Probability Puzzles for Finance: 10 Questions with Step-by-Step Solutions

Understanding the Role of Probability Theory in Modern Finance

Probability theory forms the backbone of modern finance, influencing a wide array of financial activities and decision-making processes. This mathematical framework is essential for quantifying uncertainty and making informed predictions about future events, which is crucial in the volatile environment of financial markets. From risk management to derivatives pricing, probability theory provides the tools necessary to navigate complex financial landscapes.

Brain Teasers Questions Covered Type

• Random Walk Stock Pricing - Understanding how stock prices move probabilistically

• Portfolio Default Risk - Using binomial distributions for credit risk

• Options Probability - Normal distributions in derivatives pricing

• Value at Risk (VaR) - Risk management using probability distributions

• Credit Scoring with Bayes' Theorem - Updating probabilities with new information

• Perpetual Bonds - Geometric series in bond valuation

• Portfolio Correlation - How correlation affects risk

• Monte Carlo Simulation - Setting up probabilistic simulations

• Black-Scholes Probability - Implied volatility calculations

• Kelly Criterion - Optimal position sizing using probability theory

Question 1: The Stock Price Random Walk

Problem: A stock price follows a simplified random walk. Each day, it either goes up by $1 with probability 0.6 or down by $1 with probability 0.4. Starting at $100, what is the probability that the stock price will be exactly $102 after 4 days?

Step-by-Step Solution:

Step 1: Define the problem parameters

• Starting price: $100

• Target price: $102

• Time period: 4 days

• Up probability: p = 0.6

• Down probability: q = 0.4

• Net gain needed: +$2

Step 2: Determine the required moves

• Let U = number of up moves, D = number of down moves

• We need: U + D = 4 (total days)

• Net change: U - D = +2 (to reach $102 from $100)

Step 3: Solve the system of equations

• U + D = 4

• U - D = 2

• Adding equations: 2U = 6, so U = 3

• Substituting: D = 4 - 3 = 1

Step 4: Calculate the probability

• Number of ways to arrange 3 ups and 1 down in 4 days = C(4,3) = 4

• Probability = C(4,3) × (0.6)³ × (0.4)¹

• Probability = 4 × 0.216 × 0.4 = 0.3456 or 34.56%

Question 2: Portfolio Default Risk

Problem: A portfolio contains 3 bonds, each with a 20% probability of default within one year. Assuming defaults are independent, what is the probability that exactly 2 bonds will default?

Step-by-Step Solution:

Step 1: Identify the distribution

• This follows a binomial distribution: B(n=3, p=0.2)

• We want P(X = 2) where X is the number of defaults

Step 2: Apply the binomial formula

• P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

• P(X = 2) = C(3,2) × (0.2)² × (0.8)¹

Step 3: Calculate each component

• C(3,2) = 3!/(2!×1!) = 3

• (0.2)² = 0.04

• (0.8)¹ = 0.8

Step 4: Compute final probability

• P(X = 2) = 3 × 0.04 × 0.8 = 0.096 or 9.6%

Question 3: Options Expiration Probability

Problem: You own a call option on a stock with a strike price of $50. The stock currently trades at $48. Historical data shows the stock price changes follow a normal distribution with daily volatility of 2%. What is the probability that the option expires in-the-money after 10 trading days?

Step-by-Step Solution:

Step 1: Set up the normal distribution parameters

• Current stock price: S₀ = $48

• Strike price: K = $50

• Daily volatility: σ_daily = 2% = 0.02

• Time period: t = 10 days

• Volatility over 10 days: σ_t = σ_daily × √t = 0.02 × √10 = 0.0632

Step 2: Model the stock price distribution

• Assuming geometric Brownian motion (simplified)

• Stock price after 10 days ~ N(S₀, S₀ × σ_t)

• S₁₀ ~ N(48, 48 × 0.0632) = N(48, 3.034)

Step 3: Calculate the probability

• We need P(S₁₀ > 50)

• Standardize: Z = (50 - 48)/3.034 = 0.659

Step 4: Use standard normal table

• P(Z > 0.659) = 1 - Φ(0.659) = 1 - 0.7549 = 0.2451 or 24.51%

Question 4: Risk Management - VaR Calculation

Problem: A trading desk has a portfolio with daily returns that follow a normal distribution with mean 0.1% and standard deviation 1.5%. What is the 1-day Value at Risk (VaR) at the 95% confidence level for a $10 million portfolio?

Step-by-Step Solution:

Step 1: Understand VaR concept

• VaR represents the maximum expected loss over a given time period at a specific confidence level

• 95% confidence means we want the loss that occurs 5% of the time or worse

Step 2: Identify distribution parameters

• Mean daily return: μ = 0.1% = 0.001

• Standard deviation: σ = 1.5% = 0.015

• Portfolio value: $10,000,000

Step 3: Find the 5th percentile of returns

• For 95% confidence, we need the 5th percentile (left tail)

• Z₀.₀₅ = -1.645 (from standard normal table)

• Return at 5th percentile = μ + Z₀.₀₅ × σ = 0.001 + (-1.645) × 0.015 = -0.01368

Step 4: Calculate VaR

• Expected return = 0.001 × $10,000,000 = $10,000

• 5th percentile return = -0.01368 × $10,000,000 = -$136,800

• VaR = Expected return - 5th percentile return = $10,000 - (-$136,800) = $146,800

Question 5: Credit Scoring Bayes' Theorem

Problem: A bank's credit model shows that 5% of all loan applicants default. A credit score test correctly identifies 90% of defaulters (sensitivity) and correctly identifies 95% of non-defaulters (specificity). If an applicant receives a "high risk" score, what is the probability they will actually default?

Step-by-Step Solution:

Step 1: Define the events and probabilities

• P(Default) = 0.05, so P(No Default) = 0.95

• P(High Risk | Default) = 0.90 (sensitivity)

• P(Low Risk | No Default) = 0.95 (specificity)

• So P(High Risk | No Default) = 0.05

Step 2: Apply Bayes' Theorem

• We want P(Default | High Risk)

• P(Default | High Risk) = P(High Risk | Default) × P(Default) / P(High Risk)

Step 3: Calculate P(High Risk) using law of total probability

• P(High Risk) = P(High Risk | Default) × P(Default) + P(High Risk | No Default) × P(No Default)

• P(High Risk) = 0.90 × 0.05 + 0.05 × 0.95 = 0.045 + 0.0475 = 0.0925

Step 4: Calculate final probability

• P(Default | High Risk) = (0.90 × 0.05) / 0.0925 = 0.045 / 0.0925 = 0.4865 or 48.65%

Question 6: Geometric Series in Perpetual Bonds

Problem: A perpetual bond pays $50 annually forever. If there's a 2% probability each year that the bond defaults (and payments stop), what is the expected total payout from this bond?

Step-by-Step Solution:

Step 1: Set up the survival probabilities

• Annual survival probability: p = 1 - 0.02 = 0.98

• Probability of surviving n years: p^n = (0.98)^n

Step 2: Express expected payout as a series

• Year 1 payout: $50 × 0.98 (survive first year)

• Year 2 payout: $50 × (0.98)² (survive first two years)

• Year n payout: $50 × (0.98)^n

• Expected total = $50 × [0.98 + (0.98)² + (0.98)³ + ...]

Step 3: Recognize this as a geometric series

• Expected total = $50 × 0.98 × [1 + 0.98 + (0.98)² + ...]

• The series in brackets is: 1/(1-0.98) = 1/0.02 = 50

Step 4: Calculate final answer

• Expected total payout = $50 × 0.98 × 50 = $2,450

Question 7: Correlation and Portfolio Risk

Problem: You have two assets in equal weights. Asset A has a standard deviation of 20%, Asset B has a standard deviation of 30%, and their correlation is 0.4. What is the standard deviation of the portfolio?

Step-by-Step Solution:

Step 1: Set up the portfolio variance formula

• For two assets with equal weights (w₁ = w₂ = 0.5):

• σ²_portfolio = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂

Step 2: Substitute the given values

• σ₁ = 0.20, σ₂ = 0.30, ρ₁₂ = 0.4, w₁ = w₂ = 0.5

• σ²_portfolio = (0.5)²(0.20)² + (0.5)²(0.30)² + 2(0.5)(0.5)(0.20)(0.30)(0.4)

Step 3: Calculate each term

• First term: 0.25 × 0.04 = 0.01

• Second term: 0.25 × 0.09 = 0.0225

• Third term: 2 × 0.25 × 0.06 × 0.4 = 0.012

Step 4: Find portfolio standard deviation

• σ²_portfolio = 0.01 + 0.0225 + 0.012 = 0.0445

• σ_portfolio = √0.0445 = 0.2109 or 21.09%

Question 8: Monte Carlo Simulation Setup

Problem: You want to simulate the price of a stock after 252 trading days (1 year) using Monte Carlo methods. The stock starts at $100, has an expected annual return of 8%, and annual volatility of 25%. Set up the simulation formula and calculate one sample path.

Step-by-Step Solution:

Step 1: Convert annual parameters to daily

• Annual return: μ = 8% = 0.08

• Daily return: μ_daily = 0.08/252 = 0.000317

• Annual volatility: σ = 25% = 0.25

• Daily volatility: σ_daily = 0.25/√252 = 0.01575

Step 2: Set up the geometric Brownian motion formula

• S(t+1) = S(t) × exp[(μ_daily - σ²_daily/2) + σ_daily × Z]

• Where Z ~ N(0,1) is a standard normal random variable

Step 3: Simplify the drift term

• Drift = μ_daily - σ²_daily/2 = 0.000317 - (0.01575)²/2 = 0.000317 - 0.000124 = 0.000193

Step 4: Generate one sample path (using Z = 0.5 as example)

• Day 1: S(1) = 100 × exp[0.000193 + 0.01575 × 0.5] = 100 × exp[0.008068] = 100.81

• Continue this process for 252 days to get the final stock price

Question 9: Black-Scholes Probability

Problem: Using the Black-Scholes framework, a stock trading at $100 has a 60% probability of finishing above $105 in 30 days, given a risk-free rate of 3% annually and no dividends. What is the implied volatility?

Step-by-Step Solution:

Step 1: Set up the Black-Scholes probability formula

• P(S_T > K) = N(d₂) where N is the cumulative standard normal distribution

• d₂ = [ln(S₀/K) + (r - σ²/2)T] / (σ√T)

Step 2: Input known values

• S₀ = $100, K = $105, T = 30/365 = 0.0822 years, r = 0.03

• P(S_T > $105) = 0.60

• Therefore: N(d₂) = 0.60, which means d₂ = N⁻¹(0.60) = 0.2533

Step 3: Set up equation for volatility

• 0.2533 = [ln(100/105) + (0.03 - σ²/2) × 0.0822] / (σ√0.0822)

• 0.2533 = [-0.04879 + (0.03 - σ²/2) × 0.0822] / (σ × 0.2867)

Step 4: Solve for σ (requires iterative methods or approximation)

• Rearranging: 0.2533 × σ × 0.2867 = -0.04879 + 0.002466 - σ² × 0.0411

• This leads to a quadratic equation in σ

• Solving numerically: σ ≈ 0.25 or 25% (implied volatility)

Question 10: Kelly Criterion for Optimal Betting

Problem: A trader has identified a trading strategy with a 55% win rate. When it wins, the average gain is 1.2 times the bet. When it loses, the entire bet is lost. What fraction of capital should be allocated to maximize long-term growth using the Kelly Criterion?

Step-by-Step Solution:

Step 1: Define Kelly Criterion parameters

• Kelly fraction = (bp - q) / b

• Where: b = odds received on winning bet, p = probability of winning, q = probability of losing

Step 2: Identify the parameters from the problem

• p = 0.55 (55% win rate)

• q = 1 - p = 0.45 (45% loss rate)

• When winning: gain 1.2 times the bet, so b = 1.2

• When losing: lose entire bet

Step 3: Apply the Kelly formula

• Kelly fraction = (bp - q) / b

• Kelly fraction = (1.2 × 0.55 - 0.45) / 1.2

• Kelly fraction = (0.66 - 0.45) / 1.2 = 0.21 / 1.2 = 0.175

Step 4: Interpret the result

• The optimal allocation is 17.5% of capital per trade

• This maximizes long-term logarithmic growth

• Note: This assumes the strategy parameters remain constant and ignore transaction costs

Key Takeaways

These probability puzzles illustrate several essential concepts in finance:

1. Random walks model the movement of stock prices and aid in options pricing

2. Binomial distributions are vital for modeling credit risk and defaults

3. Normal distributions form the basis of many risk management models, including VaR

4. Bayes' theorem is important for updating probabilities with new data

5. Geometric series are useful in valuing perpetual securities and annuities

6. Correlation greatly affects portfolio risk and the benefits of diversification

7. Monte Carlo simulation offers flexible tools for complex financial modeling

8. Black-Scholes framework links probabilities to options pricing

9. Kelly Criterion optimizes position sizing for repeated investments

Understanding these probabilistic foundations enables better decision-making in trading, risk management, and portfolio optimization. Practice with these types of problems builds the intuition necessary for advanced financial modeling and analysis.