experimenting random price movement

Question

Random price movement

Consider the following:

The price of an apple starts at 1 dollar. On each day, the price will change -10% or +10%, with equal probability. You buy this apple on day 1, and sell on day N. What is your probability of losing money at the end?

Suppose N = 2, then we have 4 possible outcomes with equal probability:

• 0.9 * 0.9 = 0.81
• 0.9 * 1.1 = 0.99
• 1.1 * 0.9 = 0.99
• 1.1 * 1.1 = 1.21

Thus the probability of losing money is 3/4.

Let's run an experiment with different delta values (like 10%) and steps (days).

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import matplotlib.pyplot as plt
def main():
    for delta in [0.01, 0.1, 0.2, 0.4, 0.5, 0.8]:
        steps = list(range(1, 20))
        p_lose = [find_p_lose(n, delta) for n in steps]
        plt.figure(figsize=(5, 3))
        plt.title(f"delta = {delta}")
        plt.xlabel("number of steps (days)")
        plt.ylabel("probability of losing")
        plt.xticks(steps)
        plt.tight_layout()
        plt.plot(steps, p_lose)
        plt.show()
def find_p_lose(steps, delta):
    outcomes = find_outcomes(steps, delta)
    p_lose = len([x for x in outcomes if x < 1]) / len(outcomes)
    return p_lose
def find_outcomes(steps, delta):
    if steps == 1:
        return [1 - delta, 1 + delta]
    outcomes = []
    for a in [1 - delta, 1 + delta]:
        for b in find_outcomes(steps - 1, delta):
            outcomes.append(a * b)
    return outcomes
print("First let's verify what we have at the beginning (2 steps, delta = 0.1)")
print("Outcomes", find_outcomes(2, 0.1))
print("p(lose)", find_p_lose(2, 0.1))
main()

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Observations

Case 1: delta in [0.01, 0.1]

p(lose) oscillates

• It is always 0.5 for odd number of steps.
• You are always more likely to lose money for even number of steps, with the probability decreasing as number of steps increases, and it follows a nice curve.

Case 2: delta in [0.2, 0.4, 0.5, 0.8]

p(lose) increases while oscillating

Questions

1. We can see that there seems to be a bifurcation occurring, what is the value of delta where this bifurcation occurs?
2. What are the analytical expressions for the curves that the trends are following in both cases?