We’re rewarding the question askers & reputations are being recalculated! Read more.
43

Consider the standard error, and in particular the distance between the upper and lower limits: \begin{equation} \Delta = (\bar{x} + SE \cdot \alpha) - (\bar{x} - SE \cdot \alpha) = 2 \cdot SE \cdot \alpha \end{equation} Using the formula for standard error, we can solve for sample size: \begin{equation} n = \left(\frac{2 \cdot s \cdot \alpha}{\Delta}\...


22

A similar question for put option has been discussed in this question: Finding Arbitrage in two Puts. Basically, the call option payoff is a convex function of the strike. Then the call option price is also a convex function of the strike. Specifically, let $C(K)$ denote the call option price with strike $K$. Then for $ 0 < K_1 < K_2$, \begin{align*} ...


15

Setting aside, that it's not pure riskless arbitrage, but rather statistical arbitrage: You can extract the profit by performing continuous delta hedging. If you constantly adjust your hedge position you gain/lose money by delta hedging. Being long option (gamma long), you sell at higher prices and buy at lower ones. Over the course of time you realize ...


11

Neither. Black--Scholes says nothing about the parameter values: $\mu$ and $\sigma.$ A very large $\mu$ and very small $\sigma$ is very unlikely to actually occur in the market and if it did you could make money with high probability without using option contracts. BS simply says that if the market follows a certain process then a certain option price is ...


10

The "price protection" refers to RegNMS in the US. A stock exchange that does not have the best price must route all order flow to the exchange that does. The SIP in the figure is a consolidated feed that lists the best price among all exchanges. Consider this example: a broker sends a market order to buy JNJ to NYSE where the best offer is \$86.97. However,...


10

In the derivatives context, "arbitrage free" means almost surely for the probability measure under consideration. This is in opposition with statistical arbitrage used at high frequencies for example. More precisely the assumption is that there is no $T\geq 0$ and self-financed portfolio $V$ such that $V_0 = 0$, $P(V_T < 0) = 0$ and $P(V_T > 0) > ...


9

The option is a contract that gives you the right to buy the stock in one year for 18. Today people are trading the stock for 20, so you can sell the stock short for 20 today, meaning, someone gives you 20 cash today in return for a stock IOU, where you are obligated to deliver the stock to them on a later date. So you get 20 cash upfront but you need to ...


9

1) Why would you trade the error on the residual instead of creating a long/short factor model and trade expected returns? I would posit that the biggest reason people do this is for orthogonality of return. There are about 2,000 incredibly mature firms trading value, momentum, vol, etc. You would be competing with the likes of AQR, LSV Asset Management, ...


8

The original Nelson Siegel paper describes a parsimonious model of the term structure using only four or three (if $\lambda_t$ is fixed). Filipovic (1999) proves that this model can never be used in a arbitrage free context, paraphrasing the abstract: We introduce the class of consistent state space processes, which have the property to provide an ...


8

This depends a little bit on your definition of volatility arbitrage but in general what is meant is a strategy that takes advantage of the difference between implied volatility and realized volatility. Normally you receive implied variance and pay realized variance. This strategy is the classical example of picking up nickles in front of a steamroller ...


8

This is called on the run/off the run arbitrage, a type of convergence trade. The basic idea is that as the liquidity premium disappears for the on-the-run issue, the price will fall and converge to the price of previous issues. Here are a couple papers - http://people.stern.nyu.edu/lpederse/courses/LAP/papers/SearchBargaining/VayanosWeill.pdf http://...


8

Let $K_1=0$, $K_2=80$, and $K_3=90$. Then \begin{align*} K_2 = 1/9 \, K_1 + 8/9 \, K_3. \end{align*} Moreover, \begin{align*} Put(K_2) &= Put(1/9 \, K_1 + 8/9 \, K_3)\\ &< 1/9 \, Put (K_1) + 8/9\, Put(K_3)\\ &= 8/9 \, Put(K_3). \end{align*} Taking $K=K_3$ and $\lambda = 8/9$, we have that $$ Put(\lambda K) < \lambda Put(K).$$


8

The following link has a good summary of a typical pair trading strategy: https://www.quantstart.com/articles/Backtesting-An-Intraday-Mean-Reversion-Pairs-Strategy-Between-SPY-And-IWM It actually has full python code as well. It doesn't include a cointegration check though. Edit: if X and Y are cointegrated: calculate Beta between X and Y ...


8

Generally speaking, let us consider a problem where you have a series of simple payoffs $f_{K_i}(S_T)$ of strike $K_i$, $i \in I$, that depend on the value of $S_T$ at time $T$, as well as a more complex, laddered payoff $P_L(T)$ which pays a quantity $g_i(S_T)$ on regions of the form $\{K_i \leq S_T < K_{i+1}\}$ $-$ regions are delimited by the strikes ...


8

No. The dirty price is the market's estimate of fair value for the bond. The clean price is just a quoting convention (so that the price doesn't jump when you pass over a coupon date). The market doesn't try to estimate the clean price and then get the all-in (dirty) price wrong. The market estimates the all-in price, and then applies the accrued interest ...


8

As a practical aside on a large scale, I have heard the rumours of European banks and even a consortium of banks considering plans to build an ultra secure deposit facility for cash, and also the ECBs push back for doing so based on an unwillingness to actually provide physical currency. I have never heard about the actual realisation of any of these rumours ...


7

The main problem is that you cannot achieve Libor in the markets. So the old-fashioned method of discounting at Libor doesn't work any more. As an example, if you compound up the 3m Libor with today's price on a 3x6 FRA, you won't get 6m Libor. Traditionally, that would mean arbitrage, but these days it's just a fact of life. You cannot achieve 3m Libor for ...


7

In my mind, there are two questions here: 1) How does DB make money given a zero expense ratio? This is covered by Dirk and Lliane. Basically, DB gets cheap funding and stock loan fees in return for paying marketing / index / hedging costs. The ETF investor gets zero expense ratio in return for taking DB credit risk. 2) Why does it look like the etf ...


7

Are there any other mechanisms at play here which might explain this kind of tracking error? Dirk is right, you often lend the titles internally or not, etc. You can also write calls for your index, this is not orthodox, but it's ETF, there is no orthodoxy there... Edit : With the graph and given the outperforming is seasonnal (around May), I think we can ...


7

The answer by @HenriK is certainly correct. However, for justification, technique such as the Jensen inequality is needed. For example, since $x^+$ is a convex function, assuming zero interest and zero divdiend, \begin{align*} E\big((S_{T_{2}}-K)^+ \mid \mathcal{F}_{T_1} \big) &\ge \big(E(S_{T_{2}} \mid \mathcal{F}_{T_1})-K\big)^+\\ &=(S_{T_1}-K)^+. \...


7

If you imagine you have two risk-less assets that have a unit payoff at maturity $V_1(T) = V_2(T) = 1$ but their present value is not equal, e.g. $V_1(t) < V_2(t)$. You buy the cheaper, sell the more expensive, have a strictly positive cash-flow today and at maturity the cash-flows cancel out with certainty. This is a free lunch arbitrage. The same ...


6

On Bloomberg. Go to ETF -> holdings and type "97 Enter".


6

To see the connection between put-call parity and option price you should read this highly insightful paper by Espen Gaarder Haug & Nassim Nicholas Taleb: Option traders use (very) sophisticated heuristics, never the Black– Scholes–Merton formula It shows how you can heuristically derive option pricing formulas by adapting the tails and skewness by ...


6

You could compute index dividend yield from ATM options using linearized put-call parity (assuming index options are European.) The present value of the dividend payment is: $PV(div) = P - C + (S - K) + K(e^{rT} - 1)$ where $r$ is interest rate to the option expiration and $T$ is time to maturity in years. Then the implied dividend is: $d = \frac{PV(div)}{...


6

In three bullet points: Efficiency: the obtained prices maximize assumed utilities of different agents. In their paper "The Valuation of Option Contracts and a Test of Market Efficiency", Cohen, Black and Scholes compare the theoretical value of options to their market price. The efficiency is in this sense: can agents obtain more or less in practice than ...


6

In a pure diffusion setting, you can equivalently write no calendar arbitrage constraints: In terms of implied volatility: total implied variance should be non decreasing in time, and that, for any given forward moneyness level, see Gatheral top of page 4. In terms of European option prices: see Gatheral end of page 3. The price-based constraint builds on ...


6

Short rate models are broadly divided into equilibrium models and no-arbitrage models. The models from Vasicek, Dothan and Cox, Ingersoll and Ross are examples of equilibrium short rate models. The models from Ho-Lee, Hull-White and Black-Karasinski are no-arbitrage models. Take Vasicek and Hull-White as an example. The short rate processes are $\mathrm{d}...


5

You are correct that this cannot happen under America's RegNMS. Of course, TSX is in Canada. This "strategy" also makes generous assumptions about the likelihood of getting filled on passive orders. I doubt the author of this paper ever had much success with this. If a passive order were to lock the market in RegNMS, then that order is usually routed to the ...


5

Yes, there is a software application that you can purchase for $39.99 which stores all your tick data in a highly compressed format while still allowing maximum throughput and lowest latency data queries that I have ever seen. The package provides APIs to all languages under the sun but because they have a special sale going on it comes with the complete ...


Only top voted, non community-wiki answers of a minimum length are eligible