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37 votes

Explaining 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 $M$...
Kevin's user avatar
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24 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 ...
Patrick S Hagan's user avatar
20 votes
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Explaining the Risk Neutral Measure

Intro: Great answer given by Kevin. I would like to contribute an additional perspective. My experience with and my understanding of the Risk Neutral measure is entirely based on "no arbitrage&...
Jan Stuller's user avatar
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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 ...
Phil H's user avatar
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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 ...
Gordon's user avatar
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10 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 ...
Daneel Olivaw's user avatar
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 ...
dm63's user avatar
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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{...
Quantuple's user avatar
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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 ...
Bob Jansen'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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7 votes

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

Unless I am missing the obvious, I do not see the question being answered? In my opinion, trying to understand in simple language what $\alpha, \beta, \rho$ mean requires an explanation what these ...
AKdemy's user avatar
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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 ...
oliversm's user avatar
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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
  • 8,969
7 votes

What are some interesting recent machine learning related developments in the QF domain?

Sirignano, J., & Cont, R. (2019) (High-frequency stock forecasting): The authors apply a large-scale deep learning model (recurrent neural network with Long Short-term Memory units) to high-...
Pleb's user avatar
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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 ...
LocalVolatility's user avatar
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 ...
RRL's user avatar
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6 votes

Pricing In Real Life vs Theory

1. Let me first reconcile the Black-Scholes pricing formula with the idea of prices being determined by supply-and-demand. Even if it is not explicitly said this way, from an equilibrium perspective, ...
Stéphane's user avatar
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6 votes
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FX Forward rate agreement valuation in quantlib

You are not giving the constructor a discountCurve. The constructor is: ...
David Duarte's user avatar
  • 5,825
6 votes

Explaining 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) ...
Arshdeep's user avatar
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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 ...
ryc's user avatar
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6 votes
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If I have the present value of an amortizing bond's cashflows, how do I figure out price?

Bullet bond prices are quoted as a percentage of face value (par). For most amortizing bonds that have already amortized part of the initial principal (face value), the price is a percentage of the ...
Dimitri Vulis's user avatar
6 votes
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Why do we need to split market and default information into 2 separate filtrations?

I think you are absolutely correct if the hazard rate is deterministic, although I think you are forgetting a discounting factor in your example. But sometimes the hazard rate cannot be assumed to be ...
mmencke's user avatar
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6 votes

What are some interesting recent machine learning related developments in the QF domain?

Empirical Asset Pricing via Machine Learning (2020) by Gu, Kelly and Xiu
TwoII's user avatar
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6 votes
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What does implied volatility say about the underlying?

A vol surface displays implied volatilities (IVOL) for various tenors and strikes. It can be displayed in several ways, with the two most common being: Moneyness Delta Interest rate options are a ...
AKdemy's user avatar
  • 8,969
6 votes

Discounted price of an option

The process $Y_t:=(S_t-K)^+$ cannot be the price of a traded asset because of Jensen's inequality. Instead, it is the price of the option which is a martingale. In the Black-Scholes model, the ...
Daneel Olivaw's user avatar
5 votes

Calculate Average Price, Cost, (Un)Realized P&L of a position based on executed trades

Using Andy Flury answer and bit polishing it gives following Python class for PnL calculator: ...
mde's user avatar
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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 ...
Chris Taylor's user avatar
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5 votes
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Economics of spoofing

Layering is a spoofing of buy(sell) orders sometimes complemented by higher(lower) sell(buy) orders that push the market up(down). Execution of a limit order above(below) would not be a concern, as ...
rrg's user avatar
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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 ...
Alexandr  Proskurin's user avatar

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