Science topic

Mathematical Finance - Science topic

Mathematical Finance is an imerging subject in which we search the opportunities to find the solution of financial problems with the application of mathematics. After the commencing the two noble prices in economics, its appear a bonanza itself.
Questions related to Mathematical Finance
  • asked a question related to Mathematical Finance
Question
5 answers
I am trying to calculate the idiosyncratic volatility for stocks from 2010-2016. The regression window is 30 days, my dependent variable is excess return (return - RF), and my independent variables are F&F's MKTRF (Rm-Rf). SMB, and HML. The R-squared I am receiving is about 10%, but I am expecting 80-90%. Does anybody have any advice, or can share a sample of data so I can verify that my approach is correct?
Thanks,
Ossama
Relevant answer
Answer
Hi, In my case I am getting extremely high r2 for all 2x3 sorted portfolios for the five factor model. The 6 size-bm, 6 size-op, 6 size-inv portfolios are my LHS variables whereas the factor data is my RHS variable. The data for both is downloaded from Kenneth French data library. The regressions were done in Excel. The R2 ranges from 97-99% for all portfolios for developed and emerging market datasets. This doesnt seem right
  • asked a question related to Mathematical Finance
Question
1 answer
Mathematics or finance masters or doctoral students can look at decomposing credit rating matrices from market prices as research project.
the project page is given below. Current research questions/ hypotheses are stated on the project page.
Relevant answer
Answer
The relationship between artificial intelligence and financial performance
  • asked a question related to Mathematical Finance
Question
7 answers
I am doing a research on FDI , with six independent variables , my objective is to see long term relationship of IV with DV. FDI  is stationary at I(1) , and other five IV are also at I(1) , but one IV is I(0) stationary at level. Can cointegration be applied in such case or some other test is used? 
Relevant answer
Answer
I am doing a research on the impact of FDI on Total factor Productivity. Can I apply Bayer-Hanck cointegration test for a mixed integrated series.... I(0) and I(1)?
  • asked a question related to Mathematical Finance
Question
3 answers
Hi All,
Suppose I have a Yield Curve (assume Semi-Annual Compounding), at term 1M, 3M... 1Y, 2Y... 10Y, 15Y ...30Y (x-axis is maturity / term).
How should I parameterize this yield curve? Any recommendations? Is there any known formula of Yield as a function of Maturity (or any approximations)?
And I have some questions about the property of this yield curve:
First of all, is yield curve (strictly) monotone increase? Does the first order derivative have any meaning?
Secondly, does yield curve has an asymptote, as x -> Inf, y -> constant? Is the y-values bounded by a lower bound when x=0?
Thirdly, what can we say about the second order derivative f''? Does f'' has an upper-bound? Should the f'' be strictly non-negative? Or should we expect f'' change sign? If f'' did change sign, what does it tell us?
Finally, does the Area-Under-Curve has any meaning? (like those ROC curve has a meaningful AUC)
Thank you so much for your help!
All the best,
Kathy Chenying Gao
Relevant answer
Answer
One accepted way is to use the semi parametric model from Nelsen where you can simply fit in values of B1 and B2 to find out the inflection points of the fitted curve.
For more analyzable phenomena beyond the graphical form you can either go for cochrane piazzesi factor which uses bond yields up to 5 years , here
You can also look at the work of Diebold et al, including this one: https://www.econstor.eu/bitstream/10419/25435/1/515150223.PDF
Here is another one using bond yields to tease out Currency and FX implications
You can also use log functions and polynomial factorizations of the fitted curve to start off with your own analysis of the curve .
  • asked a question related to Mathematical Finance
Question
2 answers
I'm an unassigned researcher and educator in the mathematics subject. I completed PDF in 2017, under the topic of Mathematical Finance. I seek position currently in the SA institutions of study and learning. I would like to find scholar assigns in the mathematics which to do with the mentioned skills.
Relevant answer
Answer
>Laudable goal. I can't help but I wish you well.
  • asked a question related to Mathematical Finance
Question
9 answers
Hi,
During my master's risk-neutral probability measure was the standard in advanced finance theory, with Stanley Pliska's book as the (only) reference.
But nowadays it seems this technique, or, should I say, this paradigm is no longer used, and I wanted an opinion about it. 
More specifically:
- is it still though and used in advanced finance courses?
- what could have caused its demise? was it because it was pretty hard to grasp and use?
- what replaces it? if not, is it still relevant and used by a narrow elite?
All relevant inputs, thoughts, advice, opinion is appreciated.
Relevant answer
Answer
The risk-neutral probability measure has nothing to do with pure risk-neutrality but rather is a powerful tool used in pricing contingent claims. It is a sophisticated tool which allows you to move from a multi risk environment to a controlled risk environment where you can easily price a derivative. It is still used extensively by the elite who knows and understands it, namely, the asset pricing academic community. I wrote several papers using the risk neutral approach. Some of the papers are posted on ResearchGate.net. The risk neutral probability measure is still on and will remain actively used forever.  
  • asked a question related to Mathematical Finance
Question
9 answers
Doing research in the field of finance and investment, I found it difficult to understand various models, such as ARCH; GARCH and many more used in the contemporary research articles.
I am searching such a book(s) which can explain these models and methods in a very simple manner accompanied by example and data set. As I am not an expert in these areas of analysis, I want to learn those model from the very basic and beginners level.
Relevant answer
Answer
wooldridge introductory econometrics is also good
  • asked a question related to Mathematical Finance
Question
7 answers
Economic Growth and Macroeconomic Analysis
Relevant answer
Answer
Indeed macroeconomic predictions are meaningful although their possibility may not be 100%. Aggregate predictions of the macro economy are based on the micro behavior of economic agents which if captured well in the data used for such macro predictions, under certain assumptions truth is held. But of course with some level of conventionally accepted error. 
  • asked a question related to Mathematical Finance
Question
9 answers
please can anyone help me with possible implications for policy makers, investors and regulators. thank you
Relevant answer
Answer
Thank you guys for your replies. Much appreciated.
  • asked a question related to Mathematical Finance
Question
5 answers
Dear all, if one return series is positively skewed or negatively skewed, how one investor can benefit out of this. Whether positively skewed is good or negatively skewed is good for an investor. Kindly justify your answer with an example.
Relevant answer
Answer
Other things being equal and assuming rationality, investors in general would prefer assets with positive skewness, so these would be expected to trade at higher prices (or offer lower expected returns). The reason is that investors would be willing to pay a premium for assets whose return distribution are concentrated on the positive side.
Theoretically, if asset returns are skewed or leptokurtic, then beta alone is not sufficient to price assets and one needs to include higher moments in the pricing model. Then what matters is the amount of skewness an asset adds to the portfolio as the market moves (coskewness measure). This leads to the Higher Moment CAPM.
There is some evidence that higher moment, including skewness, matter in asset pricing. The following references and references therein might be useful:
  • Kraus, A., & Litzenberger, R. (1976). Skewness Preference and the Valuation of Risky Assets. Journal of Finance, 21(4), 1085-1094.
  • Lim, K. (1989). A new test of the three-moment capital asset pricing model.
Journal of Financial and Quantitative Analysis, 24 (2), 205-216. http://dx.doi.org/10.2307/2330772
  • Hwang, S., & Satchell, S. (1999). Modelling Emerging Market Risk Premia using Higher Moments. International Journal of Finance and Economics, 4 (4), 271-296.
  • Harvey, C., & Siddique, A. (2000). Conditional skewness in asset pricing tests.
The Journal of Finance, 55 (3), 1263-1295.
  • asked a question related to Mathematical Finance
Question
6 answers
Folks,
Company ABC trades on two exchanges in India (BSE and NSE). I have total traded volumes and number of trades from each of the Exchanges for company ABC daily.
If i want to calculate overall Volumes per trade, is the below formula the right way to do it?
Volumes per trade = (volumes at BSE + Volumes at NSE)  / (Number of trades at BSE + Number of Trades at NSE)
Any thoughts?
Relevant answer
Answer
Hey Surya,
I agree to your arguments. In today's so called high-end technology backed trading infrastructure measuring the liquidity based on the per trade ticket or trade size is misleading. Partner in crime is high frequency trading. Another aspect is, in today's trading terminal there is an option for the trader to disclose his qty in pre-set multiples.
Then added complexity as you mentioned is more day or swing traders are more active on NSE than BSE causing high turnover on NSE. I feel it can easily be explained by the technology friendly real time data feed provided by NSE, while BSE provides it very expensive that makes BSE more institution friendly. Most trading and charting platforms are ready to run NSE while BSE needs lot of hacking. thats the reason Indian traders are active on NSE than BSE,
Now back to my objective, I feel in some cases Per trade volume has predictive power of accumulation/ distribution. One needs to really master the art of spoting which "per trade volume" is real and which one is manipulated by those high frequency guys. Thats the real art. It can not be only explained by the qunts but really require experience in interprting the whole scenario in that script at that point of time, the information, disclosures, index movement, movement of similar scripts etc.
Taking trade size of total market is baseless and is a fake measurement of overall liquidity. It only misleades.
  • asked a question related to Mathematical Finance
Question
9 answers
... if you are not an expert in data mining (or similar) techniques?
Relevant answer
Answer
Julia,
I am not sure what numerical experiments you are attempting to do; however, I can share some experience that might help.  We use a variety of statistical packages and although it is not as robust, Excel has had the same results as the others when we do preliminary analysis on something.  So, it seems reliable.
For example, running Ordinary Least Squares (OLS) regression in Excel is relatively simple if one arranges the data in a format that is consistent with what Excel requires. You may need to add the Analysis ToolPak; it is easy and free.  From an Excel sheet, just click in File, then Options, then Add-Ins, then Manage  Excel Add-Ins at the bottom; click go and then select Analysis ToolPak. Click OK and then you should see Data Analysis as part of the toolbar in the Data tab.  
You will note that Excel has limitations and does not do as much as dedicated statistical and econometric packages.  It quick and easy to use.
I hope this helps.
Rob 
  • asked a question related to Mathematical Finance
Question
7 answers
Would be grateful, if researchers can give suggestions on methodology that can be used
Relevant answer
Answer
hello
I feel event study could be right tool as it is specific to understand the January effect.  Plenty of literature available on this topic.
All the best. 
  • asked a question related to Mathematical Finance
Question
4 answers
Hey,
I'm currently looking for a scenario tree modeling the return of some stocks or indices. As scenario tree generation is not the focus of my work, but I need some to test my portfolio model, it would be nice, if someone could send me his work. There is no need that the data is up-to-date. Ideally the tree has 3-5 stages and 3-5 stocks/indices.
Thank you very much!
Jelto
Relevant answer
Answer
I am working on it and I am almost done with it just wait for few months. Other scenarios are there in literature which can guide you for asset  return. You can  use bound and bound algorithm and can try yourself.
  • asked a question related to Mathematical Finance
Question
4 answers
I tests Independent Sample T Test to check the mean difference between pre and post Break period. Each of the period comprises 10 observations, 20 observations and 30 observations. From the descriptive statistics I find mean difference between pre and post break period. But the coefficient of T Test is insignificant. Can you please suggest what to do? average mean differs from period to period but it is not proved from T Test.
Relevant answer
Answer
Thank you all for your answer. Dear all , I consider closing stock price and volume data for comparing the mean difference of pre and post circuit breaker period. All data have been estimated with 10, 20 and 30 observations separately.
As of Now from all of you I get two suggestion. One is paired T Test and another is One way Annova. I will try. If some more suggestion i will get , than I will be glad to employ that in my data series.
  • asked a question related to Mathematical Finance
Question
11 answers
Folks,
I am researching on the impact of a Corporate disclosure on future abnormal returns (in short term). 
After reading few whitepapers, i realized one of the accepted way to measure the impact is removing the Normal Return portion from total return, the residual is Abnormal Return.
Theory sounds good to me, but my question is, how can i calculate the Abnormal return for a Stock? ( I am not well versed with understanding mathematical notation in whitepapers)
I have Closing Price, volume, Corporate news and its disclosure date.
Any guidance on that please?
Relevant answer
Answer
Hello Ankur,
The calculation of abnormal returns is simple. Firstly, you should understand that abnormal returns are the returns over the benchmark returns. In simple words, abnormal returns are stock returns minus the benchmark returns. 
The benchmark could be any market index or portfolio against which you want to measure the performance of the stocks you are studying. 
For calculation purposes, you needs to calculate the stock returns and the benchmark returns separately. Then subtract the benchmark returns from the stock returns.
Returns = (Closing price - opening price) / opening price
Abnormal returns = Stock returns - benchmark returns
The file attached by Faris would also be of help.
Please feel free to get back if you need further clarification.
Best,
  • asked a question related to Mathematical Finance
Question
1 answer
I want to calculate the marginal abatement cost by using directional distance function and then distribute that cost among the player by shapely value
Relevant answer
Answer
Dear Luqman Muhammad
Greetings,
Please see the attached file may be useful for you ,
Best Regards
  • asked a question related to Mathematical Finance
Question
3 answers
I need help with the syntax.
Also, some of my variables are stationary as I(0) and some as I(1).
Please can you also guide me with how I should couple variables for this test?
Examples of my variables are:
I(0):
adol_fertrate
gdpgrowth_annual
adj_income_percap
gdp
gni
hlthexp_percap_ppp
I(1):
pop_density
pop_growth
school_sec
pop_total
pop_growth_urban
birth_rt
impvdsani_rur
child_hiv
impvdsani_urb
mobsub
oop_pvt
pop_65
oop_total
school_prim
Thank you.
Best regards,
Soumya
Relevant answer
Answer
Please you help me with the syntax for the same in SAS?
  • asked a question related to Mathematical Finance
Question
3 answers
I am interested in seeing if anyone has applied the Mahalanobis Distance measurement to public pension funds and public-private partnerships. Please contact me and thank you in advance for your time and thoughtfulness. I am interested in collaborative research for academic publication. Cheers, Daniel G Bauer, Florida Atlantic University
Relevant answer
Answer
I had tried to see whether stock prices are influenced by market signals looking at equal centroids while making cluster analysis. 
  • asked a question related to Mathematical Finance
Question
3 answers
Assume that stock price of a given stock has its own distribution F(X). The industry (sector) of the stock is distributed G(X). The stock market where the stock is traded is distributed H(X). All three arrays are correlated.
What is the effective method of forecasting stock price X?
Relevant answer
Answer
Depends on what the forecast is based on. If x, y,z represent stock price, sector index value and stock index value, are you trying to find prob (x|y,z), i.e. forecast stock price given numerical values of indices? If so - you can calibrate a tri-variate copula from the time series data of the three variables, and sample the necessary conditional distribution from it. My understanding of the problem might be incomplete, however, from the problem statement above.
  • asked a question related to Mathematical Finance
Question
14 answers
I would like someone to discuss the following hypothesis:
The Black-Scholes formula is not a valid optionpricing model.
When backtesting S&P stock options using B&S and the real volatility (= standard deviation) ex post the costs exceed the payoffs by 4 percent, using the VIX (=Volatility of the S&P500) by 26 percent. 
The method is: buy fictitious call options day by day over 15 years  at the money and at the price of fair value - compare the sum with the cumulated payoffs. The rationale:
The payoffs should somehow match the amounted procurement costs at least.
(For puts it's even worse - 18 / 46 percent overpricing.)
Any comment appreciated.
Relevant answer
Answer
Added a working paper that contains more detailed experiments applying the BS-formula to S&P and VIX data. 2 questions:
o Does the revers application of B&S induce a systematic error with a tendency to increase the index volatility?
o The payoffs of options on a broad portfolio like S&P don't match their fair value at emission time. Is the B&S formula still applicable - even for financial reports and balance sheets? 
  • asked a question related to Mathematical Finance
Question
5 answers
I want guidance or literature from anybody that can guide me to measurement items (questionnanires items) on measurement of pension/hedge/mutual fund or firm performance. The model I developed requires primary data and not secondary data for firm's of fund's performance. Thanks
Relevant answer
Answer
Hi! I am also working on mutual fund performance but using secondary data. I have seen one thesis on this topic with primary data in Delhi University. You can search that thesis.
  • asked a question related to Mathematical Finance
Question
3 answers
Details relevant to the question are in attachment. Thank you for your answers. 
Relevant answer
Answer
Dear Mantas Gabrielaitis,
Thank you very much for your answer to my question, I investigated your answer and this answer is enough for now, but associated with other versions of the FP equation will give you information, see you as soon as possible...
  • asked a question related to Mathematical Finance
Question
2 answers
Can anybody help me in getting this software? I have research work on MGARCH modelling.
Relevant answer
Answer
You should contact Tom Doan at Estima Software.
I'm sure he could help.
  • asked a question related to Mathematical Finance
Question
9 answers
I'm working with a database of longitudinal data with an unbalanced panel, data with monthly frequency. I'm using the "plm" package of R to analyze and run the models fixed and random.
NOTE: Data presented serial autocorrelation and heteroscedasticity, and I am correcting with the covariance of Arellano (1987) coming in plm package (R software).
1nd - It is redundant to run FE / RE and GMM as robust proof?
2nd - What is the impact of using the GMM if the results come out different?
3nd - What other suggestions would be to perform a robust test?
The analysis is very important.
Relevant answer
Answer
Dear Melquiades
1) Linear models, with fixed or random effects, are a case in GLMM models. GLMM allows to analyze data with a different distribution than normal distribution.
2) GLMM also allows you to use random or fixed effect.
3) I do not agree to use a test to compare a fixed an a random effect model. One effect is random if the levels of the effect are a random sampling of a population of effects. If you select the levels of the factor then you have a fixed effect model. However I know that in practice a lot of people  interchange the effect of some factor in order to obtain results or based in tests.
4) In the context of GLMM you can model the autocorrelation and the heterogeneity of variances.
Answer to your questions:
1) If you use different models you will have different results and I there is a risk to select the model which answer what we know. Nowadays it is very easy to run models, but in my opinion we need to think more on the data and the problem and then select the model. The questions that you have to answer are: Are my effects random or fixed? What is the distribution of my data? What kind of autocorrelation model has sense in my data?
2) Because you have counts you have to use GLMM with the correct specification of the distribution (Poisson for counts and Binomial for percentages) and an appropriate link function.
3) What do you mean with robust test?
  • asked a question related to Mathematical Finance
Question
6 answers
This text follows up our recent article „Consentaneous Agent-Based and Stochastic Model of the Financial Markets“ published in open access interdisciplinary journal PLoS ONE [1]. This article is a result of the ongoing research at the Institute of Theoretical Physics and Astronomy of Vilnius University implementing the ideas of econophysics. Though our research is mostly related to the modelling of return statistics in financial markets implementing ideas from statistical physics, the concepts behind this work and conclusions are related to the much more extensive interdisciplinary understanding of the social and physical sciences. The desire to extend conventional boundaries and achieve more understanding between researchers of physical and social sciences is a strong motivation for us to deal with econophysics.
The price is a key concept in economics as it enables general quantitative description in economy and theoretical economics. Market price plays a central role as it is assumed to precisely reflect the real exchange values. Therefore a belief that market price is the most objective one lies in the background of mainstream economics, based on the rational expectation and efficient market concepts. These concepts lie in the background of huge financial industry (stock exchange, other securities, derivatives, currency exchange, etc.), making a vast impact on the overall health of the global economy. However, periodically emerging local and global economic crises give rise to the alternative views opposing mainstream concepts of economics.
Econophysics much more often than econometrics criticizes mainstream theory of economics. The observed fluctuations of the market price are larger than it should be accordingto the equilibrium view of the efficient market theory. Alternative views arise even in circles of the economists as behavioral finance and economics receive much more attention. Nobel Prize winner of 2013 professor of Yale University Robert Shiller is an outstanding representative of the behavioral alternative. The decision to award a Nobel price of economics to the most outstanding advocate of efficient market theory prof. E.F. Fama and representative of the alternative view serves as a proof that economics with its concepts is in the crossroad. From our point of view behavioral finance criticism towards econometrics and mathematical finance serves as an obstacle to positively evaluate contribution of econophysics. Nevertheless, the understanding that unstable financial and economic processes have to be considered on the bases of statistical physics is taking place [2].
From the point of view of representatives of behavioral economics and finance the behavior of agents acting in the market is much more like the behavior of realistic personalities with inherent intellectual and psychological bugs than like the behavior of extraordinarily capable individuals with ability to evaluate and account for all of the surrounding circumstances and information. For example, they pay much more attention to the tendency of imitation than to the individual capabilities of agents to make independent decisions. In our work we aim to demonstrate that it is possible to build a consentaneous model of financial markets, where choice of agents between three different trading strategies is based only on the transition probabilities between these choices. From the mathematical point of view these transition probabilities between two of the choices are the same as proposed by Alan Kirman to describe the herding interactions of agents [3]. This way we propose an adaption of the herding model to the financial markets, which can be solved by using method from statistical physics to shape macroscopic description of financial markets by the set of two stochastic differential equations. The main objective of this model is to reproduce general statistical properties of price movements observed in the real stock exchanges from Vilnius to New York.
The detailed simulation of market return statistics, reproducing power law probability density functions, power spectral densities and autocorrelations of absolute return as well as reproducing very details of these statistics, shows that herding of market participants is the most general property dominating their very heterogeneous and less meaningful rationality. We think that rationality is so heterogeneous and so ambiguous, that in the final macroscopic view of the whole agent society only the most general statistically meaningful property – herding –is observed. Rationality as very diverse can be neglected in a same way that physicists neglect trajectories of separate particles in thermodynamic consideration.
Our proposed structure of agent groups is based on a conventional choice considering three opportunities [4]: 1) intrinsic (fundamental) value oriented market traders – fundamentalists, which buy stocks, when market price is lower than fundamental value and sell when market price is higher than fundamental value; 2) speculative traders, who forecast price movement and believe that market price will go up – optimists and 3) speculative traders, who believe that market price will go down, pessimists. Permanent dynamical change of traders’ choices impacts the demand and supply ratio and so forms a long term dynamics of market price. In order to make such agent population dynamics comparable with real financial markets we had to combine it with permanent exogenous impact – external information flow or order flow noise. These are all necessary assumptions to reproduce main general properties of market price dynamics, observed for all markets and all stocks. Though the model proposed has few independent parameters, the same choice of parameter values is appropriate for all markets and all stocks is the main it‘s advantage.
Proposed model provides evidence that price dynamics can be reproduced by the memory-less Markov transitions of traders between possible choices of behavior. From our point of view the proposed model suggests new interpretation of market price, which may exhibit very large deviations from fundamental value. In this new interpretation market price highlights herding based drifts of agent-based societies, neglecting intrinsic (fundamental) understanding of value and surrendering to the imitational waves of collective wandering. Such wandering can be supported by the public tales about unexpected economic opportunities, emerging in the context of new financial, technological, social and political tendencies. As the proposed model is based on the agent opinion dynamics, we ask a question – whether the market price is economic or sociological category? Answering the question we would prefer to assume that fundamental value is more likely to be economic concept and market price is more likely to be sociological concept. To make practical distinction between different market price constituents might be a hard task, nevertheless, the new market price interpretation can be helpful looking for the opportunities to diminish observed huge market price movements, responsible for the local or global economic crises.
From our point of view, the herding as a statistically dominating behavioral property of agents can be used for the stabilization of undesirable market price fluctuations. It appears that only a small number of agents trading exceptionally based only on fundamental values is required to make a considerable influence on other market participants leading to the much more stable movement of the market price [5]. Certainly, it is obvious that for the implementation of such mechanism a new and more comprehensive understanding of fundamental price is needed. It should help to define a new reference point in economics instead of the currently used market price.
References
1. Gontis V, Kononovicius A (2014) Consentaneous Agent-Based and Stochastic Model of the Financial Markets. PLoS ONE 9(7): e102201. doi:10.1371/journal.pone.0102201
2. Castellano C, Fortunato S, Loreto V (2009) Statistical physics of social dynamics. Reviews of Modern Physics 81: 591–646. doi: 10.1103/revmodphys.81.591
3. Kirman AP (1993) Ants, rationality and recruitment. Quarterly Journal of Economics 108: 137–156. doi: 10.2307/2118498
4. Lux T, Westerhoff F (2009) Economic crysis. Nature Physics 5: 2–3. doi: 10.1038/nphys1163
5. Kononovicius A, Gontis V (2014) Control of the socio-economic systems using herding interactions. Physica A 405: 80–84. doi: 10.1016/j.physa.2014.03.003
Relevant answer
Answer
We do share the view of Markus Kreer. Nevertheless, our model reveals the significance of endogenous agent dynamics more or less independent of exogenous noise. The agent system may experience phase transitions just as a result of endogenous dynamics, which is extremely unstable. 
  • asked a question related to Mathematical Finance
Question
13 answers
Which is better when considering financial data?
Relevant answer
Answer
The normal distribution is used when the population distribution of data is assumed normal. It is characterized by the mean and the standard deviation of the data. A sample of the population is used to estimate the mean and standard deviation.
The t statistic is an estimate of the standard error of the mean of the population or how well known is the mean based on the sample size.
Which to use in financial data depends entirely on the question you are trying to answer. If the question concerns the entire population as it is distributed, then the normal distribution should be used. If the question concerns the mean of the population then the t statistic may be used. For the use of either, a larger sample size gives a better result.
The decision to use one or the other requires a clear, logical statement and argument to establish whether to use the t-distribution or the normal distribution. Use of either assumes a normally distributed population.
  • asked a question related to Mathematical Finance
Question
11 answers
Searching for the present value for a linear interest rate series.
Relevant answer
Answer
Dear Joao,
according to what I now, no way!
I suggest a semi-lexical distinction between "sum" and "series". They are different animals.and they must be kept distinct.
A sum is only a sum: the number of addenda is finite.
A series is a sum with an infinity of addenda and it is something else.
The widespread notion of "finite series" is simply pernicious.
Here are significant cases:
(1) - You have a sum, you have a closed form for it, you try to compute the limit of the sum for n -> + infinity. This is not the case.
(2) - You have a series (an infinity of addenda), but you are able to compute differently the sum(*), very well, but this, once again, this is not the case.
Of course I could be wrong. I'm Italian and hence Latin. A well known sentence in Latin, which is here relevant is "Onus probandi incumbit ei qui dicit".
I translate for people that don't practice Latin: "athe burden to prove an assertion concerns who asserts".
(*) - In the case of interest I can provide the indication of some interesting books (for instance H.G. GARNIR (1965): "Fonctions de variables réelles", Tome II, Librairie Universitaire - Louvain et Gauthier-Villars - Paris: it's in French, but nobody's perfect).
  • asked a question related to Mathematical Finance
Question
4 answers
There are many ways to generalise the O-U model. What is done in this paper? The abstract does not give sufficient information for a guess.
Relevant answer
Answer
Okay, so the answer is that this is a single-factor Ornstein-Uhlenbeck process with additive noise and where the mean-reversion level is a deterministic exponential, which is a function that gets along well with the structure of the O-U process.
  • asked a question related to Mathematical Finance
Question
2 answers
We use the Wiener process for modeling stock prices. What are the differences between this model and time series models when the observations are stock prices? Let time series model be AR model, MA model, or ARIMA model.
Relevant answer
Answer
The main difference is that the Wiener process (or Brownian motion) is indexed by R_+, that is, the positive line: it can be observed at any time. Time series models are discrete, that is, they are indexed by N, the natural numbers. For instance, the random walk with Bernoulli jumps is observed for t=0,1,2,3,... However using a time and space scaling, it can be shown that the Brownian motion is a limit of the random walk, see for instance
In Finance, the use of continuous time models is justified by the fact that high frequency trading makes time almost continuous. But more importantly, these models are tailored for log-returns, which aggregate very well in time. Most results of option pricing theory are given in continuous-time, but they often have discrete time counterparts.
In practice, one can only observe discrete points. Usually, people consider that is is simply a finite sample of a process that has a continuum of values which cannot be all observed. But these points can also be considered as a discrete process.
In the end, it all depends on the purpose of what you want to do. For option pricing, usually, it's continuous time processes, but for prediction for instance, very complex econometric (discrete) models are probably very competitive. For portfolio optimization, both exist.
  • asked a question related to Mathematical Finance
Question
2 answers
I am using a barrier option, in particular a Down-And-Out call option, to simulate default probability of a firm where the asset values follow a geometric brownian motion. For reasonable values of volatility, drift, barrier and strike price, the code returns me default probability larger than one. The formula for PD is that of Black,Cox (1976) but with constant barrier (Reisz, Perlich 2007).
Relevant answer
Answer
Hello Markus,
many thanks for your answer. Yes, the formula that I am working is (A5). It should be the same as in Black,Cox (1976) but with a constant barrier instead of a time dependent barrier. Is not it? I checked also other sources and it should be right, so I do not know if I need to add some boundary condition (for example, PD=min(PD,1)), but I thought that as probability measure, it was constrained to be between 0 and 1 by construction.
  • asked a question related to Mathematical Finance
Question
3 answers
Is there any difference between financial mathematics and mathematical finance?
Relevant answer
Answer
I'll say:
- financial mathematics are mathematics and use financial objects in mathematical reasoning to obtain theoretical results of mathematical significance.
- mathematical finance is the use of mathematical tools in finance in order to serve realistic goals, focus on adequacy with data or solve numerical issues.
But in practice I'll admit there is just a degree of "theoricalization" (i would not dare say complexity). What need for a real difference? The two terminologies would only divide in communities, cut the link between practicionners and research and rise epistemological problems by isolating theory from practice.