42 votes
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How to estimate real-world probabilities

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 ...
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13 votes

Theoretical limits for contango and backwardation

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 ...
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  • 9,077
13 votes
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Derivation of VIX Formula

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 ...
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  • 1,487
12 votes
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Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$

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 ...
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11 votes

How to use the stock as a numeraire to price a derivative with payoff of the form $(S_T f(S_T))^+$?

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*} ...
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11 votes

Present and future role of pricing quants

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 ...
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11 votes
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Relationship between Vega and Gamma in Black-Scholes model

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 ...
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10 votes

Innovative ways of visualizing financial data

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&...
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10 votes
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Impact on DV01 of cbot bond futures by changing coupon from 6% to 4%

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 ...
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9 votes
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When would open interest equal trading volume?

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 ...
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8 votes
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Option on an Option

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 $t_e$...
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7 votes
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Why Must Dividends Be Reinvested to Use Risk-Neutral Pricing?

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 ...
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What is a Constant Maturity Swap (CMS) rate?

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 ...
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7 votes

Heston Model Integration Oscillations

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 ...
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7 votes

Using a Constant as a Numeraire

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 ...
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7 votes
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Black-Scholes formula for Poisson jumps

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, \...
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7 votes

Why discounted derivative price is a martingale?

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 ...
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7 votes
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The dice game and derivatives trading

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 ...
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7 votes

CMS Pricing - Convexity Adjustment by Replication

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 ...
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7 votes
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Static vs Dynamic Hedging: when is each one used?

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 ...
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7 votes

How do market makers calculate the IV for options?

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 ...
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7 votes
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What is the Radon-Nikodym derivative in the Heston model?

Let \begin{align*} \mathrm{d}S_t&=\mu S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}, \\ \mathrm{d}v_t&=\kappa(\bar{v}-v_t)\mathrm{d}t+\xi\sqrt{v_t}\mathrm{d}B_{v,t}, \end{align*} where $\...
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7 votes
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Risk Neutral Valuation, Drifts and Calibration

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, ...
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7 votes
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Value of contingent claim at a given time

This is related to the valuation of a forward start option. Let's assume Black-Scholes world (constant vol) and for simplicity byt without loss of generality I'll set $r=q=0$. Notice that $$ \min(S_T,...
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6 votes
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Curve Euribor - Euribor 3M

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 ...
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6 votes

What is a Constant Maturity Swap (CMS) rate?

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 ...
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6 votes
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Derive vega for Black-Scholes call from this formula?

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}{\...
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6 votes

Using a Constant as a Numeraire

A Numeraire must be a tradeable asset. If you can find a constant tradeable asset, then yes a constant can be used as a numeraire.
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6 votes
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What are the books in which to study the basics of the derivative financial instruments?

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 ...
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