Questions tagged [finance]
Questions related to the various aspects of financial mathematics. Topics include option pricing, arbitrage theory, market completeness and stochastic analysis.
2,563
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117
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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:
...
34
votes
3
answers
16k
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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 ...
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25
votes
1
answer
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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 ...
- 1,454
21
votes
2
answers
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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 ...
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17
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3
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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)^...
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16
votes
2
answers
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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; ...
- 1,713
16
votes
1
answer
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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,...
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14
votes
2
answers
54k
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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 ...
- 141
13
votes
1
answer
1k
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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 $Y_t=-\int_t^s{f(t,u)du}$ (...
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11
answers
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Can purchase of insurance be justified mathematically? [closed]
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 ...
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12
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7
answers
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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 ...
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2
answers
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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
...
- 173
12
votes
1
answer
686
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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 ...
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12
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0
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How useful actually is financial calculus? [closed]
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 &...
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11
votes
4
answers
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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 ...
user40167
11
votes
5
answers
2k
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Why using compound interest formula gives (potentially) wrong answer in this instance
I was doing some catch up exercise on Khan academy and was given this seemingly simple looking problem
Find the compound interest and the total amount after 4 years and 6 months if the interest is ...
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11
votes
2
answers
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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)...
11
votes
1
answer
19k
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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(...
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10
votes
4
answers
3k
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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 ...
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10
votes
7
answers
745
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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 ...
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10
votes
2
answers
38k
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Average percent increase not equal to total percent increase?
I tried searching around for this but it was difficult to boil down the search terms. Plus nothing seemed to be showing up anyway.
What's an easy way to show that the average percentage increase of n ...
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10
votes
1
answer
1k
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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, ...
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9
votes
4
answers
35k
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No-Arbitrage Principle
I hope you do not mind me asking a financial question in this section.
I am having trouble understanding the concept of the no-arbitrage principle for a particular example in my notes:
Suppose a ...
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9
votes
3
answers
26k
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Proof of Continuous compounding formula
Following is the formula to calculate continuous compounding
...
- 231
9
votes
3
answers
2k
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Book request: Mathematical Finance, Stochastic PDEs
I'm a math student, starting a PhD in the near future. My field of research will be mostly in the field of applied mathematics / numerics. Topics will deal with Kinetic Theory, Moment Equations, ...
- 4,315
9
votes
2
answers
10k
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Showing that the square of Brownian motion, minus time, is a martingale
What exactly are we supposed to do to show what they have given is a martingale.
If I try to follow through I'm getting a bit confused. In the third to last line I don't understand how they have ...
- 1,107
9
votes
1
answer
230
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Name for this kind of derivative?
In a problem I was working on, I found it convenient to use the notation $ dX/dA_{i \rightarrow j} $ to represent the marginal change in $X$ from redistributing a marginal amount of $A_i$ to $A_j$. Is ...
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8
votes
6
answers
8k
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Understanding Black-Scholes
Assume I have only basic math knowledge, what specific areas of math would I need to learn in order to understand the following webpage:
Black-Scholes
Many thanks.
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8
votes
4
answers
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How do you solve equations of any degree?
I have stuck solving this problem of financial mathematics, in this equation:
$$\frac{(1+x)^{8}-1}{x}=11$$
I'm stuck in this eight grade equation:
$$x^{8}+8x^{7}+28x^{6}+56x^{5}+70x^{4}+56x^{3}+28x^{...
8
votes
2
answers
412
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How can I calculate the optimal frequency of moving funds between two compound interest accounts?
The Problem
I have two compound interest accounts, Account A, which has 0.5% yearly interest, but I have to keep \$30,000 there to unlock some bonus; and Account B, which has 2% yearly interest. ...
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8
votes
1
answer
217
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An example of high dimension (financial) integrals?
Introduction
This question mainly arises out of the context of [Quasi Monte Carlo integration][1].
Which uses "quasi-random" numbers, (i.e. deterministic) with low discrepancy to reduce the ...
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8
votes
1
answer
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Novikov condition for CIR porcess
I want to apply the Girsanov theorem for change of measure for geometric Brownian motion under real world measure $\mathbb{P}$ to risk-neutral probability measure $\mathbb{Q}$ where the drift is given ...
- 111
8
votes
2
answers
625
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Heavy-tailed distributions
I have encountered the following two definitions of heavy-tailedness (right tail) for a $[0,\infty)$-valued random variable $X$ satisfying $\mathbb{E}[X]<\infty$:
(i) $\limsup_{x\to\infty}\frac{\...
- 1,454
8
votes
1
answer
611
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History of the power series for $e^x$ and compound interest
As discussed in How did Bernoulli approximate $e$?, Bernoulli showed that $2\frac{1}{2} < e < 3$ in this paper:
https://books.google.com/books?id=s4pw4GyHTRcC&pg=PA222#v=onepage&q&f=...
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2
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Market Making Card Bet Game
In an interview I received the follow question: We have 3 cards face down, and we give each card in a deck of 52 a numeric score ( A = 1, 2=2, .... , J=11, Q=12, K = 13). The interviewer asked me to ...
- 5,809
8
votes
0
answers
309
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First moments of Geometric Brownian Motion-like process with non-normal shocks
First consider a standard GBM process of the form,
$$\frac{dS_t}{S_t} = \mu \, dt+ \sigma \, dW_t$$
but instead of the normal $W_t \sim N(0,1)$ , instead we have that, $$W_t \sim \operatorname{EMG}^...
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7
votes
1
answer
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The so-called rule of 72 (or rather, 69)
This BBC article discusses the 'rule of 72' - essentially along the lines that questions to do with economic growth and inflation and so forth can be approximated by a simple formula using the number ...
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votes
5
answers
3k
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Paying off a mortgage twice as fast?
My brother has a 30 year fixed mortgage. He pays monthly. Every month my brother doubles his principal payment (so every month, he pays a little bit more, according to how much more principal he's ...
- 173
7
votes
2
answers
2k
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Black Scholes PDE and its many solutions
I know the general Black-Scholes formula for Option pricing theory (for calls and puts), however I want to know the other solutions to the Black-Scholes PDE and its various boundary conditions. Can ...
user754
7
votes
1
answer
287
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Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?
Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete.
(a) $\beta_{t}=e^{t}, S_{t}...
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7
votes
2
answers
502
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Calculate return on investment (ROI)
Before coming to the question, I'll quickly explain how to calculate NPV (Net present value) with an example
Information Available
Equipment Cost - \$20,000
Annual Benefit - \$6,000
Scrap Value - \$...
- 175
7
votes
1
answer
339
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Locate proof of Second Fundamental Theorem of Asset Pricing
Where can I find a rigorous proof of the Second Fundamental Theorem of Asset Pricing?
That is:
A arbitrage-free market is complete if and only if it has a unique risk neutral measure.
Please do not ...
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7
votes
1
answer
933
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Arbitrage opportunity
Given odds $o_i$ for $i=1,2,\dots,n$ and the possibility to bet the amount $b_i \in \mathbb{R}$ on each event such that if event $i$ occurs you receive $b_i o_i$ and if it doesn't you receive $-b_i$. ...
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7
votes
0
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430
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Has Mathematical Finance education become unnecessarily inaccessible? [closed]
Background
I had a very good and bright student with some decent exposure to markets and the standard college math curriculum who got overwhelmed during a mathematical finance course due to ...
- 1,001
6
votes
2
answers
221
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Ito integral under an expectation.
I am trying to apply Ito's formula to
$$P(t, X_t) = \mathbb{E}^\mathbb{Q}\left[\exp\left(-\int_t^{T}r(X_s)ds\right)\middle|X_t\right]$$
where $T$ is a fixed constant and
$$
dX_t = \mu\left(X_t\right)...
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6
votes
1
answer
3k
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Why is there a difference between life-insurance and general (non-life) insurance mathematics?
I know that actuarial mathematicians specialize in either life or non-life insurance. My question is, what is the difference between these two fields mathematically? Why is there a need for ...
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6
votes
2
answers
1k
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Show that the market is arbitrage free if $a < S_0^1(1+r)< b$
Exercise :
We consider a market of one period $(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$, where the sample space $\Omega$ has a finite number of elements and the $\sigma-$algebra $\mathcal{F} = 2^\...
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6
votes
1
answer
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Why is the drift of the stock price not important for options pricing?
This question is motivated by MSE question 4199364: Bachelier model option pricing.
There, one considers the price of a stock depending on time $t$, given by the family of random variables $(S_t)_{t\...
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6
votes
3
answers
443
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Bachelier model option pricing [closed]
Consider a Brownian motion $W_t$ and Bachelier model $S_t = 1 + \mu t + \sigma W_t$ for the stock price $S_t$. Find the value of an option that pays $1(S_1 > 1)$.
Attempt: As I understand it, the ...
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6
votes
1
answer
3k
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Ornstein–Uhlenbeck SDE.
I am trying to understand the solution to the following exercise, however it is kind of poorly written. Can someone please explain it to me?
For $V = (V_t)$ the solution to the Ornstein-Uhlenbeck SDE ...
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