Q1. What methods can be used for computing greeks given a method for computing the price?

A: There are many different ways to compute the greeks, but we will focus on 3 popular methods:

1. Finite-difference approximations

2. Pathwise method

3. Likelihood ratio method

1. Finite-difference approximations

They involve calculating the price for a given value of a parameter, then changing the parameter value slightly, by , and recalculating the price.

Let f be the payoff function and the parameter we are interested in, an estimate of the sensitivity will be: - easy to implement / does not require too much thought

- biased estimator of the sensitivity

This can be reduced by using a central-difference estimator: - slower because 3 prices to calculate.

- issue with discontinuous payoffs.

Ex: with digital call.

If we want to estimate the delta for a digital call, we will get it being zero except for the small number of times when it will 1. For these paths, our estimate of delta will be very big, approximately of order -1.

2. Pathwise method

It gets around the problem of simulating for different values of by first differentiating the option's payoff and then taking the expectation under the risk-neutral measure: where is the density of in the risk-neutral measure.

- unbiased estimate

- only requires simulation for one value of and is usually more accurate than a finite-difference approximation.

- becomes more complicated when payoff is discontinuous but to around this we can write f = g + h, with g continuous and h piecewise constant.

3. Likelihood Ratio method

It is similar to the pathwise method, but instead of differentiating the payoff we differentiate the density  where is the derivative of by - only one value of needs to be simulated to calculate both the price and the sensitivity.

- do not need to worry about discontinuities in the payoff function as we are differentiating the density.

- need to explicitly know the density.

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