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
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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
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Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
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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
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What is a Hedge?Gamma
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Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
”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.
!
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Black & Scholes FormulaGamma
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Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
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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.)
Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
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“Simplicityis the ultimate sophistication”
Leonardo da Vinci
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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.
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ANALYSISLikewise, the value of an option changes in respect to changes in
Underlying Stock Price, Delta
Time to maturity, Theta
Volatility, Vega
Etc.,
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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.
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Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
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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.
GammaDynamic Delta HedgingΓ
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Δ Delta GammaDynamic Delta HedgingΓ
Mathematical DefinitionΔ= “Slope of Curve at Current Price”
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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
3.
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Error of Delta GammaDynamic Delta HedgingΓ
Static Hedging: Unrealized Profits
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Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
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Γ Gamma GammaDynamic Delta HedgingΓ
Mathematical DefinitionΓ= “The rate of change in Delta for changes in spot price”
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Meaning & InterpretationHow volatile is the option relative to spot price Gamma measures curvature
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Assumptions GammaDynamic Delta HedgingΓ
No Dividends1.Volatility (Vega) is constant2.Interest Rate (Rho) is not considered3.No Transaction Costs4.
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P&L: Spot Price Increase GammaDynamic Delta HedgingΓ
Portfolio: Call-ΔShares
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P&L: Spot Price Decrease GammaDynamic Delta HedgingΓ
Portfolio: Call-ΔShares
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P&L: Positive Gamma GammaDynamic Delta HedgingΓ
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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 ++
2 200 ⬆ None + Short More Shares ++
Portfolio: Call-ΔShares
Assumption: Ignore Time Decay
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P&L: Stock Price Decrease 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
0 200 0 None 0 None 0
1 100 ⬇ None + Long Shares ++
2 50 ⬇ None + Long Shares ++
Portfolio: Call-ΔShares
Assumption: Ignore Time Decay
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P&L: Varying Stock Price GammaDynamic Delta HedgingΓ
Portfolio: Call-ΔShares
Assumption: Ignore Time Decay
Portfolio: Call-ΔShares 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
1 50 ⬇ None + Long Shares ++
2 100 ⬆ None + Short More Shares ++
3 150 ⬆ None + Short More Shares ++
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Implied Volatility GammaDynamic Delta HedgingΓ
SXTσr
B/S C
Scenarios about the Maturity
Implied Volatility = Realized Volatility1.Implied Volatility < Realized Volatility2.Implied Volatility > Realized Volatility3.
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P&L: Stock Price Increase GammaDynamic Delta HedgingΓ
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⟹∞
Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
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P&L: Stock Price Increase GammaDynamic Delta HedgingΓ
Portfolio: Call-ΔSharesAssumption: Implied Volatility>Realized Volatility
Portfolio: Call-ΔShares No Rebalancing (Static 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 - ? ?
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Time Decay GammaDynamic Delta HedgingΓ
Effect of Time on the Option PriceC
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Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
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Δ, Γ, θ GammaDynamic Delta HedgingΓ
C
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P&L: Stock Price Increase GammaDynamic Delta HedgingΓ
Portfolio: Call-ΔSharesAssumption: Implied Volatility<Realized Volatility
Portfolio: Call-ΔShares No Rebalancing (Static 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 50 ⬇ - + None - Long Shares ++2 100 ⬆ - + None - Short More
Share ++3 150 ⬆ - + None - Short More
Share ++θ<Γ , P&L>Premium
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P&L: Stock Price Increase GammaDynamic Delta HedgingΓ
Portfolio: Call-ΔSharesAssumption: Implied Volatility=Realized Volatility
θ<Γ , P&L=Premium
Portfolio: Call-ΔShares No Rebalancing (Static 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 - Long Shares +2 100 ⬆ - + None - Short More
Share +3 101 ⬆ - + None - Short More
Share +
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P&L: Stock Price Increase GammaDynamic Delta HedgingΓ
Portfolio: Call-ΔSharesAssumption: Implied Volatility>Realized Volatility
θ>Γ , P&L<Premium
Portfolio: Call-ΔShares No Rebalancing (Static 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 - Long Shares -2 100 ⬆ - + None - Short More
Share -3 101 ⬆ - + None - Short More
Share -
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QUESTIONS & ANSWERS
Q&APLEASE, DON’T BE AFRAID!
GammaDynamic Delta HedgingΓ
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Delta Neutral PositionGamma
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(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.
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Delta Neutral PositionGamma
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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.
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Gamma Hedging - Example
GammaDynamic Delta HedgingΓ
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.
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Basketball ExampleGamma
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Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
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%
!!
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Basketball ExampleGamma
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Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
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 !
!!
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Basketball ExampleGamma
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Basics Gamma Simplified Dynamic Hedging Volatility & Time Decay Extras
60 days before maturity
1 day before → bigger change in Delta, bigger Gamma
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MoneynessGamma
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Relations of GreeksGamma
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Time Delta Theta Gamma
Long Call + - +Long Put - - +Short Call - + -Short Put + + -
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BondsGamma
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THANKSFOR YOUR ATTENTION
GammaDynamic Delta HedgingΓ