Equity Option Pricing and Hedging, Volatility Smile Flashcards
List some assumptions of the BSM model
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
List three advantages of using financial models
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
Describe static replication and dynamic replication
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
State some limitations of pricing financial derivatives through replication
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
Describe the creation of a collar
Own the stock S
Buy an OTM put with strike L < S
Sell an OTM call with strike U > S
State how to replicate an arbitrary piecewise payoff
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
Compare implied volatility with realized volatility
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
State the dynamic hedge P&L from realized volatility
Profit = 1/2gammaS^2(realvol^2 - impliedvol^2)dt
- profit when realized volatility is greater than the implied volatility of the option
State the stock price level at which the variance Vega of an option is maximized
S* = Ke^{0.5(sigma*sqrt(t))^2}
Compare using options vs variance/volatility swaps to speculate purely on volatility
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
Approximate the variance swap payoff with a volatility swap
Variance_R - Variance_K = 2Volatility_K(Volatility_R - Volatility_K)
Describe how to replicate a variance swap’s Vega profile only using options
Infinite number of options with weight 1/K^2
List sources of error from replicating variance swaps with options
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
State the key difference when hedging with realized volatility vs implied volatility
Realized volatility - incremental P&L stochastic, but total P&L deterministic
Implied volatility - incremental P&L deterministic, but total P&L stochastic
What is the impact of increasing the rebalancing frequency when using a hedging volatility that is different from the realized volatility
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
What is the impact of increasing the rebalancing frequency when hedging with the realized volatility
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
Suggest a hedging strategy to balance hedging too much and not hedging enough
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
Define the volatility term structure
Illustration of how the implied volatility varies with time to expiration for a fixed strike or moneyness
Define the volatility surface
Illustration of how implied volatilities vary with both strike and time to expiration
Comment on the four ways of plotting the volatility smile
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
Describe the empirical volatility smiles of foreign exchange options
Tend to be roughly symmetric for equally powerful currencies
Developing currencies have a negative skew because emerging market economies are less stable
List common facts of equity index volatility smiles
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
Describe the typical shape of individual equity smiles
Single stock smiles tend to be more symmetric because they experience both large positive and negative shocks
State the bounds for change in implied volatility
Lower bound (put): change >= -1.25/sqrt(tau)*dK/K Upper bound (call): change <= 1.25/sqrt(tau)*dK/K
Describe local volatility models
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
Describe stochastic volatility models
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
Describe jump diffusion models
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
List two ways the Black-Scholes model can be misused in the presence of volatility smiles
Produce incorrect hedge ratios
Inaccurate pricing of hedging options
State the Breeden-Litzenberger formula for the implied distribution of the stock price
P(S_t, t, S_T, T) = e^{rtau}(second derivative of the call price with respect to the strike)
State key properties of the implied distribution
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
State the replication of option payoff wrt calls and puts using the Breeden-Litzenberger formula
[Placeholder]
State difficulties with local volatilities and binomial trees
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
State the formula for an up or down move in odd numbered levels
S_u, d = F +- S^2sigma^2dt/F - S_d (S_u - F)
State the relationship between local volatility and implied volatility
The implied CRR probability is approximately the average of local volatility between the current stock price level and the stock price of the option
State the relationship between local and implied volatilities
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
State advantages of using the local volatility model
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
State disadvantages of using the local volatility model
Needs to be frequently recalibrated
Future volatility skew for short term option tenors is too flat
Compare the volatility of P&L hedging with local volatility vs Black-Scholes
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
Compare Delta with the Black-Scholes Delta under different implied volatility rules
Sticky strike: Equal
Sticky moneyness: Delta > BSM Delta
Sticky delta: Delta > BSM Delta
Local volatility: Delta < BSM Delta
Describe the sticky strike rule
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
Describe the sticky moneyness rule
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
Describe the sticky delta rule
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
Describe the sticky local volatility model
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
Describe hedging under stochastic volatility models
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
Describe the impact of volga on option pricing
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
Describe the impact of vanna on option pricing
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
