# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### ABM Crossing Times

Suppose I have a process that follows an arithmetic brownian motion $dX_t = \sigma dW_t$ How do I calculate, within a certain interval $\Delta t$ , the expected number of times that the process will &...
1 vote
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### Kolmogorov's backward equation with initial value

I am refreshing basic financial mathematics concepts and self-learning from the text, A first course in Stochastic Calculus, by Louis Pierre Arguin. I understand that, the transition probability ...
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### Fitting a multidimensional Ornstein-Uhlenbeck pProcess

If I have a dataset X, where each row is a time point and we have several variables, say 100, (so this is a multivariate time series), what is the best way to fit a multidimensional Ornstein-Uhlenbeck ...
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### Resources to understand derivative pricing and simulation models and aggregation of risk measures

I am looking for a book or set of videos which explains theorems and models of quantitative finance and stochastic calculus required for understanding derivative pricing and simulation in a very ...
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### Constant cancellation model for volume of LOB queues

I have been reading Jean-Philippe Bouchaud's book on stochastic models of LOB queues in Chapter 5, which starts with the simplest model. In this model, market/limit/cancel orders are assumed to be of ...
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1 vote
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### ARCH-Vasicek model solution

I understand how we can obtain the solution of Vasicek model $dr_t=\alpha(\mu-r_t)dt+\sigma dW_t$: $$r_t=r_0e^{-\alpha t}+\mu(1-e^{-\alpha t})+\sigma\int_0^te^{-\alpha(t-s)dW_{s}}$$ This easily ...
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### Parameters bounds for Heston model calibration

Still working on my master thesis and I have a question I have been looking at for some time but can't find a good reason. I am looking to follow the steps of Horvath et al. (2019) in order to ...
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### About Hedging of One-touch Options

The pricing of American Digital Call (one-touch Calls) has the following formulas, taken from P13, the textbook \begin{aligned} C_{\mathrm{d}}^{\mathrm{Am}}(S, t ; E) & =\left(\frac{S}{E}\right)^{\...
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### Orthogonalizing brownian path

I want to improve the stability of my SDE sample (statistical properties do not change much when using a different seed). I am using a sobol brownian bridge to generate the brownian path increments dw....
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### Model for markets with friction

Is there a stochastic model for describing how equities behave in markets with trading fees, and if so what model is most commonly used? I'm envisioning something similar to the Black Scholes model, ...
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### Three mathematical mistakes in Black-Scholes-Merton option pricing?

In this preprint on arXiv (a revised version of the one discussed in a post here) we show that there are three mathematical mistakes in the option pricing framework of Black, Scholes and Merton. As a ...
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1 vote
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### multivariate geometric brownian motion equivalent martingale measure

Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by dS(t) = S(t)\big((\mu - ...
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### Did I derive the Kelly criterion correctly?

$$\frac{dX_t}{X_t}=\alpha\frac{dS_t}{S_t}+(1-\alpha)\frac{dS^0_t}{S^0_t}$$ where $\alpha$ is proportion of the investment in the risky asset $S_t$ and $S^0_t$ is the risk-free asset. $S_t$ follows a ...
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### Equivalent definition of brownian motion

I'm having a question about this characterization of Brownian Motion : Theorem : If a process : $\big( X_t \big)_{t\geq 0}$ satisfies these conditions, $\big( X_t \big)_{t\geq 0}$ is a Gaussian ...
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### Deriving an Analytical Expression for Standard Deviation of Log Returns

I am looking to find an expression for the standard deviation log returns of a stock price process. I have a stock price which follows the following dynamics: $dY(t) = Y(t)(r(t)dt + η(t)dW(t))$ Here,...
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### Method of conditional expectations for basket

I am reading paper "An analysis of pricing methods for baskets options". Unfortunatly, I can not find the working paper "Beisser, J. (1999): Another Way to Value Basket Options, Working ...
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### If the price of a stock follows a Geometric Brownian motion, then does stock return depends on past stock returns? [closed]

Got this question from my homework. I think if past returns are keep raising then current return should also be positive, but the answer is it's not related to past returns, why? I tried to ask ...
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1 vote
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### Bloomberg FXFM: what is the point of knowing risk neutral probabilities?

Among other things, Bloomberg FXFM function allows you to check risk neutral probabilities for currencies. For instance, you can check the probability of the euro depreciating 5% vs the dollar in 6 ...
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