Option Probability

Eric Hartford February 2007

The Creative Destruction Company All Rights Reserved

Lazy Calculations of Option Probability Finishing ITM©

Instead of using a proprietary calculation of the probability of an option expiring in-the-money, there are many substitutes, each with drawbacks, primarily what estimate of volatility to use. I was short S&P March 1500 calls, so I'll use those as examples after each, and let you decide validity...

- Probability, based on standard deviation - 1.6 standard deviations away, where 1 std dev is 68% confidence, and 2 std dev is 95% confidence. Except that stock prices don't follow a normal distribution, they have fat tails, and a higher peak in the center (leptokurtosis!) so std dev models overprice near-the-money options and underprice far out-of-the-money options. A cheap and easy approximation to find the 1 std deviation based on the implied volatility of the options prices, is to add the ATM put and call together and multiply that result by 1.25. The formal calculation of std dev = stock price x volatility (I use the 20-day historical or statistical vol) x square root of time to expiration, expressed as fraction of a year. Alternately, 1 std dev can be figured substituting implied vol for statistical vol = stock price x implied vol (use ATM call) x the square root of the time to expiration, expressed as a fraction of a year.

- Probability, based on log normal distribution - 9.2 %. Better, but still relies on an input for volatility - again, using 20-day historical volatility. See attached doc for the maths, including the 5th order polynomials... I put together an excel spreadsheet for these.

- Delta - 8.9% It is an approximation ( +/- 3% ) of the probability of a call option finishing ITM, when volatility is less than 50% and there are less than 70 days to expiration. The lazy man's estimate, as it can be found at futuresource.com for S&P 500 futures options.

- Monte Carlo simulation - 18%. There is one proprietary calculator that I like. I use the Larry McMillan calculator which tells you the probability that your option strike will ever be crossed before expiration- which is what you really care about, because prudent money management dictates that you close losers at some point before they wipe you out. So, while the probability at expiration is definitely useful, the probability for any time between now and then is more useful. This probability generally doubles the probability of finishing ITM.



Delta vs. Probability - Black-Scholes Option Math©

OPt = p N(d1) – s e

– r t N(d2) where d1 = ln (p/s) + ( r + v

2 / 2)t v√t and d2 = d1 - v√t OPt = Theoretical option price p = Underlying current price s = Option strike price t = time to expiration as a percentage of a year = days to expiration / 365 r = risk free interest rate (30-yr bond) assumed to be 5% for simplicity v = volatility as annual standard deviation, known volatility used to calculate OPt

e = 2.71828 ln = natural logarithm √ = square root iv = implied volatility, found using actual option price and solving for v N(x) = cumulative normal density function of x Where N(σ) = x if σ > 0 and N(σ) = 1 – x if σ < 0 and y = 1 1 + .2316419 | σ | and z = .3989423 e – (σ * σ) / 2

and x = 1 – z (1.330274y5 - 1.821256y4 + 1.781478y3 - .356538y2 + .3193815y)

∆ = Delta = N(d1) is cumulative normal density of d1 and is actually three things; 1) equivalent number of shares of the underlying (hedge ratio) 2) the probability* that a given option will finish in-the-money (ITM) 3) amount an option will change in price as a ratio to the underlying

Calls have positive deltas, Puts have negative deltas *For options less than 70 days to expiry, with v < 50%, not including dividends



Equations in excel programming formats Probability of stock finishing above strike price =1-NORMSDIST(LN(Strike/Tradeprice)/(historicalvol*SQRT(Daystoexp/365))) Note it is debatable whether to use in the above formula historicalvol or impvol Delta of a put =NORMDIST((LN(Tradeprice/Strike)+(Rate+Impvol^2/2)*Daystoexp/360)/(Impvol*SQRT(Daystoexp/365)))-1 Delta of a call =NORMDIST((LN(Tradeprice/Strike)+(Rate+Impvol^2/2)*Daystoexp/360)/(Impvol*SQRT(Daystoexp/365))) Differences between Delta & Probability Options finishing ITM

The actual probability of a given underlying instrument finishing above a particular strike price is;

Probability = 1 - N ln (s/p) which is not that different than d1 v√t

Delta can be a useful approximation of the likelihood that a given strike will finish in the money. This has been one of my key assumptions about options trading, so it is useful to examine it in detail

Let me cite some references to back up my impression that delta is an approximate probability that a given option will finish in the money at expiration. I picked up the idea from Jay Kaeppel when he came to our Atlanta Options Investors meeting in November of 2003. I had previously read one of his books that touched on this and he reinforced it at the meeting.

He states in The Option Trader's Guide to Probability, Volatility, and Timing first edition, page 89, "The delta value for a given option provides an estimate of the probability that the option will expire in the money. Thus an option with a delta of 20 currently has a roughly 20% probability of expiring in the money (although this is not mathematically correct, it does provide a useful estimate and a valuable frame of reference). "

On page 97, Jay Kaeppel writes, "As a simple tool, an option's delta value serves as a handy estimate of the likelihood that it will expire in the money."

Furthermore, McMillan acknowledges the applicability to calls. In McMillan's Options as a Strategic Investment , 4th edition, page 967 under the definition of Delta, we find " (2) the percent probability of a call being in-the-money at expiration." He references this only briefly in the Chapter 40: Advanced Concepts text on page 848. "In another context, the delta of a call is often thought of as the probability of the call being in-the-money at expiration. That is, if XYZ is



50 and the January 55 call has a delta of 0.40, then there is a 40% probability that XYZ will be over 55 at January expiration."

In context, both these references have the air of hearsay that is being passed on, but not refuted. Neither claims that the delta is the equivalent of the probability for all options, and one should not make that claim, without additional explanation. There is an ease of use (lazyness?) issue in taking one of the existing greeks that is provided by the brokers and using it as an approximation. One should favor doing the actual probability calculation. And it is an easier calculation in that it has fewer terms.

The probability formula from McMillan is a little fuzzy, but using the same divisor as the delta formula seems permissible, and I like it for giving the instantaneous market price implied value, rather than the historical.

Numerically, I would note that the interest rate risk plus half of the implied volatility squared all multiplied by the time remaining to expiration could throw off a number that could be overlooked in an approximation that was plus or minus 3%, as long as:

1) the options in question are near term (within 70 days). Obviously the closer one gets to expiration, the more accurate this approximation becomes.

2) the implied volatility is below some limit (.5). This is the dangerous assumption, because as options traders, we are looking for extreme volatilities.

If my memory of natural log functions isn't too faulty, ln (a/b) = - ln (b/a), so the transposition of the the natural log of trade price over stock price in the delta calculation versus the stock price over trade price in the probability calculation is solved by stating that probability should be an absolute value or always positive.

While the following doesn't make the case at all (because 4+1=5 and 3+2=5 does not mean that 4=3 or 1=2) it is worth noting that,

Probability ITM (above strike) + Probability ITM (below strike) = ~1

and

Call Delta + abs val [Put Delta] = ~1

Black Scholes option maths have many limitations, chief among them that they primarily apply to European-style options (only exercised or assigned at expiration), not options exercise-able/assignable any time (American style). Other models superceded Black-Scholes in the 15 years since these calculations were the state of the art. However, it is useful to know this work and the contributions of Merton, Cox, Rubenstein and others.

Special thanks and credit to Larry McMillan, Jay Kaeppel and Joe Murad for their work in the field, particularly Joe Murad for his efforts sustaining Atlanta Options Investors.

Option Probability

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