Questions tagged [finance-mathematics]
Financial mathematics, or mathematical finance, is a set of mathematical tools allowing to express use cases on financial markets a way that can (or could) be solved using mathematics.
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Option pricing under distribution assumption
For simplicity assume zero interest rates in the following.
Given the price of a (European) put option with strike K and maturity T at time point t. $P_t(K, T)$ for a given underlying S with values $...
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Fama-French Regression Output Interpretation (Intercept/Alpha)
I am currently doing a report regarding Fama and French 3 and 5 factors model. I was provided 3 companies with each of its daily stock return from 2015-2020, and the values of all 5 factors during ...
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Is sorting stocks into portfolio mandatory in Fama-French model?
I am currently doing a report regarding Fama and French 3 and 5 factors model. I was provided 3 companies with each of its daily stock return from 2015-2020, and the values of all 5 factors during ...
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How to calculate VaR given mean and sd?
Sarah manages a hedge fund with a portfolio valued at \$2,000,000. The portfolio's daily returns have a standard deviation of \$3,000 and an average daily return of \$1,200. Calculate the five-day VAR ...
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Standard practice to round values in ARM loans
I have an application that calculates payments schedule of ARM (Adjustable Rate Mortgage) loans, where these loans are in the books of commercial banks.
It seems to work fine, with the exception of ...
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Estimating implied probability based on prediction betting odds
I am attempting to estimate prediction betting market efficiency for a project, and I am hoping for assistance with a couple of questions.
The prediction market makers add a commission to the betting ...
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Straddle Approximation - Directly from Integral
The ATMF straddle approximation formula, given by
$V_\text{Str}(S, T) \approx \sqrt{\frac{2}{\pi}} S_0 \sigma \sqrt{T}$
where $S_0$ is the current underlying spot price, $T$ is the time remaining ...
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Machine learning techniques for small datasets
I am dealing with financial data which is available on a Monthly basis. I am planning to apply machine learning techniques like LSTM but issue here is that overall I have very limited training dataset ...
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Lopez de Prado Advances in Financial Machine Learning- entropy for adverse selection
In chapter 18: Entropy Features, Lopez de Prado discusses how entropy can be used to estimate adverse selection. He suggests a method where order imbalance is mapped to quantiles and entropy is ...
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Advances in retail modelling
The risk-neutral modelling framework leads to very advanced and mathematically rich approach to contingent claims modelling. However, in my experience, retail modelling in Banks is done using generic ...
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Characteristic Function for Wishart Heston Model
I don't know if this is the right place (at most they will close the post). Anyway, I am trying to implement the characteristic function of the Heston Wishart Stochastic Volatility model illustrated ...
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Commercial bank mortgages schedule calculation
I need to calculate the schedule of a fixed rate mortgage and an adjustable rate mortgage.
Is there an open source library, preferable in python, that already makes these calculations? I tried ...
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Expectation of the realized volatility
I was reading Zhang and Wang 2023 and I have some doubts regarding it. The realized Stochastic Volatility Model is expressed as follows:
$$\begin{matrix}
y_t = \exp \big( \frac{h_t}{2} \big) \...
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Can anyone help me to understand why the GMV point is not on the efficient frontier?
I am following a course about portfolio construction with Python. I am able to successfully draw the efficient frontier and capital market line (CML), and the global minimum variance (GMV) point using ...
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Hypothesis Test Contradiction?
I have a question regarding hypothesis testing. I used the t-test (2-tailed) for these hypotheses:
Whether the (monthly) mean return of company A's stock is different from 0
Whether the (monthly) ...
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Risk Neutral Pricing Exercise
I have the following exercise: A financial security pays off a dollar amount of $S_T^2$. Using Ito`s Lemma, what is the price today $V_t$ of this security? (S follows a Geometric Brownian Motion $dS = ...
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Why is the stochastic process of the volatility of a stock price square integrable?
I am taking a course in financial mathematics(Ito-Integrals, Black-Scholes,...) and there is something that is not immediately clear to me. When constructing our stock price model, the integral $\...
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References/Direction on what functional of wealth to optimize for a given goal?
I seem to have gotten stuck trying to approach trading strategy development from a financial mathematics(?) perspective.
To start, let:
$T \gt 0.$
$\mathcal{T}$ be a closed non-empty set of $\mathbb{...
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American Contingent Claim vs European Option pricing
Suppose $Y$ is an American Contingent Claim (ACC) defined as $Y = \{Y_t, t \in 0,1,...,T\}$ and asssume $U_t$ is its fair price. Also suppose $C_t$ is the arbitrage-free price at time $t$ of a ...
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Check for arbitrage - European calls with same strike price, different duration and price
I tried a lot of different things to check for arbitrage on the following calls but didn't succeed.
Let's suppose we have a stock that is currently valued at 40. The interest rate is 0.05 and the ...
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Is the self-financing condition necessary/"useful" in practice outside of replication/valuation?
I know that the need for a portfolio/strategy to be self-financing (the purchase of a new asset needs to be funded by selling of an older one/ones) is very helpful when attempting to price derivatives ...
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Incorporating the I-Spread and Parallel Shift for Accurate Bond Pricing
I am currently working on pricing bonds and intend to utilize the S490 curve sourced from Bloomberg. This curve is constructed exclusively using swap rates. However, I have encountered challenges when ...
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Discrete self financing strategy
Let $H$ be an investment strategy in a discrete price model. Proof $H$ is self financing if and only if the following holds for the portfolio process $P_t$: $$P_t = P_0 + \sum_{s=1}^tH_{s-1}(X_s-X_{s-...
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Follow-up Fixed Payment at Default - Pricing
This question is a follow-up of this question Fixed Payment at Default - Pricing for more clarity.
Starting from the following expression of payoff:
$$
D(0,T) = E(\exp(-\int_{0}^{\tau} r(t)dt) \cdot \...
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Fixed Payment at Default - Pricing
Let us consider a product paying an amount of 1 if default (τ<T that is the time of default arrives before maturity time), and 0 otherwise.
The payoff of such a product would be given by:
$$D(0, T) ...
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Law of iterated expectation for the pricing of a Zero Recovery Risky Zero Coupon Bond
I am currently reading "Modelling single-name and multi-name credit derivatives" by Dom O'Kane but I struggle at one point that should be relatively easy.
Let us consider a Zero Recovery ...
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Proper way to backtest strategy using bootstrap method
Should I back-test in a single (original) price series and bootstrap the strategy returns to get statistics of interest?
Or should I create bootstrapped price series using bootstrapped returns from ...
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Quantifying the impact of rates change on bond prices
How can I quantify the impact of a change in interest rates on bond prices?
I know that in a classical textbook setting the answer would be to compute the modified duration of the bond and, to account ...
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Parameters in Nelson-Siegel model and Nelson-Siegel-Svensson model
I am trying to determine the parameters for the Nelson Siegel and Nelson Siegel Svensson model and try to solve
SE=$\sum_{i=1}^{n_{i}}(y_{t_{i}}-\hat{y}_{t_{i}}(X))^{2}$
where $y_{t_{i}}$ denotes the ...
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Hurst Exponent and Smoothed Hurst Exponent values are the same and incorrect plotting
I'm working on a script to calculate and plot the Hurst Exponent and Smoothed Hurst Exponent for a stock's historical price data using Python. When I run the script, I face two major issues:
The ...
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How to analyse the resilience of banks during financial crises using linear regression and other statistical methods?
I am a student in finance and have to work on a project for the semester.
I have to study the difference of resilience during financial crises between the 5 biggest US banks and the 5 biggest Canadian ...
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Roll Critique - CAPM and mean variance tautology?
Wikipedia introduces the Roll Critique mean-variance tautology:
Any mean-variance efficient portfolio $R_p$ satisfies the CAPM equation exactly:
$$
E(R_i) = R_f + \beta_{ip}[E(R_p) - R_f]
$$
A ...
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Proving that the return from the butterfly spread is nonnegative [closed]
The butterfly spread satisfies the inequality
c(X1) - 2c(X2) + c(X3) >= 0
Where call strikes satisfy X1<X2<X3 and X2 - X1 = X3 - X2.
There is a “proof” that was provided here https://quant....
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QuantLib: How to bootstrap Yield Curve using 3M futures - Python
I need to bootstrap a yieldcurve with 3M futures, using a cubic spline if possible.
Using, for example 3M Euribor, how do I bootstrap the yield curve using python?
I have a vector of dates and a ...
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Calculation of accruals using Actual/Actual AFB day count convention in QuantLib library
I am using the QuantLib library to calculate accruals for a fixed rate leg, using the "Actual/Actual AFB" day count convention. The payment period is annual, and the cash flows occur between ...
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How to find interesting open math problems in quantitative finance that I could publish articles about?
Which books on financial mathematics would you recommend for people with good background in probability, statistics and stochastic processes but without any background in financial mathematics?
The ...
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How does a mispricing affect CAPM/MPT statistical parameters?
In the CAPM/MPT context, would a mispricing affect the various statistical parameters? For instance, if Alpha is 2% and the CAPM E(R) is 10% (in equilibrium) and the E(Ra) = 12%, when calculating all ...
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What is the closed-form solution for the PV of the following series? [closed]
I have the following exercise, where a closed-form solution is needed for the PV of the cash flows.
The teacher's solution is the following:
But I fail to understand how exactly we get to it in the ...
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Questions on constructing WML factor (Fama French)
https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
We can see that the Winner and Loser portfolios are determined by the cumulative return from t-12 to t-2.
To construct the WML ...
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Can I extend the private information model of Kyle in in a continuous analogue, e.g. the Ornstein–Uhlenbeck process?
Taking into account an old post of maths.stackexchange, I recall the following:
On the one hand, we know that the Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of ...
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I can’t understand why the premium of two butterflies with same strike but different broadness are approximately the same
Consider the following premiums of calls option with different strikes.
C90 = 57.35
C95 = 52.55
C100 = 47.3
C105 = 42.9
C110 = 38.25
In this case, the butterfly 90-100-110 cost 1 and the 95-100-105 ...
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Discounted expectation of generic $\mathbb{C}^2$ function
Consider a standard geometric Brownian motion $V_t$ with drift $\mu<r$ and standard deviation $1$.
It holds that the discounted expectation is
$$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right]...
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Is "extreme CVaR" (CVaR from extreme value theory) elicitable or conditionally elicitable with some other statistical mapping (like VaR)? [closed]
I am not able to find loss function (scoring function) extreme CVaR (CVaR from extreme value theory) which is a conditionally elicitable statistical mapping (conditioned on VaR).
In this regard, can ...
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In the derivation of the Black-Scholes PDE, using delta hedging, how is this linked to the risk neutral valuation? [closed]
I was reading this paper:
http://www.columbia.edu/~mh2078/FoundationsFE/BlackScholes.pdf
I don't understand the paragraph here:
"The most interesting feature of the Black-Scholes PDE (8) is that ...
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Optimal consumption process [Munk (2011)]
I'm trying to solve problem 4.4 in Munk (2011). The problem is as follows:
Assume the market is complete and $\xi = (\xi_{t})$ is the unique state-price deflator.
Present value of any consumption ...
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Dynamics of discounted prices (multi-dimensional)
My objective is to find the dynamics of the discounted prices, given by $\mathbf{y}_{t} = \mathbf{P}_{t}\mathrm{e}^{-\int^{t}_{0} r_{s} ds}$. I know the dynamics should be $d\mathbf{y}_{t} = \mathrm{...
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Munk (2011) exercise 3.6
I'm trying to solve the exercise in Munk (2011). The exercise reads:
"Find the dynamics of the process: $\xi^{\lambda}_{t} = \exp\left\{-\int^{t}_{0} \lambda_{s} dz_{s} - \frac{1}{2}\int^{t}_{0} \...
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Why do companies trade options?
Companies buy options to reduce the variability in future cash flows.
Institutional investors invest in portfolios to maximize return for a fixed amount of risk. If an investor owns stock in company A ...
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Martingales and Arbitrage in Multiperiod Securities Markets
I have been reading the paper "Martingales and Arbitrage in Multiperiod Securities Markets".
The paper works in the probability space $(\Omega, F, \mathbf{P})$. $X$ is defined as the set of ...
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Should we use the conditional expectation to write the value of an option?
So, I've just started looking into financial mathematics and the following question keeps bugging me. From what I understood, if the market is arbitrage-free and a given contingent claim of value $h$ ...
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