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by EdisonCruise
September 7th, 2023, 3:23 am
Forum: Technical Forum
Topic: Is there any professional name for this stochastic process?
Replies: 2
Views: 3156

Re: Is there any professional name for this stochastic process?

I think it should be called something like  Levy processes driven by Ornstein-Uhlenbeck processes.
I find many research papers on Ornstein-Uhlenbeck processes driven by Levy processes, but none on this one.
by EdisonCruise
August 28th, 2023, 4:52 pm
Forum: Technical Forum
Topic: Is there any professional name for this stochastic process?
Replies: 2
Views: 3156

Is there any professional name for this stochastic process?

If the jump time of a price follows a Poisson process, but the jump size follows normal distribution with constant sigma_1, but a time dependent mean \alpha_t, which follows a mean-reversion process.

$$ d\alpha_t=-\xi\alpha_tdt+\sigma_2 dW_t $$
by EdisonCruise
August 25th, 2023, 3:36 am
Forum: Numerical Methods Forum
Topic: How to solve this HJB equation more stably?
Replies: 5
Views: 3437

Re: How to solve this HJB equation more stably?

Thank you so much for your suggestions Cuchulainn. It seems that I can solve the equation below instead.   $$ \frac{dV(t,x)}{dt}+max(H1(V,x),H2(V,x)-e,H3(V,x))=0 $$ Where e is a very small number like 1e-12. In practice, action H2 indeed has marginal cost e compared with H3, though e is difficult to...
by EdisonCruise
August 17th, 2023, 6:47 am
Forum: Numerical Methods Forum
Topic: How to solve this HJB equation more stably?
Replies: 5
Views: 3437

How to solve this HJB equation more stably?

For an HJB equation like this,   $$ \frac{dV(t,x)}{dt}+max(H1(V,x),H2(V,x),H3(V,x))=0 $$ I try to solve it with Euler scheme until the actions becomes steady. I find when x<=x0, the optimal action is H1, but when x>x0, at each integration time step, the optimal action switches between H2 and H3. I a...
by EdisonCruise
December 15th, 2022, 12:21 pm
Forum: Trading Forum
Topic: How to implement this limit order fill rate model in practical trading?
Replies: 0
Views: 9597

How to implement this limit order fill rate model in practical trading?

In paper Optimal make-take fees for market making regulation, Euch, Mastrolia, Rosenbaum, Touzi, the fill rate of limit order (Eq.2 of the paper) is \( \lambda(x)=Aexp(-k(x+c)/ \sigma) \), where k indicates the impact of market order, \( \sigma\) is volatility, c is taker fee and x is the maker’s qu...
by EdisonCruise
May 10th, 2022, 3:25 am
Forum: Technical Forum
Topic: How to estimate this stochastic model?
Replies: 7
Views: 4119

Re: How to estimate this stochastic model?

Thank you so much for your suggestions. Actually this model appears frequently in Prof. Alvaro Cartea’s high frequency trading paper. I need to figure out a way to calculate \(\alpha_t\) and estimate parameters in real trading efficiently. Maybe I can some approximations of this model by following t...
by EdisonCruise
May 9th, 2022, 11:04 am
Forum: Technical Forum
Topic: How to estimate this stochastic model?
Replies: 7
Views: 4119

Re: How to estimate this stochastic model?

Is the left hand of your second equation correct? Did you mean to include dt in it?
sorry, that's a typo. You are right. How can I edit the equation after submission?
by EdisonCruise
May 9th, 2022, 9:53 am
Forum: Technical Forum
Topic: How to estimate this stochastic model?
Replies: 7
Views: 4119

How to estimate this stochastic model?

$$dS_t=\alpha_tdt+\sigma_1dW^1_t$$ $$d\alpha_tdt=-\xi\alpha_tdt+\sigma_2dW^2_t$$ where \(S_t\) is the observable price; \(\alpha_t\) is the unobservable trend; \(dW^1_t\) and \(dW^2_t\) are two Brownian motions. I think \(S_t\) can be sampled with regular discrete time, so that this model can be fo...
by EdisonCruise
February 14th, 2022, 6:45 am
Forum: Technical Forum
Topic: How to get this limit order fill probability?
Replies: 0
Views: 5590

How to get this limit order fill probability?

I find this power class limit order fill probability $$h(\delta)=\frac{1}{1+(\kappa\delta)^{\gamma}}$$  where \(\delta\) is the distance to mid price;\(\kappa\) and \(\gamma\) need to be calibrated from data   . in https://tspace.library.utoronto.ca/bitstream/1807/68262/1/Ricci_Jason_201411_PhD_thes...
by EdisonCruise
August 11th, 2021, 4:51 am
Forum: Numerical Methods Forum
Topic: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?
Replies: 6
Views: 7716

Re: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

I can think of one more problem. Ornstein-Uhlenbeck process has been used to study interday mean-reversion. Usually daily data is used to estimate model parameters. But can we use millisecond level or minute level data to study interday mean-reversion? If not, it seems that, for practical financial ...
by EdisonCruise
August 10th, 2021, 2:58 am
Forum: Numerical Methods Forum
Topic: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?
Replies: 6
Views: 7716

Re: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

\(\alpha_t\) still tends to be infinity even in the case of volume weighted mid-market price.
by EdisonCruise
August 9th, 2021, 6:53 am
Forum: Numerical Methods Forum
Topic: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?
Replies: 6
Views: 7716

Re: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

Thank you all for your suggestions. I can calibrate the model to get close parameters with simulated data sampled at difference frequencies. So I think the parameter estimation should be correct. A weighted mid-market prices based on available volumes on best bid and ask seem improve the result slig...
by EdisonCruise
August 5th, 2021, 11:20 am
Forum: Numerical Methods Forum
Topic: Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?
Replies: 6
Views: 7716

Is it necessaet to resample data before Maximum likelihood estimation of mean reverting processes?

I am working on high frequency data and try to implement an Ornstein-Uhlenbeck mean reverting process to model short-term momentum \(\alpha\).  $$ {d\alpha_t=\eta(\theta-\alpha_t)dt+\sigma dB_t} $$ The timestamp interval between each best bid-ask sample is 0 - 30 milliseconds, irregularly.  I am con...
by EdisonCruise
June 29th, 2021, 3:21 am
Forum: Numerical Methods Forum
Topic: How to calibrate this stochastic model?
Replies: 4
Views: 7309

Re: How to calibrate this stochastic model?

Thank you so much. I find the problem. Eq (4.5)  in your reference is clear. 
by EdisonCruise
June 28th, 2021, 12:23 pm
Forum: Numerical Methods Forum
Topic: How to calibrate this stochastic model?
Replies: 4
Views: 7309

How to calibrate this stochastic model?

I meet some problems when calibrating the stochastic model, given in Eq(3.2) and appendix (E.2) of thesis: https://tspace.library.utoronto.ca/bitstream/1807/68262/1/Ricci_Jason_201411_PhD_thesis.pdf I give a simplified version here. $$d\lambda_t=\beta(\theta-\lambda)dt+\eta dM$$ where M is a poisson...
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