1. Black and Scholes
Professor Brooks
BA 444
01/23/08

2. Chapter 6 – Using BSOPM First a Call Option
Then Put – Call Parity (from arbitrage)
Then Put Formula
Problems with BSOPM
Good for at the money options
Volatility Smiles…

3. Black and Scholes OPM
Continuous pricing of binomial multiple period model
Five variables, four observable
Current Stock Price
Current Risk-free Rate
Strike Price of Option
Time to Maturity
Volatility of Stock Returns (not observable)
Developed formula for Call
Developed formula for Put

4. Finding the Call Price with OPM
Co = SoN(d1) - Ke-rTN(d2)
d1 = [ln(So/K) + (r + 2/2)T] / (T1/2)
d2 = d1 - (T1/2)
N(d) = probability that a random draw from a normal distribution will be less than d.

5. Variables in OPM C = Current call option value.
S = Current stock price
K = Strike or Exercise price.
e = , the base of the natural logrithm
r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option.
T = time to maturity of the option in years.
ln = Natural log function
Standard deviation of annualized continuously compounded rate of return on the stock

6. Application of OPM So = 100 K = 95 r = .10 T = .25 (quarter of a year)
= .50 (estimated…)
d1 = [ln(100/95) + (.10+(5 2/2)).25]/ (5.251/2)
d1 = .43
d2 = ((5.251/2) = .18
N (.43) = .6664
N (.18) = .5714
You can use Excel Function to find normal distribution values for N(d1) and N(d2)

7. Call Value…Put Formula
C = SN(d1) - Ke-rTN(d2)
C = $100 x $95 e- .10 X .25 x .5714
C = $13.70
Solve Put Price with Put-Call Parity?

8. Put-Call Parity Formula: Put in Values from previous example,
This is a pricing relationship based on arbitrage
Any mispricing is quickly recognized and exploited
There are transaction costs to exploiting put-call parity
Formula:
C – P – S + Ke-rT or P = C – S + Ke-rT
Put in Values from previous example,
Call = $13.70, S = $100, K = $95, e-rT=
Put = $ $100 + $95 x = $6.35

9. Put Formula P = Ke-rTN(-d2) - SN(-d1)
And again we can see the K – S for the value of the put…
This is on a non-dividend paying stock.
Use CBOE calculator…its fast and easy…

10. Application of Put Formula
So = 100 K = 95
r = T = .25 (quarter of a year)
= .50 (estimated…)
d1 = [ln(100/95) + (.10+(5 2/2)).25]/ (5.251/2)
d1 = 0.43
d2 = ((5.251/2) = 0.18
N (-.43) =
N (-.18) =
Put = $95 x x $100 x
Put = $6.35

11. Implied Volatility
The only variable we can not directly observe is the volatility of the underlying asset’s returns…
Estimate based on historical returns?
Estimate based on belief of future volatility
If we know price of the call or put…
Imply Volatility from the BSOPM
Unfortunately can not isolate σ so we have to iterate for the answer…or use a calculator

12. Implied Volatility Estimate of volatility from observable variables
Price of call
Price of put
Strike price
Risk-free rate
Time to maturity

13. Volatility Smiles The BSOPM works well for at-the-money options…
Implied volatility is good for at-the-money options
Deep out-of-the-money options have higher volatilities
Deep in-the-money options have higher volatilities
If you plot volatilities on the y-axis and strike prices on the x-axis you get a smile…

14. Sensitivity of the Call Option
Call Premium = f(S,K,T,r, σ, dividends)
As S increases Call premium increases (+)
As K increases Call premium decreases (-)
As T increases Call premium increases (+)
As r increases Call premium increases (+)
As σ increases Call premium increases (+)
As dividends increase Call premium decreases (stock price falls by size of the dividend) (-)

15. Sensitivity of the Put Option
Put Premium = f(S,K,T,r, σ, dividends)
As S increases Put premium decreases (-)
As K increases Put premium increases (+)
As T increases Put premium increases (+)
As r increases Put premium decreases (-)
As σ increases Put premium increases (+)
As dividends increase Put premium increases (stock price falls by size of the dividend) (+)