Volatility and Complex Strategies — Volatility, Correlation, and Dispersion Products and Strategies

Volatility and Complex Strategies — Volatility, Correlation, and Dispersion Products and Strategies

CAIA Level II Study Guide

What Is It?

1. What is this topic?
   Volatility, correlation, and dispersion products (e.g., variance swaps, correlation swaps, dispersion trades) are structured derivatives that isolate and trade on market volatility, asset relationships, or cross-sectional risk.
2. How is it tested, applied, or used?
   Tested via pricing models, risk decomposition, and strategy backtests. Used by hedge funds, proprietary traders, and risk managers to hedge tail risk, exploit mispricing, or express macro views.

Why Does the Exam Ask This?

CAIA tests this to assess: - Pricing judgment: Can you decompose payoffs and reverse-engineer implied parameters (e.g., implied volatility, correlation)? - Risk awareness: Do you recognize how these products behave in stressed markets (e.g., correlation breakdowns, volatility skew)? - Strategy logic: Can you distinguish between directional, relative-value, and tail-risk strategies? - Compliance lens: Are you aware of liquidity risks, model risk, and regulatory constraints (e.g., Dodd-Frank, EMIR)?

What Do I Need to Know First?

1. Option Greeks (vega, gamma, volga) and their role in volatility trading.
2. Stochastic calculus basics (Itô’s Lemma, Girsanov’s Theorem) for pricing.
3. Correlation vs. covariance and how they drive multi-asset derivatives.
4. Variance swaps mechanics (log contracts, replication via options).
5. Volatility surfaces and skew dynamics.

Topic Snapshot

This topic bridges quantitative finance and alternative investments in CAIA Level II. It explains how volatility and correlation are traded as assets, not just measured as risk metrics. Mastery is critical for: - Evaluating hedge fund strategies (e.g., volatility arbitrage, dispersion trading). - Stress-testing portfolios under extreme market conditions. - Auditing model risk in exotics desks.

Exam / Job / Audit Weighting

• Frequency: 5–8% of Level II exam (1–2 questions per sitting).
• Difficulty Rating: Advanced (requires synthesis of options theory, statistics, and strategy logic).
• Question Type:
• Exam: Multi-step calculations (e.g., pricing a variance swap), scenario-based strategy evaluation, or model risk critiques.
• Job/Audit: Pricing validation, risk decomposition, or regulatory reporting (e.g., "Explain why this dispersion trade blew up in March 2020").

Difficulty Level

Advanced

Must-Know Rules, Formulas, Standards, or Principles

1. Variance Swap Pricing

• Fair strike = Replicated via a continuum of options (Breeden-Litzenberger): [ K_{var} = \frac{2}{T} \left( \int_0^{F_0} \frac{P(K)}{K^2} dK + \int_{F_0}^\infty \frac{C(K)}{K^2} dK \right) ]
• (P(K)) = Put price, (C(K)) = Call price, (F_0) = Forward price.
• Key insight: The payoff is the realized variance over the swap’s life, not volatility.

2. Correlation Swap Payoff

• Payoff = Notional × (Realized correlation – Strike correlation).
• Realized correlation = Average pairwise correlation of returns (e.g., for 3 assets: (\rho_{12}, \rho_{13}, \rho_{23})).
• Trap: Correlation is bounded [-1, 1], but realized correlation can "break" (e.g., crash to -1 in crises).

3. Dispersion Trade Mechanics

• Long dispersion = Buy single-stock options (high vol), sell index options (low vol).
• Short dispersion = Sell single-stock options, buy index options.
• Rationale: Exploits the volatility spread between idiosyncratic and systematic risk.

Misconceptions

1. "Variance swaps are just volatility bets."
2. Reality: They trade variance (volatility squared), not volatility. A 20% vol market has 400% variance!
3. "Correlation swaps are symmetric."
4. Reality: Correlation can crash to -1 but maxes at +1, creating asymmetric payoffs.
5. "Dispersion trades are market-neutral."
6. Reality: They’re volatility-neutral but exposed to correlation risk (e.g., all stocks crash together).
7. "Implied volatility = expected volatility."
8. Reality: Implied vol embeds risk premiums (e.g., demand for tail hedges).

Common Mistakes

1. Ignoring skew in variance swap replication.
2. Error: Using ATM options only → underprices tail risk.
3. Fix: Include OTM puts/calls to capture skew.
4. Confusing correlation with covariance.
5. Error: Using covariance in correlation swap payoffs → units mismatch.
6. Overlooking funding costs in dispersion trades.
7. Error: Assuming zero cost of carry for short options.
8. Misapplying the "variance swap = log contract" identity.
9. Error: Forgetting the ( \frac{2}{T} ) scaling factor.
10. Assuming correlation is stable.
11. Error: Backtesting with historical correlation → ignores regime shifts.

The Common Trap

Assuming "diversification works" in dispersion trades. - Trap: Short dispersion (sell single-stock vol, buy index vol) profits when stocks move independently but blows up when correlation spikes (e.g., COVID-19, 2008). - Why it’s tempting: Historical correlation is often low, lulling traders into complacency. - How to avoid: Stress-test for correlation breakdowns (e.g., -80% correlation scenarios).

Terms to Remember

1. Volatility skew: Implied vol’s dependence on strike (e.g., OTM puts have higher vol).
2. Variance swap: OTC derivative paying realized variance vs. a fixed strike.
3. Correlation swap: Derivative paying realized correlation vs. a fixed strike.
4. Dispersion trade: Strategy exploiting the vol spread between single stocks and the index.
5. Volga (volatility gamma): Second-order sensitivity of vega to volatility changes.

Step-by-Step Process

1. Pricing a Variance Swap

1. Gather inputs: Option chain (strikes, prices), forward price (F_0), risk-free rate (r), time to maturity (T).
2. Compute the integral:
3. For each strike (K), calculate ( \frac{P(K)}{K^2} ) (puts) and ( \frac{C(K)}{K^2} ) (calls).
4. Numerically integrate (e.g., trapezoidal rule) over all strikes.
5. Apply the formula:
   [
   K_{var} = \frac{2e^{rT}}{T} \left( \int_0^{F_0} \frac{P(K)}{K^2} dK + \int_{F_0}^\infty \frac{C(K)}{K^2} dK \right)
   ]
6. Adjust for skew: Ensure OTM options are included to capture tail risk.

2. Evaluating a Dispersion Trade

1. Decompose vol:
2. Single-stock vol = Idiosyncratic vol + Systematic vol.
3. Index vol = Systematic vol (since idiosyncratic risk diversifies away).
4. Calculate the spread:
5. Long dispersion: Buy single-stock vol (high), sell index vol (low).
6. Short dispersion: Sell single-stock vol, buy index vol.
7. Stress-test correlation:
8. Simulate correlation spikes (e.g., -80%) to assess tail risk.
9. Check funding costs:
10. Short options require collateral → adjust for cost of carry.

3. Auditing a Correlation Swap

1. Verify the payoff formula:
2. Ensure it uses correlation (not covariance) and is bounded [-1, 1].
3. Check the reference basket:
4. Are the underlying assets liquid? Are weights fixed?
5. Model risk:
6. Is the correlation model (e.g., Gaussian copula) appropriate for the assets?
7. Regulatory compliance:
8. Is the swap cleared (Dodd-Frank)? Are margin requirements met?

Exam Answer Builder

1-Mark Question (MCQ)

What it tests: Recognition of variance swap payoff mechanics. Example: A variance swap with a notional of $1M and a strike of 25% (variance) pays out if the realized variance over 1 year is 30%. What is the payout? A) $50,000 B) $200,000 C) $500,000 D) $1,000,000 Correct Answer: B) $200,000 Explanation: Payoff = Notional × (Realized variance – Strike variance) = $1M × (0.30 – 0.25) = $50,000. But variance is in %², so 30%² = 0.09, 25%² = 0.0625 → $1M × (0.09 – 0.0625) = $27,500. Wait—this is a trap! Key Tip: Variance swaps use variance (volatility squared), not volatility. The question likely expects you to square the percentages: 30%² = 0.09, 25%² = 0.0625 → $1M × (0.09 – 0.0625) = $27,500. But the options don’t match. Recheck the question: If the strike is 25% (variance), then 25% = 0.25, not 0.0625. The correct calculation is $1M × (0.30 – 0.25) = $50,000. The question is poorly worded—assume the strike is in variance units (not %²). Trap Option: A) $50,000 (forgets to square the percentages).

3-Mark Question (Calculation)

What it tests: Variance swap pricing via replication. Example: Given the following option prices for a 1-year variance swap on Stock X (forward price = $100, risk-free rate = 2%): - $90 put: $5 - $100 put: $8 - $110 call: $7 - $120 call: $4 Estimate the fair variance swap strike using the trapezoidal rule (assume no other strikes). Key Tip: 1. Use the formula:
[
K_{var} = \frac{2e^{rT}}{T} \left( \int_0^{F_0} \frac{P(K)}{K^2} dK + \int_{F_0}^\infty \frac{C(K)}{K^2} dK \right)
] 2. Approximate the integral with the given strikes:
- Puts: ( \frac{5}{90^2} + \frac{8}{100^2} )
- Calls: ( \frac{7}{110^2} + \frac{4}{120^2} ) 3. Multiply by ( \frac{2e^{0.02 \times 1}}{1} ).

5-Mark Question (Scenario)

What it tests: Dispersion trade risk assessment. Example: A hedge fund runs a short dispersion trade: sells 1M vega of single-stock options (avg. vol = 35%) and buys 1M vega of index options (vol = 20%). The fund’s risk system shows a P&L of +$500K over 3 months. A junior analyst flags that the trade lost money in March 2020. Explain the discrepancy and propose a fix. Key Tip: 1. Identify the issue: Correlation spike in March 2020 → single-stock vol and index vol converged → short dispersion loses. 2. Decompose the P&L:
- Normal periods: Single-stock vol > index vol → profit.
- Crisis: Correlation → -1 → single-stock vol ≈ index vol → loss. 3. Propose a fix:
- Add correlation triggers (e.g., unwind if correlation > 80%).
- Buy tail hedges (e.g., OTM index puts).

Case Study (10-Mark)

What it tests: Correlation swap valuation and model risk. Example: A bank sells a 1-year correlation swap on 3 stocks (A, B, C) with a strike of 50%. The realized correlation is calculated as the average of pairwise correlations. In 6 months, Stock A crashes 50%, while B and C rise 10%. The bank’s model assumes Gaussian copula correlation. An auditor questions the valuation. What are the key risks, and how would you validate the model? Key Tip: 1. Key risks:
- Tail dependence: Gaussian copula underestimates extreme correlation (e.g., A crashes, B/C don’t).
- Non-stationarity: Correlation is not constant (e.g., pre- vs. post-crash).
- Liquidity: Can the bank hedge the swap dynamically? 2. Validation steps:
- Backtest the copula with historical data (e.g., 2008, 2020).
- Stress-test for correlation breakdowns (e.g., -90%).
- Compare with alternative models (e.g., t-copula, vine copula).

This vs That

Time-Saver Hack

Variance swap strike ≈ ATM implied vol² + skew premium. - Shortcut: If ATM vol = 20% and skew is steep (OTM puts expensive), the variance swap strike will be higher than 4% (20%²). - Why it works: The integral in the pricing formula is dominated by OTM puts (tail risk).

Mini Scenarios

1. Basic

Scenario: A trader buys a 1-year variance swap on the S&P 500 with a strike of 25% (variance). At maturity, the realized volatility is 30%. What to notice: The payoff depends on variance, not volatility. 30% vol = 9% variance → Payoff = Notional × (0.09 – 0.25) = negative (trader loses).

2. Applied

Scenario: A hedge fund is long dispersion (buys single-stock vol, sells index vol). The index drops 10%, but single stocks move independently. What to notice: The trade profits because: - Single-stock vol spikes (idiosyncratic risk dominates). - Index vol rises less (diversification effect).

3. Tricky

Scenario: A bank sells a correlation swap on 2 stocks with a strike of 70%. Stock A rises 20%, Stock B drops 20%. The bank’s model shows a small loss, but the auditor flags a large discrepancy. What to notice: The model likely assumes linear correlation, but the realized correlation is nonlinear (e.g., -100% if one stock crashes). The bank underpriced tail risk.

Diagnostic MCQ Bank

Easy

Question: Which product pays the difference between realized variance and a fixed strike? A) Volatility swap B) Variance swap C) Correlation swap D) Dispersion trade Correct Answer: B) Variance swap Explanation: Variance swaps pay realized variance vs. a fixed strike. Volatility swaps pay realized volatility. Trap Option: A) Volatility swap (