# Tag Info

### 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 $M$...

### 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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### 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&...

### 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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### 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 ...

### 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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### 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 ...

### 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 ...

### 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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### 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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### 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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### 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 ...

### 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) ...
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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 ...

### 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-...

### 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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### 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 ...

### 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 ...
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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 ...

### 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, ...
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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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### 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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### 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 ...
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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 ...

### 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
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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 ...

### 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 ...