Analytical Option Pricing: Black-Scholes –Merton model; Option Sensitivities (Delta, Gamma, Vega, etc) Implied Volatility Finance 30233, Fall 2010 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 (  ) reflect limited liability -100% is lowest possible stable return variance over option life

Simulated lognormal returns Lognormal simulation: visual basic subroutine : logreturns

Simulated lognormally distributed asset prices Lognormal price simulation: visual basic subroutine : lognormalprice)

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 N( x) = Standard Normal [~N(0,1)] Cumulative density function: N(x) = area under curve left of x; e.g., N(0) =.5 coding: (excel) N(x) = NormSdist(x) 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

Function scm_d1(S, X, t, r, sigma) scm_d1 = (Log(S / X) + r * t) / (sigma * Sqr(t)) * 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 VBA Code for Black-Scholes-Merton functions To enter code: tools/macro/visual basic editor at editor: insert/module type code, then compile by: debug/compile VBAproject

Historical 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 Or: 1) Compute monthly returns 2) calculate variance 3) Multiply monthly variance by 12 4) take square root annualized standard deviation of asset rate of return  =

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

Implied volatility (implied standard deviation): VBA code: Function scm_BS_call_ISD(S, X, t, r, C) high = 1 low = 0 Do While (high - low) > If scm_BS_call(S, X, t, r, (high + low) / 2) > C Then high = (high + low) / 2 Else: low = (high + low) / 2 End If Loop scm_BS_call_ISD = (high + low) / 2 End Function

Call Theta: Time decay