Derivatives Analytics - Options Greeks¶

Author: Gabriele Pompa: gabriele.pompa@unisi.com

Table of contents¶

Executive Summary

1. Option Greeks
   1.1. Numeric Derivatives
   1.2. NumericGreeks class\

        1.2.1. Numeric $\Delta$
        1.2.2. Numeric $\Theta$
        1.2.3. Numeric $\Gamma$
        1.2.4. Effect of $\epsilon$ reduction
        1.2.5. Numeric Vega
        1.2.6. Numeric $\rho$
        1.2.7. Put options
1.3. Analytic Vs Numeric greeks 2. Implied volatility
2.1. Newton method to find the root of $\sin(x)$ for $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$
2.2. Newton method to find $\sigma_{imp}$

Resources:¶

• Python for Finance (2nd ed.): Chapter 6 Object-Oriented Programming.

Executive Summary ¶

In this notebook we introduce option greeks. That is, the first-order derivatives of an option price with respect to its pricing parameters. We provide both a numeric computation using finite-difference methods implemented in NumericGreeks class and their analytic expression under the Black-Scholes models.

The following sections are organized as follows:

• In Sec. 1 we introduce NumericGreeks class implementing finite-difference methods to compute the main greeks of an EuropeanOption. We make also comparisons between the numeric evaluation of greeks and their analytic expression under the Black-Scholes model.
• In Sec. 2 we introduce the concept of implied volatility, the Newton root-finding method and the .implied_volatility() method that implements it (and Least-Squares constrained method) for an EuropeanOption.

All features introduced in this and previous notebooks are collected in pyBlackScholesAnalytics Python package.

These are the basic imports

1. Option Greeks ¶

Consider a plain-vanilla option of strike $K$, expiration date $T$ with pricing function at time $t$

$$c_t = c(S_t, K, t, T, \sigma, r)$$

The greeks of this option are the first-order derivatives w.r.t. its pricing parameters: $S$ (the $\Delta$), $t$ (the $\Theta$), $\sigma$ (the $Vega$) and $r$ (the $\rho$).

We first introduce finite-difference methods to compute them numerically and then compare them with their analytical expression under the Black-Scholes model.

1.1. Numeric Derivatives ¶

Let's start with a non-financial example. Let's compute the numeric derivative $f'(x)$ of $f(x)=\sin(x)$ at $x_0=0$. We use finite differences, that is we approximate the exact derivative $f'(x_0)=\cos(x_0)$ with the central finite-difference

$$f'_{num}(x_0) = \frac{f(x_0 + \epsilon) - f(x_0-\epsilon)}{2 \epsilon} + \mathcal{O}(\epsilon^2)$$

where $\epsilon$ is a tolerance threshold and the order of the remaining part (that we omit in our approximation) is second-order in $\epsilon$. More on finite-difference methods (which are very general methods) on Wikipedia (or this post).

1.2. NumericGreeks class ¶

Let's go back to our options world and use finite-differences methods to compute the numeric derivatives of a plain-vanilla call option.

This has been implemented in the NumericGreeks class.

1.2.1. Numeric $\Delta$ ¶

$\Delta$ is the derivative w.r.t. to underlying level $S$

1.2.2. Numeric $\Theta$ ¶

$\Theta$ is the derivative w.r.t. to time $t$

1.2.3. Numeric $\Gamma$ ¶

$\Gamma$ is the second derivative w.r.t. to underlying level $S$. Being a second-order derivative, its finite-difference approximation is the following (omitting other pricing parameters)

$$\Gamma_{num}(...,S_0) = \frac{c(...,S_0 - \epsilon) - 2 c(...,S_0) + c(...,S_0 + \epsilon) }{\epsilon^2} + \mathcal{O}(\epsilon^2)$$

1.2.4. Effect of $\epsilon$ reduction ¶

Reducing $\epsilon$ does not guarantee increased accuracy. On the contrary, it can severely decrease the stability and overall quality of numeric approximation. Let's consider this effect on the $\Gamma$ greek.

1.2.5. Numeric Vega ¶

$Vega$ is the derivative w.r.t. to underlying volatility $\sigma$

1.2.6. Numeric $\rho$ ¶

$\rho$ is the derivative w.r.t. to short-rate level $r$

1.2.7. Put options ¶

Let's make an example using a Vanilla_Put option

1.3. Analytic Vs Numeric greeks ¶

Under the Black-Scholes model, analytical expressions for greeks of plain-vanilla (as well as for digital) options are available. We have implemented them in PlainVanillaOption and DigitalOption classes, respectively. It's interesting to compare the numeric and analytical implementation of the same greek, to better appreciate the numerical error.

Let's define a simple utility function to better display plot titles

Comparison of $\Delta$:

Comparison of theta:

Comparison of gamma:

Comparison of vega:

Comparison of rho:

2. Implied volatility ¶

Black-Scholes implied volatility $\sigma_{imp}$ of an option is that value of the underlying asset volatility $\sigma$ such that the Black-Scholes price of the option matches a target price $c_{*}$ (typically either coming from a market quote for the same option or from another model used to evaluate it). That is $\sigma_{imp}$ is the solution of the equation:

$$c(S_t, K, t, T, \sigma, r) = c_{*}$$

where $c(S_t, K, t, T, \sigma, r)$ is the Black-Scholes price of an option of strike $K$ and expiratio $T$ at time $t$. This parameter is what is tipically quoted for options (instead of prices) and is, therefore, fundamental. Its centrality has lead to the development of very sofisticated root-finding methods. We'll start our exploration of implied volatility using the simplest one: the Newton method that we now introduce.

2.1. Newton method to find the root of $\sin(x)$ for $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ ¶

Let's quickly refresh Newton method. It is a numerical method to find roots of a function $f(x)$ through an iterative procedure. The goal then is to find a solution for $f(x)=0$. Given the n-th approximating root $x_n$, the subsequent is found as

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

that is, it uses the old approximation $x_n$, the function $f$ evaluated at $x_n$ as well as its first-derivative $f'$ with respect to $x$ evaluated at $x_n$. Typically, a stopping criterion involving differences between subsequent approximations

$$|x_{n+1} - x_{n}| < \epsilon$$

or relative differences

$$\frac{|x_{n+1} - x_{n}|}{|x_{n}|} < \epsilon$$

is used.

Start from Wikipedia if you never heard about this fundamental method.

Let's see an example, trying to find a solution for the equation $\sin(x) = 0$ for $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.

Here is the solution found (should be $x=0$ of course)

And the solution has been found in 7 iterations (very quick!)

2.2. Newton method to find $\sigma_{imp}$ ¶

We now specialize the Newton method to find the implied volatility of the option. The iterative updating procedure becomes

$$\sigma_{n+1} = \sigma_n - \frac{c(...,\sigma_n) - c_{*}}{Vega(..., \sigma_n)}$$

where we have considered that

$$\frac{\partial c(...,\sigma) - c_{*}}{\partial \sigma} = \frac{\partial c(...,\sigma) }{\partial \sigma} = Vega(..., \sigma)$$

by definition.

An implied volatility calculation using the Newton method is implemented in the .implied_volatility() method of EuropeanOption class.

So let's consider our plain-vanilla call

Let's use the Black-Scholes price of this option as our target price $c_*$

By construction, this option has been generated assuming a MarketEnvironment with an underlying volatility of 20% per year

Therefore, we expect that the Black-Scholes implied volatility of this option to be exactly $\sigma_{imp} = 0.2$.

This can be easily proved usnig the .implied_volatility() method and providing (is optional) an initial guess iv_estimated for the volatility, as well as the target price $c_*$ as the target_price parameter

The solution is found in 5 iterations and is 20% per year, as expected.

One last example. Let's compute our target price $c_*$ with another volatility value: $\sigma = 0.15$

We thus expect in this case to be $\sigma_{imp}=0.15$. And this is the case: