I am doing a PhD in applied maths. For the moment my research has been quite theoretical (think Lévy copula + Multilevel Monte-Carlo) and I am looking for a pratical problem to solve.
A few topics have gained quite a lot of traction in the recent years, like Machine Learning or rough volatility, but to my knowledge these are not heavily used in real life. I would like to focus on something that has concrete pratical consequences, maybe something based on new regulation (e.g. FRTB) which requires new modelling or which computation is numerically challenging. It might be a new problem or one that has been there for decades but lacks a satisfying solution.
I am posting here in order to get a few inputs from practionners.
Feel free to message me privately.
Serving the Quantitative Finance Community
Re: Let me solve your problem
Since no takers yet, here is a problem that is just starting to interest me.
I am interested in it for a book project of mine. Here it is in a nutshell.
In finance dynamical models, there is "real-world" evolution (P-models) and "risk-neutral" evolution (Q-models).
Vanilla option quotes give us some information -- take SPX options to be concrete.
A long-standing topic in finance is to construct dynamical evolution models that are consistent with given (perhaps smoothed) option quotes. This is a very practical problem with lots of applications. It is well known that there may be many evolutions consistent with a set of option quotes. Nevertheless, it is still useful to construct at least one.
Dupire's local volatility model was a prototype. However, option quotes are discrete in strike and maturities, often with *very lengthy* intervals between maturities. For example, see Buehler and Ryskin. In general, the result of this literature is a Q-model.
I want to develop something similar for the associated P-model, using the "exponential tilt" measure transformation discussed in the slides at the book project link above. In other words, the self-consistent P-model evolution will reproduce the same equity risk premia values seen in the slides.
I am interested in it for a book project of mine. Here it is in a nutshell.
In finance dynamical models, there is "real-world" evolution (P-models) and "risk-neutral" evolution (Q-models).
Vanilla option quotes give us some information -- take SPX options to be concrete.
A long-standing topic in finance is to construct dynamical evolution models that are consistent with given (perhaps smoothed) option quotes. This is a very practical problem with lots of applications. It is well known that there may be many evolutions consistent with a set of option quotes. Nevertheless, it is still useful to construct at least one.
Dupire's local volatility model was a prototype. However, option quotes are discrete in strike and maturities, often with *very lengthy* intervals between maturities. For example, see Buehler and Ryskin. In general, the result of this literature is a Q-model.
I want to develop something similar for the associated P-model, using the "exponential tilt" measure transformation discussed in the slides at the book project link above. In other words, the self-consistent P-model evolution will reproduce the same equity risk premia values seen in the slides.
Re: Let me solve your problem
How can we identify parameters in the P-model, Alan?Since no takers yet, here is a problem that is just starting to interest me.
I am interested in it for a book project of mine. Here it is in a nutshell.
In finance dynamical models, there is "real-world" evolution (P-models) and "risk-neutral" evolution (Q-models).
Vanilla option quotes give us some information -- take SPX options to be concrete.
A long-standing topic in finance is to construct dynamical evolution models that are consistent with given (perhaps smoothed) option quotes. This is a very practical problem with lots of applications. It is well known that there may be many evolutions consistent with a set of option quotes. Nevertheless, it is still useful to construct at least one.
Dupire's local volatility model was a prototype. However, option quotes are discrete in strike and maturities, often with *very lengthy* intervals between maturities. For example, see Buehler and Ryskin. In general, the result of this literature is a Q-model.
I want to develop something similar for the associated P-model, using the "exponential tilt" measure transformation discussed in the slides at the book project link above. In other words, the self-consistent P-model evolution will reproduce the same equity risk premia values seen in the slides.
Re: Let me solve your problem
Everything is determined by the option prices and one-extra parameter, [$]\kappa[$], the coefficient of relative risk aversion. You can fix [$]\kappa[$] from an estimator I provide; it's around 3 with say a confidence interval of 2-4. (See section 2. here)
The P-model is then determined from [$]\kappa[$] and the coefficients of the Q-model.
I was just working today on the simplest case, the Dupire setup where you pretend to have a complete set of option prices [$]\{ C(K,T;S_0) \}[$]. Then, if the P-evolution is [$]dS_t = \alpha(S_t,t;S_0) S_t dt + \sigma(S_t,t;S_0) S_t dW^P_t[$], then [$]\sigma[$] is the same as the Dupire Q-model local vol, and I have an expression for [$]\alpha[$] that I'm still checking.
The P-model is then determined from [$]\kappa[$] and the coefficients of the Q-model.
I was just working today on the simplest case, the Dupire setup where you pretend to have a complete set of option prices [$]\{ C(K,T;S_0) \}[$]. Then, if the P-evolution is [$]dS_t = \alpha(S_t,t;S_0) S_t dt + \sigma(S_t,t;S_0) S_t dW^P_t[$], then [$]\sigma[$] is the same as the Dupire Q-model local vol, and I have an expression for [$]\alpha[$] that I'm still checking.
Re: Let me solve your problem
Did I hear the word 'estimator'? Statistics does not work in finance because there are no repeatable experiments, the market mechanism changes over time as some players make money and others lose. We can assume the existence of a measure P for some theoretical arguments, but we have no way of determining any parameters.Everything is determined by the option prices and one-extra parameter, [$]\gamma[$], the coefficient of relative risk aversion. You can fix [$]\gamma[$] from an estimator I provide; it's around 3 with say a confidence interval of 2-4. The P-model is then determined from [$]\gamma[$] and the coefficients of the Q-model.
Re: Let me solve your problem
In principle, you're correct if you insist on a strictly frequentist interpretation. But, I am willing to assign probabilities to one-time future events like the chance of rain tomorrow. Then, what is our best estimate of that?
Related, do you believe in Q-probabilities from options? (Breeden-Litzenberger?) I am happy to declare a set of probabilities as simply a list of non-negative numbers that add up to one. With that, we can make progress.
Related, do you believe in Q-probabilities from options? (Breeden-Litzenberger?) I am happy to declare a set of probabilities as simply a list of non-negative numbers that add up to one. With that, we can make progress.
Re: Let me solve your problem
I don't, to be honest I don't like the frequentist interpretation of probability. But in physics, in biology and to some extend in finance it makes sense. Not in finance.In principle, you're correct if you insist on a strictly frequentist interpretation.
It is called subjective probability, isn't it?But, I am willing to assign probabilities to one-time future events like the chance of rain tomorrow. Then, what is our best estimate of that?
I thinkt that is exactly what option prices are. Subjective probabilities of Mr Market.Related, do you believe in Q-probabilities from options?
Re: Let me solve your problem
Erratum. Should be: But in physics, in biology and to some extend in economics it makes sense. Not in finance.
Re: Let me solve your problem
I don't, to be honest I don't like the frequentist interpretation of probability. But in physics, in biology and to some extend in finance it makes sense. Not in finance.In principle, you're correct if you insist on a strictly frequentist interpretation.
It is called subjective probability, isn't it?But, I am willing to assign probabilities to one-time future events like the chance of rain tomorrow. Then, what is our best estimate of that?
I thinkt that is exactly what option prices are. Subjective probabilities of Mr Market.Related, do you believe in Q-probabilities from options?
The terminologies can get confusing. Suppose we take a one-time event like the chance of a category 5 hurricane making landfall in Miami on a particular day. Say, a few days before, the best weather forecasting algorithm known estimates the probability of that event as 55%. Now, open up a binary betting market and say a lot of Miamian's use it for insurance and the net effect is that Mr. Market prices the event at 67%. Which one is the subjective probability? That's why I like P and Q, as it's usually clearer. Here Q=67% and P=55%.
Re: Let me solve your problem
If it is calculated it is logical probability, neither frequentist nor subjective. There are four types of probability: logical (Carnap), frequentist (Mises), subjective (Keynes) and propensity (Popper). There's even a good book on https://www.amazon.pl/Search-Certainty-Science-Philosophy-Probability/dp/9814273708
Re: Let me solve your problem
Yes, thanks for the link. Like the Eskimos and words for snow, I suspect one could write a book titled "100 types of probability".
I did a sidebar in my second book (on the topic of prediction), highlighting an interesting blog piece by Peter Cotton. He introduced P,Q, and R probabilities, with of course, R being non-standard. His taxonomy was:
P -probability: estimates of "true" probability, attempting to exploit all relevant information.
Q-probability: market implied risk-neutral probability.
R-probability: estimates of probability deliberately eschewing market information.
For example, if you attempt to predict SPX volatility, but deliberately ignore the VIX, you create a (sub-optimal) R-probability estimate. I thought the concept was useful.
I did a sidebar in my second book (on the topic of prediction), highlighting an interesting blog piece by Peter Cotton. He introduced P,Q, and R probabilities, with of course, R being non-standard. His taxonomy was:
P -probability: estimates of "true" probability, attempting to exploit all relevant information.
Q-probability: market implied risk-neutral probability.
R-probability: estimates of probability deliberately eschewing market information.
For example, if you attempt to predict SPX volatility, but deliberately ignore the VIX, you create a (sub-optimal) R-probability estimate. I thought the concept was useful.
Re: Let me solve your problem
Hehe -- I just had a thought for a new type of probability in that list of 100: political probability.
Defn: 'political probability' is a subjective probability that is strongly influenced by political beliefs.
Example. What is the probability that the Covid-19 pandemic was started by a lab leak from the Wuhan Inst. of Virology?
If your answer is strongly influenced by your politics, you've created a political-probability estimate.
Defn: 'political probability' is a subjective probability that is strongly influenced by political beliefs.
Example. What is the probability that the Covid-19 pandemic was started by a lab leak from the Wuhan Inst. of Virology?
If your answer is strongly influenced by your politics, you've created a political-probability estimate.
Re: Let me solve your problem
Let's be serious, subjective probability influenced by political beliefs is still subjective probability. Entia non sunt multiplicanda praeter necessitatem. Some people (including Carnap himself!) deny the existence of logical probability claiming that all calculations must be based on input probabilities, and these must be either frequentist, subjective or propensity. Either way - statistics can only be used in a frequentist approach.
Re: Let me solve your problem
Frankly, I don't really understand your general objection. Let's take a common exercise that will perhaps clarify.
We want to characterize a historical long-run empirical distribution of market returns, say SPX daily log-returns post WWII-date for definiteness. Call this historical set of returns [$]\{x_i\}_{i=1}^N[$].
There are standard formulas for low moments, skewness, and kurtosis. We use those and then have point estimates. But, thinking of our data as just a random sample from some possible sets of histories, we would like to have confidence intervals and sampling distributions for these moment estimates. A way to get those is to treat our data sample as a draw of [$]N[$] terms from the empirical distribution [$] p(x) = \frac{1}{N} \sum_{i=1}^N \delta(x - x_i)[$]. By doing a Monte Carlo bootstrap (i.e., making M IID draws of N returns from [$[p(x)[$], where M is large), we then construct sampling distributions and confidence intervals for our moment estimates.
Note that this procedure would certainly be suspect for statistics involving serial dependencies, like low order correlations. But this can likely be remedied by block sampling.
So, your objections to this are?
We want to characterize a historical long-run empirical distribution of market returns, say SPX daily log-returns post WWII-date for definiteness. Call this historical set of returns [$]\{x_i\}_{i=1}^N[$].
There are standard formulas for low moments, skewness, and kurtosis. We use those and then have point estimates. But, thinking of our data as just a random sample from some possible sets of histories, we would like to have confidence intervals and sampling distributions for these moment estimates. A way to get those is to treat our data sample as a draw of [$]N[$] terms from the empirical distribution [$] p(x) = \frac{1}{N} \sum_{i=1}^N \delta(x - x_i)[$]. By doing a Monte Carlo bootstrap (i.e., making M IID draws of N returns from [$[p(x)[$], where M is large), we then construct sampling distributions and confidence intervals for our moment estimates.
Note that this procedure would certainly be suspect for statistics involving serial dependencies, like low order correlations. But this can likely be remedied by block sampling.
So, your objections to this are?
Re: Let me solve your problem
Still the same: Statistics does not work in finance because there are no repeatable experiments, the market mechanism changes over time as some players make money and others lose. There is only one Microsoft, there was only Black Monday. You may imagine all real world moments and other measures: mean, variance, skewness, kurtosis etc. That is all what you may do, you cannot measure them.So, your objections to this are?