ACFI310 - Lecture 8 - The Greek letters and volatility surface

What is a naked position in options trading?

Taking no hedging action after selling an option. The risk is that if the option is exercised, the institution must buy shares at the prevailing market price.

What is a covered position strategy?

Buying the underlying shares immediately to cover the option sold. This works well if the option is exercised but can lead to significant losses in other circumstances.

What is a stop-loss strategy?

• Buy shares as soon as the price reaches the strike price

• Sell shares as soon as the price falls below the strike price

What are two problems with the stop-loss strategy?

• Cash flows occur at different times and must be discounted

• Purchases and sales are not made at the same price

Why do neither naked nor covered positions eliminate all risks?

Naked positions expose the seller to unlimited loss if exercised, while covered positions can lead to significant losses if the option is not exercised and the stock price falls.

What is Delta (Δ)?

The rate of change of the option price with respect to the underlying asset price.

What is the delta of a European call option on a non-dividend-paying stock?

N(d₁)

What is the delta of a European put option on a non-dividend-paying stock?

N(d₁) - 1

If an option has a delta of 0.6, what hedged position would be required for someone who has sold 100,000 options?

Buy 60,000 shares (short 100,000 options, long 60,000 shares creates a delta-neutral position).

What is delta hedging?

Maintaining a delta-neutral portfolio where gains/losses on the option position are offset by losses/gains on the stock position.

Why must delta hedges be rebalanced?

Because delta changes as the stock price changes and time passes.

What does delta hedging a written option involve in terms of trading?

A "buy high, sell low" trading rule.

What is Theta (Θ)?

The rate of change of the option value with respect to the passage of time.

Is theta for call or put options usually positive or negative?

Usually negative.

What does negative theta mean for a long option position?

If time passes with the price and volatility of the underlying remaining the same, the value of a long call or put option declines (time decay).

When is theta most negative for an option?

When the option is at-the-money.

What is Gamma (Γ)?

The rate of change of delta (Δ) with respect to the price of the underlying asset.

When is gamma greatest?

For options that are close to the money (at-the-money).

When is gamma greatest?

For options that are close to the money (at-the-money).

What problem does gamma address in delta hedging?

Delta hedging errors caused by the curvature of the option price relationship with the underlying asset.

How can gamma be adjusted?

By taking a position in another option or derivative (cannot be adjusted by taking a position in the underlying asset alone).

What is the relationship between delta, gamma, and theta for a portfolio of derivatives on a non-dividend-paying stock?

Θ + rS₀Δ + ½σ²S₀²Γ = rΠ

What is Vega (ν)?

The rate of change of the value of a derivatives portfolio with respect to volatility.

When is vega greatest?

For at-the-money options.

How can vega be adjusted?

By taking a position in another option or derivative (cannot be adjusted by taking a position in the underlying asset alone).

What is Rho (ρ)?

The rate of change of the value of a derivative with respect to the interest rate.

A portfolio has Delta=0, Gamma=-5000, Vega=-8000. Option 1 has Delta=0.6, Gamma=0.5, Vega=2.0. What position in Option 1 and the underlying makes the portfolio delta and gamma neutral?

Long 10,000 of Option 1, short 6,000 of the underlying asset.

For the same portfolio, what position makes it delta and vega neutral?

Long 4,000 of Option 1, short 2,400 of the underlying asset.

If you need to make a portfolio delta, gamma, AND vega neutral, what must you use?

Positions in two different options plus the underlying asset.

Given the portfolio (Δ=0, Γ=-5000, ν=-8000), Option 1 (Δ=0.6, Γ=0.5, ν=2.0), and Option 2 (Δ=0.5, Γ=0.8, ν=1.2), what positions make the portfolio delta, gamma, and vega neutral?

• Long 400 of Option 1

• Long 6,000 of Option 2

• Short 3,240 of the underlying asset

How often do traders typically ensure their portfolios are delta-neutral?

At least once a day.

What happens to hedging costs as a portfolio becomes larger?

Hedging becomes less expensive (economies of scale).

What is the difference between hedging and creating an option synthetically?

• Hedging: Take positions that offset delta, gamma, vega, etc.

• Creating synthetically: Take positions that match delta, gamma, vega, etc.

What is a volatility smile?

A curve showing the variation of implied volatility with strike price for options with a certain maturity.

Is the volatility smile the same whether calculated from European calls or European puts?

Yes (this follows from put-call parity). It's also approximately the same for American options.

What is a volatility term structure?

The variation of implied volatility with the time to maturity of the option.

When does the volatility term structure tend to be downward sloping vs upward sloping?

Downward sloping when volatility is high; upward sloping when volatility is low.

Describe the volatility smile for foreign currency options.

A U-shaped curve - implied volatility is higher for both low-strike and high-strike options compared to at-the-money options.

What are the properties of the implied distribution for foreign currency options compared to lognormal?

• Both tails are heavier than the lognormal distribution

• More peaked than the normal distribution

What are two possible causes of the volatility smile for foreign currencies?

• Exchange rates exhibit jumps rather than continuous changes

• Volatility of the exchange rate is stochastic

Describe the volatility smile for equity options.

A downward sloping curve (sometimes called a "smirk" or "skew") - implied volatility is higher for low-strike (out-of-the-money put) options and decreases as strike price increases.

What are the properties of the implied distribution for equity options compared to lognormal?

• The left tail is heavier than the lognormal distribution

• The right tail is less heavy than the lognormal distribution

What are two reasons for the volatility smile in equity options?

1. Leverage (as stock price falls, leverage increases, increasing volatility)

2. Crashophobia (fear of market crashes)

What are three ways to characterize volatility smiles?

• Plot implied volatility against K/S₀

• Plot implied volatility against K/F₀

• Plot implied volatility against the delta of the option

How do traders often define "at-the-money" for options?

In multiple ways:

• When K equals the forward price F₀ (not just the spot price S₀)

• As a call with delta of 0.5 or a put with delta of -0.5 (called "50-delta options")

What is a volatility surface?

A three-dimensional representation showing how implied volatility varies with both strike price and time to maturity.

Looking at the volatility surface table with maturities from 1 month to 5 years and strikes from 0.90 to 1.10, what general pattern emerges?

• Volatility smile is present at all maturities (lowest at K=1.00)

• Short-term options show more pronounced smile

• Longer-term options have flatter volatility across strikes

• Volatility term structure varies by strike price

If the true distribution has a heavier left tail and lighter right tail compared to lognormal, what would the volatility smile look like?

Downward sloping (equity options pattern) - higher implied volatility for low strikes, lower for high strikes.

If the true distribution has both tails heavier than lognormal, what would the volatility smile look like?

U-shaped (foreign currency pattern) - higher implied volatility at both low and high strikes.

If the true distribution has both tails less heavy than lognormal, what would the volatility smile look like?

Inverted U-shape (frown) - lower implied volatility at extreme strikes, higher in the middle.