Question

Consider a futures contract on a non-dividend paying stock with futures price $20 and time to expiration 4 months. Assume that in four months the stock price will be either $22 or $19. Calculate the risk-neutral probabilities for the future stock prices.

Answer 1

Futures Price = $20 | Upstate Price = $22 | Downstate Price = $19

In Risk-neutral environment, the Futures price for a 4 months contract should equal the expected price of stock in 4 months.

Let probability of Upstate be p and probability of downstate be (1 - p)

Expected Stock Price in 4 months = p * Upstate Price + (1 - p) * Downstate price

Risk-Neutral: Price of Futures contract of 4 months maturity = Expected Stock Price in 4 months

=> 20 = p * 22 + (1 - p) * 19

=> 20 = 22p + 19 - 19p

=> 3p = 1

p = 1 / 3 or 0.33

(1 - p) = 1 - 1/3 = 2 / 3 = 0.67

Hence, Probability of stock price reaching price of 22 in 4 months is 33.33% or 1/3 and probabilty of stock price reaching price of 19 in 4 months is 66.67% or 2 / 3