Equity Option Pricing and Hedging, Study Notes Flashcards
Briefly describe different volatility models
Econometric models - use time series analysis to estimate future volatility (GARCH)
- decompose modeling into components
Deterministic models - assume volatility is deterministic
- BSM or local volatility
Stochastic volatility - assume volatility is random
- Heston model, better at capturing the dynamics of traded option prices than deterministic models
Poison processes - model jumps between periods of low and high volatility
Uncertain volatility - use ranges instead of probabilities
- generate a range of option prices
List three methods of building volatility smiles into options pricing
Deterministic volatility surface
Stochastic volatility
Jump diffusion
Give examples of how supply and demand can impact the shape of the volatility curve
Lots of OTM puts for insurance, increase price and implied volatility for OTM puts
Sell OTM calls to earn premium, creates oversupply of calls and decreases price and volatility for OTM calls
Describe stochastic volatility modeling
Stochastic volatility models have two sources of randomness: stock and volatility
- one parameter is the correlation between the stock price and the volatility
- correlation typically negative and results in a negative skew
Assuming constant volatility, vomma is greatest for ITM options
- OTM options have a convexity price that can cause the smile
Need to find a good model and calibrate it properly
Greater potential to capture the dynamics and evolution of the volatility surface over time
Describe the four key tools to analyze the shape of a volatility surface
Supply and Demand - people purchasing lots of OTM puts and selling lots of OTM calls
Kurtosis of fat tails
Correlation between stock prices and volatility - negative correlation so OTM puts have higher volatility
Volatility gamma - OTM puts have large vomma which is associated with larger implied volatilities
Describe how vomma affects vega and implied volatility
Positive vomma - vega and volatility positively related, increases in implied volatility associated with increases in vega
Negative vomma - opposite of positive vomma
List assumptions that can be adjusted and still fit within the Black-Scholes framework
Discrete hedging - no bias even when hedging discretely even though there is not exact replication
Transaction costs - adjust the volatility as a proxy, can be modeled as a term that depends on gamma
Time dependent volatility- use the root mean square average volatility for the Black-Scholes parameter
Arbitrage opportunities - can still use to value the profit from the arbitrage
Non-lognormal underlying - still a good approximation
Borrowing costs - adjust the risk neutral drift rate (like dividends)
Non-normal returns - can still accommodate with finite variance because of the Central Limit Theorem
State the pros and cons of hedging with actual volatility
Pros
- know exact profit at expiration
- best for a Mark to Market strategy
- strategy has more leeway than implied volatility
Cons
- daily P&L fluctuations can be substantial
- need to use the value of your volatility forecast for the hedging delta
State the pros and cons of hedging with implied volatility
Pros
- minimal fluctuation in P&L, continual profit
- do not need a precise estimate of actual volatility, just need to be on the correct side
- calculating delta is easy since the implied volatility is observable
- most reasonable for a market value approach
Cons
- do not know the final profit, only that it is positive
Describe how delta changes as a function of time
ITM - delta converges to 1 (call) or -1 (put)
OTM - delta converges to 0
ATM - discontinuity because I’d uncertainty about ITM or OTM
Describe how delta changes as a function of volatility
OTM - higher volatility increases probability of expiring ITM, so delta increases
ITM - lower volatility increases probability of expiring ITM, so delta increases
Describe gamma as a function of the spot price
Gamma is maximized close to the strike price
As options become deeply ITM or OTM, gamma converges to 0
Analyze how much the underlying spot price can move for the long gamma and short theta affects to balance
Sigma / sqrt(252)
- realized volatility for the day must be approximately equal to the option implied volatility
- if realized > implied, long gamma position gains
- if realized < implied, long theta position gains
Describe how gamma changes as a function of time
Gamma approaches 0 closer to maturity
- ITM = own asset
- OTM = don’t own asset
ATM case to expiration, gamma can be large and unstable
Describe how gamma changes as a function of volatility
OTM/ITM - higher volatility means higher gamma
ATM - lower volatility means higher gamma
- decrease probability of flipping between OTM and ITM
- gamma higher because change in price can trigger change in delta
Describe how vega changes as a function of time
Vega approaches 0, and the tails become thinner
Describe how vega changes as a function of volatility
OTM/ITM - increasing volatility increases vega
- higher probability of the underlying price crossing the strike before expiry
ATM - little sensitivity to changes in volatility
Describe how theta changes as a function of the spot price
Theta approaches 0 far from the strike
Theta is maximized when the option is ATM
Describe how theta changes as a function of time
Theta generally approaches 0 as the option tends towards expiration
If the option is ATM, the theta can be large due to uncertainty about whether the option will expire ITM or OTM
Describe how theta changes as a function of volatility
When volatility increases, theta increases
- higher time value of the option due to uncertainty about whether the option will expire OTM or ITM
Briefly describe call and put spreads
Long call spread - long low strike call, short high strike call
Long put spread - long high strike put, short low strike put
List pros and cons of using empirical interpolation to build a volatility smile
Pros - match market quotes exactly
Cons - extra effort to handle extrapolation, introduce discontinuities in the smile shape
List pros and cons of using a functional form to build a volatility smile
Pros - provides a naturally smooth smile, provides a convenient mechanism for extrapolation
Cons - may not reproduce market quotes exactly
Describe straddles and strangles
Straddle - long ATM put, short ATM call (same maturity)
Strangle - long OTM put, long OTM call (same maturity)
Useful for vega hedging
Describe risk reversals
Buy a call option at one strike, sell a out option at a lower strike
Useful for vanna hedging
- reversals have high vanna because their vega is sensitive to the underlying stock price
- highly sensitive to the volatility skew of the underlying asset
Describe butterfly spreads
Short position K_2 straddle, long position K_1 and K_3 strangle
- short inside, long outside
Way to take position in the convexity of the volatility skew
Useful to hedge volga risk
- significant amount of sensitivity to volatility convexity
- short straddle has little volga, but long strangle has meaningful volga
- higher volatility increases net vega, while lower volatility decreases net vega
