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

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 ...


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 ...


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

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

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

My thoughts: Risk-neutral probability measure ${Q}$ is a convenient mathematical tool that is used primarily for pricing derivatives The price of a derivative is essentially the price of the replicating portfolio. So to price a derivative, one can attempt to build a portfolio that replicates the derivative pay-off at maturity and then work backwards in ...


5

It would make total sense for you to quote the price of a financial instrument by discounting its future cash flows according to your own risk aversion ($\Bbb{P}$ measure, stochastic discount factor). However, it would make it complicated for me and you to agree on that price hence developing a liquid market. That being said, if we are both rational and ...


5

I answer from a general discrete time/discrete state model point of view. This includes the binomial tree model as a special case. In finite dimensions, you can interpret asset payoffs and returns as vectors and retreat to linear algebra. Suppose you have $N$ states of nature and $J$ assets. Your payoff matrix is \begin{align*} A=\begin{pmatrix} X_1(\omega_1)...


4

I am not sure why the question you link to does not provide an answer. I’ll try to answer it but it is really similar to what has already been said there. Bottom line is: if the value $K$ is reachable by the underlying asset $S$, that is $K$ belongs to the domain of process $S$, then the butterfly should be strictly positive. First note that the butterfly is ...


4

Drifts under $\mathbb{Q}$ and $\mathbb{P}$ Some good answers already. Let me just repeat for clarity: under the risk neutral measure $\mathbb{Q}$, the drift of all assets has to equal to the rate at which the Numeraire appreciates, i.e. typically this is the risk-free rate $r$ of the money market. The reason for this is the "no-arbitrage" argument: ...


4

There is no uncertainty. Assume at $t=0$ I buy one unit of asset $1$ and sell $100$ units of asset $0$. Moreover, at $t=1$, I close both positions. At $t=0$ my payoff is $100-100=0$ and at $t=1$, it is $101-100=1$. Hence there is arbitrage. Assume I want to attain a payoff of $\xi(T)$ at period $T$. I can obtain this e.g. by investing $\xi$ in asset $0$ at $...


3

I feel this is not a duplicate of a question asking about applications of graph theory as this goes the other way. If you're talking purely about currency arbitrage, the quickest way seems to be finding a negative cycle in a graph of currency where the vertices are the currencies and the nodes the exchange rate.


3

as it was stated correctly in the question all long butterfly options have to have a non-negative premium in order for No-arbitrage to hold. So we can say that: No-Arbitrage holds implies All Butterfly spreads have a non-negative premium. However, the reverse is not true. Just because all butterfly spreads have non-negative premiums does not mean that ...


3

For $r=q=0$ and $t\leq T'\leq T$: $$ C_t(T)=E_{t}[(S_T -K)^+] = E_{t}[E_{T'}[(S_T -K)^+] \geq E_t[(S_{T'} -K)^+]=C_t(T'),$$ where we used the tower property of conditional expectation and the sub-martingality of $(S_{T'}-K)^+$ they mentioned (which is a consequence of Jensen inequality for conditional expectation). A calendar spread (one long call with ...


3

I might have misunderstood the question; but the index is just a weighted-average of its constituent stocks. As such, it does not trade, and thus does not incur any transaction costs. The forward price on said index is just the spot, adjusted for interest-rate versus (expected) dividend basis. Lest there be arbitrage. Trading all of the stocks to replicate ...


3

The family politics of the Porsche dynasty (one of whom was VW's CEO at the time) were obviously a factor in the background of all of this; but the story doesn't need any warring cousins. They just add a bit of journalistic "human interest" spice. So the cashflows from VW's Ords and Prefs were identical. The difference between the two was thus a ...


3

Almost all bonds have a "minimum amount" and "minimum increment", in the thousands of dollars, which is a lot if you're a retail investor working with thousand-dollar notionals, but is effectively zero if you're an institutional investor working with million-dollar notionals. As I recall, U.S. treasury is now USD 100 minimum, most ...


3

What @noob2 said: Actually there is empirical evidence of the opposite, i.e. the existence of a Term Premium. But this is not evidence of arbitrage, just that a more complicated risk model than assumed here is needed. And the simpler theory is still useful in many ways I feel it's helpful to unpack this a little. Let's say you are buying a 10 year Treasury/...


3

See the graph below. Let's define the PNL as the position's payoff at expiry plus accrued initial investment, i.e. collected / paid option premia. Assuming $K_1=95,K_2=100,K_3=105$ (i.e. $\lambda=0.5$), the orange payoff diagram below belongs to a setup where $C_2<\lambda C_1 + (1-\lambda) C_3$: You paid some net fee initially, and you obtain a position ...


3

Too long for a comment - so I add this here as an answer. Not sure what delta hedging arbitrage is but I think you define delta as a difference in price? While cross listing is not uncommon, I think AAPL is actually Nasdaq only as opposed to say IBM which is cross-listed. This is verified by Reuters when looking at IBM vs Apple If you ever were to see ...


3

The context of weak$^*$ topologies and no free lunch is often the proof of the first fundamental theorem of asset pricing. All the ideas below are from Delbaen and Schachermayer (1994). Notation Suppose the price process is a semimartingale $S$. Let $K_0$ represent the space of all claims generated by admissible trading strategies (self-financing and zero ...


2

Suppose the risk-free rate of return is $R^f$, the rate of return of the portfolio here is $R^p$, and the market return is $R^m$. We know $$\mathbb{E}[R^p-R^f]=0.04+1.4\mathbb{E}[R^m-R^f]$$ If we put \$1 into the portfolio, short \$1.4 the market portfolio, and invest the \$0.4 at the risk-free rate, then the expected wealth will be $$\mathbb{E}[X]=R^f+0.04+...


2

Supply and demand is your answer. Porshce wanted to acquire influence over the company, therefore they had significant demand for ordinary shares (with voting rights), and drove their price up significantly. Other investors recognised the distortion between these and placed a medium term bet on mean reversion, i.e. that the value of voting rights versus non-...


2

I've worked in this industry for a while and have run ETF market making for quite a few years. It's very difficult to perfectly lock in profit as you detailed above. With fast equipment it can be done sometimes. But most of the time you really are just hedging to model - and there is risk in that case. For example, you might sell ETF X and then hedge ...


2

It sounds like you haven't filtered away static arbitrage strategies (such as butterfly spreads, call spreads and calendar spreads) from your data. To keep my answer short and concise, there's a paper by Carr & Madan (2005) that establish the structure of a finite set of tests (a filtering procedure) on your option quotes. When you are left with options ...


2

Short answer: a. Ask yourself what distributions are implied by the options you have calibrated to? b. Any other product priceable using at most a subset of those distributions can be safely priced by that model without a second thought. Also, any payoff statically replicable by the calibration instruments can be safely priced. c. Any other product to be ...


2

NB: IPO investors are not the same as sponsors. "Sponsor" refers to the entity putting up the risk capital. "The IPO investors could just sell their common shares for at least $10 right after the IPO" - no, they cannot. The unit (stock + warrant) doesn't split until a certain number of days after IPO (often but not always 52). "For ...


2

There are two different papers published by Freddy Delbaen and Walter Schachermayer in 1994. A general version of the fundamental theorem of asset pricing They prove a general version of the first fundamental theorem of asset pricing. Published in Mathematische Annalen They prove that NFLVR is equivalent to the existence of at least one EMM (``First FTAP'')....


2

This is an opinion-based question. In practice, sometimes one needs to calculate mark to market and also various risk measures under risk scenarios that stress/perturb market data so much that arbitrage becomes possible. (See (*) below for more information.) Hence, 1 it's a perfectly reasonable question, provided there was some class discussion of heuristics ...


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