# 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$per troy ounce S = SQUAREROOT(2*THETA/GAMMA) S = SQUAREROOT(2*25.62/9.342) S = 2.34 = (2.34/1078.2)*100 = 0.21% , The market must move 0.21% during one day to breakeven Unfortunately, this number is much smaller than the numbers I am getting using the MARS model on bloomberg, or than a number that would make sense given the history of the option. Each day I do the calculation its off by a factor of 3-5. I'm expecting a number in the 0.7% to 1.0% range. Any ideas? I'm assuming the problem is units related, but I worry it could be equation related. Completely stumped. • the formula you wrote down looks correct$\Delta S = 2( \frac{\theta}{\gamma})^2$. Probably indeed, units. How about you list all the details of this option. I can quickly check for you. Spot, Strike, rate, delta, gamma, theta. Dec 21 '15 at 19:40 • Thanks mbision! I'll grab the latest variables on bloomberg OVML for ATM options. Strike: 1074.7 (ATM) Spot: 1074.7 delta: 50.8044% Gamma: 9.7177% or 9.71 Theta: -25.28 Rate: ? (Are you asking about the risk free rate? can assume it's zero) For the unit definitions see my original post. My confusion is to how to be modifying the units. Theta is clearly in$$, Underlying spot is in CME Gold contract units (100$?), and I've tried using both the gamma value as is, and as a percentage. Dec 22 '15 at 16:10
• just realized you wrote a different formula down than what I entered. Did you make a mistake on the syntax or is that the correct formula I should be using? Dec 22 '15 at 17:32
• I forgot to check on OVML today when i was in the office. But if you are still behind your pricer you could write down what your call price is with 60days to maturity. Then in OVME you bump the days to 59 days to maturity keeping all other parameters the same. Write down the call price. Compare the 2 prices. This should roughly be the theta you saw with 60days (up to potential scaling). I will set a reminder to look at the gold contract tomorrow when i m in the office. Dec 22 '15 at 17:41
• syntax is what i am personally used to. We can use your notation if you prefer. does not really matter. Dec 22 '15 at 17:42

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%. • First of all, I can't thank you enough for this. The reason the numbers don't add up is the values for theta and gamma from you're linked BLS pricer are different than the ones generated from bloomberg OVML Using your BLS pricer instead of bloomberg, then dividing BLS pricer theta by the number of trading days (not sure why I have to do this, as I thought theta was defined as day decay, but shrug) the answer make sense. Frustrating that I don't understand the concepts needed to know the differences between this black scholes calculator and bloomberg OVML. But I really appreciate it. Dec 28 '15 at 14:36 • Glad to be of help. I went over your original post and think why OVML was confusing. I updated my answer such that you can directly use the bloomberg results. Dec 28 '15 at 20:11 • perfect, almost reached full understanding! haha, you identified the problem. But I don't understand your modification to the gamma. "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." Don't understand the logic leap, Why does the spot price effect gamma here? Why is it a factor of 10 smaller than the gamma displayed by bloomberg? Are we dividing the gamma by the spot price and you did a quick approximation of the answer? Dec 29 '15 at 21:32 • Reason why i divide your USD9.71 by the spot 1075 is because that is the "format" your formula needs the gamma number to be. The formula you use comes from:$0.5 \gamma dS^2 + \theta * dt = 0$. In this formula$\gamma$is not a USD amount but a "percentage". Multiply the percentage gamma by dS and you have a USD delta. Multiply that USD delta with dS and you got a USD pnl (hence the$dS^2$). You can not use the USD gamma directly in this formula, it has to be in the "percentage" format in order to make sense when you multiply with$dS^2\$. Dec 30 '15 at 10:41