Option Gamma - Dynamic Delta Hedging

ΓGammaBy Group 1

GammaDynamic Delta HedgingContents Γ

Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras

The BasicsHedging

Binomial Model to Black & Scholes

Black & Scholes Formula

Trader’s Perspective of view

2. GammaThe 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 HedgingP&L: Stock Price Increase

P&L: Stock Price Decrease

P&L: Varying Stock Price

4. GammaVolatility?

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. ExtrasQuestions

Basketball Example

Moneyness

Relations of Greeks

Gamma01Dynamic Delta HedgingObjectives

To understand “Dynamic Delta Hedging”

To understand what gamma is and how it interacts with delta

HedgingWe 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 portfolioPortfolio is hedged against a certain risk if the portfolio value is not sensitive to that risk

GammaDynamic Delta HedgingΓ

What is a Hedge?Gamma

Dynamic Delta HedgingΓ

”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.

Black & Scholes FormulaGamma

Trader’s Angle GammaDynamic Delta HedgingΓ

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 GammaDynamic Delta HedgingΓ

“Simplicityis the ultimate sophistication”

Leonardo da Vinci

Ferrari Example GammaDynamic Delta HedgingΓ

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.

ANALYSISLikewise, the value of an option changes in respect to changes in

Underlying Stock Price, Delta

Time to maturity, Theta

Volatility, Vega

ANALYSISHowever, 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.

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.

Δ Delta GammaDynamic Delta HedgingΓ

Mathematical DefinitionΔ= “Slope of Curve at Current Price”

100 1 2 3 4 5 6 7 8 9

Spot Price (S)

Meaning & InterpretationSuppose 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

Error of Delta GammaDynamic Delta HedgingΓ

Static Hedging: Unrealized Profits

10 1 2 3 4 5 6 7 8 9

Spot Price (S)

100 1 2 3 4 5 6 7 8 9

Spot Price

Γ Gamma GammaDynamic Delta HedgingΓ

Mathematical DefinitionΓ= “The rate of change in Delta for changes in spot price”

Meaning & InterpretationHow volatile is the option relative to spot price Gamma measures curvature

Assumptions GammaDynamic Delta HedgingΓ

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

P&L: Spot Price Increase GammaDynamic Delta HedgingΓ

Portfolio: Call-ΔShares

S S1 2 5 6 7 8

Spot Price

{}Gain on Long

Loss on Short

P&L: Spot Price Decrease GammaDynamic Delta HedgingΓ

S S1 2 5 6 7 8

Spot Price

0 {}Gain on Short Loss on Long

P&L: Positive Gamma GammaDynamic Delta HedgingΓ

10 1 2 3 4 5 6 7 8 9

Spot Price (S)

ΔSΔ𝚷

P&L: Stock Price Increase GammaDynamic Delta HedgingΓ

Portfolio: Call-ΔShares 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 ++

Assumption: Ignore Time Decay

P&L: Stock Price Decrease GammaDynamic Delta HedgingΓ

Rebalancing (Dynamic Hedging)

0 200 0 None 0 None 0

1 100 ⬇ None + Long Shares ++

P&L: Varying Stock Price GammaDynamic Delta HedgingΓ

Rebalanced (Dynamic Hedging)

0 100 0 None 0 None 0

Implied Volatility GammaDynamic Delta HedgingΓ

SXTσr

Scenarios about the Maturity

Implied Volatility = Realized Volatility1.Implied Volatility < Realized Volatility2.Implied Volatility > Realized Volatility3.

Portfolio: Call-ΔSharesAssumption: 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⟹∞

Portfolio: Call-ΔSharesAssumption: Implied Volatility>Realized Volatility

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 - ? ?

Time Decay GammaDynamic Delta HedgingΓ

Effect of Time on the Option PriceC

Δ, Γ, θ GammaDynamic Delta HedgingΓ

Portfolio: Call-ΔSharesAssumption: Implied Volatility<Realized Volatility

0 100 0 None 0 None 0

1 50 ⬇ - + None - Long Shares ++2 100 ⬆ - + None - Short More

Share ++3 150 ⬆ - + None - Short More

Share ++θ<Γ , P&L>Premium

Portfolio: Call-ΔSharesAssumption: Implied Volatility=Realized Volatility

θ<Γ , P&L=Premium

0 100 0 None 0 None 0

1 99 ⬇ - + None - Long Shares +2 100 ⬆ - + None - Short More

Share +3 101 ⬆ - + None - Short More

Share +

Portfolio: Call-ΔSharesAssumption: Implied Volatility>Realized Volatility

θ>Γ , P&L<Premium

0 100 0 None 0 None 0

1 99 ⬇ - + None - Long Shares -2 100 ⬆ - + None - Short More

Share -3 101 ⬆ - + None - Short More

Share -

QUESTIONS & ANSWERS

Q&APLEASE, DON’T BE AFRAID!

Delta Neutral PositionGamma

(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.

Delta Neutral PositionGamma

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.

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.

Basketball ExampleGamma

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%

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 !

60 days before maturity

1 day before → bigger change in Delta, bigger Gamma

MoneynessGamma

100 1 2 3 4 5 6 7 8 9

Time to Maturity

At the Money

Out of the Money

In the Money

Relations of GreeksGamma

Time Delta Theta Gamma

Long Call + - +Long Put - - +Short Call - + -Short Put + + -

BondsGamma

THANKSFOR YOUR ATTENTION

Option Gamma - Dynamic Delta Hedging

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