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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Intuitive Understanding of d1 in Black-Scholes [duplicate]
I am reading about the derivation of Black-Scholes using Future prices. I understand the mathematical steps of the derivation, which starts by separating it into the difference of two integrals,
the ...
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Recover value function from its dual
I take as my reference Optimal Investment by Rogers (2013). Denote the agent's value function, which is convex, by $V(w)$, where $w$ is the state variable. Define the dual variable $z = V_w$, where ...
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Fitting a multidimensional Ornstein-Uhlenbeck pProcess
If I have a dataset X, where each row is a time point and we have several variables, say 100, (so this is a multivariate time series), what is the best way to fit a multidimensional Ornstein-Uhlenbeck ...
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Simple arbitrage pricing of bond option
This is Tuckman fixed income security textbook. The text here is trying to price a 990 six month call on a six month zero bond. When we replicate the portfolio, where is the F_.5 coming from?
My ...
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Predatory trading as a game of size
Predatory trading has been addressed in literature frequently. I have read for example Brunnemeier (2005) but that paper mostly addresses predatory trading surrounding a preexisting distressed trader. ...
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Calibrating CIR to bond prices
Consider the Hull-White model -
$$dr_t = (\theta_t - kr_t)dt + \sigma_tdw_t$$
We can/have to calibrate $\theta_t$ to the current bond prices $P(0,t)$ and make it consistent with the HJM framework. For ...
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Preferred Option pricing model [closed]
I am at Uni studying mathematical finance and wanted to know which is most preferred /widely used model by Finance Industry Practitioners from the list below.
Fourier Transform for option pricing
...
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Multicurrency holiday calendar math and combinations
Trade Example
I have recently observed a trade which calculated spot (T+2) from Friday 16th June 2023 for a EURUSD FX product and derived the date Tuesday 20th June 2023.
In EUR under the target ...
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What is the proper way to derive risk definitions from utility functions?
In typical mean-variance analysis, the risk-adjusted relative value of an individual asset takes the general form
$\frac{\mu}{\sigma^2}$,
with further weighting and normalization depending on the ...
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Calculating spread on a par rate curve given bond’s coupon and yield
In Tuckman and Serrat’s Fixed Income Securities, they give an example of a bond and state its coupon and yield.
They also provide an HQM par rate curve and quote the bond’s spread to this curve.
How ...
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Showing a basic market admits no arbitrage
I'm learning the fundamentals of financial mathematics and came across the following problem I cannot solve
Setting
We work in $\left(\Omega, \mathcal{F},\left(\mathcal{F}_t\right)_{t=0}^1, \mathbb{P}\...
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Learning resources - Any good websites for learning about quantitative finance with a focus on the underlying mathematics? [closed]
Typically when I have ventured into a new subject in my sparetime, I have been able to find an interactive website with reading materials, exercises and the like. To name a few there is https://...
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What are the downsides of using Kim's integral equation (1990) to determine the exercise boundary of an American option?
I'm new to the industry and trying to wrap my head around American options pricing.
The integral equation(1) from Kim (1990) doesn't seem to make any strong assumptions, and approximating the integral ...
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Possibility of obtaining a positive mathematical expectation in a quoted currency
There is a currency pair C/USD = 1. C - currency in which I want to invest in order to make a profit in USD.
Suppose its price changes discretely: 50% - increases by 20%, 50% - decreases by 20%. This ...
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Can someone please help me answer this question about Black-Scholes model? (risk-neutral & true probability of the call option) [closed]
I don't even know where to get started with this question...can someone please help me? How do I answer it?
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Is this arbitrage? Infinite payoff / infinite loss (energy generation investment problem)
I'm a student using stochastic optimization in energy systems and I have a particular phenomena in an optimization problem that I think must occur in finance aswell, so I have been trying to find ...
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Two-period binomial model probability question
I have started to work with given two period binomial model S(0)=100 u=1.25 d=0.8 r=0.05 and the market probability of stock going up each period is p=0.55.
I am trying to calculate two probabilities;
...
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IFRS 9 - Amortized cost - Daily or Annual EIR
Let's consider a scenario where a bank issued a loan of USD 80,000 at some point in 2022 and expects to receive USD 17,809.33 in interest and the 80k on October 9, 2023. The Effective Interest Rate (...
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Cost of debenture using IRR method
Question:
A company issued $10000$, $10\%$ debentures of $\\\$100$ each on 1.4.2020 to be matured on 1.4.2025. Market price of the debenture is $\\\$80$. Tax rate is $35\%$. Then what would be the ...
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Breaking down gamma PnL by time
Let's say in one day time, underlying price S goes up by $2.4. Then the gamma PnL should be
$\frac{1}{2}\gamma_0 (2.4)^2$. Let's assume that in every hour the underlying goes up by $0.1. Then how do ...
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How to solve for Kyle's $\lambda$ that emerges in the demand functions of the informed and uninformed traders in the $1989$ model?
I will restate here a problem that I am finding a bit difficult to solve and I have already posted here. I summarized the problem as it follows.
From Albert S. Kyle's 1989 model.
Suppose that the are ...
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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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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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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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