31 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 $M$...
  • 14k
22 votes
Accepted

Which process is the most commonly used for modeling stock prices?

I give you a brief outline about some key properties of Lévy processes. Lévy processes have stationary and independent increments but do not necessarily have continuous sample paths. In fact, ...
  • 14k
19 votes
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Finding distribution of $\int_0 ^T uW_u du$

Using the Ito Formula The general approach that often works for these kinds of question is to search for functions such that their Ito differential contains the terms that we are interested in. In ...
18 votes
Accepted

Processes used in quant finance

Here is a short list (to be edited and improved - community wiki) : Standard brownian motion (also called Wiener process) for which: $d\, W_t \sim \mathcal N(0, \sqrt{d t})$ Geometric brownian ...
17 votes

Strictly local martingales: what is the intuition behind them?

I think to understand the martingale/local martingale distinction, it helps to bring in a third class of processes, the uniformly integrable martingale. I would argue that the local martingale and ...
  • 1,780
16 votes
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Why do we usually model returns and not prices?

Basically, prices usually have a unit root, while returns can be assumed to be stationary. This is also called order of integration, a unit root means integrated of order 1, I(1), while stationary is ...
  • 5,311
15 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&...
  • 5,306
13 votes

Why the expected return rate of a stock has nothing to do with its option price?

Because you can hedge. Once you have delta hedged, the pay-off is symmetric about up and down moves so drift doesn't matter. Also the delta-hedged call and the delta hedged put have to have the same ...
  • 6,763
13 votes
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open problems in mathematical finance

If you want to address interesting problems that are interesting for financial mathematics, I do not believe you have the good list. Pricing. For instance, most of explicit formulas for pricing that ...
13 votes
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Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$

Let \begin{align*} Y_t = e^{(a+\frac{c^2}{2})t-cW_t}. \end{align*} Then \begin{align*} dY_t = Y_t\left[\big(a+c^2\big)dt -c dW_t \right]. \end{align*} Moreover, \begin{align*} d(X_tY_t) &= Y_t ...
  • 20.5k
12 votes
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Relationships between white noise and random walk

I will assume a white noise is a process $(\varepsilon_t)$ with zero mean, no autocorrelation and constant variance $\sigma^2 > 0$ while a random walk is a process $(x_t)$ defined by $$ x_{t+1} =...
  • 3,856
12 votes
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Can I always use quadratic variation to calculate variance?

Quadratic variation and variance are two different concepts. Let $X $ be an Ito process and $t\geq 0$. Variance of $X_t$ is a deterministic quantity where as quadratic variation at time $t $ that ...
  • 2,372
12 votes

Finding distribution of $\int_0 ^T uW_u du$

Another approach consists in using the Fubini theorem to write that \begin{align} \int_0^T u W_u du &= \int_0^T \int_0^u u\, dW_v\, du \tag{$W_u = \int_0^u dW_v$} \\ &= \...
  • 14.1k
11 votes
Accepted

Correlation coeffitiont between two stochastic processes

if you talk about correlation then: compute expectation: $$\mathbb{E}(W_t)=0\text{ and }\mathbb{E}(\int_0^tW_d ds)=0$$ variance: $$\text{Var}(W_t)=t\text{ and }\text{Var}(\int_0^tW_s ds)=\frac{t^3}{...
  • 2,372
11 votes
Accepted

How to find the formula for the half-life of an AR(1) process (using the Ornstein–Uhlenbeck process)

Convenient rewriting Let $$X_t = c + \phi_1 X_{t-1} + \epsilon_t, \quad \vert \phi_1 \vert < 1 \tag{1} $$ denote a weakly stationary AR(1) process. Weak stationarity notably implies that $$\Bbb{E}[...
  • 14.1k
11 votes
Accepted

What is an adapted process

Let $\{X_t\}$ be a stochastic process and $\mathcal{F}$ be a filtration. The intuitive idea is that for $\{X_t\}$ to be adapted, it can't reveal what's unknowable (according to the filtration). By ...
  • 6,384
11 votes
Accepted

Does numeraire have to be a tradable asset

This is an interesting question that I have asked myself. Below is my take. Let us consider an economy $(\Omega,\mathcal{F},P)$ equipped with a filtration $(\mathcal{F})_{t \geq 0}$ consisting on a ...
11 votes

Two papers - two different solutions of the Ornstein-Uhlenbeck process

Note that the Ito integral of a deterministic integrand $f: \mathbb{R}_+ \rightarrow \mathbb{R}$ is normally distributed \begin{equation} \int_0^t f(u) \mathrm{d}W_u \sim \mathcal{N} \left( 0, \...
11 votes
Accepted

Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$

By construction, the Itô integral, $I_t=\int_0^t X_s\text{d}W_s$, is a martingale if $\int_0^t \mathbb{E}[X_s^2]\text{d}s<\infty$. The martingale property, $\mathbb{E}_s[I_t]=I_s$ implies $\mathbb{...
  • 14k
11 votes

Expectation of exponential of 3 correlated Brownian Motion

You need to rotate them so we can find some orthogonal axes. A simple way to think about this is by remembering that we can decompose the second of two brownian motions into a sum of the first ...
  • 2,856
11 votes
Accepted

Expectation of exponential of 3 correlated Brownian Motion

Besides @StackG's splendid answer, I would like to offer an answer that is based on the notion that the multivariate Brownian motion is of course multivariate normally distributed, and on its moment ...
  • 5,903
11 votes
Accepted

Expected value and Variance of a stopped random process

Although Math SE might be a bit more suited for this one, I wanted to give it a try. The answer relies on the law of total expectation, the law of total variance, and the relationship between Euler's ...
  • 5,903
11 votes

If the spread between two assets is an OU process, what processes do the two assets follow?

If we allow the mean reversion speeds to be identical, we could assume OU processes for the two components: Let $$ \begin{align} dx_1&=\kappa_1(\theta_1-x_1)dt+\sigma_1dW_1\\ dx_2&=\kappa_2(\...
  • 5,903
10 votes

Why should we expect geometric Brownian motion to model asset prices?

To provide a straight forward answer: It is not a good model. It never was, it never will be. Until we all do not come up with a better model that provides better modeling accuracy while it is equally ...
  • 14.1k
10 votes
Accepted

List: Behavioural characteristics of key Ito processes used in finance

I will provide some references such that you can see where the different processes are used. These papers typically motivate their models and show which effect the single paramaters have and what ...
10 votes

Heston stochastic volatility, Girsanov theorem

Consider the Heston (1993) model under the real world measure ($\mathbb{P}$) \begin{align*} \mathrm{d}S_t&=\mu^\mathbb{P} S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}^\mathbb{P}, \\ \mathrm{d}v_t&...
  • 14k
10 votes
Accepted

How to simulate Levy processes

You have many different options. Firstly, you know the characteristic function for the log stock price and, using inversion, you can recover the (inverse) distribution and density function and ...
  • 14k
10 votes
Accepted

Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?

First, let's check if these models are abritrage free. The first fundamental theorem of asset pricing says that if there exists an equivalent probability measure under which $\frac{S_t}{\beta_t} = e^{...

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