From Binomial to Black-Scholes-Merton Option Sensitivities (Delta, Gamma, Vega , etc) Implied Volatility Finance 30233, Fall 2017 The Neeley School S. Mann S.Mann, 2017

Historical Volatility Computation annualized standard deviation of asset rate of return 1) compute weekly returns 2) calculate variance 3) multiply weekly variance by 52 take square root Or: 1) Compute monthly returns 3) Multiply monthly variance by 12 4) take square root S.Mann, 2017

Calibrating Binomial Tree: 1) Set “up” and “down” factors to reflect volatility: U = 1 + s√T D = 1 - s√T Adjust so expected future price is forward price: E[S(T)] = S0 ( 1 + (r – d)T ) Simple form: U = 1 + (r – d)T + s√T D = 1 + (r – d)T - s√T Or, in continuous time (McDonald): U = exp((r – d)T + s√T) D = exp((r – d)T - s√T) See “Fin30233-F2017_Binomial Tutor.xls” (available on Mann’s course website)

S.Mann, 2017

S.Mann, 2017

VBA code for binomial European Call (for later reference) Function Binom(n, X) Binom = Application.Combin(n, X) * 0.5 ^ X * 0.5 ^ (n - X) End Function Function scm_bin_eur_call(S, X, rf, sigma, t, n) dt = t / n up = Exp((rf - (sigma ^ 2) / 2) * dt + sigma * Sqr(dt)) down = Exp((rf - (sigma ^ 2) / 2) * dt - sigma * Sqr(dt)) r = Exp(rf * dt) p = (r - down) / (up - down) q = 1 - p scm_bin_eur_call = 0 For Index = 0 To n scm_bin_eur_call = scm_bin_eur_call + Exp(-rf * t) * Application.Combin(n, Index) * p ^ Index * q ^ (n - Index) * Application.Max((S * up ^ Index) * (down ^ (n - Index)) - X, 0) Next Index Note: code in red has to be on one line for vba to compile the code

S.Mann, 2017

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, 2017

Simulated lognormal returns Lognormal simulation: visual basic subroutine : logreturns

Simulated lognormally distributed asset prices Lognormal price simulation: visual basic subroutine : lognormalprice) S.Mann, 2017

Black-Scholes-Merton Model C = S N(d1 ) - KZ(0,T) N(d2 ) ln (S/K) + (r + s2/2 )T d1 = s T d2 = d1 - s 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(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, 2017

S.Mann, 2017

S.Mann, 2017

S.Mann, 2017

S.Mann, 2017

S.Mann, 2017

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, 2017

VBA Code for Black-Scholes-Merton functions (no dividends) 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 VBA code for implied volatility: Function scm_BS_call_ISD(S, X, t, r, C) high = 1 low = 0 Do While (high - low) > 0.0001 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 To enter code: tools/macro/visual basic editor at editor: insert/module type code, then compile by: debug/compile VBAproject S.Mann, 2017

Call Theta: Time decay S.Mann, 2017