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

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…

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

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.

Variables in OPM C = Current call option value. S = Current stock price K = Strike or Exercise price. e = 2.71828, 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

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 = .43 - ((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)

Call Value…Put Formula C = SN(d1) - Ke-rTN(d2) C = $100 x .6664 - $95 e- .10 X .25 x .5714 C = $13.70 Solve Put Price with Put-Call Parity?

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= 0.9753 Put = $13.70 - $100 + $95 x 0.9753 = $6.35

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…

Application of Put Formula 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 = 0.43 d2 = .43 - ((5.251/2) = 0.18 N (-.43) = .333535 N (-.18) = .428508 Put = $95 x 0.9753 x 0.4286 - $100 x 0.3336 Put = $6.35

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

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

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…

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) (-)

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) (+)