39
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 $M$...
- 16.3k
25
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 ...
- 501
24
votes
Accepted
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&...
- 6,490
18
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 ...
- 3,719
10
votes
Accepted
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 ...
- 8,219
10
votes
Accepted
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, ...
- 16.3k
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 ...
- 17.9k
8
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 ...
- 9,884
8
votes
Accepted
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 ...
- 8,621
7
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, ...
- 2,536
7
votes
Accepted
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 ...
- 1,409
7
votes
Accepted
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 ...
- 9,884
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-...
- 4,911
6
votes
Accepted
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 ...
- 6,094
6
votes
Accepted
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 ...
- 3,730
6
votes
Accepted
FX Forward rate agreement valuation in quantlib
You are not giving the constructor a discountCurve. The constructor is:
...
- 5,895
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)
...
- 2,561
6
votes
Accepted
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 ...
- 401
6
votes
Accepted
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 ...
- 13.2k
6
votes
Accepted
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 ...
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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
- 61
6
votes
Accepted
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 ...
- 9,884
6
votes
How to price very short dated options?
In interest rate options you can observe the behavior of listed options on bond futures on the last day before expiration. What I’ve noticed:
(A) the most important consideration is whether there are ...
- 17.9k
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 ...
- 8,219
5
votes
Accepted
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 ...
- 979
5
votes
Accepted
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 ...
- 5,971
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:
...
- 221
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 ...
5
votes
How could Renaissance Technologies have near real-time prices on corporate bonds and other debt?
Outside of ETFs, corporate bond markets are driven by institutional flows. Over the course of a single day, multiple dealers will send multiple "runs" messages to their institutional clients. These "...
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