Risk-Neutral Probabilities

Single Period Model Again

• Suppose a stock priced at $100 will go up by 10% or down by 10%.

• Consider a call with a strike of $105. Using the delta argument, we obtained

• If there were no risk premium, the call value would be the expected value discounted at the risk-free rate:

• Solve the stock equation for \(p\) and substitute into the call option equation.
  
• Stock equation is solved as:

• \(p\) and \(1-p\) are called the risk-neutral probabilities.
• Why does this work?
  • Delta hedge argument didn’t depend on risk preferences, so we can act as if investors don’t require risk premia.
  • Simple algebra!

Two-period example again

• Suppose a $100 stock goes up by 5% or down by (1/1.05-1) = -4.76% in each of two periods.

• Suppose a call option with a strike of $105 expires at the end of the 2nd period.
• From the delta argument, we obtained

Risk-neutral probability

• The risk-neutral probability of “up” is

• Discounting the expected call value (using prob of up = 0.795) at the risk-free rate yields

Call with a strike of $95

Example code

import numpy as np
S = 100             # stock price
K = 95              # strike
u = 0.05            # up return per period
r = 0.03            # interest rate per period
n = 2               # number of periods
d = 1/(1+u) - 1     # down return per period
p = (r-d) / (u-d)   # risk-neutral prob
x = [S*(1+u)**(n-2*i) for i in range(n+1)]
v = np.maximum(0, np.array(x)-K)
while len(v)>1:
    v = (p*v[:-1]+(1-p)*v[1:]) / (1+r)
v[0]

10.623403551775402