Questions tagged [mathematics]

Used for question on application of mathematics in finance - from interest calculation to mathematical description of random processes.

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86 views

Mathematical proof that the covariance between two portfolios is $w_A^\top\Sigma w_B$

How to prove in a line-by-line derivation that the covariance between two mean-variance efficient portfolios is equal to $$w_A^\top\Sigma w_B$$ where $w_i$ is a unique portfolio weight vector, and $\...
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15 views

Is the feasible set of portfolios an epigraph?

In mathematics, the epigraph of a function is the set of points lying on or above its graph, in this case a convex function: The efficient frontier from mean-variance portfolio analysis encloses an ...
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34 views

Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance

The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample. ...
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32 views

Entropy-implied volatility requires itself to be calculated?

\begin{align} H &= \frac{1}{2} \ln (2\pi\sigma^2) + \frac{1}{2}\\ &= \frac{1}{2} \ln (2\pi e \sigma^2) \end{align} is the analytical solution for the entropy of a Gaussian random variable, ...
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29 views

Estimating the XIRR of a very non uniform cash flows

It's my second post, so please bear my lack of experience in this field. I've a very irregular cash flow (here you can see the set of date - cumulative cash flow) The XIRR, calculated with Excel, is ...
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1answer
94 views

Does Value-at-Risk have any mathematical equivalence to copulas?

Portfolio Value-at-Risk estimated using the copula approach often just means generating artificial data sampled from a parametric copula('s joint multivariate distribution) as a model fit over the ...
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1answer
76 views

Question on the use of a limit in a proof

I ran into a step in an argument that I can't quite figure out. It's basically how they use a limit that I don't seem to understand. The context is local-to-unity asymptotics in vector autoregressions,...
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1answer
70 views

Option proofing: Analytical solution for option math

How do I prove the following equation: P(X=100)≤(P(X=110)-P(X=90))/2 I am not sure how to start and whether it involves using the Black-Sholes formula or not (something like this: https://www.youtube....
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1answer
77 views

help with derivation of equation 8 in Derman and Kani's binomial tree for local vol

in this paper "The Volatility Smile and Its Implied Tree" - Derman and Kani 1994 i understand the derivation of all equations up to 7. But eq 8 i cannot figure out how to derive! i have ...
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70 views

Why does the Hurst exponent pseudo code not match the Python implementation?

I am working on understanding the Hurst exponent calculation by Ernest Chan; however, the description of the algorithm does not match the Python implementation. Chan [Algorithmic Trading: Winning ...
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56 views

The distribution of mean reversion time from the OU process

I was reading the paper Statistical Arbitrage in the US Equity Market and I couldn't understand the figure that plots the histogram of the empirical distribution of characteristic time to mean ...
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2answers
80 views

Is there a way to formulate a Martingale series that will never explode?

Martingale's betting method can be seen here:https://www.investopedia.com/articles/forex/06/martingale.asp My question is if there is a way to put a non-exploding martingale, [There is one attempt to ...
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54 views

What is the differential Value-at-Risk?

I am currently working on a Machine Learning Project, implementing portfolio optimization algorithms according to different risk measures. I have found sufficient information on Sharpe Ratio ...
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0answers
115 views

Dynamic programming and Bellman equation to obtain the maximum

This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM" Suppose an endowment economy where the representative ...
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1answer
108 views

Is the variance calculation correct in the book?

I'm reading the book "Financial Markets Under the Microscope" for my market microstructure studies. In the book, the variance of the market maker's gain is calculated as follows: Assume that with ...
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33 views

Stock returns: Exponential time decay

I am replicating some research that uses two years of single stock returns (i.e. N= 250*2= 500) and then applies an exponential decay with a half-life of one year to these returns. Does this mean ...
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1answer
39 views

Contingent Claim Bounds

In my course on discrete-time finance we derived the following equality for a lower bound for the value of a not necessarily replicable contingent claim $D$. Here we are looking at a single period ...
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35 views

How to find the derivative for a multi-factor geometric brownian motion model

Does anyone know how to find the derivative for a multi-factor geometric brownian motion model $ \frac { dS_{i}}{S_{i}} $. I have seen solutions for the standard GBM model however I suspect that the ...
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1answer
40 views

How can I convert rolling annual returns back to quarterly returns?

I have a series of rolling annual returns and would like to convert these back to quarterly returns, which have not been provided. Is this possible formulaically, or is something like Excel's solver ...
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1answer
478 views

Abstract algebra in economics and finance

Are there any applications of abstract algebra (group theory, rings, fields etc.) in any branch of either economics or finance?
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32 views

Linear programming and minimum cost network flows vs nonlinear and discrete optimization

At my college I have an option: To take either of these two classes. My intended career pathway is into quantitative finance and I wanted to know which one would have more use as a quant. Here is the ...
2
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1answer
825 views

What is the formula to calculate Implied Volatility Percentile [closed]

I googled and I am unable to find any formular . Can some one give me the formula to calculate IVP , based on sets of IV's given. Thanks.
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44 views

Classical Ruin Theory - Lundberg Model

In classical risk/ ruin theory, I see this formula crop up in my notes but my lecturer didn't explain to me why/ when it's employed: $M_X(r) = \int_{-\infty}^{\infty} e^{rx} f(x) dx$ I understand ...
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1answer
161 views

How can I express this sum in a easier way?

For instance, I know that the sum of the first $101$ natural numbers can be expressed in the following easy computation: $\sum_{i=1}^{101}i = \frac{101*102}{2}$ One of the questions is: and what ...
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1answer
142 views

What the expectation of S^2 is from GBM? [closed]

I was at an interview and was asked to write down the SDE for GBM. $$ dS = S\mu dt + S\sigma dX $$ Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
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3answers
872 views

How to calculate standard deviation of continuously compounded four-year stock returns?

Currently I am preparing for quant interview and I encounter the following question in Heard on the street. Question: If the standard deviation of continuously compounded annual stock returns is $...
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2answers
291 views

How to derive Black-Scholes equation with dividend?

Question: The Black-Scholes equation without dividend is given by $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} -rV = ...
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0answers
33 views

Estimator for Conditional value at risk (average value at risk)

I am following a book: Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization by Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi I'm learning about average value at risk. ...
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1answer
312 views

Do quants need to know bloomberg terminal and VBA? [closed]

I am a Pure Maths PhD student who will graduate in 2 years time. My aim is to land a quant job after gradauation. When collecting more information so that I can have some edges over others, I heard ...
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1answer
45 views

How is hypothesis testing work in population sampiling? [closed]

I am learning the basics of quant trading from quantconnect's tutorial Confidence Interval and Hypothesis Testing. I understood the first part of the article but I dont understand "Hypothesis Testing"...
3
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1answer
206 views

Forward Start Spread Options

Question: We have a spread option with payoff: $\max (P_{T} - HR\times G_T, 0)$, where $P$, $G$ are underlying prices and $HR$ is a constant. At time zero only contract $G$ is available for ...
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4answers
906 views

Examples of discrete math and graph theory within quantitative finance

The Wikipedia article on quants mentions discrete mathematics as a possible piece of their mathematical background. Are there good examples of problems within quantitative finance that are heavily ...
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0answers
45 views

Stochastic integral representation of $F(T-s,X_s)$-type equations

For $T\in R$ given and fixed consider: $$ {\rm d}F(T-t,X_t)=g(T-t,X_t)\,{\rm d}W_t. $$ where $g(t,x)$ is a given functions and $X_t$ is a given process driven by a brownian motion ($dX_t=(...)dt+(...)...
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0answers
47 views

Transformation of random variables and second-order stochastic dominance

Suppose $X$ and $Y$ are two random variables where $X$ SOSD* $Y$. Let $g(\bullet)$ be a monotonic function and $X'=g(X)$ and $Y'=g(Y)$. Under what conditions of $g$ is $X'$ SOSD $Y'$? I know if $g$ ...
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1answer
64 views

Why do most interest rate formulas, and indeed finance in general, add 1 to a rate and then subtract afterwards? [closed]

For example, in the formula that shows the relationship between the nominal and effective interest rate shown below, 1 is first added to in/m and then 1 is subtracted from the result. What is the ...
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1answer
173 views

Approximation of Forward Rates in discrete time

The forward rate from time $t$ to $T$ ($f_{t,T}$) can be approximated by: $$ f_{t,T}= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{{T-t}}}-1 \sim \frac{(1+r_T)^T-(1+r_t)^t}{T-t} $$ Why is ...
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1answer
57 views

standard brownian vs brownian motion

We say Xt with paramters (µ,σ) is brownian process if (Xt-s - X t) ~N (µs,σ2 s) AMONG other conditons . Here we don't speak about any particular distribution for X t. We only say it is a brownian ...
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1answer
129 views

Finding the process of $X/Y$

This comes from Mark Joshi's concepts of mathematical finance exercise 4 chapter 11. If $$dX_t = \alpha X_t dt + \beta X_t dW_t$$ $$dY_t = \alpha Y_t dt + \gamma Y_t d\tilde{W}_t$$ with $W$ ...
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3answers
303 views

Is there an intuitive explanation for why Kelly gambling ignores odds?

I have just learned about Kelly gambling from Chapter 6 of Cover & Thomas' Introduction to Information Theory. The mathematical setup is that we have a horse race, with horse $i$ winning with ...
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1answer
81 views

4-point Trapezium rule for numerical integration

Background: This is in reference to Mark Joshi's concepts of mathematical finance ch.7 problem 11. Question: We have in the Black-Scholes model: $S_0 = 1, T = 1, \sigma = 0.1, r = 0$. A ...
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0answers
114 views

Anti-thetic sampling and second moment matching

Background: This is in reference to ch 7 problem 10 of Mark Joshi's concepts of mathematical finance. Question: A normal random generator produces the following draws: $$0.68, -0.31, -0.49, -0....
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1answer
501 views

Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 6 [duplicate]

Suppose a stock allows a geometric Brownian motion in a Black-Scholes world. Develop an expression for the price of an option that pays $S^2 - K$ if $S^2 > K$ and zero otherwise. What PDE will this ...
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1answer
473 views

Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 4

Let an asset follow a Brownian motion $$dS = \mu dt + \sigma dW$$ with $\mu$ and $\sigma$ constant. The constant interest rate is $r$. What process does $S$ follow in the risk-neutral measure? ...
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1answer
183 views

Mark Joshi, Chapter 5 Problem 2 of The concepts and practice of mathematical finance

If $$dX_t = \mu(t,X_t)dt + \sigma(X_t)dW_t$$ with $\sigma$ positive, show there exists a function $f$ such that $$d\left(f(X_t)\right) = v(t,X_t)dt + V dW_t$$ where $V$ is constant. How unique is $f$...
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1answer
130 views

Can someone please verify or disprove this Sharpe Ratio math logic for me

I want to start by stating a problem that I wanted to figure out initially so that this all ties in somehow. I initially wanted to figure out if individual securities in an efficient portfolio all ...
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2answers
153 views

I need liquidity metrics of a portfolio (2-5 bonds) that takes into consideration difference in size of bonds and maturity profile

Context: I have bond A from say Apple, Apple also issued different types of bonds , namely B , C, D, E bonds. Bonds A B C D E are all same, except, they were issued at different times, have ...
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2answers
854 views

Mark Joshi, Quant Interview Question problem 2.34; replicating a digital option on a 4-step symmetric binomial tree

Question: Team $A$ and team $B$, in a series of $7$ games, whoever wins $4$ games first wins. You want to bet $100$ that your team wins the series, in which case you receive $200$, or $0$ if they ...
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0answers
145 views

Why do we have to use discretization methods for SDE?

I haven't found the answer for the question above in google. Why can't we just discretize the equation instead of using methods like euler or milstein for the discretization.
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1answer
192 views

How to run optimization to achieve an equal active weight portfolio?

I am trying to build an equal active weight portfolio, while minimizing the total risk. However, my constraint of equal active weight always leads to 0 active weight for everything. I know 0 active ...
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1answer
113 views

CDS protection/contingent leg pricing, taking expectation of interest and hazard rates

The Pricing and Risk Management of Credit Default Swaps, with a Focus on the ISDA Model Screenshot: Pricing protection leg of a CDS, by OpenGamma In the screenshot above, I am having trouble ...