Greeks of Options on Non-Interest Rate Instruments

Overview

To value an option one needs to calculate not only the option’s fair value, but also various risk statistics, such as delta, gamma, vega and so on. These risk statistics are also known as greeks. Greeks measure sensitivities of an option’s value to certain variables and are mostly used for hedging purposes. The most important greek is the delta. Hedging an option with a delta is called delta hedging. To delta hedge an option, one rebalances his/her portfolio dynamically by taking a short or long position on delta units of the underlying per one unit of the option in which he/she is long or short, respectively. A good estimate of the delta is essential in constructing a quality delta hedging portfolio. Good estimation means a good balance between accuracy and stability.

This document explains in general how greeks are calculated in the FINCAD math library for options on non-interest rate instruments, such as equity options, commodity options and FX options.

Formulas & Technical Details

Definitions of Greeks

Delta	The rate of change in the fair value of the option with respect to the current value of the underlying asset when other variables remain constant. This is the derivative of the option price with respect to the current value of the underlying.
Gamma	The rate of change in the value of delta with respect to the current value of the underlying asset when other variables remain constant. This is the second derivative of the option price with respect to the current value of the underlying.
Theta	The rate of change in the fair value of the option per one day decrease in the option time when other variables remain constant. This is the negative of the derivative of the option price with respect to the option time (in years), divided by 365.
Vega	The rate of change in the fair value of the option per 1% change in volatility when other variables remain constant. This is the derivative of the option price with respect to the volatility, divided by 100.
rho of rate	The rate of change in the fair value of the option per 1% change in the risk-free rate when other variables remain constant. This is the derivative of the option price with respect to the risk-free rate, divided by 100.
rho of holding cost	The rate of change in the fair value of the option per 1% change in the holding cost (or dividend yield) when other variables remain constant. This is the derivative of the option price with respect to the holding cost, divided by 100. If the underlying is futures, this statistic is not available.

Estimation of Greeks

Greeks that have closed-form solutions

Based on the Black-Scholes lognormal model a standard call or put option has a closed-form formula. For such an option, closed-form formulas can also be derived for its greeks. In this case the greeks can be calculated directly without any numeric approximation other than the calculation of a cumulative normal distribution. For the closed-form formulas of greeks, see Haug [1].

Greeks that are taken from a binomial tree.

Some options, particularly, Bermudan or American style options, are often valued with the Rubinstein binomial tree method. With this method the greeks, delta, gamma and theta, can be taken directly from the tree that is built to calculate the option’s fair value. This way no revaluation is needed and thus computational time is saved.

Recall that a Rubinstein binomial tree is a tree structure on the price of a financial instrument. For a given time horizon, e.g., time to maturity of an option, the time interval between 0 and the time horizon is divided into equally-spaced periods. Each period, called a time step, is associated with several price scenarios, called nodes. A node has two branches, one of which goes up and the other goes down. At a node with a price of , the underlying will increase to , where , or decrease to at the next time step. An option is valued by first calculating the values of the option at each of the nodes at the option’s expiry, and then iterating backwards to calculate the values of the option at other nodes. At a given node the option’s value is determined by taking the maximum of the option’s intrinsic value and its discounted expected future value for a Bermudan or American style option and simply its discounted expected future value for a European option. The value of the option at the root of the tree is then the value of the option in consideration.

Delta

Let denote the spot price of the underlying and the option’s fair value. An estimate of the option’s delta is:

where is the change in the option’s value when the price of the underlying changes by In a binomial tree let and b e the option’s values at the nodes and , respectively. Then an estimate of the option’s delta can be calculated as:

Gamma

A gamma is the change of the delta divided by the change in the price of the underlying. To estimate it, the fair values of the option at the nodes in time step 2 are also needed. Suppose the fair values at time step 2 are ( corresponding to the three possible underlying prices , noting that . To estimate a gamma, consider the difference of deltas when the underlying price is half way between and and half way between and. Since the two deltas are:

and the underlying change is:

A gamma can then be estimated as:

Theta

Theta is the rate of change in the option’s value with respect to the change in time, when the underlying price and other parameters are kept the same. Since at time step 2 the middle node has the same underlying price as the spot price at time 0, an estimate of theta is:

For more details of a Rubinstein binomial tree and estimation of greeks from a binomial tree, see Hull [2].

Greeks that are taken from a finite difference grid

Bermudan or American options can also be valued by solving the underlying no-arbitrage equation. This is a second-order partial differential equation in two variables (the underlying asset price and time). It can be solved numerically by constructing a grid of points in those two variables, and using finite differences to approximate the derivatives. The greeks delta, gamma and theta can be taken directly from the grid and no revaluation is needed.

The first step is to set up a grid of points in the two variables, asset price and time, labeled by i and j. The time interval between 0 and the time horizon is divided into periods, which need not be equally spaced (time-steps can be added on important dates, say those on which cash flows occur). The likely final range of the underlying asset price is also divided into periods, giving a two-dimensional grid of points. The present value of the underlying price can also be arranged to fall exactly on one of the grid points. An option is valued by first calculating the values of the option at each of the grid points at the option’s expiry, and then iterating backwards to calculate the values of the option at other time-steps. Various finite difference schemes can be used, the most popular and accurate being the Crank-Nicolson scheme. At each time-step, a vector of option prices at each of the asset steps is constructed, and a matrix equation is solved for this vector of option prices. The present value of the option is then the component of the time-0 vector corresponding to the present value of the underlying price.

Delta

Let denote the spot price of the underlying, the asset step size and the option price at the grid point . An estimate of the option’s delta is:

where is the option’s present (time-0) value at the grid point . This is the central difference approximation for delta, and is more accurate than either the forward of backward difference: the error is .

Gamma

A gamma is the change of the delta divided by the change in the price of the underlying. The natural approximation for this is:

The error in this approximation is also .

Theta

Theta is the rate of change in the option’s value with respect to the change in time, when the underlying price and other parameters are kept the same. There are various approximations we can use to estimate the value of theta, the one used in the Crank-Nicolson scheme being:

where is the option value at time , one time-step out from today, and at the grid point . For more details of a Rubinstein binomial tree and estimation of greeks from a binomial tree, see Wilmott [3].

Greeks that are calculated with the bumping method

Greeks of most exotic options don’t have simple closed-form solutions. If such options are not valued with a tree method, or if the greeks cannot be taken directly from the tree, then a numeric approximation is needed to estimate them. In the FINCAD math library, a generic approximation method, the so-called bumping method, is used. This is a standard numeric method for the calculation of a function’s derivative. The formulas are given below for the estimation of the first and second order derivatives of a function

Estimation of first order derivatives

A.For a one-sided approximation:

B.For a two-sided approximation:

Estimation of second order derivatives

Selection of the bumping size

One natural question to ask is that what the best choice of a bumping size is. Mathematically, if the derivative of a function exists, then the smaller the bumping size, the more accurate the approximation is. However, in implementation, a small bumping size is not always a good solution. A small bumping size may make an estimate unstable. As the bumping size decreases, the resulting derivative may become volatile as the underlying variable changes slightly. Unstable greeks may bring difficulties to the dynamic hedging of an option.

In FINCAD functions two types of bumping sizes, absolute and relative, are used. The approximation method can be one-sided or two-sided. The use of a particular bumping size or an approximation method is based on the consideration of accuracy, stability and simplicity. The following are the bumping sizes used in FINCAD math library.

Delta and Gamma

Type 1: An absolute bumping size

Type 2: A relative bumping size

where is a constant. In most cases and in other cases . Adjustments to the bumping size are used for some cases. For example, in a barrier option function, if the bumped price crosses a barrier, the bump size will be reduced so that the bumped price will not cross the barrier.

Vega and Rhos

An absolute bumping size is always used:

where for most option functions the constant and in other cases it is 0.001.

Note that vegas and rhos are always scaled by 1/100.

Theta

The theta is bumped with an absolute bumping size of one day (=1/365, approximately) and is scaled by 1/365. Rigorously:

where is the value of an option at the date

References

[1] Haug, E. G., (1997), The Complete Guide to Option Pricing Formulas, McGraw-Hill.

[2] Hull, John, (1997), Options, Futures, and Other Derivatives, 3rd ed., Upper Saddle River, Prentice Hall.

[3] Wilmott, P. (2006), Paul Wilmott on Quantitative Finance, 2nd ed., Chichester: John Wiley & Sons Ltd.

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