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32

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


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 ...


12

E.g. on Monday you get forced to buy some Friday expiry OTM puts, say 95% strike S&P weeklies. Of course, you go and buy some delta against them to "hedge" yourself. Next thing you know, the the market tanks. Unfortunately, by Friday it's only down 3.5%, so it's does not fall far enough to reach the strike. So, on Friday expiration, you are out your ...


11

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 ...


10

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 ...


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 ...


9

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 ...


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

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 ...


7

You are absolutely right, I would say that how the interview question was posed and the example given is very misleading, if not outright incorrect. Here is why: Hedging does not increase your risk in this particular example: You take on delta exposure by buying the short dated option outright. Thus buying/selling underlying (put/call) in any case will ...


7

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 ...


7

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 ...


7

You are long a vanilla option, so long gamma (positive gamma). If the stock price decreases, so does the delta of your option. Since you short-sold the stock to hedge, you now have short-sold too much since delta has decreased. As a consequence, you must buy back some stock.


7

You would be over hedged in your call position if it was delta neutral before the stock cratered. Since you are long delta on the call, you would have shorted stock to make the original position delta neutral. When the stock fell, your long call delta would have fallen, and you would buy to cover some of your short stock hedge. However, being long the ...


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

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

Your simulation is basically fine, though you need to discount in USD. For hedging purpose, you need to use the instruments available in USD. Let $S=\{S_t, \, t\ge 0\}$ be the stock price process in EUR, $X=\{X_t, \, t\ge 0\}$ be the exchange rate process from one unit EUR to units USD, $r_f$ and $r_d$ be interest rates in EUR and USD. Moreover, let $B_t^f=...


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

I would do as follows: A) First do PCA on an arbitrage-free monthly curve (assuming the most granular contract you will use is individual months). To ensure no arbitrages, you will need to drop out certain contracts, I would drop the most illiquid ones. To give you an example, if you are in Dec, you might see Jan, Feb and Mar quoted, but also Q1. In this ...


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

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}(\...


5

The empirical relationship between the futures price $F$ and the spot price $S$ is $$ F = S e^{b\tau} $$ where $\tau$ is the time to expiry, and $b$ is the empirical basis, i.e. the number that makes the equation hold, given by $$ b = \frac{1}{\tau}\log(F/S) $$ It can be compared to the theoretical basis, $$ b_{\rm theor} = r - q $$ where $r$ is the ...


5

This is a slightly extended version of my comment that summarizes the main result of the reference that I provided. This problem is discussed in detail in Chapter 12 of Wilmott (2006), which is based on the paper Ahmad and Wilmott (2005). See also the related Question 9 in Carr (2005). In your case, you are selling and delta hedging the option using the ...


5

The quoting convention must be explained somewhere in your book. For Eurodollar futures, this convention is 100 - yield, 92 means the yield is 8% per annum, so for one quarter you need to divide this discount by 4 to get the price (100% - (8% × (3month/12month)) = 100% - 2% = 98%


5

Yes, you can say they are traded on listed options, but only for a few limited markets, and not that liquid relative to options on a single asset. For instance, the commodity futures space, there are options on commodity spreads listed, and a strike of 0 would be the same as an exchange option. These options have some liquidity in energy and grain markets,...


5

If it is a single name CDS, the transaction leaves the bank short the credit spread of that bond vs a risk-free bond in the same currency. To go long the spread, the bank would i) buy the same CDS from another bank or ii) sell short the same bond, and get rid of general interest rate risk by going long a risk-free bond (or interest rate swap) of the same ...


4

You discuss the behavior of stock prices after an earnings announcement. There is a significant amount of academic research on this topic (called post-earnings-announcement drift). It basically finds that stock prices tend to move sharply initially, but continue to gradually follow in the same direction as the initial move for several weeks thereafter. I'm ...


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