Questions having to do with financial mathematics. Please note that for questions in quantitative finance, quant.stackexchange.com is perhaps a better site.

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15
votes
2answers
2k views

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 ...
10
votes
1answer
429 views

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 ...
9
votes
1answer
428 views

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 ...
8
votes
4answers
801 views

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: ...
8
votes
1answer
128 views

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) ...
7
votes
1answer
873 views

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 ...
7
votes
5answers
1k views

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 ...
7
votes
2answers
913 views

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 ...
7
votes
1answer
13k views

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 ...
7
votes
2answers
490 views

Significant digits

We use currency conversion rates for financial calculations. Our currency conversion table stores conversion rates to and from each currency (about 150 world currencies) for each day, going back 20 ...
7
votes
1answer
95 views

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: ...
6
votes
5answers
5k views

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.
5
votes
1answer
6k views

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; ...
5
votes
1answer
323 views

How to determine annual payments on a partially repaid loan?

A 10-year loan of $500 is repaid with payments at the end of each year. The lender charges interest at an annual effective rate of 10%. Each of the first ten payments is 150% of the amount ...
5
votes
2answers
423 views

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) = ...
5
votes
1answer
245 views

Arbitrage opportunity

Given odds $o_i$ for $i=1,2,\ldots,n$ and the possibility to bet the amount $b_i\in \mathbb{R}$ on each event such that if event $i$ occurs you receive $b_io_i$ and if it doesn't you recieve $-b_i$. I ...
5
votes
0answers
337 views

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": $(dt)^2=0$ ...
5
votes
1answer
61 views

$dX_t/X_t=\mu+\sigma \, dZ_t$, does this notation make sense?

I understand that the notation $$dX_t=\mu X_t \,dt + \sigma X_t \,dZ_t,$$ where $Z_t$ is Brownian Motion, is a shortcut to $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...
4
votes
1answer
201 views

How can I calculate what my credit card balance will be next month?

If my current credit card balance in July is \$1,000 USD, my credit card's APY is 20% and this month I made a payment for \$100 on time (to avoid late fees)... What will my balance be in August? I ...
4
votes
3answers
537 views

Independent math learning

I'm an undergraduate math and econ major and I'm planning on graduating relatively soon and I am very limited to the number of math classes I have left (very sad about this fact). So far I have ...
4
votes
2answers
5k views

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 ...
4
votes
1answer
73k views

Finding Revenue Function and Max Revenue

Studying for a midterm. The demand function for a manufacture's product is $p=1000-\frac1{80} q$ Where $p$ is the price (in dollars) per unit when $q$ units are demanded (per week) by consumers. ...
4
votes
1answer
222 views

Maximizing gambling performance over the long run

Background. We can play a game in which we can put one dollar and get out $X$ dollars, where $X$ is 2 dollars with probability $p>1/2$, or zero dollars with probability $1-p$. We also assume that ...
4
votes
2answers
631 views

Calculating credit card charges based on provided APR, balance amount and monthly payment amount?

Please first see the below sums and then the info below it to get a better understanding of my question: My input: ...
4
votes
1answer
124 views

Calculating interest rate for a payment plan

I really should know this from high school so I'm a bit ashamed to ask... :) I've been offered several cars for lease and want to compare the offered interest rates. E.g. I have: ...
4
votes
1answer
62 views

Annuity that pays $t^2$ at time $t$ in arrears annually.

I am asked to show that such an annuity for $n$ years will be expressed as, $$\frac{2(Ia)_{\bar n|} - a_{\bar n|}-n^2u^{n+1}}{1-u}$$ where $u=\frac{1}{1+i}$ and $i$ is the annual effective ...
4
votes
1answer
526 views

Applications of Compound Poisson Processes

I'm reading the book Non-Life Insurance Mathematics, an introduction with Stochastic Processes by Thomas Mikosch and I'm interested in applications of the Cramer-Lundberg Process to concrete examples ...
4
votes
0answers
73 views

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 ...
4
votes
0answers
62 views

Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta ...
4
votes
2answers
206 views

Speculating on the stock exchange

Imagine you model each stock as a random walk (fractal) and also that you can buy and sell at any price. Suppose also that it 'walks' with the pace of 1. If you buy, for example, 1000 shares of ...
4
votes
2answers
60 views

Can someone explain what a portfolio is in financial math?

I took mathematical probability last semester and now I am taking financial mathematics, but only probability was a pre requisite for financial math (no finance classes were required). These types of ...
3
votes
4answers
33k views

What is the reasoning for the Discount Factor formula?

I've previously come across dicount factors in my high school education but we had formula sheets so I never bothered actually learning it. In my university I have a business class in which we are now ...
3
votes
2answers
259 views

Measuring Financial Investment Performance

I'm trying to understand how the performance of financial investments are measuring when the asset has multiple points of investment. For example, say I invest ...
3
votes
2answers
330 views

Partial Differential equations and applications- Reference request

I will be taking up a PDEs course next semester and would like to find some good references. The topics covered in the syllabus is given below. Partial differential equations: Conservation laws, ...
3
votes
2answers
616 views

Put Options and Arbitrage

I came across the following problem on put options: A European put with strike price $100$ expiring in $1$ year has premium $ \$ 1$ and a European put with strike price $K$ expiring in $1$ year ...
3
votes
1answer
932 views

What is the definition of a “predictable process”?

I am reading a book on financial mathematics, and frequently encounter the phrase "predictable process", which I haven't seen definition of, and cannot find the definition online. At first I thought ...
3
votes
3answers
86 views

Rule of 72 doubling time

I need some help understanding this. So as far as I can tell. The rule of 72 is used to determine when prices will double in years. This is done by 72 divided by the rate, or interest. So it would ...
3
votes
2answers
238 views

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, ...
3
votes
1answer
68 views

Isolating for i

So this might seem a bit fundamental, but in financial math the following equation gives you the price for a bond $$ P = C \frac {1-(1+i)^{-n}} {i} + B(1+i)^{-n} $$ where $P$ is the price of the ...
3
votes
1answer
221 views

Reference Request - Introductory book on Mathematical Modelling in Economics and Business

I have to take a compulsory course named Mathematical Modelling in Economics and Business this semester and have absolutely no background on the subject. I also noticed there is no post on this site ...
3
votes
1answer
498 views

Compound interest -like calculation, but with increasing rate

Let's say an organization has 100 employees in the beginning of 2013 and grows to 110 employees by the beginning of 2014. This implies a growth rate of 10% for 2013. Now, let's say a hiring manager ...
3
votes
4answers
143k views

What is the formula to calculate Profit Percentage?

Let cost price of an item be $C$, selling price be $S$. Assume the seller makes a profit. Then profit would be: $P = S - C$. Now, what is the formula for calculating Profit Percentage? $P \% = ...
3
votes
2answers
2k views

APR Calculation

I'm hoping someone can clarify this for me. The model/example is this: We lend an amount of 1498.50 (loan amount). Other fees total 39.95. The term of the loan is for 12 months. There is no ...
3
votes
1answer
63 views

What's the purpose of this unknown (financial) math formula?

I am maintaining an old piece of financial software. In the source code I have found an implementation of the following formula: $$p2 \over (p1 + 1) - (p1 * p2)$$ The formula is used as part of some ...
3
votes
1answer
90 views

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 ...
3
votes
2answers
122 views

Book Recommendation for mathematical finance

Does anyone know a book which covers topics on: Brownian Motion Martingales Stochastic Calculus Stochastic Differential Equations Options pricing. Black-Scholes model Fundamental Theorems. ...
3
votes
2answers
207 views

Zero-coupon vs. $10\%$ coupon problem

I am working on Bonds and I am having trouble solving this problem. A zero-coupon bond pays no coupons and only pays a redemption amount at the time the bond matures. Greta can buy a zero-coupon ...
3
votes
2answers
2k views

Financial Linear Programming Problem

I'm very new at linear programming and I'm trying to figure out a way to approach this problem below: ...
3
votes
2answers
103 views

Accrued interest

I know how to calculate accrued interest over time on an initial amount. However, my assignment has me artificially adding additional sums intermittently. I'm curious if there is a formula to do that. ...
3
votes
1answer
5k views

Confused about Effective Rate of Discount- Theory of Interest

I'm currently reading Kellison's book, The Theory of Interest. I've reached the chapter on Effective Rate of Discount and it's somewhat confusing. The book explains it as a loan where interest is paid ...