An Option Theta

8/3/2019 An Option Theta

http://slidepdf.com/reader/full/an-option-theta 1/3

Copyright (c) Technical Analysis Inc.

Stocks & Commodities V. 14:4 (160-162): On Option Theta by Lawrence D. Cavanagh

OPTIONS ANALYSIS

On Option Theta

AOptions traders use various measurements to calculate an

option’s risk, the calculations of which are denoted by Greek

letters. One example is theta, which is the measure of how

much an option’s price decreases for each day that passes.

by Lawrence D. Cavanagh

nyone who has ever traded options quickly

becomes aware that if all things remain equal,

then over time, the price (or premium) of the

option declines. Theta (θ) is the Greek letter

used to mark the decay of an option premiumover time. This price decay is nonlinear (that is, it changes

over time), and most of the time, it accelerates as the expira-

tion date approaches. For an option buyer, theta can indicate





the daily cost of holding the option. For an option writer, theta

can indicate the daily profit that accrues from being short the

option. Understanding the whys and wherefores of theta thus

becomes important, and here we explore why option premi-

ums decay the way they do.

AT-THE-MONEY THETAAn option’s total premium consists of its intrinsic value and

its time value. Intrinsic value is what the option is worth if

exercised immediately. If positive, it is the difference be-

tween the price of the underlying tradable (stock, bond or

commodity) and the strike price. Time value is the part of

option premium that exceeds intrinsic value, based on the

probability of the underlying stock going through the strike

price and from the profit or loss potential caused by the

assumed dispersion of future price outcomes.

For options that are at-the-money (in which the stock and the

strike price are equal), theta is easy to estimate because

regardless of how much time remains, the option has basically

50/50 odds of ending up in-the-money (the price of the stock above the strike price if a call or below the strike price if a put).

Future premiums can be estimated by taking the square root

of the fraction (in decimal terms) of the remaining time that the

option has. For example, to estimate the value 30 days hence

of a 90-day at-the-money option priced at $2.00, one takes the

square root of two thirds (60 days divided by 90 or 0.667) and

multiplies the result (0.816) by the original premium to get

$1.63. Subtracting this number from the original premium, we

observe that the 30 days’ time decay (or loss in value) of this

option is $0.37 or 18.4% of its original value of $2.00.

IN- AND OUT-OF-THE-MONEY THETA

Both out-of-the-money premiums and in-the-money premi-

ums decay in a manner different from at-the-money premi-

ums because as time passes, the probability of the stock

ending up in the money increases or decreases, respectively.

At some point before expiration, an out-of-the-money option

will, theoretically at least, have little chance of ever ending at

an in-the-money price level. Similarly, an in-the-money

option will have little chance of expiring worthless at a

certain point and will trade on its tangible value. For example,

with a week to go and the stock trading at $30, a call option

with a strike price of $35 is 15.4% out of the money (calcu-

lated by taking the natural log of 30/35). If the weekly

standard deviation is 4.4245% (that is, the annual volatility is32%), then the call is more than three standard deviations out

of the money. The implication is that the stock has only about

1/10 of 1% chance of ever hitting the strike price.

In Figure 1, the theta of an out-of-the-money option is

compared with the theta of an at-the-money option. The at-

the-money option loses value at a progressively faster rate as

expiration approaches, while the out-of-the-money option’s

time decay actually tends to slow down at a certain point once

the likelihood of the stock reaching the strike price dimin-

ishes. Because an option’s rate of time decay changes over

time, one day’s theta is going to differ from another day’s. As

Figure 1 demonstrates, these differences in theta can vary

from day to day depending on the time and the distance of the

stock from the strike price.

A SPREADSHEET ROUTINE

Figure 2 is a spreadsheet routine to estimate the theta, using

as an example data for the WMX Technologies [WMX] Janu-

ary 30 call option. The formulas for the cells in column B are

presented just to the right of each cell. Besides the formulas,

we need to enter the following into the first seven cells in

column B:

1 Stock price: $28.88

2 Strike price: $30

3 Number of days to expiration: 108 (calculated

FIGURE 2: OPTION THETA SPREADSHEET. The formulas to calculate theta are

presented to the right of each cell in column B.

An option’s total premium consistsof its intrinsic value and itstime value. Intrinsic value is whatthe option is worth if exercisedimmediately.

EXCEL(MICROSOFT)

FIGURE 1: OPTION TIME DECAY. Here are two examples of the impact of theta

for a call option. An at-the-money option experiences a dramatic rise in the daily loss

as expiration approaches, compared with an out-of -the-money option.





using the spreadsheet’s date function)

4 Option premium: $1.13

5 Per annum volatility: 90%

6 “Risk-free” interest rate: 5.22%

7 Number of days’ decay we want to estimate: Inthis case, 1.

The result is displayed on line 16 of the spreadsheet, and the

theta for this option is $0.008 per day. Thus, an investor can

see the value of the option on 100 shares’ decay by $0.80 per

day. A writer of the option would expect to keep this much,

provided the stock stands still.

The formulas in this spreadsheet calculate the partial

derivative of the option premium with respect to time. One of

the key variables is the distance of the strike price from the

stock price in terms of standard deviations, which in turn is

calculated from annualized volatility.

While this equation can tell you how much the option is

likely to decay, its accuracy is greatly diminished if the

number of days to be estimated is large in proportion to thelife of the option. In this example, you can get a reasonable

estimation of one week’s time decay by entering 7 in cell B7.

If the option were much closer to expiration (say, with 14

days to go), then extrapolating one day’s theta out to seven

might give a very misleading result.

†See Traders’ Glossary for definition

FINIS

Other measures of an options risk, such as delta and gamma,

exist, but theta is one that most traders learn about the hard

way, having picked the direction right of a stock but not

accounting for the time decay properly. Now, with the use of

the spreadsheet mentioned here, you can more accurately

estimate the impact of theta on your option trades.

Lawrence D. Cavanagh is a senior analyst with the Value

Line Daily Options Survey.

RELATED READING

Hartle, Thom [1996]. “Options as a strategic investment:

Options strategist Lawrence G. McMillan,” interview,

Technical Analysis of STOCKS & COMMODITIES, Volume

14: February.

McMillan, Lawrence G. [1995]. “Put-call ratios,” Technical

Analysis of STOCKS& COMMODITIES, Volume 13: October.

Schinke, Steven [1995]. “Options spread psychology,” Tech-

nical Analysis of STOCKS & COMMODITIES, Volume 13:October.

S&C

Documents

An Option Theta