Choosing a Time Clock for Option Pricing and Volatility Calibration
Summary
The document explains how to represent time to maturity in option pricing and why a trading-day fraction is only one possible convention. It describes a general clock that maps calendar time into a year fraction or trading time. In the Black–Scholes framework, volatility enters through its product with the square root of that clock, so changing the time convention changes the implied volatility associated with a market price.
The practical guidance is to use a consistent clock when calibrating a model to market prices and when pricing with that model. The clock can reflect weekends and specific events such as earnings announcements or macroeconomic news, with the aim of keeping implied volatilities comparatively stable as time passes. When volatility quotes come directly from a data provider, the convention used to produce them must be known. The document does not prescribe one universal day count or provide a specific convention for every market; the appropriate choice depends on the market inputs and modeling framework.
Key ideas
- Time to maturity can be measured using different clocks that map calendar time into model time.
- In Black–Scholes, volatility affects value through total volatility over the chosen time clock.
- A time convention used for calibration must also be used consistently for pricing.
- A clock can account for weekends and scheduled events that affect expected market activity.
- Volatility quotes from data providers require knowing the time convention behind them.
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Full text
# When pricing options, which day counting conventions should be used to calculate time to maturity?
# When pricing options, which day counting conventions should be used to calculate time to maturity?
In most option pricing textbooks, time to maturity is given as a convenient figure such as 6 months (T=0,5).
In practice how do you effectively calculate time to maturity given today's date and the expiration date? Knowing that there are about 252 trading days in one year, do you consider T as the ratio between the number of trading days between "today" and the expiration date over 252?
Thanks!
Edit: a related question (and answer) can be found here: Ways of treating time in the BS formula
## Answer by LocalVolatility (score 6, accepted)
https://quant.stackexchange.com/a/32039
There are different ways to define your clock. No matter how you do it, the key is that you use the same one for the calibration of your model to market data and for pricing.
Consider for example the Black-Scholes model. When calibrating the model to the market prices of European plain vanilla options, you obtain the generally strike and time-to-maturity dependent implied volatility $\sigma_{\text{IV}}(T, K)$.
Now let $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ be a non-decreasing function that that maps actual time into trading time. To get the intuition think of it as mapping a maturity instant into a year fraction - so something similar to the day counter that you refer to in your question. To not overcomplicate things, lets for the moment assume that rates are zero, i.e. $r = 0$. You can then write the Black-Scholes formula in terms of the clock as
\begin{equation} V_0 = \phi \left\{ S_0 \mathcal{N} \left( \phi d_+ \right) - K \mathcal{N} \left( \phi d_- \right) \right\}, \end{equation}
where
\begin{equation} d_\pm = \frac{1}{\sigma \sqrt{f(T)}} \left( \ln \left( \frac{S_0}{K} \right) \pm \frac{1}{2} \sigma^2 f(T) \right) \end{equation}
and $\phi \in \{ -1, +1 \}$ indicates a put or call option. The diffusion coefficient $\sigma$ never appears alone but only as the total volatility-to-maturity $\sigma \sqrt{f(t)}$. Using the trading time clock, you immediately see that different choices for $f$ yield different implied volatilities.
In the real world, you model $f$ to incorporate things like weekends and special events like earning announcements or macroeconomic news. Again - the key is to be consistent in calibration and pricing. A well-defined clock allows you to obtain implied volatilities that are relatively stable as time passes.
When you directly receive implied volatility quotes instead of prices from a data vendor, it is crucial to know under which clock (usually a day count convention) these were computed.
As a reference for further reading: See Chapters 4.3 and 4.4 in Clark (2011) "Foreign Exchange Option Pricing - A Practitioner's Guide".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.