Equity Option Pricing and Hedging, Volatility Smile

Question 1

List some assumptions of the BSM model

Answer 1

Stock returns are normally distributed and follow GBM
Market for options and futures of the stock have unlimited liquidity
One can continuously hedge (dynamically replicate) a position in a financial option with futures
Market has no transaction costs
Volatility is constant

Question 2

List three advantages of using financial models

Answer 2

Interpolate/extrapolate from the prices of liquid securities to value illiquid securities
Rank securities in terms of value
Quantify intuition (linear quantities) into nonlinear dollar values

Question 3

Describe static replication and dynamic replication

Answer 3

Static Replication - reproduces payoffs of the target security over its entire lifetime with an initial portfolio whose weights will never need to be changed
Dynamic Replication - components and weights of the replicating portfolio must change over time

Question 4

State some limitations of pricing financial derivatives through replication

Answer 4

Adjusting weights by trading in the market can become problematic
Bid ask spreads, illiquidity, and market impact can push up a security’s price when we need to buy more of it
Financing costs, transaction costs, and operational risks may vary from firm to firm
Dynamic hedging often requires us to estimate the future values of certain parameters that are difficult to observe in the market such as future volatility

Question 5

Describe the creation of a collar

Answer 5

Own the stock S
Buy an OTM put with strike L < S
Sell an OTM call with strike U > S

Question 6

State how to replicate an arbitrary piecewise payoff

Answer 6

Linear combination of options with a bond and the underlying stock
V(t) = Ie^{-r(T-t)} + lambda_0S_t + (lambda_1-lambda_0)*C(K_0) + …
- lambdas are the slope between the piecewise payoffs

Question 7

Compare implied volatility with realized volatility

Answer 7

Implied volatility - volatility in BSM that allows the model to replicate the market option price
- use BSM to calculate the appropriate hedge ratio
- market expected value of future volatility
Realized volatility - statistical standard deviation of stock returns per unit time

Question 8

State the dynamic hedge P&L from realized volatility

Answer 8

Profit = 1/2gammaS^2(realvol^2 - impliedvol^2)dt

- profit when realized volatility is greater than the implied volatility of the option

Question 9

State the stock price level at which the variance Vega of an option is maximized

Answer 9

S* = Ke^{0.5(sigma*sqrt(t))^2}

Question 10

Compare using options vs variance/volatility swaps to speculate purely on volatility

Answer 10

Volatility/variance swaps are better than options for speculating purely on volatility

• option price is sensitive to both stock and volatility
• sensitivity of the option price depends on the moneyness of the option

Question 11

Approximate the variance swap payoff with a volatility swap

Answer 11

Variance_R - Variance_K = 2Volatility_K(Volatility_R - Volatility_K)

Question 12

Describe how to replicate a variance swap’s Vega profile only using options

Answer 12

Infinite number of options with weight 1/K^2

Question 13

List sources of error from replicating variance swaps with options

Answer 13

Perfect replication of variance swaps requires knowledge of option prices at all possible strikes
- gaps between option prices in the market
- strikes are limited in range
Variance swap formulas are only valid if no jumps are assumed

Question 14

State the key difference when hedging with realized volatility vs implied volatility

Answer 14

Realized volatility - incremental P&L stochastic, but total P&L deterministic
Implied volatility - incremental P&L deterministic, but total P&L stochastic

Question 15

What is the impact of increasing the rebalancing frequency when using a hedging volatility that is different from the realized volatility

Answer 15

Increasing the number of rebalancing does not reduce the error in P&L
Replication has a random component proportional to (Delta_I - Delta_R) that does not go away

Question 16

What is the impact of increasing the rebalancing frequency when hedging with the realized volatility

Answer 16

Hedging increases uncertainty in the outcome, but the expected P&L is still zero
Volatility of the hedging error decreases proportionally to the square root of n

Question 17

Suggest a hedging strategy to balance hedging too much and not hedging enough

Answer 17

Hedging too much results in high transaction costs, while not hedging enough creates hedging error
One option to strike a balance is to rebalance the portfolio based on a trigger
- for example, rebalance when delta changes by 0.02

Question 18

Define the volatility term structure

Answer 18

Illustration of how the implied volatility varies with time to expiration for a fixed strike or moneyness

Question 19

Define the volatility surface

Answer 19

Illustration of how implied volatilities vary with both strike and time to expiration

Question 20

Comment on the four ways of plotting the volatility smile

Answer 20

Function of strike
- difficult to compare options on stocks with very different prices
Function of moneyness
- less useful for comparing options with different times to expiration or significantly different volatilities
Function of normalized log moneyness
- provides a comparison across tenors and volatilities
Function of BSM delta
- advantages: every option has a delta, x axis standardized 0 to 1, gives the number of shares to hedge an option in the BSM world, can be approximately equal to the risk neutral probability
- disadvantages: formula for delta itself depends on implied volatility, validity of the BSM model is questionable

Question 21

Describe the empirical volatility smiles of foreign exchange options

Answer 21

Tend to be roughly symmetric for equally powerful currencies

Developing currencies have a negative skew because emerging market economies are less stable

Question 22

List common facts of equity index volatility smiles

Answer 22

Smiles typically have a negative slope as a function of strike
During a financial crisis with high volatility, the term structure of volatility is typically downward sloping
Negative skew/slope is steeper for short expirations
Implied volatility and index returns are negatively correlated
Implied volatility tends to be greater than realized volatility
Volatility of implied volatility is greatest for short expirations
Implied volatility is mean reverting
Shock on the implied volatility surface are highly correlated and characterized by a few principal components: level of the surface, term structure, skew

Question 23

Describe the typical shape of individual equity smiles

Answer 23

Single stock smiles tend to be more symmetric because they experience both large positive and negative shocks

Question 24

State the bounds for change in implied volatility

Answer 24

Lower bound (put): change >= -1.25/sqrt(tau)*dK/K
Upper bound (call): change <=  1.25/sqrt(tau)*dK/K

Question 25

Describe local volatility models

Answer 25

Stock evolution: dS/S = mu(S_t, t)dt + sigma(S_t, t)dZ

• sigma(S_t, t) is the local volatility function that varies deterministically as a function of future time t and the future random stock price S
• can generate an implied volatility function that varies with strike and expiration
• can calibrated sigma(S_t, t) to generate implied volatilities that are consistent with the market

Question 26

Describe stochastic volatility models

Answer 26

dS = muSdt + sigmaSdZ, d(sigma) = psigmadt + qsigmadW, E[dZdW] = rho*dt

• don’t know the appropriate SDE for volatility
• correlations between the stock and the volatility may also be stochastic

Question 27

Describe jump diffusion models

Answer 27

Models allow stock to make an arbitrary number of jumps in addition to GBM diffusion

• jumps capture fear of stock market crashes
• helps fit the steep short term skew of equity index market implied volatilities
• can combine jump diffusion with stochastic volatility models

Question 28

List two ways the Black-Scholes model can be misused in the presence of volatility smiles

Answer 28

Produce incorrect hedge ratios

Inaccurate pricing of hedging options

Question 29

State the Breeden-Litzenberger formula for the implied distribution of the stock price

Answer 29

P(S_t, t, S_T, T) = e^{rtau}(second derivative of the call price with respect to the strike)

Question 30

State key properties of the implied distribution

Answer 30

Not the true distribution of the stock price at time T
Can only be used to value options whose payoff depends on the terminal stock price S_T
- cannot be used to value path dependent options
- implied distribution does not give any information about the stock price on the way to expiration
Implied distribution may be jagged
- can occur if options price quotes are stale
- for a smoother function, approximate the call price function with a time differentiable continuous function
Equation for the implied distribution is not dependent on any assumed model of the stock price or volatility skew

Question 31

State the replication of option payoff wrt calls and puts using the Breeden-Litzenberger formula

Answer 31

[Placeholder]

Question 32

State difficulties with local volatilities and binomial trees

Answer 32

If the local volatility varies too rapidly, no arbitrage constraints will be violated
- stock prices will violate no arbitrage
- transitional probabilities less than 0 or greater than 1
Try to mitigate with smaller time steps
- finite number of options and implied volatilities, so there may not be enough information without interpolation or extrapolation

Question 33

State the formula for an up or down move in odd numbered levels

Answer 33

S_u, d = F +- S^2sigma^2dt/F - S_d (S_u - F)

Question 34

State the relationship between local volatility and implied volatility

Answer 34

The implied CRR probability is approximately the average of local volatility between the current stock price level and the stock price of the option

Question 35

State the relationship between local and implied volatilities

Answer 35

Local volatilities are twice as sensitive to the stock price as implied volatilities when the volatilities only depends on the stock price and not on time

Question 36

State advantages of using the local volatility model

Answer 36

Simplest extension of the BSM model that can accommodate the volatility smile
Can calibrate the local volatility function to any market implied volatility surface
Provides arbitrage free option values and hedge ratios for standard and exotic options
Notion that BSM implied volatility is the average of local volatilities from the initial stock price is intuitive

Question 37

State disadvantages of using the local volatility model

Answer 37

Needs to be frequently recalibrated

Future volatility skew for short term option tenors is too flat

Question 38

Compare the volatility of P&L hedging with local volatility vs Black-Scholes

Answer 38

BSM will like have more volatility in typical regimes

• typical regimes: down market high volatility, up market low volatility
• atypical regimes: down market low volatility, up market high volatility

Question 39

Compare Delta with the Black-Scholes Delta under different implied volatility rules

Answer 39

Sticky strike: Equal
Sticky moneyness: Delta > BSM Delta
Sticky delta: Delta > BSM Delta
Local volatility: Delta < BSM Delta

Question 40

Describe the sticky strike rule

Answer 40

Assumes options with a fixed strike will always have the same implied volatility
Linear approximation: sigma(S, K) = sigma_0 - Beta*(K - S_0)
Negative skew decreases implied volatility when S increases

Question 41

Describe the sticky moneyness rule

Answer 41

Assumes that an option’s implied volatility only depends on its moneyness K / S
Linear Approximation: sigma(S, K) = sigma_0 - beta*(K - S)
Negative skew means dsigma/dS > 0

Question 42

Describe the sticky delta rule

Answer 42

Assumes the implied volatility is a function of the BSM Delta
Linear Approximation: sigma(S, K) = sigma_0 - beta(ln(K/S)/sigma_ATM(S)sqrt(tau))
Shape of the smile tends to be more stable because delta depends on moneyness, volatility, and time to expiration

Question 43

Describe the sticky local volatility model

Answer 43

Assumes the calibrated local volatility never changes
Linear Approximation: sigma(S, K) = sigma_0 + 2betaS_0 - beta*(S + K)
Increase in the index has the same impact on implied volatility as an equal increase in the strike
- opposite of what was assumed for sticky moneyness and sticky delta

Question 44

Describe hedging under stochastic volatility models

Answer 44

Four possible future states due to volatility and stock being stochastic
Trading just the stock and cash is no longer enough to hedge an exotic option
- must also use securities whose price is sensitive to volatility to remove volatility risk

Question 45

Describe the impact of volga on option pricing

Answer 45

E[d(sigma)^2] is always positive, so stochastic volatility increases the value of an option when the Volga is positive
Volga is highest above and below the ATM strike
Results in higher implied volatilities for OTM options relative to ATM options, which results in a symmetric smile

Question 46

Describe the impact of vanna on option pricing

Answer 46

Vanna is positive when the call is OTM and negative when the call is ITM
- at high strikes, vanna will decrease the value of the call option relative to BSM value
- at low strikes, vanna will increase the value of the call option relative to BSM value
Consistent with negative skew in volatility smiles observes in the market