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Re: negative transition probability

March 25th, 2022, 9:47 pm

How Much Do Negative Probabilities Matter in Option Pricing?: A Case of a Lattice-Based Approach for Stochastic Volatility Models

"it seems unlikely that one could exploit negative probabilities consistently as some practitioners may hope."
 
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Cuchulainn
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Re: negative transition probability

January 10th, 2023, 3:16 pm

How Much Do Negative Probabilities Matter in Option Pricing?: A Case of a Lattice-Based Approach for Stochastic Volatility Models

"it seems unlikely that one could exploit negative probabilities consistently as some practitioners may hope."
I read this article up to and including section 3 (CIR model).
A lot of hullabaloo .. 
Lattice methods are numercal approximation based on discretisation with parameters [$]h[$] and [$]\Delta t[$]. There are constraints, otherwise we get nonsense results. The article uses some tricks to ensure well-posednes but I would avoid talking about negative probabilities (a kind of Fata Morgana) ... they are getting the run of themselves in the article.
All these numerical issues are well known in PDE/FDM, e.g. explicit schemes, monotonicity (large correlation), stability.

Extensive numerical tests show that this optimized lattice model is a reliable and robust approach for financial option valuations.
So, the model is not even wrong.

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Cuchulainn
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Re: negative transition probability

January 10th, 2023, 3:42 pm

Not needed. The delta function is handled by the jump condition and never appears to the solver under my stop/re-start scheme.
The situation is different with hand-crafted PDE with delta payoff. They are generalised functions (Sobolev/Schwartz) and care is needed to avoid negativity. 
e.g. taking the pulse [$]u_0 = 1/h[$] and zero everywhere else will cause a spike with Crank Nicolson.