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

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Gamal, amicus: don't ask what Wilmott can do for you, ask what you can do for Wilmott.

marco, if you send me a .pdf of your article -- I don't have access to SSRN -- I'll take a look at it and see if I can make enough sense of it to comment. Probably I won't be able to, but I'm game to take a shot.

I don't have access to SSRN

How can one access the internet but not the SSRN?

I don't have access to SSRN

How can one access the internet but not the SSRN?

Gamal, every day I do the impossible. Often before coffee in the morning.

Plus I have an aggressive pop-up blocker.

Anyway, marco, I have the paper now, at least on my phone.

I looked briefly. Here is my understanding of the paper in my own notation. Suppose a simplified setup where today is time 0, and there are observed plain vanilla option prices at times 1 and 2. The class of models considered are Markov models, where future stock price [$]S_t[$] is the only source of randomness. Risk-neutral prob. transition densities are generally written [$]Q_{i,j}(S_j | S_i)[$], where [$]T_i < T_j[$]. By the Chapman-Kolmogorov relation:

(*) [$]  Q_{0,2}(S_2 | S_0)  =  \int_0^{\infty}   Q_{0,1}(S_1 | S_0) Q_{1,2}(S_2 | S_1) \, dS_1  [$]

Plain vanilla option prices more-or-less determine [$]Q_{0,1}[$] and [$]Q_{0,2}[$], so the goal is to determine all possible unknowns [$]Q_{1,2}[$] consistent with (*).  The paper discretizes the stock prices and also imposes some very strong approximations to reduce the problem to a linear programming program.  This leads to a band of admissible prices for exotic option prices -- say barrier options.

If my account is right, I'd say that's fine and interesting. The main weakness, in addition to the approximations, would seem to be: what if the "true" risk-neutral transition functions contain other state variables (such as stochastic volatility). Then, the "true" exotic option prices could lie outside the previously determined bands.

First of all, thank you for reading my paper and for the insightful comments!

The recap by Alan is absolutely correct. Compared to the paper, the only change is the language used. In my paper, I prefer to refer to the concept of the forward transition matrix, but it's almost the same (but see point 2 below).

The two objections raised by Alan are very relevant and go to the heart of the methodology. My responses to these points are: 
1) On the topic of linear approximation, by using a transition matrix, all constraints are always linear (e.g., butterfly spread or others); what is nonlinear is only the relationship that links prices to forward volatility. The adopted solution is to iteratively solve the linear programming problem several times, gradually narrowing the search range for solutions (e.g., initially 3% -> 80%), until final convergence. The result is that the error on MIN/MAX forward volatility is 2 or 5 bps in my graphs, with about twenty iterative cycles. It can even be brought below one basis point. So, the solution to the nonlinearity problem exists and leads to a precise solution as desired. 
2) The objection, considering stochastic processes with stochastic volatility, do they fit into the proposed framework? A keen observation that gets to the heart of the method. In my opinion, the answer is yes because the transition matrix can always be defined, even for a process with stochastic volatility. This is because additional random variables, hidden like stochastic volatility or jumps in a Merton model, can always be integrated out. In other words, the transition probability is calculated by integrating over all possible values of hidden variables. Accepting the fact that for any process there is a forward transition matrix, the simplex method will ensure that the minimum and maximum limits are respected by all.

Re "can always be integrated out". 

It's not clear to me what is being asserted. 

It's true that, in Hidden Markov Models, for example, there is indeed a lot of integrating out. I have a whole chapter on "Stochastic Volatility as a Hidden Markov Model" in my "Option Valuation under Stochastic Volatility II" book. 

Separately, Dupire/Gyongy theory does a lot of integrating out of univariate or multivariate stochastic volatility in the construction of a local volatility surface for diffusions. 

However, there remains the problem of "re-calibration". 

In any event, if you can make/prove a careful statement about the application of your theory to data generating processes with additional state variables (hidden or not), my suggestion is to include that in your paper.    

I have understood and absorbed your comment. Thanks very much for that. 
I have a clear understanding of your observation, which seems to me a crucial point. It's a question that needs to be clarified thoroughly, and I agree. I will certainly follow your suggestion. In fact, if the framework proves effective, the objective is to reach a stage where we can extract from the set of all models the one that precisely replicates the prices of all derivatives (not only plain vanilla but every exotic deal) in the Heston model, for example. Many thanks for all

marco, this is mostly a stylistic comment.

Any paper making an argument should probably pretty closely follow this structure:

• Introduction
• Assumptions/background
• Thesis
• Proof
• Discussion

You use the word "theory" in your paper repeatedly, but when I looked for a clear statement of your theory -- which is how I tend to start reading mathematics-related papers; if you have an idea of what the destination is, you can often more easily make sense of the foreign notations and the alien reasoning -- but I couldn't find it. Maybe it's there and I missed it, but it should be almost impossible to miss. It is really the entire point of a paper.

Somewhat conjecturing here now, I think your theorem probably should be stated something like

Given incomplete sets of prices for derivatives of underlying security S maturing at two or more times in the future, a financial model must satisfy conditions [x,y,z] in order to avoid arbitrage.

Maybe I have completely missed your point, but if it's not something like that, you should at least have whatever it is succinctly stated prominently in your paper.

And beyond that, simplify, simplify, simplify. You're not trying, I wouldn't think, to summarize all or even a small part of financial mathematics; set your goal and chart a straight line to reach it, and leave all else on the side of the road.

I mean this all constructively, and I hope you receive it as such. From the glimmer of understanding I take from your paper and from Alan's synopsis of it, it does seem like you're likely on to something worthwhile.

But the final step of intellectual progress is communicating what you have found to other people, so that your insight doesn't die with you. And this tends to be painful drudgery; there's probably no way around that.

I think it generally starts, though, with clearly stating your thesis.

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