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The negative price that was all over the news was the front contract for WTI (West Texas Intermediate) futures that went to -40 and had a last trade date of 21.04.2020, so today. This movement was connected to derivatives and among other explanations was the fact that traders were exiting positions in order to avoid the risk of taking delivery of physical ...

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Usually stock returns are assumed to be normally distributed: $R\sim N(\mu,\sigma)$ If market goes up 1%, the expected stock return is $\mu = \beta\cdot 0.01 = 0.02$ (since β is the senstivity to market). Stock price going from 100 to over 103 requires a return $R$ of at least 103/100 – 1 = 0.03. As we have from the question σ = 0.02, we get: $$P(R\geq ... 8 In the academic literature it is extremely widely applied in the last 20 years. I would estimate maybe 200 empirical papers, or more. For example a common finding is that higher frequency (daily) wavelet correlations have been high since 2007, attributable either to increasing financial interation or the financial crisis. It is also popular to estimate the ... 7 Ask minus bid has nothing to do with the mid price - it is the spread. Generally you see a collection of bid/offer orders resting on different price levels. In the simplest case, you just see one bid at price p_b and one offer at price p_a. In this case the mid price is$$ p_m = \frac{p_a + p_b}{2} That's all there is to it - you don't need to "... 6 I'm not sure how deep of a question you are asking. The dog that did not bark is from a Sherlock Holmes murder mystery. The dog at the house did not bark at the intruder, so Holmes believed the dog knew the intruder. Therefore, the lack of evidence like barking, was itself the evidence. In the Cochrane paper, the introduction mentions that the lack of ... 5 Pring was (probably) simply referring to the fact that most indicators are function of price -- lots of different ways to twist and contort prices to define trends, reversal points, etc. Volume is another parameter entirely, as it doesn't depend on price; the market or share price can have an up day on average, high, or low volume, it can have a down day on ... 4 Art markets typically have huge transaction costs of the order of 10%, caused by buyers premium and auction fees. Therefore long holding periods are unavoidable, with long-term returns somewhere between those of bonds and equities. By its very nature, art is not easily replicated so arbitrage or derivatives are out. The rationality of agents (aka collectors) ... 4 Perhaps construct a Brownian Bridge between the day's open and close, then scale it according to the day's high and low. 4 Since you're asking on a quant finance forum, the mathematical approach would be Decide on a model that the stock price follows, and Compute the expected value of the price, conditional on the most recent price. A famous model, made ubiquitous by Black, Scholes and Merton, is a geometric Brownian motion. Under this model, the stock price S_T at time T ... 4 Assume we knew the density function f of the FX price that we observe in the market. Then the market price of a call option C(K) with strike K would be \begin{align*} C(K)&=e^{-rT} \int_0^{\infty}(s-K)^+ f(s)ds \\ &=e^{-rT} \left( \int_K^{\infty} s f(s)ds - K \int_K^{\infty}f(s)ds \right) \tag*{(1)}. \end{align*} C(K) is a market price and ... 3 Why not just use Geometric Mean Returns? Each time you buy/sell an ETF calculate the holding period return as a percentage and plug into the formula. The answer is a percentage that you can use to calculate the approximate money appreciation (or loss) against your "fixed notional" 3 This mean that the reason why apple stock price went from 3 to 100 in 10years is the overnight variation in price. This is quite unexpected, if there was no overnight variation the stock price would have died a long time ago... Why is that ? Have we been lying to us ? This is because many business and financial news are reported at market close, either pre-... 3 I would say the financial- and the art market is very different, only the roots of the market / auctions is the same. As the art market is unique and very illiquid, alot of the strategies from the modern financial market simply does not apply. I have been building (and still maintains) a toolbox of models, which mostly try to find trends based on multiple ... 3 Yes I would recommend you to plot the log of prices instead of prices. It will re-scale the data while preserving the hierarchy of prices, and more importantly it allows to compare easily the growth among several stocks because a vertical move of 0.01 corresponds to a 1% change of the price at any point in the figure (not matter the price level and the ... 3 By definition, modified duration is D_\text{mod} = \frac{1}{P} \frac{dP}{dy} where P is the dirty price of a bond. Clean price is the standard quoting convention for the vast majority of bond markets (though not all), but nearly all analytics, be it yield to maturity, DV01, or duration, are all computed using dirty price. 3 That simply means that a bond pays one unit of the currency in any state (regardless what happens in the future, i.e. there is no default risk about the payoff of a bond). So you will receive 1 in the next period (regardless what you paid for it). Of course, today you probably pay less than 1 due to time value of money... 3 You can extract the risk neutral density implied by option prices and have a look at that. The implied probabilities are given by the prices of butterfly spreads in the market. This is common knowledge. Page 241 of this book explains how you could go about doing it in Excel: https://gaussiandotblog.files.wordpress.com/2018/02/wiley-trading-giles-peter-jewitt-... 3 Step 1: Know your distribution Since \int_0^t W_s\mathrm{d}s\sim N\left(0,\frac{1}{3}t^3\right), we have \begin{align*} S_t &= S_0 \exp\left( rt-\frac{1}{6}\sigma^2 t^3 + \sigma \int_0^t W_s\mathrm{d}s \right) \\ &\overset{d}{=} S_0 \exp\left( rt-\frac{1}{6}\sigma^2 t^3 + \sigma \sqrt{\frac{1}{3}t^3} Z \right) \\ &\overset{d}{=} S_0 \exp\left( ... 3 Hi: Based on your question, it sounds like the Diebold-Mariano test might be perfect for your case. It doesn't require any sophisticated assumptions about models or processes etc. All one needs are the two sets of forecasts and the actuals. I can't find the actual paper but below is the reference to it. I imagine that, if you google hard enough, the paper ... 2 On exchanges, there is such orderbook with sufficient amount of limitorders, so when you place an order (market or limit), the "best" limitorders for you will be hitted and change the price last traded price. The price you see is actually just the midpoint between the currently best available bid and ask prices in the orderbook. Therefore, this price might ... 2 price went from \200 to \202, this is "one percent change", because \frac{\2}{\200}100=1 2 The state price vector are the prices of securities which pay \1 if and only if that state of the world occurs. This is just a question of being able to replicate the payoffs \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} $$with payoff vectors \vec{b} = [1,1,1]^T and \... 2 I think a good way to think about your problem is the example of finding an optimal VWAP trading strategy. You basically have a finite point in time by which you must have performed your transaction and you trade a similar asset than the one you are considering, one with the same underlying assumptions of mean-reversion (I make such assumption in the same ... 2 This is an example of minimum price variation (also known as the minimum price increment or the minimum price fluctuation). All public quotes for US equities are displayed to the nearest penny. (Hidden quotes may be entered at sub-penny increments.) US stock indices follow this convention and thus quote to the nearest penny. The oil listing is odd indeed. ... 2 The upper bound for the 80 call is C(90) + 10, or 30. At least assuming no arbitrage. Let's start by assuming the risk-free rate is 0 (this isn't a problem, but the math is clearer without it), so we don't have to discount the price. Then, the call price is given by C(K) = E_t[(S_T - K)^+], which gives: \begin{array} C(K - 10) &= E_t[max(S_T - (K - ... 2 This is a very broad question and a large number of issues have been discussed in the literature. As such, it's hard to give specific advice except that it is better to model returns instead of prices directly. What I would do if I were you: Read some of the available literature to get a good overview. This is an interesting paper but many more exist. ... 2 The average would be called the mid-price, not the best in my opinion, but that depends on your modeling. Another strategy is to weight the bid and offer prices according to size, also called the micro-price or bid-offer weighted price. This has the advantage of moving your calculated price closer to where it is traded as volume is depleted from whatever ... 2 Actually prices dont make sense as they are correlated with previous samples (prices), returns are not. Better will be difference between prices, but then you dont have reference point and comparability between assets, so eventually you need returns. At the end that is what you are interested in I think as profit is usually measured in return. 2 As it was pointed above the phrase is taken from Sherlock Holme's novel. It describes the case when the dog should have bark, but didn't. Now if we come to the Cochrane paper. He introduces the system of equations (r_{t+1} - returns, \Delta d_{t+1} - dividend growth and d_t - p_t - dividend-price ratio):$$ r_{t+1} = a_r + \beta_r(d_t - p_t) + \...

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This is an implied mid price. If an illiquid market and/or quiet time of day to snap bid and ask prices, you may have an implied mid that is skewed (consider the case that a dealer offers at his mid to quickly close a position, but fails to find a buyer unless he hits a bid, crossing full B-A). A better solution would be weighted implied mid, pick the top ...

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