The Greek Letters

Read Hull, Chapter 17

Outline

Definition of the Greeks
Why hedge?
Delta Hedging
Gamma Hedging
Vega Hedging
Note
Homework Problem

Definition of the Greeks

The chart below shows how a change in the variable changes the option's value

The American and European options are the same except for time
For example, if the stock price, S ₀ , increases by a small amount
- The value of the call option increases
- The value of the put option decreases

	Greek	c	p	C	P
S ₀	Delta	+	-	+	-
K		-	+	-	+
T	Theta	?	?	+	+
σ	Vega	+	+	+	+
r	Rho	+	-	+	-
D		-	+	-	+

The chart refers to the first derivatives of the Black-Scholes equations to respect to one of the variables
- Indicates a first derivative with respect to the Greek
Delta – the approximate change in the option’s value with respect to a one-unit change in the underlying asset's price, S ₀
- Thus, if the S ₀ increases by one unit, the call option increases while the put option decreases
- Look at the chart, call is positive and put is negative for both American and European
- Why?
- The greater the stock price, the more likely the holder exercises the call option and less likely the holder exercises the put option.
- delta ≈ Δ option price / Δ S
Vega – the approximate change in the option’s value with respect to a one-unit change in volatility.
- Thus, if the volatility increases by a small amount, the values of both calls and puts rise.
- Why? More volatility indicates greater price swings. Thus, the holder is more likely to exercise the option.
- vega ≈ Δ option price / Δ σ
Theta – the approximate change in the option’s value with respect to a one-unit increase in time.
- Why? The longer the time period for an American option, the holder has more time to exercise the option.
- Usually measured in one calendar day
- theta ≈ Δ option price / Δ T
Rho – the approximate change in the option’s value with respect to a one-unit increase in the risk-free interest rate.
- For example, the value of a call option rises while the value of a put option falls.
- Why? The risk free interest rate refers to the time value of money.
- rho ≈ Δ option price / Δ r
Second derivative
Gamma – the second derivative with respect to delta
- The approximate change of delta for a one-unit increase in the spot price of the underlying asset.
- gamma ≈ Δ delta / Δ S

Why Hedge?

Bank sells a European call option for 10,000 shares of a non-dividend paying stock
Bank sells the call options for $35,000
- To create the call options, Black-Scholes equations estimate the calls cost $30,803
- Profit = 35,000 – 30,803 = 4,197
Characteristics of the call option
- S ₀=60, K=65, r=7%, s=25%, T=0.5
- c = 3.0803 (or $30,803)
- Used program from www.ken-szulczyk.com/programs/program_black_scholes.php
Naked Position
- Bank does not buy stock
- If stock prices falls below $65
  - Holder does not exercise call
  - Bank’s position does not change and keeps the option premium
  - Profits stay at 4,197
- If stock price rises to $75
  - Holder exercises call and buys stock at $65
  - Bank must sell at $65 and takes a $10 loss on each share of stock
  - Total losses = $10*10,000 = $100,000
Covered Position
- Bank buys 10,000 shares of stock
  - Bank pays $60x10,000 = 600,000
- If stock prices falls below $65
  - Holder does not exercise calls
  - Bank’s stock portfolio loses value for stock price less than $60
  - Let S _T = 55
  - Bank loses (55 – 60)x10,000 = -50,000
- If stock price rises above $65
  - Holder exercises call and buys stock at $65
  - Bank must sell at $65 and sells its stock portfolio
  - Bank still made money on the stock portfolio

Delta Hedging

Every asset in a portfolio has a Δ
- Buying a share of stock (S) has Δ = 1
  - Delta definition - If the stock price increases by $1, the value of the stock increases by $1
- If you short the stock, the Δ = -1
- Gamma must be zero because delta does not change
  - gamma = 0
- Also, vega = 0
Be consistent on terminology
- Assets – stock, currency, or a commodity
- All examples use stock in lecture notes while tutorial uses a currency
- The variable S reflects the asset
Delta hedging finds a middle ground between naked and covered positions
We set up a riskless portfolio by shorting one call option:
From tutorial

A portfolio has a delta
- Portfolio _Delta = -1 Δ _option + w = 0
Delta hedging requires the portfolio’s delta to equal zero
- We must buy Δ shares of stock to make portfolio delta neutral
- w = Δ _option
- We know the call delta is positive
- A minus sign indicates a short while a plus sign indicates a long
We set up a riskless portfolio by longing one put option:
From Tutorial

Portfolio
- Portfolio _Delta = 1 Δ _option + (-1) w = 0
- w = Δ _option
- We know a put option is negative so w is negative
- We must short Δ shares of stock to make portfolio delta neutral
Example
- Trader sells 50 call options with a Δ = 0.75 for $15 each
- Each call has 100 shares
- The trader must sell 5,000 shares if holder exercises the calls
- The trader collects 5,000x15 = 75,000 in premium
- How many shares should we buy to make portfolio delta neutral?
Solution
- Delta definition - a $1 increase in the stock price increases the option’s value by $0.75
- Portfolio _Delta = -5,000 (0.75) + w = 0
- Remember the negative indicates we short the calls
- w = 3,750
- Thus, we buy (long) 3,750 shares of stock because each stock share has a delta = 1
The gain from the stock would offset the loss on the call option, or vice versa
If the stock price increases by $1
- The trader gains on the stocks by $1 x(3,750) = $3,750
- The option price increases by delta, so all call options increase by $3,750
- Change in the portfolio = $3,750 - $3,750 = 0
- Thus, the portfolio’s value does not change
Hedging Types
- Dynamic – you continuous adjust your portfolio to keep delta neutral
- Static - you adjust your portfolio once for delta neutrality
A cool result
- If we delta balance the portfolio by financing the stock purchases at r% interest, the total borrowing costs would reflect the Black-Scholes option price
- We synthetically mirror the option by hedging against it

Gamma Hedging

If gamma is a large number
- Then delta experiences large swings for a change in the stock price
If gamma is a small number
- Then delta experiences little variations for a change in the stock price
We can use gamma hedging to reduce swings in the portfolio’s value
For example, you short 100 put options for a stock
- Each option entails 100 stocks and has a
- delta = -0.65 and gamma = 0.5
- Calculate the delta and gamma for your portfolio
  - Portfolio _Delta = -100x100 (-0.65) = 6,500
  - Portfolio _Gamma = -100x100 (0.5) = -5,000
- We balance the portfolio for delta
  - Portfolio _Delta = -10,000(-0.65) + w = 0
  - w = -6,500
  - You short 6,500 shares of stock to make portfolio delta neutral
If the stock price increases by $1 (or ΔS = 1)
- The delta changes by -0.65 + (0.5)(-1) = -1.15
- You shorted the call which is why it is negative
- Then we calculate the portfolio’s delta
  - Portfolio _Delta = -10,000 x (-1.15) = -11,500
- We rebalance the portfolio to make portfolio delta neutral again
  - Portfolio _Delta = -10,000(-1.15) -6,500 + w = 0
  - w = -5,000
  - You already shorted 6,500 shares and need to short and additional 5,000 stock shares to balance delta again
We want to balance the portfolio for both gamma and delta
- You can buy a new option with delta = 0.25 and gamma = 1.25
- First, we balance the portfolio for gamma
  - Portfolio _Gamma = -100x100x(0.5) + w (1.25) = 0
  - w=4,000
  - You must long 4,000 new options to make portfolio gamma neutral
- Second, we balance the portfolio for delta
  - Portfolio _Delta = -100x100x(-0.65)+4,000x(0.25) + w = 0
  - w=-7,500
  - You must short 7,500 stock shares to make portfolio delta neutral
Note: Most likely vega will not balance

Vega Hedging

Vega reflects the asset’s volatility and can change over time
- A large vega means a portfolio is sensitive to changes in volatility
- A small vega means a portfolio is not sensitive to changes in volatility
Example
- A trader shorts 500 call options
- Each option entails 100 stock shares and has a
  - Delta = 0.75
  - Vega = 2.0
- The trader can long or short a new put
  - Delta = -0.35
  - Vega = 4.0
First, balance the vega
- Portfolio _Vega = -500x100 (2.0) + w(4.0) = 0
- w = 25,000
- You must long 25,000 units of the new option to make portfolio vega neutral
Second, balance the delta
- Portfolio _Delta = -500x100(0.75) + 25,000(-0.35) + w = 0
- w = 46,250
- You must long 46,250 stock shares to make portfolio delta neutral
Note: Gamma most likely will not balance

Note

Please be careful of the minus sign
- Gamma is positive for longs and negative for shorts for both calls and puts
- The tutorial and lecture notes places the negative sign with the position
- The textbook places the minus sign with gamma
Vega is always positive
There is no theta hedging
- You cannot hedge against time
Changes in the interest rate can change the portfolio value
- No similar portfolio balancing

Homework

Please work out this problem
You are a trader with a stock option portfolio as below:

Type	Position	Option Delta	Option Gamma	Option Vega
Call	-2,000	0.40	1.50	0.80
Call	-1,500	0.75	0.20	0.30
Put	-1,000	-0.60	0.80	1.10
Put	-3,000	-0.25	1.25	1.50

You can trade with a new option
Option: Delta = 0.5, Gamma = 1.3, and Vega = 0.75
1. Please delta hedge this portfolio
2. Please gamma and delta hedge this portfolio
3. Please vega and delta hedge this portfolio
4. Please gamma, vega, and delta hedge the portfolio at the same time
  - You have two options
    - Option #1: Delta = 0.5, Gamma = 1.3, and Vega = 0.75
    - Option #2: Delta = 0.3, Gamma = 0.6, and Vega = 0.5