Black-Scholes-Merton model assumptions Implied & Historical Volatility Finance 30233, Fall 2009 The Neeley School S. Mann 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 (m , s) reflect limited liability -100% is lowest possible stable return variance over option life S.Mann, 2009

Black-Scholes-Merton Model C = S N(d1 ) - KB(0,t) N(d2 ) ln (S/K) + (r + s2/2 )t d1 = s t d2 = d1 - s 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 S.Mann, 2009

Code for Mann’s Black-Scholes-Merton VBA functions 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)) 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 To enter code: tools/macro/visual basic editor at editor: insert/module type code, then compile by: debug/compile VBAproject S.Mann, 2009

Black-Scholes-Merton Model: Delta C = S N(d1 ) - KB(0,t) N(d2 ) 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) N(d1 ) = Call Delta (D) = 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 S.Mann, 2009

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 S.Mann, 2009

Historical annualized Volatility Computation annualized standard deviation of asset rate of return s = 1) compute daily returns 2) calculate variance of daily returns 3) multiply daily variance by 252 to get annualized variance: s 2 4) take square root to get s or: 1) compute weekly returns 2) calculate variance 3) multiply weekly variance by 52 4) take square root S.Mann, 2009