Equity Option Pricing and Hedging, Study Notes

Question 1

Briefly describe different volatility models

Answer 1

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

Question 2

List three methods of building volatility smiles into options pricing

Answer 2

Deterministic volatility surface
Stochastic volatility
Jump diffusion

Question 3

Give examples of how supply and demand can impact the shape of the volatility curve

Answer 3

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

Question 4

Describe stochastic volatility modeling

Answer 4

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

Question 5

Describe the four key tools to analyze the shape of a volatility surface

Answer 5

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

Question 6

Describe how vomma affects vega and implied volatility

Answer 6

Positive vomma - vega and volatility positively related, increases in implied volatility associated with increases in vega
Negative vomma - opposite of positive vomma

Question 7

List assumptions that can be adjusted and still fit within the Black-Scholes framework

Answer 7

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

Question 8

State the pros and cons of hedging with actual volatility

Answer 8

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

Question 9

State the pros and cons of hedging with implied volatility

Answer 9

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

Question 10

Describe how delta changes as a function of time

Answer 10

ITM - delta converges to 1 (call) or -1 (put)
OTM - delta converges to 0
ATM - discontinuity because I’d uncertainty about ITM or OTM

Question 11

Describe how delta changes as a function of volatility

Answer 11

OTM - higher volatility increases probability of expiring ITM, so delta increases
ITM - lower volatility increases probability of expiring ITM, so delta increases

Question 12

Describe gamma as a function of the spot price

Answer 12

Gamma is maximized close to the strike price

As options become deeply ITM or OTM, gamma converges to 0

Question 13

Analyze how much the underlying spot price can move for the long gamma and short theta affects to balance

Answer 13

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

Question 14

Describe how gamma changes as a function of time

Answer 14

Gamma approaches 0 closer to maturity
- ITM = own asset
- OTM = don’t own asset
ATM case to expiration, gamma can be large and unstable

Question 15

Describe how gamma changes as a function of volatility

Answer 15

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

Question 16

Describe how vega changes as a function of time

Answer 16

Vega approaches 0, and the tails become thinner

Question 17

Describe how vega changes as a function of volatility

Answer 17

OTM/ITM - increasing volatility increases vega
- higher probability of the underlying price crossing the strike before expiry
ATM - little sensitivity to changes in volatility

Question 18

Describe how theta changes as a function of the spot price

Answer 18

Theta approaches 0 far from the strike

Theta is maximized when the option is ATM

Question 19

Describe how theta changes as a function of time

Answer 19

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

Question 20

Describe how theta changes as a function of volatility

Answer 20

When volatility increases, theta increases

- higher time value of the option due to uncertainty about whether the option will expire OTM or ITM

Question 21

Briefly describe call and put spreads

Answer 21

Long call spread - long low strike call, short high strike call
Long put spread - long high strike put, short low strike put

Question 22

List pros and cons of using empirical interpolation to build a volatility smile

Answer 22

Pros - match market quotes exactly

Cons - extra effort to handle extrapolation, introduce discontinuities in the smile shape

Question 23

List pros and cons of using a functional form to build a volatility smile

Answer 23

Pros - provides a naturally smooth smile, provides a convenient mechanism for extrapolation
Cons - may not reproduce market quotes exactly

Question 24

Describe straddles and strangles

Answer 24

Straddle - long ATM put, short ATM call (same maturity)
Strangle - long OTM put, long OTM call (same maturity)
Useful for vega hedging

Question 25

Describe risk reversals

Answer 25

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

Question 26

Describe butterfly spreads

Answer 26

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