value_left = self.present_value(fixed_seed=True)

numerical volatility value for right value

vola_del = self.underlying.volatility + interval

update the simulation object

self.underlying.update(volatility=vola_del)

calculate the right value for numerical Vega

value_right = self.present_value(fixed_seed=True)

reset volatility value of simulation object

self.underlying.update(volatility=vola_del - interval)
vega = (value_right - value_left) / interval
return round(vega, accuracy)

One topic covered by the generic valuation_class class is the estimation of Greeks. This

is something we should take a closer look at. To this end, consider that we have a

continuously differentiable function available that represents the present value

of an option. The Delta of the option is then defined as the first partial derivative with

respect to the current value of the underlying S 0 ; i.e., .

Suppose now that we have from Monte Carlo valuation (see Chapter 10 and subsequent

sections in this chapter) a numerical Monte Carlo estimator for the option value.

A numerical approximation for the Delta of the option is then given in Equation 17-1.

[ 73 ]

This is what the delta method of the generic valuation class implements. The method

assumes the existence of a present_value method that returns the Monte Carlo estimator

given a certain set of parameter values.

Equation 17-1. Numerical Delta of an option

Similarly, the Vega of the instrument is defined as the first partial derivative of the present

value with respect to the current (instantaneous) volatility 0 , i.e., . Again

assuming the existence of a Monte Carlo estimator for the value of the option,

Equation 17-2 provides a numerical approximation for the Vega. This is what the vega

method of the valuation_class class implements.

Equation 17-2. Numerical Vega of an option

Note that the discussion of Delta and Vega is based only on the existence of either a

differentiable function or a Monte Carlo estimator for the present value of an option. This

is the very reason why we can define methods to numerically estimate these quantities

without knowledge of the exact definition and numerical implementation of the Monte

Carlo estimator.