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oBIN( ) Model Greeks

The simplest way to approximate the greek statistics from the binomial model involves constructing a new tree with a small increment for each of the input parameters. The option value derived from the new binomial model is then used together with the original option value to approximate the relevant greek letter. For a set of 6 risk statistics, this approach would require that the binomial tree is reconstructed six times and this gives rise to computational efficiency issues.

 

The need to reconstruct the tree can however be avoided for delta, gamma, and theta. Our approach to calculating all of the greeks is set out below.

 

Delta

Delta can be approximated as the change in option value given a small change in the underlying. That is:

Equation Template

 

where,

= the option's delta.

 

= change in option value.

 

= small change in the underlying asset value.

 

This value can be approximated from the values at each of the two nodes at the first step in the tree.

oBIN( ) Greeks 2

 

With reference to the one-step tree given above, delta is approximated as:

Equation Template

 

Note that this approach gives an estimate of delta at the end of the first step in the tree, whereas we want to estimate delta as at time zero. As long as the tree is built with a reasonable number of steps, this timing difference should not be significant.

 

Gamma

Gamma can be approximated as the change in delta given a small change in the underlying. That means that we need two estimates of delta in order to approximate a value for gamma. Two values of delta can be calculated at the second branch of the tree.

 

With reference to the two-step tree given above, two deltas can be evaluated at a stock price of Equation Template andEquation Template , as:

 

Equation Template

 

Given that the difference between the stock price at which and are evaluated is equal to:

Equation Template

the option gamma can be approximated as:

Equation Template

Note that part of the approximation error for gamma relates to the fact that we are using option values from the tree at the end of the second step.

 

Theta

Like Gamma, Theta can be also be approximated using values from the second branch in the tree. Theta is defined as the change in option value given a small change in time.

Using values in the tree for which nothing but time has changed means that theta can be approximated as:

Equation Template

where is the length of time (measured in years) between each branch in the tree.

 

Vega

Vega is approximated using the incremental approach, using option values from the original tree as well as a tree constructed using a slightly higher volatility value.

Equation Template

 

where,

Equation Template = option value at the original volatility level.

Equation Template = option value at the incremented volatility level.

= small change in the volatility level.

 

 

Rho

Rho is approximated using the incremental approach, using option values from the original tree as well as a tree constructed using a slightly higher riskless rate.

Equation Template

where,

Equation Template = option value at the original volatility level.

Equation Template = option value at the incremented volatility level.

= small change in the volatility level.

 

 

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