Option Pricing Models
The purpose of an option pricing model is to determine the theoretical fair value for a call or put option given certain known variables. In other words - to determine an option's expected return.
Basically, the expected return of an option contract is a function of two variables:
- The payoff of the option at maturity date
- The probability of the option being in-the-money at maturity
These two values multiplied together give you the theoretical price.
Calculating the option payoff is quite easy: for call options it is the maximum of either 0 or the underlying price minus the strike price. For put options it is the maximum of either 0 or the strike price minus the underlying price. More simply:
Call Option Payoff = Max (0, (Underlying Price - Strike Price))
Put Option Payoff = Max (0, (Strike Price - Underlying Price))
But it is in determining the probability of the payoff that becomes a little more difficult.
Essentially, you want to know where the underlying price is likely to be trading at by the expiration date. To determine this probability is no easy task.
For example, say that a stock is currently trading at 100 and you are trying to value a call option on this stock with a strike price of 100 and maturity date of 1 month. Imagine that you know the exact probabilities of where this stock will be trading at the maturity date:
50% chance it will be trading at 95
50% chance it will be trading at 105
If these were the only two outcomes available and you knew the probabilities of these outcomes, then pricing this option is very easy.
First, you know that for a call option, if he underlying is trading below the strike price than the call option is worthless. Second, if the underlying is trading above the strike price then the payoff of the option is the underlying price minus the strike price - i.e. 5 (105 - 100).
So now we have two outcomes and two payoffs.
A 50% chance of making 0 and a 50% chance of making 5.
Then we can construct a simple formula to describe the expected return of our option contract:
(Probability of Stock Trading at 95) x (Option Payoff at 95) + (Probability of Stock Trading at 105) x (Option Payoff at 105)
Which becomes: (0.50 x 0) + (0.50 x 5) = 2.50
Of course in the real world, there is a much larger set of price outcomes and we will never know for sure what the true probability really is. That was the challenge Fisher and Black had when they ventured into writing their paper on pricing real options.
Comments (14)
Peter
May 28th, 2010 at 7:19am
What do you mean by manually? You can check the code that I use for the spreadsheet in the VBA editor. Alternatively, you can review the Black and Scholes formula here;
optiontradingtips.com/pricing/black-and-scholes.html
Let me know if I misunderstood you.
Madhukar
May 26th, 2010 at 4:40pm
22.50 Strike Price
25.00 Underlying Price
26-May-10 Today's Date
30.00% Historical Volatility
20-Jun-10 Expiry Date
3.50% Risk Free Rate
2.00% Dividend Yield
25 DTE
0.07 DTE in Years
If I have to calculate the Theoretical price Manually without using the excel formula how do I do it, i.e. what gets multiplied by what and what get deducted to get a value that excel shows of 2.59. Please send a reply to this message. (note: I have taken this data from Options trading workbook)
Peter
November 16th, 2009 at 4:04am
Hi Sam, not sure exactly what you're asking but the expected return of the underlying at each maturity is e^(rt), where r = risk free rate - dividend yield and t = time to maturity in years.
So, you would need to calculate this twice I guess.
Sam
November 14th, 2009 at 5:23am
Dear Sir,
How can the expected return on the underlying asset be calculated using derivatives with different maturities?
Thanx
Peter
September 4th, 2009 at 6:32am
Hi Hakim, are you after stock price history? You could try http://finance.yahoo.com (free) or http://www.csidata.com
Hakim
September 4th, 2009 at 12:50am
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Admin
March 23rd, 2009 at 4:21am
Hi Will, that's impossible to answer actually. The decay of an option is not linear and cannot be simplified like that. Not only does it depend on the "in-the-moneyness" like you mentioned but also on the volatility, interest rates and dividends.
What exam is this question from? Can you please send it to me?
will
March 22nd, 2009 at 11:57pm
as an option reaches maturity does the option premium havle or quarter in price?....i know it depends on whether the option is at the money etc.
However it is a question in an exam and no other details are given...thanks
Admin
March 22nd, 2009 at 6:39am
Yes that's correct. You will notice that in-the-money options have higher premiums than out-of-the-money options.
Roger
March 19th, 2009 at 9:21pm
Is it accurate to assume that the higher the probability of stock trading above strike price, the more premium an option buyer would need to pay and vice versa if the probability is lower?
Admin
December 8th, 2008 at 3:17am
Hi Sanjib,
Not sure why the calculator isn't working for you. What do you see in the cells? Perhaps you don't have Macros enabled?
You can try http://www.bankrate.com/ for interest rate information.
sanjib sinha
November 27th, 2008 at 11:34pm
sir,
how to operate the the calculator & how can i get get interest of index
Admin
August 1st, 2008 at 6:37pm
Hi Deepesh,
That is the idea behind an option pricing model; to calculate the premium. I.e. the premium is the result of the calculation.
Deepesh
July 31st, 2008 at 11:49am
the formula given above is wrong. Payoff should also take premium in account
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