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48 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 ...
Quantuple's user avatar
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15 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 ...
Kevin's user avatar
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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 ...
Raskolnikov's user avatar
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10 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 ...
Alex C's user avatar
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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 ...
Helin's user avatar
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10 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 ...
Antoine Conze's user avatar
9 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 $t_e$...
Daneel Olivaw's user avatar
8 votes
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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 ...
compilation-error's user avatar
8 votes

Which models do Bloomberg/Reuters use to derive implied volatility for interest rate derivatives with negative forward rates?

Short Version Market standard (since the low interest rate environment after 2008) is to use Normal Vol (used in the Normal / Bachelier model) Market data comes from contributors like Tullett, ICAP ...
AKdemy's user avatar
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8 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, ...
Kevin's user avatar
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8 votes

What is meant by the funding cost of a derivative?

The theory for pricing derivatives is based on self-financing trading strategies that replicate all the payoffs of the derivative. Hence, derivatives pricing requires funding at the risk free rate, ...
AKdemy's user avatar
  • 8,924
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 ...
Mark Joshi's user avatar
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7 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 ...
Alex C's user avatar
  • 9,382
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 ...
Mark Joshi's user avatar
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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, \...
Gordon's user avatar
  • 21.1k
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 ...
Daneel Olivaw's user avatar
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 ...
Matthew Gunn's user avatar
  • 6,954
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 ...
FunnyBuzer's user avatar
  • 1,012
7 votes

Radon Nikodym derivative when changing numeraires

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}\...
siou0107's user avatar
  • 2,680
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 ...
StackG's user avatar
  • 3,026
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 ...
nbbo2's user avatar
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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 $\...
Kevin's user avatar
  • 16k
7 votes
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Differences between main classes of interest pricing derivatives models

I am not sure if you can classify it like that. Mind you, I never wrote a book. I'll write what I know below and you can decide if the classification makes sense or not. 1 ) STIR: as the term ...
AKdemy's user avatar
  • 8,924
7 votes
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Calculating DV01 for Treasury Futures with CTD switch risk

You are trying to calculate the so-called "option-adjusted DV01" (OA DV01). The nice thing about OA DV01 is that it's a smooth function of yield shifts. I'm going to be lazy here and simply ...
Helin's user avatar
  • 11.7k
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}{\...
Gordon's user avatar
  • 21.1k
6 votes

What is a Constant Maturity Swap (CMS) rate?

In a vanilla swap, the IR on the floating leg usually depends on the reset period/swap frequency. If frequency is 6m, 6m LIBOR is used for reset, 3m LIBOR for quarterly resets etc. In a floating CMS ...
quant360's user avatar
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.
user9403's user avatar
  • 1,429
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 ...
Daneel Olivaw's user avatar
6 votes
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Replicating a portfolio with a certain payoff function

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) =...
Daneel Olivaw's user avatar

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