The Volatility Smile Ch 18: Patterns of Volatility Change

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 1 / 8 Overview

∂Σ ∂Σ Various heuristic relationships between ∂K and ∂S , such as: The sticky strike rule The sticky moneyness rule The sticky delta rule The sticky local volatility model

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 2 / 8 ∂Σ How to estimate ∂S ? ∂Σ The slope of the volatility smile, ∂K , is what is observed from market data

Calculating ∆

When volatility is not constant, the ∆ of a call option is given by:

∂C ∂Σ ∆ = ∆ + BSM ∂Σ ∂S

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 3 / 8 ∂Σ The slope of the volatility smile, ∂K , is what is observed from market data

Calculating ∆

When volatility is not constant, the ∆ of a call option is given by:

∂C ∂Σ ∆ = ∆ + BSM ∂Σ ∂S

∂Σ How to estimate ∂S ?

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 3 / 8 Calculating ∆

When volatility is not constant, the ∆ of a call option is given by:

∂C ∂Σ ∆ = ∆ + BSM ∂Σ ∂S

∂Σ How to estimate ∂S ? ∂Σ The slope of the volatility smile, ∂K , is what is observed from market data

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 3 / 8 A linear approximation is:

Σ(S, K ) = Σ0 − β(K − S0)

β is a constant that determines the slope of the skew The ATM implied volatility of an option is:

ΣATM (S) = Σ(S, S) = Σ0 − β(S − S0)

Under the presence of negative skew (β > 0), the ATM implied volatility decreases when S increases

Sticky Strike Rule

Assumes options with a fixed strike will always have the same implied volatility

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 4 / 8 β is a constant that determines the slope of the skew The ATM implied volatility of an option is:

ΣATM (S) = Σ(S, S) = Σ0 − β(S − S0)

Under the presence of negative skew (β > 0), the ATM implied volatility decreases when S increases

Sticky Strike Rule

Assumes options with a fixed strike will always have the same implied volatility

A linear approximation is:

Σ(S, K ) = Σ0 − β(K − S0)

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 4 / 8 The ATM implied volatility of an option is:

ΣATM (S) = Σ(S, S) = Σ0 − β(S − S0)

Under the presence of negative skew (β > 0), the ATM implied volatility decreases when S increases

Sticky Strike Rule

Assumes options with a fixed strike will always have the same implied volatility

A linear approximation is:

Σ(S, K ) = Σ0 − β(K − S0)

β is a constant that determines the slope of the skew

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 4 / 8 Under the presence of negative skew (β > 0), the ATM implied volatility decreases when S increases

Sticky Strike Rule

Assumes options with a fixed strike will always have the same implied volatility

A linear approximation is:

Σ(S, K ) = Σ0 − β(K − S0)

β is a constant that determines the slope of the skew The ATM implied volatility of an option is:

ΣATM (S) = Σ(S, S) = Σ0 − β(S − S0)

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 4 / 8 Sticky Strike Rule

Assumes options with a fixed strike will always have the same implied volatility

A linear approximation is:

Σ(S, K ) = Σ0 − β(K − S0)

β is a constant that determines the slope of the skew The ATM implied volatility of an option is:

ΣATM (S) = Σ(S, S) = Σ0 − β(S − S0)

Under the presence of negative skew (β > 0), the ATM implied volatility decreases when S increases

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 4 / 8 A linear approximation of the implied volatility is:

Σ(S, K ) = Σ0 − β(K − S)

The implied volatility of an option with the same moneyness should always be the same, regardless of the level of the stock price ∂Σ When β > 0 (negative skew), it follows that ∂S > 0 Thus, under the sticky-moneyness rule, the correct hedge ratio for a standard option will be greater than the BSM delta

Sticky Moneyness Rule

An option’s implied volatility only depends on its moneyness, K /S

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 5 / 8 The implied volatility of an option with the same moneyness should always be the same, regardless of the level of the stock price ∂Σ When β > 0 (negative skew), it follows that ∂S > 0 Thus, under the sticky-moneyness rule, the correct hedge ratio for a standard option will be greater than the BSM delta

Sticky Moneyness Rule

An option’s implied volatility only depends on its moneyness, K /S

A linear approximation of the implied volatility is:

Σ(S, K ) = Σ0 − β(K − S)

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 5 / 8 ∂Σ When β > 0 (negative skew), it follows that ∂S > 0 Thus, under the sticky-moneyness rule, the correct hedge ratio for a standard option will be greater than the BSM delta

Sticky Moneyness Rule

An option’s implied volatility only depends on its moneyness, K /S

A linear approximation of the implied volatility is:

Σ(S, K ) = Σ0 − β(K − S)

The implied volatility of an option with the same moneyness should always be the same, regardless of the level of the stock price

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 5 / 8 Thus, under the sticky-moneyness rule, the correct hedge ratio for a standard option will be greater than the BSM delta

Sticky Moneyness Rule

An option’s implied volatility only depends on its moneyness, K /S

A linear approximation of the implied volatility is:

Σ(S, K ) = Σ0 − β(K − S)

The implied volatility of an option with the same moneyness should always be the same, regardless of the level of the stock price ∂Σ When β > 0 (negative skew), it follows that ∂S > 0

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 5 / 8 Sticky Moneyness Rule

An option’s implied volatility only depends on its moneyness, K /S

A linear approximation of the implied volatility is:

Σ(S, K ) = Σ0 − β(K − S)

The implied volatility of an option with the same moneyness should always be the same, regardless of the level of the stock price ∂Σ When β > 0 (negative skew), it follows that ∂S > 0 Thus, under the sticky-moneyness rule, the correct hedge ratio for a standard option will be greater than the BSM delta

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 5 / 8 When viewing implied volatilities in terms of delta, the shape of the volatility smile tends to be more stable This is because the delta of an option depends on the moneyness, volatility, and time-to expiration of the option (as opposed to just moneyness alone) A linear approximation of the sticky-delta rule is:   K ln S Σ(S, K ) = Σ0 − β √ ΣATM (S) τ

Sticky Delta Rule

Expresses the implied volatility as a function of the BSM√ delta, which is itself a function of the term: ln(K /S)/[Σ(S, K ) τ]

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 6 / 8 This is because the delta of an option depends on the moneyness, volatility, and time-to expiration of the option (as opposed to just moneyness alone) A linear approximation of the sticky-delta rule is:   K ln S Σ(S, K ) = Σ0 − β √ ΣATM (S) τ

Sticky Delta Rule

Expresses the implied volatility as a function of the BSM√ delta, which is itself a function of the term: ln(K /S)/[Σ(S, K ) τ] When viewing implied volatilities in terms of delta, the shape of the volatility smile tends to be more stable

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 6 / 8 A linear approximation of the sticky-delta rule is:   K ln S Σ(S, K ) = Σ0 − β √ ΣATM (S) τ

Sticky Delta Rule

Expresses the implied volatility as a function of the BSM√ delta, which is itself a function of the term: ln(K /S)/[Σ(S, K ) τ] When viewing implied volatilities in terms of delta, the shape of the volatility smile tends to be more stable This is because the delta of an option depends on the moneyness, volatility, and time-to expiration of the option (as opposed to just moneyness alone)

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 6 / 8 Sticky Delta Rule

Expresses the implied volatility as a function of the BSM√ delta, which is itself a function of the term: ln(K /S)/[Σ(S, K ) τ] When viewing implied volatilities in terms of delta, the shape of the volatility smile tends to be more stable This is because the delta of an option depends on the moneyness, volatility, and time-to expiration of the option (as opposed to just moneyness alone) A linear approximation of the sticky-delta rule is:   K ln S Σ(S, K ) = Σ0 − β √ ΣATM (S) τ

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 6 / 8 From this equation, it follows that:

∂Σ ∂Σ = = −β ∂S ∂K

Thus, an increase in the index has the same impact on the implied volatility as an equal increase in the strike This is the opposite of what was assumed for the sticky moneyness and sticky delta rules

Sticky Local Volatility Model

For the strikes of options that are close to ATM, use the following linear approximation:

Σ(S, K ) = Σ0 + 2βS0 − β(S + K )

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 7 / 8 Thus, an increase in the index has the same impact on the implied volatility as an equal increase in the strike This is the opposite of what was assumed for the sticky moneyness and sticky delta rules

Sticky Local Volatility Model

For the strikes of options that are close to ATM, use the following linear approximation:

Σ(S, K ) = Σ0 + 2βS0 − β(S + K )

From this equation, it follows that:

∂Σ ∂Σ = = −β ∂S ∂K

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 7 / 8 This is the opposite of what was assumed for the sticky moneyness and sticky delta rules

Sticky Local Volatility Model

For the strikes of options that are close to ATM, use the following linear approximation:

Σ(S, K ) = Σ0 + 2βS0 − β(S + K )

From this equation, it follows that:

∂Σ ∂Σ = = −β ∂S ∂K

Thus, an increase in the index has the same impact on the implied volatility as an equal increase in the strike

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 7 / 8 Sticky Local Volatility Model

For the strikes of options that are close to ATM, use the following linear approximation:

Σ(S, K ) = Σ0 + 2βS0 − β(S + K )

From this equation, it follows that:

∂Σ ∂Σ = = −β ∂S ∂K

Thus, an increase in the index has the same impact on the implied volatility as an equal increase in the strike This is the opposite of what was assumed for the sticky moneyness and sticky delta rules

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 7 / 8 Summary

Heuristic Linear Approximation of Σ ∆ vs. ∆BSM Sticky strike Σ(S, K ) = Σ0 − β(K − S0) ∆ = ∆BSM Sticky moneyness Σ(S, K ) = Σ0 − β(K − S) ∆ > ∆BSM   K ln S Sticky delta Σ(S, K ) = Σ0 − β √ ∆ > ∆BSM ΣATM (S) τ Local volatility Σ(S, K ) = Σ0 + 2βS0 − β(S + K ) ∆ < ∆BSM

The Volatility Smile Ch 18 The Infinite Actuary - QFI Quant 8 / 8