The Black and Scholes Option Pricing Model Made Easy
Table of Contents: Black Scholes Option Pricing
Investment Books: Black Scholes
Basic Black-Scholes: Option Pricing and Trading
Amazon Price: $40.00 (as of 12/18/2009) 
Black-Scholes and Beyond: Option Pricing Models
Amazon Price: $45.31 (as of 12/18/2009)
Probability Theory in Finance: A Mathematical Guide to the Black-Scholes Formula (Graduate Studies in Mathematics)
Amazon Price: $39.15 (as of 12/18/2009)
Option Pricing: Black-Scholes Made Easy (Wiley Trading)
Amazon Price: $61.43 (as of 12/18/2009)
Black and Scholes Option Pricing Model
The Black and Scholes Option Pricing ModelC=SN(d1)-Ke^(-rt)N(d2)
P=Ke^(-rt)N(-d2)-SN(-d1)
C=Theoretical Call Premium
P=Theoretical Put Premium
S=Current Stock Price
t=time until option expiry
K=option strike price
r=risk-free interest rate
N=cumulative standard normal distribution
e=expoential term (2.7183)
d1=(ln(S/K)+(r+s^2/2)t)/sSQRT(t)
d2=d1 - sSQRT(t)
s=standard deviation of stock returns
Delta = N(d2)
The model is divided into two parts. The first part, SN(d1) is the expected benefit from acquiring a stock outright. Derived by multiplying stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N(d1)].
The second part of the equation, Ke(-rt)N(d2), gives the current value of paying the exercise price on the expiry day. The fair market value of the call option is then calculated by calculating the difference between these two parts.
Assumptions:
1) The stock pays no dividends during the option or warrant's life
Many companies pay dividends to their share holders, so this is a limitation, but the model may be adjusted by subtracting the discounted value of a future dividend from the stock price.
2) European exercise terms are used
i.e. option can only be exercised on the expiration date. American exercise terms allow the option to be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility. Fortunately few calls are ever exercised before the last few days of their life, because exercising a call option early forfeits the remaining time value on the option.
3) Markets are efficient
Markets are random and people cannot consistently predict the direction of a market or stock.
4) No commissions are charged
5) Interest rates remain constant
The Black and Scholes model uses the risk-free rate to represent this rate e.g. U.S. Government Treasury Bills with 30 days left until maturity is often used to represent it.
6) Returns are lognormally distributed
i.e. returns on the underlying stock are normally distributed (Gaussian bell curve distribution).
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The Explanation
If the strike price is much higher than the current price and the end-date not far away the option is almost worthless, but if the remaining time is long and the strike price below the current price, then the value is high. If risk-free interest rates are low then options increase in value whereas when high they become less attractive. If the volatility of the market is high there is more chance of reaching the strike-price so the value goes up.
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Investment Books
Solutions Manual - A Primer For The Mathematics Of Financial Engineering
Amazon Price: $29.50 (as of 12/18/2009)
Principles of Financial Engineering, Second Edition (Academic Press Advanced Finance)
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A Primer for the Mathematics of Financial Engineering
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Monte Carlo Methods in Financial Engineering (Stochastic Modelling and Applied Probability) (v. 53)
Amazon Price: $48.36 (as of 12/18/2009)
Mathematics for Finance: An Introduction to Financial Engineering (Springer Undergraduate Mathematics Series)
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Please Leave Some Feedback
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Reply
- AndyPo AndyPo Aug 17, 2009 @ 9:57 am
- Thanks.
Hmmm. Yes. You have a good point about the Rich Dad Poor Dad books. They offer good advice, but the target audience of this web-page would perhaps already be aware of that.
[in reply to exchange-tradedfunds]
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Reply
- exchange-tradedfunds exchange-tradedfunds Aug 15, 2009 @ 4:57 am
- Nice work her Andy, I really like you point of view, it's interesting that you have the Rich Dad Poor Dad books on here for sale. :)
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Reply
- JaguarJulie JaguarJulie Jul 11, 2009 @ 8:01 am
- OK, allow me to freely admit this to you Andy and whomever else reads this lens, you honestly LOST me at: C=SN(d1)-Ke^(-rt)N(d2) -- ;)
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Reply
- spirituality spirituality May 15, 2009 @ 9:23 am
- Great lens - you've been blessed by a squidoo angel :)
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Reply
- spirituality spirituality May 15, 2009 @ 9:23 am
- Great lens - you've been blessed by a squidoo angel :)
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