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28 votes

What is the Risk Neutral Measure?

Life Without a Risk-Neutral Measure How would we price assets without the measure $\mathbb Q$? Well, we would start with some version of the Euler equation $P_t=\mathbb{E}_t[M_{t+1}P_{t+1}]$, where $M$...
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18 votes

What is the importance of alpha, beta, rho in the SABR volatility model?

We created the SABR model because we realized that (a) option values were nonlinear in the volatility, and (b) volatilities are stochastic. This means that if one had an option (or portfolio of ...
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17 votes

What is the importance of alpha, beta, rho in the SABR volatility model?

Let's relabel this as What (TF) is SABR? Alpha, Beta and Rho are the point of the model. So explaining them is explaining the model. A model of two processes Unlike earlier models in which the ...
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  • 3,589
13 votes
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Why does it take so many lines of code to price even the simplest of options with QuantLib

I've been using QuantLib for quite a while. Let me share some experience: QuantLib is a highly sophisticated quantitative framework. It can do much and much more than a simple pricing of European ...
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12 votes
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Free and tested optimization, statistical and visualization packages for C#

A popular open-source option for the numerics in .NET is Math.NET (https://github.com/mathnet/mathnet-numerics). It has both managed implementations and allows you to use the optimized MKL native ...
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  • 238
12 votes
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Arbitragefree Pricing: Q vs. P

In the derivatives context, "arbitrage free" means almost surely for the probability measure under consideration. This is in opposition with statistical arbitrage used at high frequencies for example. ...
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12 votes

How to derive the price of a square-or-nothing call option?

I provided an answer, based on an elementary approach, to an exactly same question yesterday. However, that question has disappeared, even though I like to keep a record for what I wrote. I would ...
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12 votes
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What is the Risk Neutral Measure?

Intro: Great answer given by KeSchn above. I would like to contribute an additional perspective. My experience with and my understanding of the Risk Neutral measure is entirely based on "no ...
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11 votes
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Numeraire correlated to the traded asset

As @ilovevolatility explains, the seminal reference for this matter is Geman, El Karoui & Rochet (1995). We assume none of the assets are dividend paying, and they are strictly positive. There are ...
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9 votes

What is the industry standard pricing model for CME-traded Eurodollar future (American) options?

Having traded these options for a number of years I have some insight. It’s my belief that those that make a living specifically out of these options do have tree-style models that take into account ...
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8 votes

Which interest rate model for which product

The model of choice depends on the purpose of the exercise. In general there are two types of models: Equilibrium models: These are general used use for "fitting" the spot curve to the discount ...
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  • 301
8 votes

How to derive the price of a square-or-nothing call option?

See this excellent paper by @MarkJoshi which defines/discusses the use of power numeraires. Starting from a dynamics specified under the risk-neutral measure $\mathbb{Q}$ \begin{align} &\frac{...
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8 votes
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what does the cover page of Guyon and Labordere's Nonlinear Option Pricing represent?

Julien Guyon was so kind as to explain the story behind the cover and gave me permission to share it with the rest of the community: There's no direct link between the contents of the book and the ...
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  • 7,598
7 votes
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Two different ways of pricing that leads to two answers

$$\frac{1}{(1+r_{02})^2} = E\left(\frac{1}{1+r_{12}}\right)\frac{1}{1+r_{01}}$$ Indeed, in the pricing measure, the distribution of $r_{12}$ has to be such that this relation holds. If you look at ...
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  • 6,743
7 votes

Black and Scholes pricing

At least two ways to price this: Use Carr-Madan Use $S^2$ as a (power) numeraire, in which case you can price the payoff $(S_T - 1)_+$ under the power numeraire measure. EDIT: Put-call symmetry. ...
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7 votes
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Why is $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for prices?

We don't model the prices, we model the returns. The stock prices aren't explicitly modelled as log-normal, but rather this is a consequence of the actual model used to describe the returns. The core ...
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  • 1,359
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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6 votes
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The effect of negative interest rates on derivative pricing

Independently if it makes economically sense or not, negative interest rates have become a reality for Europe which can no longer be neglected. (Even LIBOR became negative in the last months.) One ...
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  • 76
6 votes
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Why QuantLib computes the fixed-leg swap rate by this formula?

fixedLegBPS is the basis-point sensitivity of the fixed leg, that is, how much its NPV changes when the fixed rate changes by one basis point: it's calculated as ...
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6 votes

Why does it take so many lines of code to price even the simplest of options with QuantLib

And don't forget that there are wrappers as eq RQuantLib which I use on the command-line here: ...
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6 votes
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Pricing a double barrier option using Monte Carlo (C++ & Python code included)

Here are at least three mistakes in your code: p += s0 * exp(...) should be p *= exp(...). Your volatility and rates are per ...
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6 votes
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Equivalence of Put Pricing Formulas

The first equation expresses the option price as a discounted expected value of the payoff contingent on an asset price $S \geqslant 0$. Without loss of generality, we assume that the probability ...
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  • 3,250
6 votes
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FX Forward rate agreement valuation in quantlib

You are not giving the constructor a discountCurve. The constructor is: ...
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  • 5,285
6 votes

What is the Risk Neutral Measure?

I believe the other answers are nearly exhaustive; but here's a bit of intuition I'd like to add: Think of the decision (= equilibrium price) of a market as: Decision = f(probabilities, risk aversion) ...
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  • 1,662
6 votes
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Modelling Geometric Browian Motion price model with stochastic volatility

Let me try to answer, this topic is much deeper than my answer 1. Why are these models like this unpopular? First, these models produce marginal distributions that does not fit the market, which ...
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  • 356
6 votes
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If I have the present value of an amortizing bond's cashflows, how do I figure out price?

The price of most (not all) bonds is quoted as a percentage of face value (par). For most amortizing bonds that have already amortized, the percentage is of the face value now, after amortizations, ...
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6 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 ...
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  • 4,178
5 votes

Why does it take so many lines of code to price even the simplest of options with QuantLib

To add to Student T's answer, which I second: the complex setup starts making sense (and its cost gets amortized) once you start keeping the instruments around instead of throwing them away after the ...
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5 votes
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How to Compute Dates for Bond

To compute the cash flow dates you need to know the maturity date, the tenor, the payment frequency, the business day convention and the holiday calendar. The cash flow dates step backward from the ...
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  • 5,603
5 votes

Overlapping vs Non-overlapping returns

Actually, overlapping samples is a big problem in financial machine learning which is called concurrency. Marcos Lopez de Prado discusses this issue in Chapter 4 of his book Advances in Financial ...
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