Questions tagged [finance]

Questions having to do with financial mathematics. This is not a tag about financing, which is not within the scope of mathematics defined by the help center: http://math.stackexchange.com/help/on-topic Topics may include: option pricing, arbitrage theory, market completeness, and applications of stochastic analysis to finance. Please note that for questions in quantitative finance, quant.stackexchange.com is perhaps a better site.

Mathematical finance, also known as quantitative finance, is a field of applied mathematics, concerned with financial markets. Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock (see: Valuation of options; Financial modeling). The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results. -Wikipedia

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The math behind Warren Buffett's famous rule – never lose money

This is a question about a mathematical concept, but I think I will be able to ask the question better with a little bit of background first. Warren Buffett famously provided 2 rules to investing: Rule No. 1: Never lose money. Rule No. 2: Never…
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Stochastic calculus book recommendation

I'm a quantitative researcher at a financial company. I have a PhD in math, but I'm an algebraist, so I only took the two required analysis courses in grad school (measure theory for the first, and I don't even remember the content of the second…
Matt Samuel
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Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator and admitted integrands to be progressively…
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Farkas’ lemma: purely algebraic intuition

Here is a statement of Farkas Lemma from the Wikipedia. Let $A$ be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the following two statements is true: There exists an $x \in \mathbb{R}^n$ such that $Ax = b$ and $x…
Jyotirmoy Bhattacharya
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In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ $dZ(t)^2=dt$ $dZ(t) \, dt=0$ I tried to do some…
3x89g2
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Price of a European Call option is a convex function of strike price K

I'm trying to show that the price of a European call option (payoff function is $(S_1-K)^+$) in a no-arbitrage market is a decreasing and convex function of K. That it shall be decreasing makes sense; as $K$ increases, $S_1-K$ decreases and we make…
Marie. P.
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How useful actually is financial calculus?

I am a PhD student in math, for context. I was looking through some old PhD theses from my school, and some of that have to do with finance- specifically financial stochastic calculus. So, when I say "financial calculus" I mean stochastic calculus…
MathIsLife12
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Can purchase of insurance be justified mathematically?

When I ask people to explain why they buy insurance, I often hear vaguely of "spreading the risk", but I am not actually sure what that means nor if insurance does this. How is an insurance company any different than a casino? In a thought…
Jeff
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Analogue of Leibniz Rule for Stochastic Integrals

Suppose $$f(t,u)=f(0,u)+\int_0^t{\mu (w,u)dw}+\int_0^t{\sigma(w,u)dB_w},$$ where $B_w$ is a standard Brownian motion. I would like to calculus the drift and diffusion of $Y_t=-\int_t^s{f(t,u)du}$ (under sufficient conditions that guarantee all the…
epsilon
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Proof of the Black - Scholes pricing formula for European Call Option

I want to prove the following The price of a European call option with strike price $K$ and time of maturity $T$ is given by the formula $\Pi(t) = F(t,S(t))$, where $$F(t,s) = sN[d_1(t,s)]-e^{-r(T-t)}KN[d_2(t,s)]$$ $$d_1(t,s) =…
Teodor Fredriksson
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What's the math formula that is used to calculate the monthly payment in this mortgage calculator?

What's the math formula that is used to calculate the monthly payment in this mortgage calculator? I would like to know this math formula so that I can plug in the following values Mortgage Amount: $100,000 Rate Type: Fixed Interest Rate: 6% …
burnt1ce
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A (mathematically) sound investment strategy

It is common wisdom in the investment community that a long-term investor saving for his future would do well to invest in high-risk/high-return assets when he is young, slowly switching his portfolio over to low-risk/low-return assets as he grows…
Chris Taylor
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Modelling risk when market making

I'm interested in learning about algorithmic trading, particularly in bitcoin. Looking at this chart, I can see that I could simultaneously offer a bid that was slightly higher than the highest bid, and an ask that was slightly lower than the…
Tom Busby
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A winning wager that loses over time

This problem was posted in Scientific American (vol. 321.5, Nov 2019, p. 73), and it was troubling. The game: We flip a fair coin. If we flip heads we gain 20% of our bet If we flip tails we lose 17% of our bet. Starting bankroll:…
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Is vector geometry useful within economics?

I'm going to be taking a semester of math after my bachelor's in economics before I go on to do a master's, and one of the mandatory courses in that semester is linear algebra with a focus on vector geometry. This is how they describe it: The…
Chisq
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