26

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 ...


12

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) > ...


12

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 ...


11

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 ...


11

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, ...


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

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 $$ Put(\lambda K) < \lambda Put(K).$$


10

For example, Thomas H. Cormen, Charles E. Leiserson, Ronald Rivest, Clifford Stein. Introduction to Algorithms, problem 24-3 says: 24-3 Arbitrage Arbitrage is the use of discrepancies in currency exchange rates to transform one unit of a currency into more than one unit of the same currency. For example, suppose that 1 U.S. dollar buys 49 Indian rupees, 1 ...


9

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 ...


9

hope I am not too late to the party. tl;dr Taleb's paper draws incorrect conclusions from a set of wrong assumptions. In practice, the movements of the forecast at 538 are very much in line with what can be defined an "arbitrage free prediction" based on a binary option model. The gist of the article is summarized in its incipit. A standard ...


9

I get this question frequently from academic types, and happily for you, the path does not involve any of those books. The major gaps in your knowledge, from the point of view of statistical arbitrage, are not mathematical. Most or all of them are not even statistical. Rather, they are gaps in knowledge about arbitrage, and how to take part in it. PhDs ...


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

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

Taleb argues that under uncertainty, election forecasts should be seen as a Binary option. A similar thought is presented by De Finetti's principle that probability should be treated like a two-way "choice" price. Therefore, under high levels of volatility, forecast should not have extreme variation across time (equivalently, the price of the binary option ...


8

Assume a store is fairly pricing a bottle of water at \$1. Now another store is pricing the same bottle of water for \$1.2. Assuming it is possible, you can buy the water at the first store, end sell it to the second store for a \$0.2 risk-less profit, as a consequence forcing him to lower the price to \$1. At this point you no longer arbitrage. In this ...


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

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 ...


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 ...


7

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}...


7

Let's carefully distinguish which exercise type we consider. European-style call option $$ \max\{S_0-Ke^{-rT},0\}\leq C_E \leq S_0.$$ European-style put option $$\max\{Ke^{-rT}-S_0,0\}\leq P_E\leq Ke^{-rT}.$$ American-style call option $$\max\{S_0-K,C_E\}\leq C_A\leq S_0.$$ American-style put option $$\max\{K-S_0,P_E\}\leq P_A\leq K.$$ Because ...


7

You and the paper are both correct. Funding was not free before the GFC, but the funding cost of both positions then was almost equal, generating almost-zero basis. Since then, holding physical bonds has become relatively more expensive, because these tie up bank balance sheet, which has become a scarce resource since the GFC. Or at least a scarcer resource ...


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

On 10/24/17, Wells Fargo announced that they would pay a dividend of 0.39 to holders of record on 11/3/17. Thus, if you buy the stock after this date (through the exercise of the call) you do not get the dividend. This means that the potential arbitrage, instead of being 0.30, is -0.09. This is now sufficiently close to zero that there is likely no ...


6

Conceptually, an arbitrage gives you something for nothing. This is a different idea than making or losing money almost surely. A risk free bond allows you to make money almost surely, but it isn't an arbitrage. What's going wrong, the source of confusion in your example? You've implicitly assumed the existence of a cash security, with interest rate 0 (but ...


6

Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral ...


6

You're right. Euler's equation states $$p_t=\mathbb E^\mathbb P_t[M_{t+1}X_{t+1}],$$ that is pricing under $\mathbb P$ requires you to know the stochastic discount factor (SDF, aka pricing kernel) $M$. $M$ is (typically) found in a general equilibrium setting, depending on the marginal utility of investors. (Note: a strictly positive $M$ exists if the market ...


5

You need to read papers about market impact. You will find a lot of information about transaction costs. Two recent ones: Market impacts and the life cycle of investors orders, by Emmanuel Bacry, Adrian Iuga, Matthieu Lasnier, CAL Beyond the square root: Evidence for logarithmic dependence of market impact on size and participation rate, by Elia Zarinelli, ...


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