How Dividends Can Produce Positive Theta for an At-the-Money Call
Summary
The document presents a historical index-option example in which a European call near the money has positive reported theta. The author uses the observed discount factor to infer a negative risk-free rate, then uses spot, forward, and maturity to infer a high dividend yield. An options pricing library returns a delta near one-half alongside positive theta, prompting questions about the calculation, the dividend’s role, and whether a delta-hedged buyer can obtain free gamma.
The material is useful as a setup for examining how carry inputs affect option Greeks, but it contains no answer or explanation of the result. It therefore does not establish that the data or calculations are correct, explain the theta convention used by the software, or show whether positive theta survives hedging costs and realized price moves. The example alone cannot support a claim of a profitable or risk-free gamma strategy.
Key ideas
- A historical near-the-money call example reports positive theta.
- The author infers rates and dividend yield from discount factor, spot, forward, and maturity inputs.
- The example raises the possibility that unusually high dividend yield affects the reported Greek.
- The document does not provide an answer or establish whether the calculation is correct.
- Positive theta by itself does not demonstrate a profitable delta-hedged strategy.
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Full text
# Positive Theta for an At The Money option (with real data)
# Positive Theta for an At The Money option (with real data)
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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.