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EdisonCruise
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Posts: 117
Joined: September 15th, 2012, 4:22 am

### How to solve this quasi-variational inequality (QVI) numerically?

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I found this equation from Eq (10) in the paper Hedge and Speculate: Replicating Option Payoffs with Limit and Market Orders by Alvaro Cartea, Luhui Gan, and Sebastian Jaimungal.

I think I can solve the first equation which is for the use of limit order, but I have difficulty is solving the last two equations, due to the Expectation operator, the utility function and an interpolation operator may also require.
This may be like the American option pricing problem, but for American option pricing, a direct comparison of the option price can be implemented quite straightforward. I am not sure if there is any general numerical scheme can solve this problem. Any reference is appreciated.
Last edited by EdisonCruise on April 22nd, 2020, 1:04 am, edited 1 time in total.

Alan
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### Re: How to solve this quasi-variational inequality (QVI) numerically?

what do you see when you look at your post?

EdisonCruise
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Posts: 117
Joined: September 15th, 2012, 4:22 am

### Re: How to solve this quasi-variational inequality (QVI) numerically?

Sorry,I have updated with a full link

Alan
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### Re: How to solve this quasi-variational inequality (QVI) numerically?

Sorry I asked. It's too ugly to even think about reading the associated paper. Unless you can repose a simpler question, my prediction is that this thread will end here.

EdisonCruise
Topic Author
Posts: 117
Joined: September 15th, 2012, 4:22 am

### Re: How to solve this quasi-variational inequality (QVI) numerically?

Maybe the QVI above can be simplified as below:
$$0=min( \frac{\partial{h}}{\partial{t}}+\phi_1 h; \phi_2 h; \phi_3 h)$$
where $\phi_1,\phi_2$, and $\phi_3$ indicate some combinations of  operators (e.g. differential operator or expectation operator) on an unknown function $h$.
Is there any reference on general numerical scheme to solve this QVI?

Cuchulainn
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### Re: How to solve this quasi-variational inequality (QVI) numerically?

Maybe the QVI above can be simplified as below:
$$0=min( \frac{\partial{h}}{\partial{t}}+\phi_1 h; \phi_2 h; \phi_3 h)$$
where $\phi_1,\phi_2$, and $\phi_3$ indicate some combinations of  operators (e.g. differential operator or expectation operator) on an unknown function $h$.
Is there any reference on general numerical scheme to solve this QVI?
I don't understand this notation; can you give a simple (but not too simple) example?
And is it a minimisation problem at each time level?
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EdisonCruise
Topic Author
Posts: 117
Joined: September 15th, 2012, 4:22 am

### Re: How to solve this quasi-variational inequality (QVI) numerically?

Yes, it is a minimisation problem at each time level.
I hope this is a non-trival example:
$$0=\min[ \frac{\partial{h(t,s)}}{\partial{t}}+ \min \limits_{a\in A} (\mathbb{E}(h(t,s+\xi a)-h(t,s-\xi a))) ; h(t,s)-b; -h(t,s)-b]$$
where $\mathbb{E}$ is the expectation operator with respect to the Bernoulli random variable $\xi$. $a$ is the action that can be taken in the policy space $A$. $b$ is a constant.

Actually, I found Eq (10) in Alvaro Cartea's paper can be solved by simply using the formulation below:
$$0= \frac{\partial{h}}{\partial{t}}+\phi_1 h+\min(0,\phi_2 h)+\min(0, \phi_3 h)$$
with explicit Euler scheme.

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