1. Analytical Option Pricing Models: introduction and general concepts Finance 70520, Spring 2002 Risk Management & Financial Engineering The Neeley School S. Mann

6. Black-Scholes-Merton model assumptions Asset pays no dividends European call No taxes or transaction costs Constant interest rate over option life Lognormal returns: ln(1+r ) ~ N (  ) reflect limited liability -100% is lowest possible stable return variance over option life

7. Black-Scholes-Merton Model C = S N(d 1 ) - K B(0,t) N(d 2 ) d 1 = ln (S/K) + (r +    2 )t  t t d 2 = d 1 -  t Note that B(0,T) = present value of $1 to be received at T define r = continuously compounded risk-free rate find r by: exp(-rT) = B(0,T) so that r = -ln[B(0,T)]/T e.g. T = 0.5 B(0,.5) = 0.975 r = -ln(.975)/0.5 = 0.02532/.5 = 0.05064

9. Function scm_d1(S, X, t, r, sigma) scm_d1 = (Log(S / X) + r * t) / (sigma * Sqr(t)) + 0.5 * sigma * Sqr(t) End Function Function scm_BS_call(S, X, t, r, sigma) scm_BS_call = S * Application.NormSDist(scm_d1(S, X, t, r, sigma)) - X * Exp(-r * t) * Application.NormSDist(scm_d1(S, X, t, r, sigma) - sigma * Sqr(t)) End Function Function scm_BS_put(S, X, t, r, sigma) scm_BS_put = scm_BS_call(S, X, t, r, sigma) + X * Exp(-r * t) - S End Function Code for Mann’s Black-Scholes-Merton VBA functions To enter code: tools/macro/visual basic editor at editor: insert/module type code, then compile by: debug/compile VBAproject

10. N( x) = Standard Normal (~N(0,1)) Cumulative density function: N(x) = area under curve left of x; N(0) =.5 coding: (excel) N(x) = NormSdist(x) Black-Scholes-Merton Model: Delta C = S N(d 1 ) - K B(0,t) N(d 2 ) N(d 1 ) = Call Delta (   call hedge ratio = change in call value for small change in asset value = slope of call: first derivative of call with respect to asset price

12. Call and Delta over time

14. Call gamma (curvature)

17. Implied volatility (implied standard deviation) annualized standard deviation of asset rate of return, or volatility.  = Use observed option prices to “back out” the volatility implied by the price. Trial and error method: 1) choose initial volatility, e.g. 25%. 2) use initial volatility to generate model (theoretical) value 3) compare theoretical value with observed (market) price. 4) if: model value > market price, choose lower volatility, go to 2) model value < market price, choose higher volatility, go to 2) eventually, if model value  market price, volatility is the implied volatility

19. Historical annualized Volatility Computation 1)compute daily returns 2) calculate variance of daily returns 3) multiply daily variance by 252 to get annualized variance:  2 4) take square root to get  or: 1) compute weekly returns 2) calculate variance 3) multiply weekly variance by 52 4) take square root annualized standard deviation of asset rate of return  =

20. Call Theta: Time decay