Multiverse Pricing Models

[marco_airoldi]

I would like to share my recent article about a theory of (pricing) models. The goal is to propose a new way to manage model risk, to find maximum and minimum limits for an exotic price (infact everything agrees on plain vanilla prices by all people have different views on exotic quotes, that is the origin of model risk at the end) and finally finding a natural way to introduce a dynamic on implied volatility surface.
Hope to receive some comments and opinions from you.
Thanks

Abstract
This article introduces a foundational framework for pricing models, specifically a multiverse model theory. The objective is to establish a new paradigm for equity option pricing. The novel theory operates on two levels: practical, where it establishes and computes price boundaries in incomplete markets, defines model risk clearly, and reprices plain vanilla market data without calibration; and conceptual, developing a comprehensive theory of models, providing insights on introducing a coherent dynamics of the forward volatility surface, and suggests a possible origin of market incompleteness where the volatility of the forward volatility surface and market incompleteness are linked. The fundamental concept involves a shift from a theory based on contingent claims, reliant on a single ideal model, to a theory encompassing all possible (potentially infinite) well-formed models. These models must adhere to a set of axioms. The approach to developing this theory of models involves: i) Using a forward transition matrix to represent a model. Models differ in their forward transition matrices but agree on the spot transition matrix. ii) Defining axioms for selecting well-formed models by imposing restrictions on the form of the forward volatility surface. iii) Translating the high-level axioms into numerous elementary low-level conditions expressed in terms of transition matrix elements. This leads to a complex linear programming problem that can be solved by simplex algorithm.
This chain of steps results in a new methodology where an infinite number of equally valid well-formed models can evaluate exotic payoffs, establishing price boundaries for exotics (a signature of market incompleteness).
In summary, this proposed approach introduces a paradigm shift in managing exotics, wherein models themselves become the focus of the theory.

Link SSRN:
Multiverse Pricing Models by Marco Airoldi :: SSRN

 

[marco_airoldi]

It seems to me that your objections are of the kind: "The problems you mention are complex, so I doubt that you have found a solution because the tools you use are simple." This is not an argument from a scientific method. Considering that the use of the simplex method is a complex matter. The overall framework is complex from a computational point of view. I am the head of financial engineering of one relevant bank, 30 years of experience in numerical simulations, several collaborations with university institutions for courses on quantitative finance. The issue is whether what I write and say is true or false. If you think it is false, present an argument that is not a superficial or subjective evaluation. In the paper, there are examples and numerous graphs with numerical evidence. Can you formulate an opinion on those? Do you have evidence from the literature of more precise calculations and estimates of the bounds on exotic instrument prices that I report? If you cannot bring clear and precise arguments, then at first glance, your observations are not founded.

 

[marco_airoldi]

Yes, of course. I'll try to summarize the key point of my proposal. There are various models in the literature. All replicate plain vanilla, but they naturally differ on exotics (so on forward implied volatility surface). My proposal is to capture any model using its forward transition matrix. However, at this level, many unrealistic models would be included (an already acknowledged issue in the literature). My solution: impose constraints on the shape of the forward volatility surface and on many other properties (continuity and constraints on the evolution over time of this surface). This leads to a problem that is nearly linear with tens of thousands of conditions to satisfy. The solution (minimum and maximum of a target, typically an exotic pay-off) is obtained with the simplex algorithm. The result is to define the bid/ask of an exotic derivative with realistic (i.e., tight) margins. In this sequence of points, what is incorrect? Which point is susceptible to criticism? Saying that everything is too simple is not an objection. However, raising doubts about other aspects, whether computational or principled, is correct, and for me, it would be a valuable perspective to have. Mine is just a proposal, an idea, to be measured against the opinions of practitioners and academics.

 

[Daniel Duffy]

I'll have a read of this article and get back (btw not much going on wilmott.com .. it's just me (Cuchulainn), or that's how feels )

 

[marco_airoldi]

It would be a great honor to have your opinion, whether positive or negative. Thank you in advance.

 

[Daniel Duffy]

Prego

Ciao Marco,
I am going through your (mega

) article and I am preparing some feedback. It is a huge endeavour and my two main perspectives soon are:

1. The scope, rationale, reader group, how the article is written (similar to Marsden's recommendation). The article does not flow yet, it is a description. Did you write it originally in Italian? It has still some Italian embedded in the text.
2. Technical stuff (density, simplex, transition matrix, Bezier etc.)

a presto

 

[marco_airoldi]

Thank you very much, Daniel. From Alan and Marsden's comments, I realized that I definitely need to reshape the article. I included too many secondary things and wrote too little about the idea that a model coincides with its forward transition matrix (at least in the case of only 2 future dates).
About style: basically I wrote the article in english apart some sentences in italian to speedup the process. At the end I used chat GPT to reshape most of the paper. I think this is one of the reasaon because the style is not sharp and clear enough.
75 pages are too many, and I greatly appreciate the opportunity you are still giving me to receive your feedback. So, thank you.

 

[Daniel Duffy]

Marco,
I use this below all the time, in maths, for C++, for books etc.
(my very first _proposed_ chapter of my very first book From Chaos to Classes (1996) had > 40 pages )

Some initial tips.

How to Solve It - Wikipedia

en.wikipedia.org

Niels Bohr say
1.tell 'em what you are gonna tell 'em
2. tell them
3.tell 'em what you told 'em

Polya again (I use them as well when supervising students and doing stuff)

“If you can't solve a problem, then there is an easier problem you can solve: find it.”

“The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.”

“It is better to solve one problem five different ways, than to solve five problems one way.”

“It is generally useless to carry out details without having seen the main connection, or having made a sort of plan.”

“Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves.”

 

[Paul Lopez]

Awesome thread!

 

[Daniel Duffy]

1. The scope, rationale, reader group, how the article is written (similar to Marsden's recommendation). The article does not flow yet, it is a description.
2. Technical stuff (density, simplex, transition matrix, Bezier etc.)


1.continued.
I think the goal should be to restructure the article in some way (I would say 40% text, 60% maths). I use data flow from input to outputs, DFDs, block diagrams and base your article around it (e.g. my MC article or Chun's ML Rough Heston model). And becomes much easier to code up in C++. And all stakeholders should be singing from the same hymn sheet..

A list: market and model data, algorithms, output, etc. each a separate module. See Figure 1 in MC Application. Divide et impera.

Then my part 2 (next) becomes easier. Can "pluck" the subsystems from the original text.

https://www.datasim.nl/blogs/29/msc-the ... nance-2020
https://onlinelibrary.wiley.com/doi/epd ... wilm.10647

 

[marco_airoldi]

Thank you very much, Daniel. I appreciate your insights, and I believe they are particularly suitable for my paper. Indeed the presented methodology involves several steps: 1) defining the forward transition matrix; 2) ensuring that the forward transition matrix satisfies a set of constraints (in C++, we could define the abstract concept of linear requirements for transition matrix and then derive concrete classes); next, applying the simplex method to the overall conditions, resulting in two forward transition matrices solutions: one that minimizes the target (e.g. an exotic payoff) and one that maximizes it. For each solution, we can calculate all the numbers for all contracts (plain vanilla contracts, of course, remain the same) beyond just the target. All of this should be presented as a block diagram with a clear workflow. In this way, it should be easier to explain each block mathematically. And following this structure, the paper writing will follow a clear path arriving to the final point: getting narrow bounds for exotics respect to the set of all well formed pricing models. Discard all not in line considerations that do not have a representation in the block diagram . Have I understood your point correctly?

Dispate the not very good writing and paper structuring, may I ask your overall impression?

Thanks for giving me the opportunity to interact with you
Marco

 

[Daniel Duffy]

Prego Marco,
This was my thinking as well. I'll go through your paper again in the light of your points 1), 2) above. I do have some (numerical) questions will pose in a couple days.

// I used Bezier stuff when in CAD (I once designed a Pirelli tire with Bezier) and for holography, but the NURBS variant. What's the compelling reason 1) nice pictures, 2) interpolation, 3) linear constraints? You described them but no formulae..

 

[Daniel Duffy]

A 101 example could be to take the simplest (not too simple) and trace/paraphrase the steps from Input to Output "meeting" Simplex, Transition along the way i.e. data flow diagram, like Figure 2 in Chun's thesis.

block diagram with a clear workflow. In this way, it should be easier to explain each block mathematically.
yep

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onlinelibrary.wiley.com

 

[Daniel Duffy]

Side remark; the design here has the same form as the project style in Baruch's Advanced C++ course. So, relevant to students here as well.

 

[Daniel Duffy]

DFD

 

[Paul Lopez]

Just bumping the thread. Two seasoned chaps here talking serious business I hope more students look at sir Marco's paper and Prof Duffy's comments. It's pure gold from both sides.

@marco_airoldi will you be publishing this paper in other languages besides English and Italian by any chance?

 

[marco_airoldi]

Thank you very much, Paul. You are very kind. Here, I feel a bit like I'm at home. I appreciate a lot what you write, even though the credit for this interesting thread goes entirely to Prof. Duffy. I observe and learn
I had never seen an article structured with data flow diagram style before. Now I'm listening and absorbing as much as possible. If I can, I'd like to share these things with my students ;-) Regarding the translation into other languages of my paper, you're very kind, but I already struggle with English :-D However, once it's rewritten decently... if you give me some help, why not?

 

[Paul Lopez]

Thank you very much, Paul. You are very kind. Here, I feel a bit like I'm at home. I appreciate a lot what you write, even though the credit for this interesting thread goes entirely to Prof. Duffy. I observe and learn
I had never seen an article structured with data flow diagram style before. Now I'm listening and absorbing as much as possible. If I can, I'd like to share these things with my students ;-) Regarding the translation into other languages of my paper, you're very kind, but I already struggle with English :-D However, once it's rewritten decently... if you give me some help, why not?

Lovely! Just out of sheer greed I'd love to see this article in Portuguese, German, and Chinese. The first two languages are for my people that would find your work fascinating and directly applicable to their projects! I don't speak English myself I leave that to Prof Duffy

 

[Andy Nguyen]

You will find a lot of members here whose first language is not English so we all communicate in technical terms.
It's great to see active discussion for something more relevant to our members.

 

[Daniel Duffy]

I will focus my feedback from now on QN
Part 2 coming up soon.