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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 ...


13

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 ...


11

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 ...


11

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 ...


11

Consider any option, vanilla or exotic. In between fixing dates it satisfies the Black & Scholes PDE (for simplicity zero interest rate and dividends) $$ \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 U}{\partial S^2}(S,t)+\frac{\partial U}{\partial t}(S,t)=0 $$ Let ${\cal V}(S,t) = \frac{\partial U}{\partial \sigma}(S,t)$ be the option vega. Differentiating ...


10

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 $P_S$. ...


10

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 ...


10

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\...


10

It's complicated. Assuming there is no CTD switches, then yes, the theoretical modified duration should be unchanged and the DV01 will be lower. For simplicity, imagine that there is only one bond eligible for delivery into the contract. We'll also ignore all the other complications (e.g., variation margins), then the theoretical futures price is simply the ...


9

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 ...


8

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 ...


8

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 :-)


8

The proof is relatively long, so I focus on displaying the reasoning and major steps. We work on a Black-Scholes model. Without loss of generality, we focus on an option with strike $P$ to buy at $t_e$ a European call option expiring at $T$, written on a stock $S$. Expectations are always taken with respect to the risk-neutral measure $Q$ unless otherwise ...


7

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 ...


7

A constant maturity swap (CMS) rate for a given tenor is referenced as a point on the Swap curve. A swap curve itself is a term structure wherein every point on the curve is the effective par swap rate for that tenor. This is analogous to a 3m LIBOR curve represents 3m forward rates for a given tenor. A swap rate can be considered as a weighted-average of ...


7

There has been a huge amount of work on this. Generally a Fourier transform approach is used. First, be careful to use the form of the characteristic function that does not wind about zero in order to avoid having to count the normal of windings. Second, using contour shifts can make the integral much better behaved. eg integrate along the line with $0.5$...


7

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 = \...


7

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 ...


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 ...


7

It depends a little bit what you're trying to do. If you can statically replicate the payoff of a position at $t=0$, then putting on that hedge will insulate you from all risk coming from the contract. Payoff doesn't need to be linear - for example, you can perfectly replicate a call option using a put option and a futures contract If you want to use only ...


7

They do not calculate it, they set it at a market clearing level based on supply and demand. It is similar to the way equity market makers set the price of a stock: a lot of buyers => raise the stock price (or the IV), a lot of sellers => lower the stock price (the IV). For a new option, not previously traded, they might look at the IV of "...


7

There are two parts to your question which I try to answer separately. The first one is about what calibration actually is whereas the second question deals with risk-neutral pricing. As an example, we can use any model. I continuously refer to the stochastic volatility model from Heston (1993) as an example for equity options. Any thoughts equally apply to ...


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

In simple terms: An ordinary swap might be a 10 year swap of Libor vs a fixed rate; this fixed rate is determined in the marketplace every day and is published by Reuters, Bloomberg etc. as the '10 year swap rate'. Once you enter into the swap this rate remains fixed for you, of course, that is why it is called a fixed rate. But every day Reuters publishes a ...


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

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 ...


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