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8 votes
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Positive theta on a long put?

If a european option value becomes lower than intrinsic value it gets negative time value. In this circumstance the theta becomes positive because as time approaches to expiry the option value has to ...
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8 votes
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Mathematical underpinnings of the square root of time rule

For any process with independent increments, by the very fact of statistical independence the variance of $x_{t3}-x_{t1}$ is going to be the sum of the variances of $x_{t2}-x_{t1}$ and $x_{t3}-x_{t2}$ ...
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8 votes

Carry calculation on an interest rate swap

I will attempt to summarise the content included in this book, which has a specific chapter dealing with carry and roll-down. There, two concepts are made completely separate. Costs-of-carry are ...
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7 votes

Carry calculation on an interest rate swap

It turns out that the two things are the same, appropriately scaled. Proof: we can construct a 5 year swap using 3 month libor combined with a 3mo-4.75yr forward swap, weighted by the dv01s of each ...
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7 votes
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Negative theta for a short put

Theta on a European Put option on a non-dividend paying stock is: $$\Theta=-\frac{S_t \sigma}{2\sqrt{\tau}}N'(d_1)+rKe^{-r\tau}N(-d_2) $$ For deep in-the-money Puts, $d_1$ and $d_2$ go to negative ...
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6 votes
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Black Scholes theta as function of time to maturity

With a long time to maturity, your options have a low theta because their time value decays quite slowly. If there are many months to go, the passage of one day does not change the exercise ...
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5 votes

is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?

if you have a portfolio of calls and puts with the same maturity then your portfolio is gamma neutral if and only if it is vega neutral. The reasons is that the BS gamma divided by the BS vega is a ...
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5 votes
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Can we trade theta?

You are neglecting the PnL from the stock position. Let us say you hold 1,000 shares at \$122 per unit. You’ve sold calls at \$0.21 per unit of stock, thus receiving \$210 in premiums. If the stock ...
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4 votes
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What is the name (Greek) for sensitivity of an option's Theta to the Time to maturity?

No Because the P&L it generates is in $O(dt^2)$. Ito's lemma tells you that you can ignore this P&L. $$PnL = \frac{\partial^2 V}{\partial t^2}dt^2 = 0$$
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  • 371
4 votes

What is FX theta in linear products?

I think your FX theta is probably not the same as the theta in black scholes sense. I think it may mean time pnl, which is applicable to all products, i.e., the PnL of time passing 1 day, but ...
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4 votes

What does "Gamma profit/loss" mean?

Think of this in terms of Taylor series. Let's say the option price today is $C\left(S,t\right)$ where S is the underlying price and t time. Let's say the underlying price changes by $\Delta S$ in a ...
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4 votes

Greeks and options hedging

I can argue your case as follows, consider a portfolio such that The value of $\Pi$ of a portfolio satisfies the differential equation given by: $$\frac{\delta \Pi}{\delta t}+rS\frac{\delta \Pi}{\...
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4 votes

What is the intuition behind a positive theta for European long puts?

It’s just the effect of interest. If you are long a deep ITM European put, it is worth the PV of K minus the stock price. But one day later the PV of K has grown a bit. That’s it. It’s the opposite ...
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4 votes

Negative theta for long OTM put?

This is possible if the option is long-dated and interest rates are high enough. For example, a five-year put struck at \$90 where the spot is \$100 (so it is in the money with respect to the spot ...
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  • 5,638
4 votes

Negative theta for long OTM put?

That is quite possible. You have negative time value and a positive theta if the option price is below the intrinsic value. Look at deep ITM put options, the stock price is basically so low, the ...
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3 votes

How does Theta benefit sellers of debit spreads?

If both options are out of the money, your higher strike put (of which you are short) has a higher theta than your lower strike put (of which you are long). Thus earn more theta than you lose.
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3 votes
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Relationship between time decay and gamma

The relationship between theta and gamma is the Black-Scholes PDE. Let's take normal B-S dynamics with $r=0$: $dS_t = \sigma S_t dW_t$ The pricing PDE for a derivative $g(S_T)$ is (with terminal ...
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  • 614
3 votes

Splitting theta from vol carry

Well it all depends how theta is calculated in the first place. Depending on your pricing scheme those could be very different things. Anyways assuming that you are dealing with european vanilla then ...
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  • 2,127
3 votes

Why doesn't the value of an in-the-money option increase approaching expiration?

The value of an option is based on its intrinsic value plus its time value. Intrinsic value is simply based on, for example for a plain option, the strike price of the option and the underlying ...
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  • 505
2 votes

Mathematical underpinnings of the square root of time rule

The reason is that in many common models including geometric Brownian motion, the variance of the logarithmic returns is proportional to time. Thus, their standard deviation/volatility is proportional ...
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2 votes

How to calculate the theta of a bond?

To answer that question you first have to define what "no change other than the passage of time" means. So you could make one of the following "no change" assumptions. the shape of the term structure ...
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2 votes

Rate of Options decay

The value of a call option that is near ATM can be approximated as $C(S,T)≈ 0.4 \sigma \sqrt T$. Therefore, under the unrealistic assumption that S does not change very much (i.e. the option stays ...
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  • 9,617
2 votes
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Estimating profit/loss of a Gold Futures option using Theta and Gamma

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. ...
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  • 1,411
2 votes

Greeks and options hedging

I dont think that people would usually use one as the substitute for the other, as: $\theta/\Gamma=-\frac{S^{2}\sigma^{2}}{2}$ which is arrived upon by neglecting the terms of the formula for $\...
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  • 1,581
2 votes

Gamma portfolio trading

There are two ways you can lose money: The actual volatility of the stock is less than the IV you assumed. For example (extreme case) let's say that the stock price does not move at all: you make no ...
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  • 9,617
2 votes

Gamma portfolio trading

If you are long gamma, your delta is increasing at an increasing rate. In order to delta hedge this position, you will be selling stock as the stock price goes up and buying stock as the stock price ...
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  • 5,335
1 vote

How do I calculate Hull White's Theta from the discount curve?

In practical situations you will never know $P^M(0,t\pm\epsilon)$ for a continuum of $t$ and $\epsilon\,.$ In other words, $\theta$ will practically always depend on an interpolation method between ...
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  • 1,508
1 vote

Buy and Hold P&L for options

Say the value of your option on day $t$ is $V(S_t, \sigma_t, \tau)$ where $S_t$ is the spot price, $\sigma_t$ is the implied volatility and $\tau$ is the number of days to expiry (it also depends on ...
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  • 5,638
1 vote

Black Scholes Theta Finite difference

First and foremost I do not agree with you Closed Form value. I get $\Theta=-8.963$. There are various of BS calculator you can use the check your results and in general you should do that. Here is ...
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  • 1,585
1 vote

theta for SPX options vs. E-mini future options

The theta for puts and calls at the same strike should be the same, so it seems the SPX theta is somehow wrong. Edit: thanks @maxim, I see now what the issue is. I think the difference is coming ...
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