# 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

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".

• many thanks for your clarifications. So if I understand correctly I have to use the day count convention which is specified in the option contract, which should also be the one used by the data vendor? Correct? – BigONotation Jan 22 '17 at 23:29
• An option contract usually doesn't specify day count conventions (unless there is some interest to be paid). A European plain vanilla call for example always pays off $\left( S_T - K \right)^+$ irrespective of how you count time. So when calibrating your model (think computing implied volatility) you need to make a choice. If you are doing the calibration, then its up to you. If you get the calibrated model then your data vendor should tell you how they arrived at those values, i.e. what day counter they used. – LocalVolatility Jan 22 '17 at 23:33
• OK got it! Many thanks for clarifying this! – BigONotation Jan 22 '17 at 23:39
• @BigLudinski Bloomberg defaults to Actual/365 if I remember correctly, but as LocalVolatility mentioned, it's all about internal consistency. – Helin Jan 23 '17 at 17:29