# Tag Info

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The risk-neutral measure $\mathbb{Q}$ is a mathematical construct which stems from the law of one price, also known as the principle of no riskless arbitrage and which you may already have heard of in the following terms: "there is no free lunch in financial markets". This law is at the heart of securities' relative valuation, see this very nice paper by ...

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Market makers, obviously, have to be willing to short an option. They will delta hedge their positions to limit risk. As for investors, they can aim for a buy-write strategy to collect extra income in lieu of unlimited upside. And lastly, someone who owns a stock he can't sell right away (such as an entrepreneur still under a vesting period after his firm ...

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This is a basic fact about futures trading and the storage of commodities. The phrase that was used by futures traders in the old days (and probably still today) was "the contango is limited by the carrying cost, there is no limit to the backwardation". This means that for example if spot gold is at 1200, gold dated one year from now cannot possibly sell ...

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Although quite simple connected scatterplots can give interesting new insights on how time series perform together: http://steveharoz.com/research/connected_scatterplot/ As an example: Gold vs. S&P 500 from 1970 till today: The green point marks 1970, the red point is today. Every point is a year, moving vertically upwards means rise in the S&P ...

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FO is shrinking across the large investment banks. The market is not developing new products that will need new pricing formulas, if anything it is reverting to more vanilla structures. Nowdays FO quants typically hack existing models around the corners to manage new market conditions (change Sabr a bit to deal with negative rates, refine the treatment of ...

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You may want to consider splitting two important, yet very different concepts: Pricing a derivative security with contingent payoff and forecasting an asset. Pricing a derivative can be achieved through setting up a hedge portfolio and track its evolution and "value" at any point in time before the derivative security pays off. Risk-neutral pricing is a ...

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Futures are in "zero net supply", or "for every long there is a short", which means that at any time there are investors who are long a certain number of contracts and other investors who are short an (exactly matching!) number of contracts. This number is called the Open Interest. It starts at zero when the exchange introduces a new contract (like Sep 2019 ...

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To me, coloring by data value is a great way to bring applications alive. If traditional ways are not enough, probably taking 3D in use would be a way: And of course 2D heatmap is a very handy for sure. I'm developing data visualization software components with 3D technologies, so definitely all feedback and ideas are welcome :-)

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And then music... Victor Neiderhoffer, in a 2001 interview: The market plays music all the time. The problem is you never know how the music of the market is going to end. But a good framework is that it will end on the tonic. Consonance to dissonance back to consonance. And whenever there's tremendous dissonance, strident moves in one direction, a good ...

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Just take something like $$\frac{\log{\frac{F_j}{F_i}}}{t_j - t_i} \times 365$$ where $t_i$ denotes the expiry (or alternatively delivery) date of future $i$. The annualization is so you can compare different futures.

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Let $P$ be the risk-neutral measure. We define the measure $P_S$ such that \begin{align*} \frac{dP_S}{dP}\big|_t &=\frac{S_t}{e^{rt}S_0}\\ &=e^{-\frac{1}{2}\sigma^2 t+\sigma W_t}. \end{align*} Then $\{\widehat{W}_t \mid t \ge 0\}$, where \begin{align*} \widehat{W}_t = W_t -\sigma t, \end{align*} is a standard Brownian motion under the measure $P_S$. ...

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The piece you are missing is an approximation via the Taylor formula of the logarithm: $$\ln(1+x) \approx x-\frac{x^2}{2} \; .$$ Apply this to the first term in the final formula of the technical paper: $$\frac{2}{T}\ln\frac{F_{0}}{S^{*}} = \frac{2}{T}\ln\left(1+\left(\frac{F_{0}}{S^{*}}-1\right)\right) \approx \frac{2}{T}\left(\left(\frac{F_{0}}{S^{*}}-1\... 7 A swap does not require a model because its price can be derived from the yield curve without any assumptions about how the yield curve may move in the future. The PFE however is an indication of by how much the swap's mark-to-market may move between now and a moment in the future. It is of course influenced by how volatile rates are. The more volatile ... 7 I will refer to "Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit" by Damiano Brigo and Fabio Mercurio. In chapter 3 (One-factor short-rate models) they have a very nice table which lists some of the properties of instantaneous short rate models. In both of your models you know the distribution of r_t. The huge difference ... 7 The interviewer meant that he's smart. Quoting Senior VP of People operations at Google, On the hiring side, we found that brainteasers are a complete waste of time. How many golf balls can you fit into an airplane? How many gas stations in Manhattan? A complete waste of time. They don’t predict anything. They serve primarily to make the interviewer ... 7 The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate. Let us start with the fair value of a swaption under the annuity measure \mathcal{A} with tenor at time \tau:$$\mathcal{A}(t)\mathbb{E}^\mathcal{A}_t[(\mathcal{S}(\tau)-k)^+]Instead ... 6 This is indeed one of the most difficult tasks to do (if not next to impossible). I would say the standard reference is the following: Expected Returns: An Investor's Guide to Harvesting Market Rewards by Antti Ilmanen An abridged (but still about 170 pages long), yet more current - and free (!) version in different formats (pdf, mobi for the Kindle and ... 6 It is incorrect to use 1m euribor or O/N euribor in a 6m Euribor forward curve. You should only use instruments based on 6M euribor, such as 1x7 FRA, 6x12 FRA or swaps v 6m Euribor, as you have done in your second example. The actual 6m euribor fixing itself can be thought of as a 0x6 FRA out of spot. Before the financial crisis basis between different ... 6 What do you mean by "so we can price as usual"? What you showed is that for every c \in \mathbb R we can find a probability measure such that the drift of S is c. But that does not really say anything about pricing. You can easily see that V_t = E^Q_t[e^{-r(T-t)} \Phi_T] does not give arbitrage free prices with your choice of Q. Indeed if \Phi_T ... 6 Note that, \begin{align*} \frac{\partial{C}}{\partial{\sigma}} &=\frac{S_0}{\sqrt{2\pi}}{e^\frac{-d_+^2}{2}}(\frac{-1}{\sigma})(d_-)-\frac{Ke^{-rt}}{\sqrt{2\pi}}e^{\frac{-d_-^2}{2}}(\frac{-1}{\sigma})(d_+)\\ &=\frac{1}{\sqrt{2\pi}}e^{\frac{-d_+^2}{2}}\left[-\frac{S_0 d_-}{\sigma} + \frac{Ke^{-rt}d_+}{\sigma} e^{\frac{d_+^2}{2} - \frac{d_-^2}{2}} \... 6 Either r=0 in which B_t is constant and is a valid numeraire (as is any multiple of it.) or  r \neq 0 in which case an asset of constant value would give an arbitrage since we could take B_t - N_t with B_0 = N_0 and get a riskless profit. (or the opposite if r<0.) and so it would be a very flawed model. 6 As mentioned by @Adam, Stochastic Calculus for Finance by Shreve is a good start if you have a reasonably strong mathematical background. Volume I is simpler, as it presents derivative pricing methods in discrete time; Volume II tackles the continuous case. Also mentioned by @noob2, Financial Calculus: An Introduction to Derivative Pricing, by Baxter and ... 6 We assume that the process \{J_t, \, t\ge 0\} is defined at the jump times of the Poisson process \{N_t, \, t \ge 0\}, and all the jump sizes are independent and identically distributed. That is, \begin{align*} Q_t \equiv \int_0^t (J_t-1) dN_t = \sum_{n=1}^{N_t} (J_i-1), \end{align*} where J_i, for i=1, \ldots, \infty, are independent and \xi_i = \... 6 A general hedging strategy Let assume that S_1(t) and S_2(t) are the price processes of your 2 stocks and that they follow a Geometric Brownian Motion (GBM):\forall \, i \in \{1,2\}, dS_i(t) =\mu_iS_i(t)dt + \sigma_iS_i(t)dW_i(t)$$We assume both stocks have an instant correlation of \rho:$$dW_1(t)dW_2(t)=\rho dt$$Let also V(t) be the value ... 6 Under a Black-Scholes framework, the dynamics of the stock price under the risk-neutral measure \mathbb{Q} are given by ...$$ S_t = r S_tdt +\sigma S_tdW^{\mathbb{Q}}_t $$... and those of the risk-free bond by:$$ \begin{align} dB_t = rB_tdt \end{align} Let us define the derivative value as V(t,S_t), which only depends on the time t and the ... 6 If by "modern C++" you mean C++11 and beyond, I'm afraid you won't find such a book at this time. If you are content with idiomatic C++ 03, as in "using the STL and smart pointers instead of managing memory by hand", I second Quantuple's suggestion of Mark Joshi's book. With a couple of caveats, I might also add my own Implementing QuantLib; the caveats ... 6 If you plot the function f, you see that you have a bear spread. You can build such vertical spreads either with call or put options. For example consider a portfolio selling one put option with strike price K_1=30 and purchasing one European-style put option with strike price K_2=35. Then, you obtain the payoff \begin{align*} \max\{35-S_T,0\}-\max\{30-... 6 Any nonnegative random variable Z with expectation 1 is a Radon-Nikodym derivative: \mathbb{E}^{\mathbb{P}} \left(Z\right) = \mathbb{E}^{\mathbb{P}} \left(\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}\right) = \mathbb{E}^{\mathbb{Q}} \left(1\right) = \int{\mathrm{d}\mathbb{Q}} = 1  \mathbb{Q} \left(A\right) = \mathbb{E}^\mathbb{P} \left(Z 1_A\...

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I provide a solution in three steps. The first step carefully outlines how to split up the expectation and what new measures are used. This first step does not require any special model assumption and holds in a very general framework. I derive a formula for the option price that resembles the standard Black-Scholes formula. In a second step, I assume that ...

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