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16

For an option with price $C$, the P$\&$L, with respect to changes of the underlying asset price $S$ and volatility $\sigma$, is given by \begin{align*} P\&L = \delta \Delta S + \frac{1}{2}\gamma (\Delta S)^2 + \nu \Delta \sigma, \end{align*} where $\delta$, $\gamma$, and $\nu$ are respectively the delta, gamma, and vega hedge ratios. Then it is clear ...


15

No, you should not expect such a relationship to hold in general. The reason is that American options have an "exercise barrier" which European options don't, and this results in different prices and greeks. In the case of put options (with interest rate $r>0$) as the spot price falls, at some point it becomes optimal to exercise early and take the cash. ...


15

Partly because it's hard to get a hold of, the Arslan et. al. paper is starting to assume mythical proportions. As said by Dimitri Vulis, the general idea of the paper is set out in (one or two of) Peter Carr's papers. For the benefit of the OP and others I will try to summarize the most salient points of the paper below and also point out the assumptions ...


13

What you have to do is to show that the dollar gamma satisfies the Black-Scholes PDE. Using Feynman-Kac it then follows that the dollar gamma is an expectation of a "payoff", just like the Black-Scholes claim price is an expectation of a payoff. And if something is the expectation of a payoff then it's a martingale. Actually, you don't need the ...


11

Under the Black-Scholes model, \begin{align*} Gamma &= \frac{N'(d_1)}{S \sigma \sqrt{T-t}}\\ Vega &= SN'(d_1) \sqrt{T-t}. \end{align*} Then, it is easy to see that \begin{align*} Vega = S^2 \sigma (T-t) Gamma. \end{align*}


11

You can't lose more than you invested by writing covered puts, because you keep enough cash to cover any potential losses from the puts. That's not to say that your losses can't be substantial, of course. The below chart shows the drawdown profile of the PutWrite index - you would have lost nearly 40% of your investment at one point. So how did the ...


10

the problem is that the pay-off has discontinuous first derivative. Try a contract with pay-off that is twice differentiable and it will probably work. The problem is that all the value comes from the tiny number of paths within $\Delta S$ of the strike, and these paths have huge value. This is a well-known problem. As the bump size goes to zero, the ...


9

Options on interest rates futures in the listed markets are always traded 1-yield (100-yield) just like the futures which are traded 1-yield. So negative rates aren't an issue and its always black volatility. In the OTC market, both normal and black volatility are quoted, but the common practice is to use black volatility is what is way more frequently used....


8

Gamma and vega have the same general shape , peaking at ATM and tapering to the tails. But gamma concentrate as the option gets closer to expiry (when vega is small). For options a long way from maturity, vega increases and gamma is small. Consequently for short dated options, if the price is close to strike, the option will have to be rehedged often (...


8

The conjecture is true when the interest rate is zero. Note that, from this question, under the Black-Scholes model, \begin{align*} \Gamma(t,S_t) &= \frac{N'(d_1(t))}{S_t \sigma \sqrt{T-t}}\\ Vega(t,S_t) &= S_tN'(d_1(t)) \sqrt{T-t}, \end{align*} where \begin{align*} d_1(t) = \frac{\ln \frac{S_t}{K} + \big(r+\frac{1}{2}\sigma^2\big)(T-t)}{\sigma \...


8

Using our good friend Taylor, we know that \begin{align*} C(S+\Delta_S)\approx C(S)+\Delta_C\Delta_S+\frac{1}{2}\Gamma_C(\Delta_S)^2, \end{align*} where $\Delta_C$ and $\Gamma_C$ are the call's sensitivities and $\Delta_S$ a small change in the price of the underlying asset. In your example, $\Delta_S=1$ and thus, \begin{align*} C(52+1) &\approx 5.057387 ...


8

$$ \frac{1}{2} \frac{\partial^2 f}{\partial S^2} dS^2 \approx \frac{1}{2} \sigma^2 S^2\frac{\partial^2 f}{\partial S^2} dt$$ (for small $dt$, ignoring $(dt)^2$ terms ) $\sigma$ is embedded in $dS = \mu S dt + \sigma S dW$ and $$ dS^2 = \mu^2 S^2 dt^2 + 2\mu \sigma S^2 dt dW + \sigma^2 S^2 dt \approx \sigma^2 S^2 dt$$ You picked up $1/2\Gamma \sigma^2$ from ...


8

Dimensional analysis is the key: The change in option price is in dollars. The change in option price is of course the sum of its changes (partial derivatives) with respect to its underlying risk factors. However you cannot add terms with different dimensions, that would literally be trying to add apples and oranges. Let's look at delta, which (in finite ...


7

Assume you buy a plain vanilla call option at the price $V$ and the spot $S$. You immediately delta hedge buy selling $\partial V / \partial S$ units of the underlying asset. The underlying asset now instantaneously jumps form $S$ to $S' = S + \Delta S$. The new value of the call option is $V'$. Your total p&l is \begin{equation} \text{P&L} = V' - ...


7

They are not the same, but they are related. Gamma is sensitivity to realized volatility. Vega is sensitivity to implied volatility. Vanilla options are always long gamma and long vega, so they are "long vol" and saying "I am a buyer of vol/gamma/vega" means that you are taking a position that benefits from a rise in volatility (either ...


6

Assuming all else remains equal (implied vol has not changed and very little time decay has occurred), Gamma scalping can best be explained by Gamma (or realized volatility) enhancing the value of a delta hedged portfolio. For example: If you are long an at-the-money call option, you are long 0.5 Delta and long Gamma. If you hedge this position, you will ...


6

Not sure this is a valid question! Gamma p/l is by definition the p/l due to realized volatility being different from implied. Vega p/l is by definition the p/l due to moves in implied volatility. The second part of the question you have answered yourself. Short dated options have more gamma exposure, long dated options have more vega exposure.


6

Put-call parity says that a call and put (worth $C$ and $P$ respectively) with the same strike $K$ have the following relationship with the spot rate $S$, risk-free rate $r$, and time to maturity $T$ -- $$C - P = S - e^{-rT} K$$ Taking the first derivative with respect to $S$, $$ \frac{\partial C}{\partial S} - \frac{\partial P}{\partial S} = 1 $$ which ...


6

Gamma is the sensitivity of the delta with respect to infinitesimal changes in the price of the underlying asset (in whatever unit your underlying is nominated, typically dollar, pounds, euros, ...). So, it is not a percentage change. Instead, the percentage change (option elasticity) equals $\Delta\frac{S}{V}$. This quantity, for instance, gives you the ...


5

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 function of $S$ and $T$ that does not vary with $K.$ So if you construct a linear combination that has zero gamma then the vega is zero too, and vice versa.


5

I think what you are missing is simply the Vega-Gamma relation in the Black-Scholes model. Namely: $$ Vega = \frac{\partial v}{\partial \sigma} = \sigma(T-t)S^2 \frac{\partial^2 v}{\partial S^2} = \sigma \tau S^2 \Gamma $$ Plugging this into your coverage error, you get its expression in terms of the Vega which is the most natural measurement of your ...


5

As long as you live in a world where implied and realized vol are the same, there is no net profit (or loss) from gamma scalping. However, if they are different, then you make a gain or loss which is not path dependent. This is all still in a hypothetical world of course with continuous trading. In reality when rehedging less frequently, pnl becomes random ...


5

Gamma scalping (being long gamma and re-hedging your delta) is inherently profitable because you make 0.5 x Gamma x Move^2 across the move from your option. (You get shorter delta on downmoves, so you buy underlying to hedge, you get longer on upmoves, so you sell on upmoves, etc.) Because it's inherently profitable across any move, you must pay for the ...


5

We work in a Black-Scholes world. Consider the following delta-hedged portfolio: $$ \Pi_t=V_t-\frac{\partial V}{\partial S}S_t$$ We assume the portfolio is self-financing$^{\text{(a)}}$, therefore: $$ \begin{align} \text{d}\Pi_t &= \text{d}V_t-\frac{\partial V}{\partial S}\text{d}S_t \\[3pt] \tag{1} & = \left(\frac{\partial V}{\partial t}\text{d}t+\...


5

The traders or practitioners’ gamma concept tries to capture the same issue. It is defined as S times gamma divided by 100: $\Gamma_P=\frac{S\, \Gamma}{100}$ Please see page 29 of this document: https://mathfinance.com/wp-content/uploads/2017/06/FXOptionsStructuredProducts2e-Extract.pdf


5

It was a Deutsche Bank Working Paper: http://refhub.elsevier.com/S0304-405X(16)00005-2/sbref0001 Unfortunately, it is very hard to find internal research published by banks. I have not seen this one myself, but as far as I know, Peter Carr's published paper has everything that this paper had.


4

I've started thinking about this, too. My gedanken conclusion turned out to be too simple once I found what I was after: http://www.investment-and-finance.net/derivatives/o/option-beta.html, which I've confirmed in Black & Scholes (1973) p10 (eq 15). In short: $$ \beta_{\text{option}} = \frac{S\cdot\Delta}{O}{\beta_S} $$ where $S$ is the underlying ...


4

Since the volatility is not changing, we can assume that the only change is the underlying asset price $S$. Then \begin{align*} C(S+\Delta) &\approx C(S) + Delta \times\Delta +\frac{1}{2} Gamma \times \Delta^2 \\ &=11.50 + 0.58 \times 0.5 + \frac{1}{2}\times 2 \times (0.5)^2\\ &=12.04. \end{align*}


4

Usually vega and gamma go in the same direction, but you can have opposite exposure in a calendar spread. For an ATM option, vega decreases closer to maturity while gamma increases. If you implement the following: -long a 1 month ATM option -short a 2 months ATM option you should be long gamma and short vega.


4

I think I've found the answer to my question (I'm waiting for confirmation from you in the comments) The intuitive difference in this negative sign correlation depends on the position taken on options in the portfolio: Gamma is always positive when you buy an option (Theta acts negatively when buying options); Gamma is always negative when selling an ...


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