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Unit 2 of 4 · MBA Sem 4

Unit 2: Volatility and risk measurement

Financial Analytics notes · PTU syllabus (MBA 965-26)

3 min read5 topics10 exam questions
On this page
  1. Unit summary
  2. ARCH and GARCH models and volatility clustering
  3. Statistical distributions in finance
  4. Value at Risk
  5. Hypothesis testing in finance
  6. Monte Carlo simulation
  7. Key terms
  8. Quick revision
  9. Important questions

Unit summary

Risk is the heart of finance, and volatility is its most common measure. This unit covers ARCH and GARCH volatility models, volatility clustering, statistical distributions and Value at Risk estimation, hypothesis testing in finance, and Monte Carlo simulation for risk analysis and option pricing.

After this unit you can

  • Model volatility with ARCH and GARCH
  • Estimate Value at Risk
  • Apply hypothesis tests to financial questions
  • Use Monte Carlo simulation for risk and option pricing

PTU syllabus topics

  • ARCH/GARCH volatility models
  • volatility clustering
  • statistical distributions and Value at Risk estimation
  • hypothesis testing in finance
  • Monte Carlo simulation for risk analysis and option pricing
Key formulasRisk measures

  • Parametric VaR

    Z × σ × portfolio value

  • Historical VaR

    Loss at the chosen percentile of past returns

  • GARCH(1,1)

    σ²ₜ = ω + α ε²ₜ₋₁ + β σ²ₜ₋₁

  • Monte Carlo

    Simulate many paths, read off the loss distribution

1

Topic 1

ARCH and GARCH models and volatility clustering

  • Volatility clustering: large changes tend to follow large changes (financial markets).
  • ARCH (Engle, 1982): variance today depends on past squared errors.
  • GARCH(1,1) (Bollerslev, 1986): σt² = ω + α e²t−1 + β σ²t−1 — variance depends on past shocks and past variance; α + β close to 1 means persistent volatility.
  • Uses: forecasting volatility for risk management, option pricing, Value at Risk.

Example

After a market crash, a GARCH model forecasts higher volatility for the following days, which then decays gradually towards its long-run average.

2

Topic 2

Statistical distributions in finance

  • Normal distribution is a convenient approximation but underestimates extreme losses; Student's t and other fat-tailed distributions fit returns better; lognormal for prices.
3

Topic 3

Value at Risk

VaR: the maximum loss over a given horizon at a given confidence level under normal market conditions.

Key formulasVaR methods

  • Parametric (variance–covariance)

    VaR = z × σ × portfolio value × √(days)

  • Historical simulation

    Loss at the chosen percentile of past returns

  • Monte Carlo

    Loss at the percentile of simulated returns

  • Expected shortfall (CVaR)

    Average loss beyond the VaR

Example

Portfolio ₹10 crore, daily σ = 1.5%, 99% confidence (z = 2.33): one-day VaR = 2.33 × 1.5% × 10 crore ≈ ₹35 lakh; ten-day VaR ≈ 35 × √10 ≈ ₹1.1 crore.

  • Limitations: says nothing about losses beyond the threshold; depends on assumptions; expected shortfall is now preferred by Basel (FRTB).
  • Backtesting: compare actual losses with VaR — too many exceptions mean the model underestimates risk.
4

Topic 4

Hypothesis testing in finance

  • Questions: Is the mean return different from zero? Does a fund beat its benchmark (alpha)? Are returns normally distributed (Jarque–Bera)? Did an event affect prices (event study abnormal returns)? Do two strategies differ?
  • Tests: t-tests, F-tests, chi-square, Jarque–Bera, ADF; care with multiple testing and data snooping.
5

Topic 5

Monte Carlo simulation

ProcessMonte Carlo simulation
  1. 1Define the model

    Price process — geometric Brownian motion

  2. 2Specify inputs and distributions

    Drift, volatility, correlations

  3. 3Generate many random paths
  4. 4Compute the outcome for each path

    Portfolio value, option payoff

  5. 5Analyse the distribution

    Mean, percentiles, probability of loss

Key formulasMonte Carlo option pricing

  • Price path

    S(t+Δt) = S(t) × exp[(r − σ² ÷ 2)Δt + σ √Δt × Z]

  • Option value

    Average discounted payoff = e^(−rT) × mean[max(S_T − K, 0)]

  • Uses: VaR for complex portfolios, path-dependent options, retirement planning, project risk; accuracy improves with more simulations.

Key terms

Volatility clustering
Periods of high volatility following each other
Value at Risk
Loss threshold at a confidence level over a horizon
Expected shortfall
Average loss beyond VaR
Backtesting
Checking model predictions against actual outcomes
Monte Carlo simulation
Random sampling to model uncertain outcomes

Quick revision

  • ARCH, GARCH(1,1); volatility clustering.
  • Normal vs fat-tailed distributions.
  • Parametric, historical and Monte Carlo VaR; expected shortfall; backtesting.
  • Hypothesis tests in finance; event studies.
  • Monte Carlo steps; GBM price paths; option pricing.

Important exam questions

Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).

Short-answer questions

  1. Q1.What is volatility clustering?
  2. Q2.Define Value at Risk.
  3. Q3.Compute one-day 95% VaR for ₹1 crore with daily σ of 2% (z = 1.65).
  4. Q4.What is expected shortfall?
  5. Q5.What is the Jarque–Bera test used for?
  6. Q6.State two uses of Monte Carlo simulation in finance.

Long-answer questions

  1. Q1.Explain ARCH and GARCH models of volatility.
  2. Q2.Explain methods of estimating Value at Risk with an example.
  3. Q3.Discuss hypothesis testing in finance.
  4. Q4.Explain Monte Carlo simulation for risk analysis and option pricing.

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