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31

Many of them are on my website at emanuelderman.com. Others I probably have anyway. Feel free to email me


16

Being short gamma simply means that you are short options regardless of whether they are puts or calls. The most common type of investor that is willing to be short gamma is someone who sells options, also known as a premium collector. These investors commonly use strategies such as short puts, covered calls, iron condors, vertical credit spreads, and a ...


15

Great question! I think the most useful starting point is Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options by Bakshi, Kapadia and Madan (2003). Their paper proposes a definition of model-free implied skewness (they originally called it risk-neutral skewness, but MFIS is more accurate), which they prove will ...


15

Short Version : Two main uses I'm doing an arbitrage/statarb strategy (volatility for instance) which should not be dependant on the Delta (I'm an arbitragist). I HAVE to keep a product in my portfolio, but I don't want to be EXPOSED to it (I'm a market maker). Long Version : The goal of Dynamic Hedging is not down the line to earn risk free rate of ...


13

You are absolutely right to point out that most proactive participants in options markets prefer to be long gamma, and it is typically reactive market makers who take the opposite side of their trades. While the typical options trader (I find it difficult to call anyone trading options an "investor") does not hedge his position, market makers will attempt ...


12

Actually there are more than just ideas and hints concerning this topic. There is an intuitive model and solution to your question already using machinery of option theory. But don't worry, it's not a surprise that you didn't find any useful literature in your search because the proposed solution actually comes from a very different topic. In addition to ...


10

Options are actually some of the least susceptible securities to the adverse impact of counterparty risk. I refer to listed options, such as those cleared through the OCC (Options Commodity Clearinghouse) in Chicago, IL. The OCC is a true central clearing counterparty (CCC) because it bears all default risk, by distributing it evenly among its members. The ...


10

I had read some of them; actually, it does not exist an on-line library that collected them (or, better, it existed here, but it seems the website does not work anymore). I reported here below some of them that you did not find: More Than You Ever Wanted To Know* About Volatility Swaps Model Risk The Volatility Smile And Its implied Tree Enhanced Numerical ...


9

You are missing the futures basis and roll cost. Futures expire, and need to be rolled into the new expiry. The basis is not static and can vary considerably, depending on the specific underlying and contract. Quants may have a hard time to appreciate this but the basis is not at all fully quantifiable at all times: It can hugely vary entirely due to shifts ...


8

The paper "Do option markets correctly price the probabilities of movement of the underlying asset? " by Yacine Aït-Sahalia, Yubo Wang, and Francis Yared should in my opinion provide many very usefull elements for this question (look in particular at section 3). Regards


8

By delta hedging you are saying that you have a view on the path and the volatility of the option you are trading, but not on its direction; in your case, that being short delta. From a theoretical perspective, all options are priced fairly and not delta hedging simply increase the variance of your payouts. In your example, selling a call and delta ...


7

Short gamma is a bet on volatility (expressed as hedging costs) not getting too large. The key concept here is that you get paid to be short gamma. Consider that any option is sold for a bit more more than its intrinsic value (the extra bit is often called volatility value.). If nothing moves, then the option ultimately expires precisely at intrinsic ...


7

I might be misunderstanding your question. My thoughts: being short gamma is being long volatility your comment re gamma increasing regardless of direction only holds for ATM options. For ITM options, being short gamma is being long the underlying. For OTM options, being short gamma is being short the underlying. Some graphs: Below, except as noted,...


7

If you get paid enough theta it absolutely makes sense to be short gamma. And the closer to expiration, the faster the time-value flees. Most of the time, most people would prefer to be gamma long though. It's simply a safer bet because of uncertainty: unexpected events can seriously damage your book if you're short vol.


7

Short gamma is being of the view that realized volatility would be less than the implied volatility for the period in which an option is valid. So if you think realized volatility in the future would be consistently lesser than implied volatility at present, then you'd be short gamma. The premium one would receive by selling an option (call or put) is a ...


7

The point is the following: Delta, $\Delta$, is defined as $\frac{\partial C}{\partial S}$, where $C$ is the value of the call option, and $S$ is the price of the underlying asset. So, given that the value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is $$C = N(d_{1})S - N(d_{2})Ke^{-rT},$$ $$\Delta ...


6

Keep in mind that most futures, equity, and index options, at least, are traded on exchanges where the counterparty risk is so tiny as to be negligible. In general, adding extra variables like this fails to invalidate the model. For example, the fact that interest rates or volatilities are not constant just ends up leading to an extended model with extra ...


6

Skew "arbitrage" is a pretty broad term. When you are trading the skew, there are 3 principal risks (or sources of P&L, if you will): (a) the actual change in the slope of the skew in the implied space. e.g. if you are trading 95% strike against 105% strike and your underlying stays in place, all of your instantaneous P&L would be due to the changes ...


6

I think there is an error implicit in your question. Dynamic delta hedging, even assuming the underlying process is a continuous martingale and trading entails zero transaction costs, only eliminates the directional risk. A number of residual risks remain, most notably volatility risk, embodied in both the gamma and vega. A dynamically hedged portfolio of ...


6

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 ...


6

The differential equation has a trend due to the interest rate. When you discount you take this trend away: $$ \frac{d}{dt} (e^{-rt}Z_t) = -re^{-rt}Z_t + e^{-rt} \frac{d}{dt}Z_t = e^{-rt}\frac{1}{2}S_t^2\Gamma_t(\hat{\sigma}^2-\beta_t^2) $$ $Z$ doesn't appear on the rhs anymore and you can integrate $$ e^{-rT}Z_T - e^{-r0}Z_0 = \int_0^T e^{-rt}\frac{1}{2}...


6

Due to the lack of a carry arbitrage, VIX futures are actually the direct hedge for VIX Index options


6

There are more ways to approach this but the method I propose should work reasonably well in practice, especially if you increase the number of assets you hold. Calculate the beta of the stocks you're holding with respect to an index Buy $N_f$ (sell when $N_f$ is negative) future contracts on that index $N_f$ can be calculated as $$N_f = \frac{\beta_T - \...


6

A general hedging strategy Let assume that $S_1(t)$ and $S_2(t)$ are the price processes of your 2 stocks and that they follow a Geometric Brownian Motion (GBM): $$\forall \, i \in \{1,2\}, dS_i(t) =\mu_iS_i(t)dt + \sigma_iS_i(t)dW_i(t)$$ We assume both stocks have an instant correlation of $\rho$: $$dW_1(t)dW_2(t)=\rho dt$$ Let also $V(t)$ be the value ...


6

Presumably the option can be exercised for intrinsic at any point. Note the interviewer asked for a static hedge using the stock, not a dynamic hedge. Hence you must find a buy and hold portfolio that will always give you at least the value of the option (if you’re short it which I suppose is the question) until it is exercised. Note that the maximum ...


5

In practice, absolute summability of hedging errors may not be applicable. Mostly, for the sequences of hedging errors, one relaxes the absolute convergence criteria and uses the squared summability of hedging errors. Note: Absolute summability is a stricter condition than squared summability. Some sequences may not be absolute summable but are only squared ...


5

You have to differentiate here between the risk-taking and the market-making side. As a risk-taker, like e.g. a hedge-fund, you are right, you could just buy the bond! But as a market-maker you sell these options but don't want to bear the risk, so you have to counterbalance it. You could of course counterbalance it with another option which would be the ...


5

If you're long the underlying and short the futures contract, then you have no risk and earn the risk-free rate. You get into the position at $S_0$ and will be able to get out of the position at $F_0$ at time $T$. By a no arbitrage argument it must be that $F_0 = S_0 \exp(r T)$. I imagine Hull has a pretty good exposition on this. The risk premium is ...


5

This is usually called Pin Risk. It's difficult because there is a high degree of uncertainty regarding the whether the options you sold are exercised or not. If you don't hedge, your short options could be exercised and you are left with risky net short position in the underlying. If you hedge and your short options are not exercised, then you have a long ...


5

$\require{cancel}$ $$\text{PnL} = -[P(t+\delta t,S+\delta S)-P(t,S)] + rP(t,S)\delta t + \Delta(\delta S - rS \delta t + q S\delta t)$$ Assuming a pure diffusion, at the order 1 as $\delta t \to 0$ $$P(t+\delta,S+\delta S) = P(t,S) + \frac{\partial P}{\partial t}\delta t + \frac{\partial P}{\partial S}\delta S + \frac{1}{2}\frac{\partial^2P}{\partial S^2}(\...


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