Examining the Volatility Skew in the South African Equity Market

1
Examining the Volatility Skew in the South
African Equity Market

• Mark de Araújo - Eben Maré
• Kruger Park - August 2005
• University of Pretoria
• Peregrine Securities

2
Outline

• What does it look like?
• Why does it exist?
• How is it estimated?
• How fair is it really?

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Introduction

• Fairly liquid index derivative market
• Developing single stock market
• Establish fair skew for illiquid counters?
• Provide transparency in derivatives decision
  making process.

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Introduction

• SAFEX options trade on implied volatility
• Option price inferred using the Black formula
  (Black 1976)
• Quoted implied volatilities depend on
• Specific views of future volatility
• Supply and demand considerations.

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The volatility skew

• Relates implied volatility to strike.

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The volatility surface

• Relates implied volatility to strike and term.

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Why does the skew exist?

• Supply and Demand
• Implied volatility independent of strike level
• Demand for OTM options for hedging
• Log returns not normally distributed
• Deep OTM put has greater chance of being
  exercised than Black-Scholes assumes
• Leverage Effects

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Why does the skew exist?

• Common justifications for skews existence
• Impact on skew difficult to measure
• (Chen, Palman, Wald 2003)
• Incorporate factors influencing option price via
  skew.

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How fair is the skew?

• Common misconception
• Skew is NOT a means for market makers to extract
  more profitability from option trades
• IV bid-offer spread reflects profit potential
• Skew reflects potential risk not margin.

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How far is the Skew?

• Typically calibrated from sample of traded option
  prices
• Many counters have few option trades
• Lack of price transparency
• Calibration for these counters unfeasible
• Utilize non-parametric option pricing theory
  (Stutzer, 1996 Duan, 2002)
• Generate implied volatility surface.

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Methodology

• Compute empirical distribution
• Require risk-neutral distribution f(x)
• Solve for f(x) that minimizes the relative
  entropy between itself and normalised empirical
  distribution
• Such that E(f(x)) r.

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Methodology

• Simulate asset price at maturity (MCMC)
• Compute expected value of the option
• Repeat pricing procedure
• Vary maturity (T) and strike (K)
• Convert option values to Implied Volatilities
  (IV)
• Plot of IV, T and K yields volatility surface.

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TOP 40 Index

• FTSE/JSE Top 40 Index
• 10 years of data
• Calculate volatility surface
• Compare
• Market implied skews (SAFEX traded options)
• Non-parametric skews (historical data)
• 3 and 6 month terms.

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Top 40 Index

• Non-Parametric Volatility Surface

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TOP 40 Index

• 6 Month Market Implied vs. Non-parametric Skew

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TOP 40 Index

• 3 Month Market Implied vs. Non-parametric Skew

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SAB-Miller

• Large-cap industrial stock

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Closing remarks

• Derive a volatility surface
• No forecasting power (historical)
• Establish if skew is expensive/cheap
• Valuable approach for illiquid contracts

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Thank you