# 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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### 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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### 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 ...

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

### 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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### 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 ...
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