Pricing of European put option with binomial model

Question

This is an exercise from Mark Joshi's book (exercise 3.6):

A stock is worth 100. Each month its value increases or decreases by precisely 10. The riskless bond is worth $e^{rt}$ at time t years with r equal to 5% Price a four-month European put option struck at 110.

At the end of the book, Joshi provides the solution 13.06. Unfortunately that's not what I find: I get 15.22. Since Joshi does not show his computation, I am wondering where the difference comes from. I use the following Python script for the computation:

import math

def get_risk_neutral_prob(S, S1, S2, r, delta_t):
    Sp = max(S1, S2)
    Sm = min(S1,S2)
    if Sm == Sp:
        return 1/2
    return (math.exp(-r*delta_t) * S - Sm)/(Sp-Sm)

def payoff(S):
    return max(110-S, 0)
r = 0.05
delta_t = 1/12


def get_price(S, N):
    if N == 0:
        return payoff(S)
    S1 = S+10
    S2 = S-10
    p = get_risk_neutral_prob(S, S1, S2, r, delta_t)
    return math.exp(-r*delta_t) * (p * get_price(S1, N-1) + (1-p) * get_price(S2,N-1))



print(get_price(100,4))

As you can see, my computation is straightforward. I first compute the risk-neutral probability, and then the discounted expected value of the payoff, recursively.

For one month, I did it by hand and my result, 10.372, agrees with what the script tells me.

Answer 1

Answer was provided by Chris Taylor: the formula for the risk-neutral probability was off by a minus sign, it should be $$ p = \frac{e^{r \Delta t} S - S_m}{S_p - S_m} $$

Answer 2

I cannot comment (low reputation) so I add an answer. In Python, you should write:

delta_t = 1.0/12