2. Option Pricing and Hedging

Question 1

Describe binary calls and puts

Answer 1

Payoff = 1 when the option is in-the-money

Question 2

Define gamma, theta, speed, vega, rho

Answer 2

1. Gamma
   
   • Second derivative of the option price V w.r.t. the underlying S
   • Represents:
     
     • Sensitivity of delta w.r.t. the underlying
     • How much or how often a position must be rehedged to maintain a delta-neutral position
2. Theta
   
   • Derivative of the option price V w.r.t. time t
   • Quantifies how much t contributes in a completely certain way
   • Option value decreases when T increases
3. Speed
   
   • Third derivative of the option price V w.r.t. underlying S
   • Quantifies the rate at which gamma changes w.r.t. the underlying S
4. Vega
   
   • Derivative of the option price V w.r.t. volatility parameter σ
   • Can add significant model risk because it relies on whether volatility is modeled correctly
   • Downfalls:
     
     • Only meaningful for options with single-signed gamma everywhere
     • Not as useful for analyzing binary options
5. Rho
   
   • Sensitivity of an option value V w.r.t. parameter for interest rate r
   • Same model risk concern as vega, but less of it because r is easier to estimate than σ
   • Typically separated into buckets and term structure interest rates so r(t) can vary over time

Question 3

Define portfolio insurance

Answer 3

• Strategy when you:
  
  • Reduce stock holdings when prices fall,
  • Increase stock holdings when prices rise
• Overall, option values due to portfolio insurance are balanced because of mean reversion

Question 4

What are two types of hedging w.r.t. models?

Answer 4

1. Model independent
   
   • Few and far in between
   • e.g., violations in Put-Call parity
2. Model dependent
   
   • Most hedging strategies
   • Requires some kind of volatility model

Question 5

What are some types of hedging w.r.t. greeks?

Answer 5

• Delta
  
  • Exploits perfect correlation between option and underlying
• Gamma
  
  • Reduces transaction costs, rebalancing needs
  • More accurate than delta hedging
• Vega
  
  • Trading strategy that results in zero vega

Question 6

Define static, margin and crash (platinum) hedging

Answer 6

• Static hedging
  
  • Buying/selling a set of more liquid contracts to reduce the CFs of the original contract
  • Positions are left to expiry
• Margin hedging
  
  • Portfolio set up such that margin calls are covered by refunds of hedging contracts
• Crash (platinum) hedging
  
  • Minimizes worst possible outcome for the portfolio

Question 7

Define implied volatility

Answer 7

• Volatility of the underlying which, when used in B-S formula, results in market prices
• Market consensus or estimate of volatility

Question 8

Define actual volatility

Answer 8

Amount of randomness of a financial quantity that actually transpires at any given point

• Amount of noise int he stock price
• Wiener process coefficient in the stock returns model

Question 9

Define historical, forward and hedging volatility

Answer 9

1. Historical volatility
   
   • Backward-looking statistical measure
2. Forward volatility
   
   • Actual or implied, for some time in the future
3. Hedging volatility
   
   • What is plugged into the detla calculation

Question 10

List types of models used for volatility

Answer 10

• Econometric
  
  • Time-series analysis to estimate current and future expected actual volatility, e.g., GARCH
• Deterministic
  
  • Deterministic volatility surface
  • Set σ(S,t) in the B-S model
  • Does not capture dynamics of volatility very well
• Stochastic
  
  • Better captures the dynamics of traded option prices compared to deterministic
• Poisson
  
  • Volatility jumps
• Uncertain
  
  • Define a range of σ ⇒ range of prices

Question 11

Compare and contrast pros/cons of hedging with actual and implied volatility

Answer 11

1. Actual volatility
   
   • PROS:
     
     • Known profit at expiration, assuming continuous hedging
     • Most reasonable if mark-to-model strategy is followed
     • More leeway than implied volatility, as long as forecast is “good enough”
   • CONS:
     
     • Daily P&L volatility can be substantial ⇒ risks
     • Need to estimate actual volatility forecast for ∆
2. Implied volatility
   
   • PROS:
     
     • Minimal local fluctuations in P&L (i.e., continual profit)
     • No need exact actual volatility estimation, just the right side of the trade
     • Easy to calculate because implied vol is observable
     • More reasonable if market value approach is used
   • CONS:
     
     • Final profit is unknown, just know that it will be positive

Question 12

What are two trains of thought for the rationale behind option pricing movements?

What are some considerations when pricing options?

Answer 12

1. Valuation (theory)
   
   • Prices are driven by B-S (theoretica, parameters, assumptions)
   • Option values are consistent with the price of the underlying
2. Pricing (practice)
   
   • Prices are driven by supply and demand

Considerations:

• OTM options sell at a premium
• American options are difficult to price becasue early exercise is seldom done optimally
• Embedded options are priced high because Σ parts > whole security

Question 13

What is the power law survival function?

Answer 13

S(x) = K / x^α

Question 14

Compare/contrast normal and fractal distributions

Answer 14

1. Normal/Gaussian
   
   • Nonscalable
   • Typical member is mediocre
   • Winner takes a piece of the pie
   • Ancestral environment
   • Not determined by a single instance
   • Tyranny of collective
   • Easy to predict from the past
2. Fractal
   
   • Scalable
   • No typical member
   • Winnter takes all
   • Modern environment
   • Determined by a few events
   • Tyranny of accidental
   • Hard to predict from the past, need large window of observation

Question 15

What is volatility smile?

How can it be built into pricing?

Answer 15

• It is the graph of strike (K) vs. implied volatility, which may result in higher implied vol for OTM calls/puts
• Can be built into pricing by:
  
  1. Deterministic volatility surface
     
     • May not describe actual dynamics very well
  2. Stochastic volatility models
     
     • Sources of randomness are stock returns and volatility
     • Greater potential to capture dynamics
  3. Jump diffusion model
     
     • Accommodate for excess kurtosis

Question 16

What is vonma?

Answer 16

• The second derivative of V w.r.t. σ
• It is negative close to ATM, and >> 0 for ITM/OTM
  
  • Results in higher price and implied volatility for OTM options
• If vonma > 0 ⇒ vega is positively related to volatility changes
• If vonma < 0 ⇒ vega is inversely related to volatility changes

Question 17

What causes volatility smiles?

Answer 17

Due to:

1. Supply and demand
   
   • ↑ OTM puts demand for insurance protection ⇒ ↑ price and σ of OTM puts
   • ↑ OTM calls supply to earn premium ⇒ ↓ price and σ of OTM calls
2. Kurtosis / fat tails
3. Correlation between stock prices and volatility
   
   • Dramatic ↓ in price ⇒ ↑ implied vol for puts with lower K
4. Volatility gamma (vonma)
   
   • OTM puts have higher vonma ⇒ ↑ implied vol

Question 18

List 10 assumptions in B-S and how to take advantage of them

Answer 18

1. Volatility is known
   
   • If σ ↑, buy a straddle/strangle
2. No jumps
   
   • If expecting symmetric jumps, buy OTM options
3. Constant rfr
   
   • If r ↑, buy calls/stocks and sell puts
4. Borrowing = lending rates; infinite borrowing
   
   • If r > lending rates + borrowing limits ⇒ buy calls
   • If r < lending (no borrowing limits) ⇒ borrow instead of buy calls
   • If implied r ↑, buy options instead of stock
5. Short sales can be invested
   
   • Instead of short stock, hold put or naked short call
6. No transaction costs
   
   • Use arbitrage bands
7. No taxes
8. No dividends
9. European options
10. No early exercise or takeover events
    
    • May affect short-term OTM options dramatically

Question 19

List 7 assumptions in B-S and how to relax them

Answer 19

1. Discrete hedging
   
   • Expected value is the same as continuous
2. Transaction costs
   
   • Use volatility range to represent bid-ask spreads
3. Time-dependent volatility
   
   • Use root-mean-square average variance over the remaining lifetime (T-t)
4. Arbitrage opportunities
   
   • Use B-S to delta-hedge and determine how much profit you would like
5. Non-lognormal underlying
   
   • Nothing to do
6. Borrowing costs
   
   • Adjust drift, similar to dividend adjustment
7. Non-normal returns
   
   • Nothing to do- only need finite variance of returns due to CLT

Question 20

What is the total PV of profit when hedging using actual volatility? And from time t to t + dt?

Answer 20

V^a - Vⁱ

e^-r(t-t0) d(V^a - Vⁱ)

Question 21

What is the total PV of profit when hedging using implied volatility? And from time t to t + dt?

Answer 21

dVⁱ = 1/2 (σ² - σ_imp²) S² Γⁱ dt

dVⁱ = 1/2 (σ² - σ_imp²) ∫_{[t0, T]} e^-r(t-t0)S² Γⁱ dt

Question 22

What is the general total PV of profit formula?

Answer 22

V(S,t;σ_h) - V(S,t;σ_i) + 1/2(σ² - σ_h²) ∫_{[t0, T]}e^-r(t-t0)S²Γ^h dt