Numeric Greeks

The "greeks" are measures of the sensitivity of an attribute (often the price) to one or more of the other attributes. For example the delta is the sensitivity of the price of the option to a change in the asset price.

Some option models provide analytical solutions for this. In particular the Black-Scholes style models have closed form solutions. Many other models, in particular tree based formulations, do not.

Finite Difference Methods

The greeks can be calculated numerically by using finite difference methods. This means calculating the price of the option multiple times, while perturbing the inputs.

This can be very intuitive. For example, to find out how the price of the option changes to a penny change in the underlying asset price, we simply recalculate the option price, adding a penny to the underlying asset price.

There are three methods that could have been used in the above example. Given the change is the difference between option prices where the underlying asset price has changed, we could:

• OptionPrice(AssetPrice + penny) - OptionPrice(AssetPrice)
• OptionPrice(AssetPrice + penny) - OptionPrice(AssetPrice - penny)
• OptionPrice(AssetPrice) - OptionPrice(AssetPrice - penny)

These are the forward, central and backward methods, and each gives a slightly different answer.

Using the central difference, the formula for calculating the delta is given below.

$$\frac{\partial V}{\partial S}=\frac{B{S}_{price}(S+\Delta S,K,T,r,\sigma )-B{S}_{price}(S-\Delta S,K,T,r,\sigma )}{2\Delta S}$$

The following is a python implementation.

def delta(is_call, S, K, T, r, v, dS):
    return (
        price(is_call, S + dS, K, T, r, v)
        - price(is_call, S - dS, K, T, r, v)
    ) / (2 * dS)

For the higher order greeks (like gamma) the maths gets a little more complicated, but it follows the same intuitive reasoning.

The NumericGreeks classes

Some classes are provided for calculating the greeks. They differ only in the way they handle carry/dividend yield, or the lack of it.

• jetblack_options.numeric_greeks.without_carry - for pricing formula with no carry or dividend yield, for example Black 76, or the original Black-Scholes formula for non-dividend paying stock.
• jetblack_options.numeric_greeks.with_dividend_yield - for pricing formulae with a continuous dividend yield.
• jetblack_options.numeric_greeks.with_carry - for pricing formulae with cost of carry, in the style of the generalised Black Scholes model.

Each of the option pricing models has a convenience method make_numeric_greeks which will choose the appropriate NumericGreeks class.

Optional Arguments

Some of the methods have an optional method parameter. This controls which finite difference is used. This can be one of: 'central', 'forward' or 'backward'.

All the methods take as an optional parameter the value of the bump being applied. For example the delta method takes a dS argument which has the default value of 0.01.

Examples

Here we calculate the delta for the Black, Scholes & Merton model with continuous dividend yield using the finite difference.

# Calculate the delta by bumping the price.
from jetblack_options.european.black_scholes_merton import make_numeric_greeks
ng = make_numeric_greeks(is_call=True)
d1 = ng.delta(is_call, S, K, T, r, q, v)

What next ?

Generalized Black Scholes