Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [finance]

Questions related to the various aspects of financial mathematics. Topics include option pricing, arbitrage theory, market completeness and stochastic analysis.

Mathematical finance, also known as quantitative finance, deals with finance and financial markets in a mathematical manner.

Some examples of mathematical finance are the fundamental theorem of asset pricing which provides the conditions for a market to be arbitrage-free and complete, and the Black–Scholes equation, which uses partial differential equations to describe the price of an option over time.

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2673 questions
120
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15 answers

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…
The Gilbert Arenas Dagger
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38
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3 answers

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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1 answer

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…
JohnSmith
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22
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2 answers

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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18
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2 answers

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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17
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3 answers

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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7 answers

How the formula for EMI is derived

I was looking for a formula to calculate EMI (Equated Monthly Installments). I have some fixed known parameters like, Principal Amount, Rate of Interest and No. Of Installments. By googling, I came across the formula, $$Installment Amount = \frac…
Rumit Parakhiya
  • 273
17
votes
2 answers

Compound Interest Formula adding annual contributions

I'd like to know the compound interest formula for the following scenario: P = Initial Amount i = yearly interest rate A = yearly contribution or deposit added. n = the deposits will be made for 10 consecutive years. F = final amount obtained. I…
user3323679
  • 179
16
votes
1 answer

An algorithm for arbitrage in currency exchange

I found a really interesting problem on currency exchange rates and I wanted to hear people's opinions. If we are given some coins $c_1, c_2, \dots, c_n$ and an array $R$ that keeps the selling price, where $R[i,j]$ is the selling price of one unit…
tasmer_k
  • 177
14
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1 answer

Brownian motion and covariance

Show that for $B = (B_t)$ Brownian motion, its covariance is $cov(B_s, B_t) = min(s, t)$. The solution I was given was: For $s ≤ t$, $B_t = B_s + (B_t − B_s)$, $B_sB_t = B_s^2 + Bs(Bt − Bs)$ $cov(B_s,B_t)=E[B_sB_t]$(as $E(B_i)=0)$ so…
vounoo
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13
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2 answers

Introduction to blockchains and cryptocurrencies for the mathematically mature.

I have spent several days off and on looking for a good introduction to blockchains and cryptocurrencies for someone with mathematical background but no specific computer science or cryptography background. I've had no real luck. All of the books…
Yly
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13
votes
1 answer

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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13
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4 answers

Definition of self-financing strategy

Consider a portfolio of two assets with prices $S_t$, $B_t$ and holdings $\Delta_t$ and $E_t$ respectively. So the portfolio value is $$ \Pi_t = \Delta_t S_t + E_t B_t$$ The portfolio is defined to be self-financing if we also have $$ d \Pi_t =…
user40167
12
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2 answers

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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12
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2 answers

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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