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18

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 ...


6

As a general principle, I would be wary of economic or financial interpretations of change of measure techniques. Changing numéraires is merely a mathematical tool to ease pricing, see for example the last part of this answer. Nevertheless, here’s my take on your question. Think of a numéraire as the basic financial asset of your economy, namely a store of ...


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

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 ...


3

Banks and arbitrageurs move money around with limited difficulty. That's one of their main functions. They move enormous volumes daily. Furthermore, I believe that for retail markets, whenever there is an opportunity to earn relatively risk free money, people will try to take advantage of it. More so if what they are doing is completely legal. If it means ...


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

The drift is the expectation of the return over an infinitesimal interval. Let $Q$ be the risk-neutral measure and $Q^S$ be measure associated with the stock price numeraire defined by \begin{align*} \frac{dQ^S}{dQ}\big|_t = \frac{S_t}{B_t S_0}, \end{align*} where $B_t=e^{rt}$ is the value at time $t$ of the money-market account. Moreover, let $E$ and $E^S$ ...


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

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...


2

You can assume two periods economy: calling them today and tomorrow is a convenient representation that is easy to relate to. Today is certain, tomorrow is not- the number of states is known, and the economy will be in one of these states tomorrow. A generic state is represented by s. $pc(s)$ is the today price of a security that will pay one unit if state s ...


2

A bond repays its notional face value (plus interest sometimes), not the original purchase price. Do not assume the the price you pay for a bond is its face value. Sometimes a law or a regulation (pretty useless, in my humble opinion:) does require that a bond newly issued in primary market be sold at exactly 100% par price (face value). Then the coupon ...


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) + \...


1

I have a take on the intuition part of the question. Isn't it a simple consequence of Jensen's inequality? Thus, assuming $r=0$ for simplicity, we have in the money market measure: $E(S_T)=S_t$, but then $E(1/S_T)>1/S_t$ by Jensen since $1/x$ is convex. Now in the stock measure, we must force $E_S (1/S_T)=1/S_t$ to create the correct martingale, but ...


1

If you are just looking at price returns, you would use: (closing price of March - closing price of Dec) / closing price of Dec. However, this would not include the returns due to dividends. If you want total returns you would need to incorporate dividends. Also, you would have to make some assumptions about the return on those dividends. For example, are ...


1

I think it also shows the pedigree of the fund manager. All else equal, if the fund manager could beat the index by 2%, that says something non-zero


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