Greeks: Options & Hedging

Question 1

The Steps of Hedging

Answer 1

1. Value of option
2. Delta of option
3. Gamma of option
4. Create the neutral portfolio

Question 2

Delta: Theory

Answer 2

The rate or change in option price w.r.t. the price of the underlying asset.

Protects against small changes in price.

Question 3

Delta: Portfolio Delta

Answer 3

Sum of weighted deltas
x_i * delta_i

Question 4

Delta: Delta of underlying asset

Answer 4

Delta of underlying asset = 1

Question 5

Delta: Delta of instruments & how to hedge

Answer 5

Linear instruments:
* Forwards
* Delta = 1
* Hedge:
* Long forward: Short 1 of underlying
* Short forward: Long 1 of underlying

• Futures
  
  • Delta = exp(rT)
  • Hedge:
    
    • Long future: Short exp(rT) of underlying
    • Short future: Long exp(rT) of underlying
• Underlying
  
  • Delta = 1

Non-linear instruments
* Call option
* Delta = N(d1) > 0
* Hedge:
* Long call: Short N(d1) of underlying
* Short cal: Long N(d1) of underlying

• Put option
  
  • Delta = N(d1) - 1 < 0
  • Hedge:
    
    • Long put: Long 1-N(d1) of underlying
    • Short put: Short 1-N(d1) of underlying

Question 6

Gamma: Theory

Answer 6

The rate or change of delta price w.r.t. the price of the underlying asset.

Small Γ
* Delta changes slowly
* Dynamic adjustments infrequently
Large Γ
* Delta changes fast
* Dynamic adjustments frequently
* Risky to leave unchanged for any length of time

Gamma captures the curvatures of the non-linear relationship between the stock price amd option price.

Protects against large changes in price.

Question 7

Gamma: Portfolio Gamma

Answer 7

Sum of weighted gammas
x_i * gamma_i

Question 8

Gamma: Gamma of underlying asset

Answer 8

Gamma of underlying asset = 0

Question 9

Gamma: Gamma of instruments & how to hedge

Answer 9

Linear instruments:
* Forwards
* Γ = 0

• Futures
  
  • Γ = 0
• Underlying
  
  • Γ = 0

Non-linear instruments
* Call option = Put option
* Γ = Appendix > 0
* Hedge:
* Long option: Short x of underlying
* Short option: Long x of underlying

Look into this

Position in traded option for gamma neutrality:
w_T = – ( Γ_pf / Γ_T )

Question 10

Theta: Theory

Answer 10

The theta is the rate of change of the value of the portfolio w.r.t. the passage of time.

Typically not hedged.
Usually negative for an option.
𝚯

Question 11

Vega: Theory

Answer 11

The ratio of the change in option price w.r.t. changes in volatility.

𝒱_call = 𝒱_put

High positive or negative vega:
High sensitivity to changes in volatility.

2 options are needed to hedge vega & gamma.

Assumption: volatility of all options in portfolio changes the same.

Question 12

Vega: Vega of instruments & how to hedge

Answer 12

Non-linear instruments
* Call option = Put option
* 𝒱 = Appendix
* Hedge:
* Long option: Short x of underlying
* Short option: Long x of underlying

Look into this

Position in traded option for gamma neutrality:
w_T = – ( 𝒱_pf / 𝒱_T )

Question 13

Rho: Theory

Answer 13

The rate of the change in option price w.r.t. changes in interest rate.

Typically not hedged, because BS model assumes r is constant.
𝝆

Question 14

Rho: Rho of instruments

Answer 14

Non-linear instruments
* Call option
* 𝝆 = Appendix > 0
* Put option
* 𝝆 = – 𝝆_call < 0

Question 15

Heding in practice

Answer 15

Ideally traders would rebalance frequently to make all greeks zero
* This is not possible and would be very expensive

Instead:
* Risk limits are often defined in terms of greeks
* Traders usually ensure their portfolios are delta neutral at least once a day
* They improve gamma and vega when possible
* You need options traded in the volume required at competitive prices

As the portfolio grows hedging is less expensive
* You hedge net positions, not individual positions