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Ive been doing some work on looking at historical options prices on a stock index using real data, and I came across an odd example that I cant really get my head around. I am aware that for extreme cases, we can have a theta > 0 for Deep ITM options, but I came across the following example that seems to imply a positive theta for an ATM call :

DF = 1.000519 #discount factor
vol = 0.1195656
fwd = 3414.933
spot = 3490.89
strike = 3414.529
Time = 63/365 #63 days to expiry

The above is an example of historical data for an ATM option. Using the discount factor, we can back out the continuous compounded risk free rate from :

$r = \frac{1}{T} * log(DF) = -0.003006125$

Using this and the forward, we can compute the continuous compounded dividend rate from :

$F = Sexp^{(r-d)*T}$

which gives $d = 0.1244475$.

Then using this and the rest of the data (and RQuantLib for pricing), we get the following:

priced = RQuantLib::EuropeanOption(type = "call",
                                   underlying = spot,
                                   strike = strike,
                                   dividendYield = div,
                                   riskFreeRate = r,
                                   maturity = Time,
                                   volatility = vol)

priced

Concise summary of valuation for EuropeanOption 
    value     delta     gamma      vega     theta       rho    divRho 
  67.9105    0.5004    0.0023  565.4711   26.1460  289.1472 -300.8429 

That is, a positive theta for an ATM (50 delta) option. So my questions are (assuming the data is itself correct - I have verified it from 2 different reputable sources):

a) Have I done something very wrong in my maths?

b) Can someone explain why this is happening - is it because of the very high dividend yield?

c) What does this mean intuitively? If I bought this option delta hedged is it "free gamma" ?

Any help would be appreciated!

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    $\begingroup$ What's the option? Where is the data from? What is the time stamp? A dividend of 12% sounds implausible (and is also the reason for the theta value you get). Is the price computed with R matching the market quote? It's not ATM, which is usually referred to ATMS (it's close to ATMF, provided the fwd is computed properly). What exactly is the discount factor? $\endgroup$
    – AKdemy
    Feb 23 at 19:35
  • $\begingroup$ The option is a vanilla european call on Eurostoxx index. The data is as of 18th April 2018. The data is from an internal library, but Bloomberg also gives pretty much the same data for the same date, so Im pretty sure the data is correct. The discount factor is as above - its just a function of the risk free and expiry $\endgroup$
    – Arron
    Feb 23 at 19:53
  • $\begingroup$ What's the BBG ticker? $\endgroup$
    – AKdemy
    Feb 23 at 21:05
  • $\begingroup$ The ticker is SX5E Index. Loading the option pricer on this and setting the trade date to 18th April 2018, with the above strike and 63 days to expiry seems to give a positive theta (as well as a large dividend yield) $\endgroup$
    – Arron
    Feb 23 at 21:08
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    $\begingroup$ The option ticker. If you load the option pricer and key in the values you have you don't look at Bloomberg's market data (just your internal data). $\endgroup$
    – AKdemy
    Feb 23 at 21:18

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