Katkat
Back

Find valid range of q for outcome probabilities

ImcData Scientist · Technical Screen
EasyData ScientistTechnical ScreenSoftware Engineering Fundamentals

Last updated

This question tests the ability to solve a system of linear inequalities derived from probability constraints, requiring an understanding of basic arbitrage-free pricing principles.

Problem Statement

You are given two outcomes, A and B, which are mutually exclusive and collectively exhaustive. A market maker provides the following prices for the outcomes:

  • Price of A: P_A = 0.6
  • Price of B: P_B = 0.5

You are considering a new outcome, C, which is also part of this sample space. The market maker offers to sell you the outcome C at a price q. For the set of outcomes {A, B, C} to represent a coherent probability space where no arbitrage is possible, the sum of the probabilities must be 1. However, since the current prices P_A and P_B already sum to 1.1, the market is currently inconsistent.

Assume that for any outcome i, its probability prob_i must satisfy 0 <= prob_i <= 1 and the sum of all probabilities must equal 1. Given the prices provided by the market maker for outcomes A and B are fixed, calculate the valid range for q (the price of C) such that there exists a probability distribution where prob_A <= P_A, prob_B <= P_B, and prob_C <= q.

Function Signature

def get_valid_price_range(p_a, p_b):
    # Returns a tuple (min_q, max_q)
    pass

Constraints

  • 0 = 1 (otherwise the outcomes are underpriced).

Example

  • Input: p_a = 0.6, p_b = 0.5
  • Output: (0.0, 0.4)
  • Explanation: Since prob_A + prob_B + prob_C = 1 and prob_A <= 0.6, prob_B <= 0.5, then prob_C = 1 - prob_A - prob_B. To satisfy prob_C <= q, we need q to be at least the minimum possible value of 1 - prob_A - prob_B given the constraints on A and B.

Hints

Hint 1

Recall that for any probability distribution, the sum of probabilities must equal 1.

Hint 2

Express the probability of outcome C as `1 - prob_A - prob_B` and substitute the upper bounds of `prob_A` and `prob_B`.

Hint 3

Consider the boundary conditions where the probabilities of A and B are at their minimum (0) or maximum (their respective prices).

Follow-up questions

  • How does the range of q change if we introduce a fourth outcome D with price p_d?
  • If the market maker adjusts the prices such that `p_a + p_b < 1`, how does this impact the existence of a valid probability distribution?
  • How would you represent this problem as a linear programming task if you had multiple constraints on the outcomes?