Option Gamma - Dynamic Delta Hedging

GammaDynamic Delta Hedging
Contents
Γ
Basics Gamma Simpliﬁed Dynamic Hedging Volatility & Time Decay Extras
The Basics
Hedging
Binomial Model to Black & Scholes
Black & Scholes Formula
Trader’s Perspective of view
2. Gamma
The Car Example
What is Delta? & What is Delta Hedging?
Issues with Delta Hedging: Why Gamma is Important
What is Gamma?
Spot Price Increase
Spot Price Decrease
Positive Gamma
1.
Simpliﬁed Dynamic Delta Hedging
P&L: Stock Price Increase
P&L: Stock Price Decrease
P&L: Varying Stock Price
3.
4. Gamma
Volatility?
Time Decay, The Role of Theta
Δ, Γ, θ
P&L: Small Change & Realized Vol.
P&L: Small Change & Implied Vol.
P&L: Small Change & Real.=Imp. Vol.
5. Extras
Questions
Basketball Example
Moneyness
Relations of Greeks

Gamma01
Dynamic Delta Hedging
Objectives
To understand “Dynamic Delta Hedging”
To understand what gamma is and how it interacts with delta
Γ

02
Hedging
We presumed that the options are needed to hedge
risks involving a position in the underlying security
Hedging = the reduction of risk
Dynamic hedging: frequently adjusting portfolio
Portfolio is hedged against a certain risk if the portfolio value is not
sensitive to that risk
GammaDynamic Delta HedgingΓ

03
What is a Hedge?
”The delta is approximately the change in the option
price for a small change in the stock price”
Please recall: Hedging in the binomial model
!
We buy h shares of stock and sold one call
We are hedged as long as we adjust the number of options per
share according to the formula for h
Delta Hedge
!
! is the above hedge in the Black-Scholes-Merton world.
A delta hedge must be done continuously to maintain our risk-
free position.
!

03
Black & Scholes Formula

05
Trader’s Angle
Only understanding the “Greeks” can help you!
!
*(This image is that of an individual trader, and is only used to illustrate that traders consider it. The accuracy of these particular numbers are not veriﬁed.)

06
“Simplicity
is the
ultimate sophistication”
Leonardo da Vinci

07
Ferrari Example
You want to buy a Ferrari!
Mileage, Age, Trends, etc. are indicators for the
value of your Ferrari.
These change the value of your car.
Example: Your Ferrari lost $1000 in its value
for every 25,000 km you drove.

08
ANALYSIS
Likewise, the value of an option
changes in respect to changes in
Underlying Stock Price, Delta
Time to maturity, Theta
Volatility, Vega
Etc.,

09
ANALYSIS
However, the change in delta,
theta, volatility etc, is not
constant. For example:
Instead of your Ferrari losing $1000
in value for every 25,000 miles,
Your Ferrari loses;
$1,000 for the ﬁrst 25,000 miles,
$2,000 for the second 25,000 miles,
$3,000 for the third 25,000 miles.

10
ANALYSIS
Likewise, since the delta is NOT constant,
An option’s Gamma tells us by how much
an option’s delta changes when the
underlying product’s price moves.

11
Δ Delta
Mathematical Deﬁnition
Δ=
“Slope of Curve at Current Price”
1.
100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
Spot Price (S)
OptionPrice(Call)
Δ}
Meaning & Interpretation
Suppose that: Δ=0.7 S(↑)=$1 C(↑)= Then, $0.7
“Hedge Ratio”
2.
Delta Hedging
“An options strategy that aims to reduce
(hedge) the risk associated with price
movements in the underlying asset by offsetting
long and short positions” Investopedia
3.

12
Error of Delta
Static Hedging: Unrealized Proﬁts
101 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
Spot Price (S)
OptionPrice

100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
Spot Price
OptionPrice
Δ
Γ
0
Γ2
1
Δ0
Δ1Δ
Δ2
13
Γ Gamma
Mathematical Deﬁnition
Γ=
“The rate of change in Delta for changes
in spot price”
1.
Meaning & Interpretation
How volatile is the option relative to spot
price
Gamma measures curvature
2.

14
Assumptions
No Dividends1.
Volatility (Vega) is constant2.
Interest Rate (Rho) is not considered3.
No Transaction Costs4.

15
P&L: Spot Price Increase
Portfolio: Call-ΔShares
S S1 2 5 6 7 8
10
0
C
C
1
2
4
5
6
8
9
Spot Price
OptionPrice
0 1
0
1
{}Gain on Long
Loss on Short

16
P&L: Spot Price Decrease
S S1 2 5 6 7 8
10
0
C
C
1
4
5
6
7
8
9
Spot Price
OptionPrice
0 1
0
{}Gain on Short
Loss on Long
1
S1

17
P&L: Positive Gamma
101 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
Spot Price (S)
OptionPrice
⦁
ΔSΔ𝚷

18
No Rebalancing (Static
Hedging)
Rebalancing (Dynamic
Hedging)
Time Stock Price Change in Δ Action P&L Action P&L
0 100 0 None 0 None 0
1 150 None +
Short More
Shares
++
2 200 None +
Short More
Shares
++
Assumption: Ignore Time Decay

19
P&L: Stock Price Decrease
Hedging)
Rebalancing (Dynamic
Hedging)
Time Stock Price Change in Δ Action P&L Action P&L
0 200 0 None 0 None 0
1 100 ➡ None + Long Shares ++

20
P&L: Varying Stock Price
No Rebalancing
(Static Hedging)
Rebalanced (Dynamic
Hedging)
Time
Stock
Price
Change in Δ Action P&L Action P&L
0 100 0 None 0 None 0
2 100 None +
Short More
Shares
++
3 150 None +
Short More
Shares
++

21
Implied Volatility
S
X
T
σ
r
B/S C
Scenarios about the Maturity
Implied Volatility = Realized Volatility1.
Implied Volatility < Realized Volatility2.
Implied Volatility > Realized Volatility3.

22
Assumption: Implied Volatility<Realized Volatility
Portfolio: Call-ΔShares Rebalancing
Time Stock Price Change in Δ Action P&L
0 100 0 None 0
1 150 Short More Share +
2 200 Short More Share +
Call Premium< P&L
IF (Realized Volatility-Implied Volatility)⟹∞
Then P&L⟹∞
++

23
Assumption: Implied Volatility>Realized Volatility
Hedging)
Rebalanced (Dynamic
Hedging)
Time
Stock
Price
Change in Δ θ Γ Action P&L Action P&L
0 100 0 None 0 None 0
1 99 ➡
- + None - ? ?
2 100 - + None - ? ?
3 101 - + None - ? ?

24
Time Decay
Effect of Time on the Option Price
C
S

25
Δ, Γ, θ
C
S
?

26
Assumption: Implied Volatility<Realized Volatility
No Rebalancing
(Static Hedging)
Rebalanced (Dynamic
Hedging)
Time
Stock
Price
0 100 0 None 0 None 0
1 50 ➡
- + None - Long Shares ++
2 100 - + None - Short More
Share ++
Share ++
θ<Γ , P&L>Premium

28
Assumption: Implied Volatility=Realized Volatility
θ<Γ , P&L=Premium
No Rebalancing
(Static Hedging)
Rebalanced (Dynamic
Hedging)
Time
Stock
Price
0 100 0 None 0 None 0
1 99 ➡
- + None - Long Shares +
Share +
Share +

29
Assumption: Implied Volatility>Realized Volatility
θ>Γ , P&L<Premium
No Rebalancing
(Static Hedging)
Rebalanced (Dynamic
Hedging)
Time
Stock
Price
0 100 0 None 0 None 0
1 99 ➡
- + None - Long Shares -
Share -
Share -

30
QUESTIONS & ANSWERS
Q&APLEASE, DON’T BE AFRAID!

31
Delta Neutral Position
(Ex) Suppose c=$10,S=$100,andΔ=0.6. 
In addition, you sold 20 options (For this, you do not have to know
the strike price.).

– To be Delta-Neutral, buy 0.6 × 2,000 shares of stock.

– If the price of the underlying security goes up by $1,

from the short position in options, 
a loss of $0.6 × 2,000 = $1,200.

– If the price of the underlying security goes up by $1, from the long
position in stocks,

a gain of $1 × 1,200 = $1,200. 
– Therefore, the gain and the loss offset each other.


32
Delta Neutral Position
Since the option Δ changes with the stock price, the as soon as the
stock price moves, the position is no longer Delta-Neutral.

• (Ex) Suppose now at S = $110, Δ = 0.65. Then to become Delta-
Neutral again, you need to buy

0.05 × 2,000 = 100 
• This is called Rebalancing, and the hedging scheme that

additional shares. 
involves rebalancing is called Dynamic Hedging Scheme.

• Notice that this hedging scheme (hedging a short position in calls)
involves a “buy high and sell low” strategy.


33
Gamma Hedging - Example

A portfolio is Delta-Neutral and has a Gamma of –3,000. The Delta
and Gamma of a traded call option are 0.62 and 1.50. How would
the portfolio become Gamma-Neutral as well as Delta-Neutral by
adding the call options?

26

– Number of options to buy = -(-3,000)/1.5 = 2,000.

– New Delta of the portfolio = 2,000×0.62 = 1,240.

– In order to make the portfolio Delta-Neutral again, 1,240 shares of
the underlying stock has to be sold.


34
Basketball Example
Although we all know that Yonsei is so much superior,
for our gamma sake,
We are going to assume that
!
!
Korea and Yonsei has equal level of strength and skill for their
basketball teams.
10min. before the end of the game With 5 point apart
!
!
!
!
Yonsei's Chance of Winning = 55%
!
!

34
Basketball Example
30sec. before the end of the game
!
!
!
!
!
With the same 5 point apart, our chance of winning jumped from
55% to 95% at the end of the game
!
!
!

34
Basketball Example
60 days before maturity
1 day before → bigger change in Delta, bigger Gamma

35
Moneyness
100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
Time to Maturity
Gamma
At the Money
Out of the Money
In the Money

36
Relations of Greeks
Time Delta Theta Gamma
Long Call + - +
Long Put - - +
Short Call - + -
Short Put + + -

36
Bonds

29
THANKSFOR YOUR ATTENTION

Option Gamma - Dynamic Delta Hedging

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