SECTION 1: LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

  • Identify key probability distributions used in finance (Normal, Log‑normal, Student’s t, Binomial, Poisson).

  • Explain why the Normal distribution is often inadequate for financial returns and why the Student’s t is preferred.

  • Understand the Log‑normal distribution and its role in asset price modelling.

  • Explain the fundamentals of Monte Carlo simulation and its applications.

  • Design a simple Monte Carlo simulation to estimate Value‑at‑Risk (VaR) for a portfolio.

  • Interpret simulation outputs to support business decisions.

  • Recognise the limitations of simulation (model risk, computational cost).

SECTION 2: KEY PROBABILITY DISTRIBUTIONS IN FINANCE

2.1 The Normal (Gaussian) Distribution

  • Symmetric, bell‑shaped, fully described by mean (μ) and standard deviation (σ).

  • Used for: modelling asset returns under the classical assumptions of the Black‑Scholes model.

  • The problem: Financial returns often have fat tails (extreme events occur more frequently than the Normal predicts) and negative skewness (more large downward moves than upward moves). The Normal distribution underestimates tail risk – a major flaw exposed by the 2008 financial crisis.

2.2 The Student’s t‑Distribution

  • Similar to the Normal but with heavier tails.

  • Controlled by a parameter ν (degrees of freedom). As ν → ∞, it approaches the Normal; smaller ν means fatter tails.

  • Financial use: Preferred for modelling return distributions in risk management. It better captures the extreme losses seen in markets.

2.3 The Log‑normal Distribution

  • A variable is log‑normally distributed if its natural logarithm is normally distributed.

  • It is strictly positive and skewed to the right – perfect for modelling asset prices, which cannot fall below zero.

  • Crucial formula: If a stock’s continuous return is normally distributed, then the stock price at a future time is log‑normally distributed. This underpins the Black‑Scholes option pricing model.

2.4 The Binomial Distribution

  • Models the number of successes in a fixed number of independent trials, each with the same probability.

  • Financial use: Pricing options via the binomial tree (Cox‑Ross‑Rubinstein model). Also used for modelling the probability of a certain number of loan defaults in a portfolio.

2.5 The Poisson Distribution

  • Models the number of events occurring in a fixed interval of time or space, given a constant average rate.

  • Financial use: Modelling the frequency of operational risk events (e.g., number of cyber‑attacks, fraudulent transactions, system outages) – required under Basel III for Operational Risk capital calculation.

SECTION 3: WHY ASSET PRICES ARE LOG‑NORMAL

Consider a stock with daily returns R that are normally distributed: R∼N(μ,σ2).

The price at time t is:
Pt=P0⋅eR1+R2+⋯+Rt

Since the sum of normally distributed returns is normal, the exponent is normal. Therefore, Pt is log‑normally distributed.

Key properties of the log‑normal distribution:

  • Always positive.

  • Right‑skewed (mean > median).

  • The variance increases as the price increases (heteroscedasticity).

Business insight: When forecasting future stock prices or option payoffs, using a log‑normal distribution is theoretically consistent with the continuous compounding of returns.

SECTION 4: MONTE CARLO SIMULATION – METHODOLOGY

Monte Carlo simulation uses random sampling to generate thousands (or millions) of possible future scenarios for a financial variable. It is incredibly flexible and can handle complex, non‑linear relationships.

General steps:

  1. Define the problem (e.g., “What is the 95% 1‑day VaR of this portfolio?”).

  2. Identify the stochastic process for the risk factors (e.g., assume asset returns follow a normal or t‑distribution with given mean and volatility).

  3. Generate random draws from the chosen distribution(s) for each risk factor.

  4. Calculate portfolio value for each simulated scenario.

  5. Repeat many times (e.g., 10,000 iterations).

  6. Analyse the distribution of portfolio values to compute risk metrics (VaR, expected shortfall).

Advantages over analytical methods:

  • Can handle options, non‑linear instruments, and complex correlations.

  • Can incorporate fat tails and other non‑normal features.

  • Easily adaptable to new assumptions or stress scenarios.

Disadvantages:

  • Computationally intensive.

  • Sensitive to the assumptions of the underlying model (model risk).

  • Requires a good random number generator.

SECTION 5: APPLYING MONTE CARLO TO VALUE‑AT‑RISK (VaR)

VaR is the maximum loss that a portfolio is expected to suffer over a given horizon at a given confidence level (e.g., 95% or 99%).

The Monte Carlo approach to VaR:

  1. Collect historical returns of the assets in the portfolio.

  2. Estimate parameters: mean return, standard deviation, and correlations (or the full covariance matrix).

  3. Choose a distribution: Normal is common, but Student’s t is better for fat tails.

  4. Generate 10,000 simulated daily returns for each asset simultaneously, respecting correlations (using Cholesky decomposition to transform uncorrelated draws into correlated draws).

  5. Calculate the simulated portfolio returns (weights × returns).

  6. Sort the simulated returns from worst to best. The 5th percentile (for 95% VaR) is the loss that we are 95% confident will not be exceeded in one day.

Business application: A bank’s trading desk uses Monte Carlo VaR to set daily risk limits and allocate capital. Regulators (Basel) accept Monte Carlo VaR for internal model approval, provided the methodology is thoroughly validated.

SECTION 6: LIMITATIONS AND BEST PRACTICES

Common pitfalls:

  • Garbage in, garbage out: If you assume returns are normal when they have fat tails, your VaR will be too low.

  • Correlation instability: Correlations between assets change during market stress (they tend to increase – “correlation breakdown”). Monte Carlo models that use historical correlations will understate risk during crises.

  • Regime changes: Parameters from the past may not apply to the future.

Best practices:

  • Use a Student’s t‑distribution for returns to capture tail risk.

  • Incorporate stochastic volatility (volatility that itself changes over time) for more realism.

  • Always back‑test your VaR forecasts – check how often actual losses exceeded the VaR estimate; this should align with your confidence level.

  • Complement Monte Carlo with stress testing – what if the market drops 20% in a week? Add those scenarios manually.

SECTION 7: SUMMARY FOR THE DATA PRACTITIONER

  • Normal distribution is a baseline but underestimates risk; Student’s t is a better choice for returns.

  • Log‑normal is the natural choice for modelling asset prices and is fundamental to option pricing.

  • Binomial and Poisson distributions model discrete events (defaults, operational losses).

  • Monte Carlo simulation is a powerful, flexible tool for valuing complex instruments and measuring risk.

  • The steps are: define the process → generate random scenarios → compute outcomes → summarise the distribution.

  • In banking, MC is widely used for VaR, credit portfolio modelling, and asset‑liability management.

  • Always validate assumptions, back‑test, and supplement with stress scenarios to avoid model risk.

SECTION 8: RECOMMENDED NEXT STEPS

  • Apply Monte Carlo to a simple portfolio of two assets to compute VaR and compare it to the analytical (normal) VaR.

  • Experiment with different distributions (Normal vs. t) to see the impact on tail estimates.

  • Explore how correlation changes affect portfolio risk.

  • Learn about Cholesky decomposition to simulate correlated variables correctly.

  • Prepare for Lesson 8, which covers Bayesian Statistics and its applications in finance (or whatever the final lesson will be, as per your curriculum needs).

[END OF LESSON 7]

 
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