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Finite-Difference Theta Depends on Day-Count Convention

Article Quant Q&A · Author: Sean Sinykin
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Summary

The document discusses why a finite-difference estimate of option theta may not match a market data provider’s reported value. A finite difference estimates the change in option value as time advances, divided by the time increment. The scale of the result therefore depends on whether time is measured in years or days and on the day-count convention used to annualize or report theta.

The response recommends checking directly with the data provider about its convention. It also suggests benchmarking the calculation against an analytic formula for a European option or a tree method for an American option. The example reports a discrepancy between a calculated theta and a provider’s figure, but does not establish its cause. Differences in model assumptions, exercise style, inputs, and time conventions can all affect comparisons, so the finite-difference step size alone may not explain the gap.

Key ideas

  • Finite-difference theta estimates option value change over a specified time increment.
  • Theta values depend on the time unit and day-count convention used in reporting.
  • Ask the data provider how its theta is defined before comparing values.
  • European option formulas and tree methods for American options can provide benchmarks.

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# Calculation of option Greek (sensitiviety) theta via finite difference


# Calculation of option Greek (sensitiviety) theta via finite difference












I am able to get good approximations for delta, gamma, and rho via finite difference method, but not theta. I believe my issue is the value of h. Theta is basically the difference between the price of the the option one time step in the future and the price today divided by the size of the time step, ie

theta (approx) = V(d_v+1) - V(d_v)/(1/365), where V(d_v+1) is the value of the option one time step (1/365) into the future

This basically comes from http://docs.fincad.com/support/developerfunc/mathref/greeks.htm

If I apply this to, for example, the call option quote on 04/18/2013 for ticker A (Agilent, I believe), strike of 40, underlying price of 41.83, expiry of 05/18/2013 (30/365 days to maturity), 1.1% Dividend Yield, 0.3% risk-free rate, I get a theta of -8.9, whereas the actual theta is approximated by a large options data reporting firm as approx -2.2. My other Greek approximations are close enough, but I cannot get a good approximation for theta. Anybody have insight into this issue? Thanks in advance for your help!

## Answer by Enrico Schumann (score 1)

https://quant.stackexchange.com/a/28317

You should ask your data provider how exactly they come up with this number. Many implementations divide theta by 365 or some other yearly day count to arrive at "theta per day".

It should be simple enough to check the value: for a European option, you can use the analytic formula; for American Options, you can use a tree. Methods to get the Greeks in the binomial method are described in the paper Implementing Binomial Trees

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.