Estimating an Option’s Break-Even Move from Theta and Gamma
Summary
The document uses a delta-hedged option approximation to estimate the underlying price move needed to offset one day of time decay. Setting the approximate gamma gain equal to theta decay gives a break-even move proportional to the square root of twice the decay divided by gamma. The question concerns a gold futures option and finds that the initial estimate is too small compared with a pricing model and historical expectations.
The answer identifies inconsistent scaling as the main issue: theta is reported for a contract multiplier, while delta and gamma are expressed on a per-underlying basis, and gamma is not itself a percentage just because it is quoted numerically. It recommends putting inputs on a consistent one-unit and percentage basis before applying the approximation, then converting the resulting move into a percentage and annualized volatility for comparison. This is a local approximation; it assumes a delta-hedged position and does not capture higher-order effects or changing Greeks.
Key ideas
- A delta-hedged option’s approximate daily P/L combines gamma gains with theta decay.
- Equating gamma gains and time decay gives an estimated break-even underlying move.
- Greeks must use consistent contract multipliers and underlying units before calculation.
- A quoted gamma value should not be mistaken for a percentage move.
- The result is a local approximation that omits higher-order effects and changing sensitivities.
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Full text
# Estimating profit/loss of a Gold Futures option using Theta and Gamma # Estimating profit/loss of a Gold Futures option using Theta and Gamma HELP! I am trying to find how much the underlying price of a gold futures option must move in order to breakeven on owning an option for a day. I was hoping someone versed in pricing options could identify a flaw in my reasoning? I am using an equation for calculating the profit and loss for a delta hedged option from. "option Trading Volatility: Trading Volatility, Correlation, Term Structure and Skew"Apr 24 2014 by Colin Bennett P/L = 1/2 * GAMMA * S^2 - TIMEDECAY S == the change in market price Assume P/L = 0 S = SQRT(2*TIMEDECAY/GAMMA) Here are some relevant variables I pulled from BLOOMBERG Option Valuation: Option expiry = Feb 16th Current Date = December 21st ImpliedVol = 13.579% Theta = -25.62 (sensitivity in option price to a decrease in 1 day of time to epxpiry) Gamma = 9.342 (sensitivity of delta to a change in spot) Price of GCH6 (Gold Future Underlying) = 1078.2 units Contract Unit = 100 Troy Ounces Price Quotation = U.S. Dollars and Cents per troy ounce Minimum Price Fluctuation = .10$ per troy ounce S = SQUAREROOT(2*THETA/GAMMA) S = SQUAREROOT(2*25.62/9.342) S = 2.34 = (2.34/1078.2)*100 = 0.21% , The market must move 0.21% during one day to breakeven Unfortunately, this number is much smaller than the numbers I am getting using the MARS model on bloomberg, or than a number that would make sense given the history of the option. Each day I do the calculation its off by a factor of 3-5. I'm expecting a number in the 0.7% to 1.0% range. Any ideas? I'm assuming the problem is units related, but I worry it could be equation related. Completely stumped. ## Answer by mbison (score 2, accepted) https://quant.stackexchange.com/a/22465 I don't understand why you think the numbers dont match up. In my opinion it all works out. Perhaps best if you first convert all numbers to percentages and for 1 underlying instead of 100 multiplier. From OVML you have - multiplier = 1 troy ounce - S = 1075 - K = 1075 - r = 0.0033 - T = 2/12 - sig = 0.12 Convert all into percentages: - S = 100 - K = 100 - r = 0.0033 - T = 2/12 - sig = 0.12 Stick into BLS pricer http://www.soarcorp.com/black_scholes_calculator.jsp - Call = 2.06 - delta = 51% - theta (daily) = -0.025 - gamma = 0.0781 Stick into your formula: dS = sqrt( 2 * 0.025/0.071) = 0.7979 Convert into percentage move = dS/S = 0.007979 = 0.8% (on daily basis) Convert to annual move: dS/S * sqrt(252) = 0.8% * sqrt(252) = 12.6% (which is very close to the implied vol you started out with, so it all makes sense). Update: I went over your original email and now see why the bloomberg pricer ovml might be confusing you. In your original email you gave the following numbers> > Strike: 1074.7 (ATM) Spot: 1074.7 delta: 50.8044% Gamma: 9.7177% or 9.71 Theta: -25.28 Notice how you say that a gamma of 9.71 equals 9.71%? Your spot was 1075, therefore a gamma of 9.71 does not equal 9.71% but about 0.97%. This is were you went wrong. Furthermore, it looks to me that the bloomberg pricer scales the theta 25.28 by the number of contracts (100 in your case) but that it does not scale the delta or gamma. The pricer simply gives you a delta of 51% and gamma 0.97%. So the theta of 1 contract is 25.28/100 = 0.2528 Sticking the number into your formula gives you $sqrt(2* 0.2528 / 0.0097) = 7.4$ (in USD). This converts into a percentage move of about $7.4 / 1075 = 0.7%. This daily percentage move of 0.7% you can convert into an annual move by multiplying with Sqrt(252). This should give you an annualized vol number of about 11%.
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