Gamma Hedging The Gobs of Money Machine Wilhelm's Warriors No Hedgetation Gamma Hedging

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Example diagnostic: why is the smart hedger stuck ?

Main parameters to tweak Frequency of hedging (HedgingToday) Family Delta threshold (NeedToHedge) Type of transaction preferences (CalcCandidateRecScores) A-E options preferences (scoring rules) Delta threshold (CalcQtyNeededToHedge) Family delta target (CalcQtyNeededToHedge) Max buy cushion (MaxPurchasePossible) Available cash cushion (AvailableCashIsLow) Max margin cushion (TooCloseToMaxMargins) Max short cushion (MaxShortWithinConstraints)

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Hedging Gamma, Vega, Theta & Rho

Delta Neutral Portfolio Decay Payoff Put portfolio Call portfolio Stock price Source: Delta Gamma Hedging and the Black-Scholes Partial Differential Equation - S. Raju, JEFE 2012

Delta & Gamma neutral Portfolios Payoff Call portfolio Put portfolio Stock price Source: Delta Gamma Hedging and the Black-Scholes Partial Differential Equation - S. Raju, JEFE 2012

Gamma Family Delta (D) measures the change in portfolio value as the underlier’s price S changes (~speed). Family Gamma (G) measures the rate of change in portfolio value as S changes (~acceleration).

Meaning of Gamma If Gamma is small (abs.), small changes in S will not affect much Delta (and your portfolio value) , so there is less need to take immediate rebalancing action. If Gamma is large, small changes in S will affect Delta (and your portfolio value) significantly, so there is a stronger need to take immediate rebalancing action.

How to Calculate Gamma Gamma = N’(d1) SsÖ t N’(d1) = e –(d1)2/2 Ö (2 p) d1 as in Black Scholes Gamma N’ is the prob density function for a standard normal distribution. Stock price S Strike

Gamma of a Portfolio of Related Securities Gfamily = S qtyi * gi

Gamma-Neutral Portfolio More stable than a delta neutral-only. Cannot use the stock to reach Gamma neutrality because the stock has Gamma = 0 1) Gportfolio <> 0 2) Gportfolio + Gx qtyx = 0 3) qtyx = -Gportfolio / Gx Warning: Acquiring qtyx will disturb Delta neutrality. You will need to rebalance.

Delta Gamma Delta It’s a financial strategy, not a sorority. Find out what you need to achieve Delta neutrality. Find out what you need to achieve Gamma neutrality. Find out what you need to re-achieve Delta neutrality. Stock is ideal because it will not affect Gamma.

Three Dark Horses Simultaneous Delta Gamma Conditional Gamma Synthetics Come and see me – not on the last day!

Simultaneous Delta Gamma Dportfolio + dx1 qtyx1 + dx2 qtyx2 = 0 Gportfolio + gx1 qtyx1 + gx2 qtyx2 = 0 All known except qty x1 and qty x2 {

Conditional Delta Gamma If below target -> delta only else Delta + Gamma

Offset the Position with a Synthetic Security Profit & Loss Perfect hedge, but costly. Synthetic Short position Long position to hedge Stock price Total Payoff

Put-Call Parity For European Ps and Cs that have the same strike K, and expire by the same time t: P + S = C + K e-rt We can solve for S, P, or C, effectively synthesizing a security with a combination of the other two and some interest- earning cash. Example: S = C + Ke-rt – P and - S = - C - Ke-rt + P

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The RoboTrader Demo

The two timers (simplified) Timer 1 (spy) Timer 2 (seconds on screen) Start Check the date on the DB New date? Decrease the seconds by 1 No Yes Run daily routine Set seconds to 59 Less than 0? Yes No