Undergraduate Research and Trading Club February 2, 2017 Gamma Scalping Undergraduate Research and Trading Club February 2, 2017

Option valuation depends on: S = stock (underlying) price E = exercise (strike) price T = time to expiration r = interest rate σ = stock (underlying) return volatility D = dividends (I’ll ignore)

Option Greeks Delta – change in option price as stock price changes Gamma – change in Delta as stock price changes (second derivative of change in option price as stock price changes) Theta – change in option price as time to expiration changes Rho – change in option price as interest rates change Vega – change in option price as implied volatility changes

Bets with options Delta is a bet on the direction of the stock price – Long positions in calls have positive deltas and long position in puts have negative deltas. Gamma – Long positions in options have positive gamma. One form of buying this gamma (it isn’t free) and hedging the directional risk (delta = 0 or delta neutral) is gamma scalping. This is bet on realized volatility (versus implied volatility in Vega). Theta – Long positions in options are hurt by the passage of time. Gamma vs. Theta is the typical trade-off with long option positions: you win if realized volatility is large (relative to implied volatility) and lose if realized volatility is small and time passes.

Picture of Call Value as a function of S Note Delta is a straight line drawn tangent to the curve at the current S Gamma is (roughly) the difference between the curve and the straight line Gamma is a good thing – price increases accelerate and price decreases decelerate Suggests that it costs How to see Theta? What is the option’s value at expiration?

Gamma Scalping Example S = $40, E = $40, T = 90 days = .2466 years, r = 0.005/yr., σ = .25/yr. Position Black-Scholes Price Delta Gamma Call $2.00 0.53 .08 Put $1.95 -0.47 Portfolio 0.06 .16

Trade To acquire Gamma, you need to buy options. Let’s buy both the put and the call to reduce the Delta of our positon. (Assume option is for a single share of stock.) That leaves us with a portfolio Delta of +0.06. How can we fix that? Sell other options – that would reduce the Gamma. Sell (short) stock – it has a Delta = 1 and a Gamma = 0. Choose the latter: sell (short) .06 shares of stock at $40 If you don’t like partial shares of stock, then multiply the number of options purchased by a big enough number to make that whatever you would like. You would end up multiplying my profit numbers by the same factor.

What happens as price changes? Suppose that the price initially rises to $42. An aside: Estimating price change in the option’s price using delta and gamma (ignoring changes in time, implied volatility and interest rates) = Change in S*Δ + (1/2)*(Change in S)^2*Γ = $2*.53 + ½*$4*.08 = $1.22 It then returns to $40. Time will play a crucial role in the example. Initially, I assume no time passes. Then, I redo the example assuming it takes 2 weeks to get to $42 and 2 more weeks to get back to $40

S = $42, T = 90 days (all other inputs constant) Position Black-Scholes Price Delta Gamma Call $3.21 0.68 .075 Put $1.17 -0.32 Portfolio 0.36 .15

Profit and New Trade (to keep Delta = 0) Call = $3.21 - $2.00 Put = $1.17 - $1.95 Stock = -.06 * $2 Portfolio = $1.21 - $0.78 - $0.12 = +$0.31 Trade to keep portfolio Delta neutral Must change -0.36 Delta to zero We are already short .06 shares Need to short .3 more shares at $42

S = $40, T = 90 days Let S return to $40 (again without passage of time). Position Black- Scholes Price Delta Gamma Call $2.00 0.53 .08 Put $1.95 -0.47 Portfolio .06

Profit and New Trade Because I assumed no time passes, the call and the put show no overall profit or loss. Because the portfolio Delta = 0.06 (as it was initially), we must repurchase 0.30 shares of our short stock position. We shorted .3 shares when S = $42 and repurchase it when S = $40. We repurchase that at a gain of .3 * $2 = $0.60. This is repeatable. We always sell when the price goes up and buy when the price goes down. Realized volatility is our friend. Would have worked if stock price fell to $38 (requiring us to buy stock to partially undo our short position) and then returned to $40 (requiring us to sell stock to return the short position to its initial level). However, time is our enemy.

Repeat the same example except allow 2 weeks to pass between each price point. S = $42, T = 76 days = .20822 year Position Black-Scholes Price Delta Gamma Call $3.05 0.69 .08 Put $1.02 -0.31 Portfolio 0.38

Profit and New Trade Profit Trade to keep portfolio Delta neutral Call = $3.05 - $2.00 Put = $1.02 - $1.95 Stock = -.06 * $2 Portfolio = $1.05 - $0.93 - $0.12 = +$0.00 (coincidence it is exactly zero) Trade to keep portfolio Delta neutral Must change -0.38 Delta to zero We are already short .06 shares Need to short .32 more shares at $42

Two more weeks pass and S returns to $40 S = $40, T = 62 days = .16986 year Position Black-Scholes Price Delta Gamma Call $1.66 0.52 .10 Put $1.63 -0.48 Portfolio 0.04

Profit and New Trade Option Positions’ Profit Call: $1.66 - $2.00 = -$0.34 Put: $1.63 - $1.95 = -$0.32 With new portfolio Delta of 0.04, we need to repurchase 0.34 of our short 0.38 share position. We repurchase the .32 shares we sold at $42 for a $2 profit We repurchase .02 shares we sold at $40 for a $0 profit Or a total profit on the stock of .32*$2 = $0.64 Now, total profit = -$0.34 - $0.32 + $0.64 = -$0.02.

More general gamma play - earnings Compare the cost of a straddle to the “typical” stock price movement around an earnings announcement. Consider a $100 stock with an earnings announcement in a week. If E = $100, r = .0025, and σ = .5, then C = $2.77 and P = $2.76. So, the cost of a straddle is $5.53 or 5.5% of the stock price. Breakeven at $94.45 and $105.53. Will stock price get out of that range? How much does the stock price usually move (in absolute terms) around an earnings announcement? If less than 5.5%, then sell the straddle. If more than 5.5%, then buy the straddle.