the objective which is to understand the technique.

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} $

where

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

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

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

 

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.