Calibrating Trees and the Black-Scholes Formula

Calibration

• Estimate \(\sigma=\) std dev of annual stock return
• Find \(r_f =\) annualized continuously compounded interest rate \(= \log(1+\text{annual rate})\).
• \(T=\) time to maturity of an option in years
• \(N=\) number of periods in a binomial model

• \(\Delta t = T/N=\) period length
• Set the up rate of return as \(u = e^{\sigma\sqrt{\Delta t}}-1\) and set \(d=1/(1+u)-1\) as before
• Set the 1-period interest rate as \(r=e^{r_f \Delta t}-1\)
• The risk-neutral probability of “up” is

Dividends

• The simplest assumption is that the stock pays dividends continuously with a constant dividend yield.
• This means that the dividend paid in each small period \(\Delta t\) is \(q \times S \times \Delta t\) where \(S\) denotes the price at the beginning of the period and \(q\) is a constant.
• Set \(q =\) annual dividend / stock price

• We can still use a binomial model with the same \(r\), \(u\), and \(d\), except that the risk-neutral probability of “up” should be

Take limit

• As the number of periods is increased, the distribution of the stock price converges to lognormal, meaning that log stock price has a normal distribution.
• The values of options converge
• The limits of European option values are given by the Black-Scholes formulas

Black-Scholes Call Formula

import numpy as np
from scipy.stats import norm

def callBS(S, K, T, sigma, r, q=0):
    d1 = np.log(S/K) + (r-q+0.5*sigma**2)*T
    d1 /= sigma*np.sqrt(T)
    d2 = d1 - sigma*np.sqrt(T)
    N1 = norm.cdf(d1)
    N2 = norm.cdf(d2)
    return np.exp(-q*T)*S*N1 - np.exp(-r*T)*K*N2

Black-Scholes Put Formula

def putBS(S, K, T, sigma, r, q=0):
    d1 = np.log(S/K) + (r-q+0.5*sigma**2)*T
    d1 /= sigma*np.sqrt(T)
    d2 = d1 - sigma*np.sqrt(T)
    N1 = norm.cdf(-d1)
    N2 = norm.cdf(-d2)
    return np.exp(-r*T)*K*N2 - np.exp(-q*T)*S*N1