# 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 $M$...
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### 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, ...
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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&...
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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 ...
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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 ...
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### 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{$W_u = \int_0^u dW_v$} \\ &= \...
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### 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&...
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### 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 ...
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### 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 \int_0^t f(u) \mathrm{d}W_u \sim \mathcal{N} \left( 0, \...
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### 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 ...
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### 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(\...
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### SABR Model Pricing Engine in Python QuantLib

Here is a simple example that might be useful. Basically finding parameters for a given section. Some of the parameters might be assumed at start instead of calibrated. ...
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