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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14
votes
2answers
1k 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
375 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 ...
8
votes
4answers
670 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
269 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 ...
7
votes
1answer
564 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
828 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
258 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 ...
6
votes
2answers
758 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 ...
6
votes
1answer
6k 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 ...
5
votes
5answers
4k 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
136 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
1answer
85 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": $(dt)^2=0$ ...
4
votes
1answer
3k 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; ...
4
votes
1answer
165 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
2answers
291 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
196 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
194 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
271 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 ...
4
votes
1answer
108 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
396 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
2answers
151 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 ...
3
votes
4answers
21k 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
245 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
446 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
631 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
2answers
409 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 ...
3
votes
4answers
60k views

What is the formula for calculating Profit Percentage?

Let cost price of an item be $C$, selling price be $S$. Assume the seller gets benefited. Then, Profit, $P = S - C$. Now, What is formula for calculating Profit Percent? $P \% = \dfrac{P}{C} ...
3
votes
2answers
816 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
2answers
43 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) = ...
3
votes
2answers
936 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
68 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
2k 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 ...
3
votes
1answer
81 views

How the banker limits his risk?

In a game like Deal or no Deal, what is the formula the banker uses to give an offer to the contestant? The banker offers the contestant an amount of money to quit the game, the offer based roughly on ...
3
votes
1answer
64 views

What is the Most Efficient Way to Calculate the Internal Rate of Return?

I have built a program that prices financial assets and it does this in part by calculating the IRR. The problem is that it does not run as quickly as I would like it to. I currently use the ...
3
votes
1answer
62 views

Why is that a risk averse consumer buys the optimum insurance when there is actuarially fair insurance?

I've asked the same question at the Quantitative Finance StackExchange. Consider the following example: "As a risk-averse consumer, you would want to choose a value of x so as to maximize expected ...
3
votes
1answer
121 views

Text on Probability Theory applied to Actuarial Science

I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life ...
3
votes
2answers
76 views

Calculating annuity payment

I have a time payment question I'm having trouble with: "Bill would like to save 60000 dollars for a deposit on his first home. He decides to invest his net monthly salary of 3000 dollars in a bank ...
3
votes
1answer
242 views

Futures pricing and futures price process under the real world measure

This is something that keeps bothering me about the Benchmark approach of Platen, which (very) shortly is as follows: Compare the development of an economic value with a growth optimal portfolio. ...
3
votes
1answer
111 views

Math for Future Value of Growing Annuity

Am I working this out correctly? I need to verify that my code is correct... $$1000 \cdot \left(\frac{(1 + 0.1 / 12)^{40 * 12} - (1 + 0.06 / 12)^{40 * 12}}{(0.1 / 12) - (0.06 / 12)}\right)$$ ...
3
votes
0answers
90 views

How to get interest in the mathematics of tax

In a similar vein to my previous thread, I will also be teaching about the mathematics behind taxation - to a lot of people, this is very mundane - but that is not true of everyone. The practicality ...
3
votes
2answers
695 views

Maximizing a function containing an integral

Problem. Let $\rho\colon[-1,\infty)\to\mathbb{R}$ be a function such that $$\int_{-1}^\infty\rho(x)\,dx=1.$$ Let $G\colon[0,1]\to\mathbb{R}$ be a function that is defined with $$G(f) := ...
3
votes
0answers
152 views

Algorithm/Formula to compute adding and/or removing compound and/or non-compound percentages from a value?

I will first start with a scenario, I have to apply some adjustments to a particular value. These adjustments are either compound or non-compounded and they can either be added or subtracted to the ...
2
votes
6answers
746 views

How to solve this problem without the assistance of a calculator in under 3 minutes?

I have 3 minutes to solve questions like these: Someone invested $\$2,000$ in a fund with an interest rate of $1\%$ a month for $24$ months. Consider it to be compounded interest. What will be the ...
2
votes
2answers
474 views

Using Black-Scholes Equation to “buy” stocks

From what I understand, Black-Scholes equation in finance is used to price options which are a contract between a potential buyer and a seller. Can I use this mathematical framework to "buy" a stock? ...
2
votes
5answers
279 views

Game Theory Question about Financial Markets

This is a recent quote from one of the outstanding bond portfolio managers: "First of all, for every buyer there is a seller. Therefore, in order for someone to sell their bonds and buy stocks means ...
2
votes
2answers
111 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, ...
2
votes
2answers
243 views

Black Scholes model

I am looking to program the Black Scholes model, but I don't understand how to write this equation as basic math. Equation => http://en.wikipedia.org/wiki/Black%E2%80%93Scholes Known inputs => ...
2
votes
2answers
2k views

Repayments of a loan with compound interest

Suppose I have a loan of M dollars. At the end of each year, I am charged interest at rate R and make a repayment of P. The loan is repaid after n years. How long (n) does it take to repay the loan ...
2
votes
1answer
334 views

Derivatives of Brownian motion or Box Options Greeks

Here's the probability (I think) that a particle in Brownian motion (w/ standard deviation $\sqrt{t}$) will exceed $m$ between times $t_1$ and $t_2$: $$\frac1{2\sqrt{2\pi}}\int_{-\infty }^m ...
2
votes
2answers
59 views

Actuarial Science FM Question

$$a(t)=Zt^2+Bt+1$$ If \$100 at $t=0$ grows to \$152 at $t=4$ and \$200 at $t=0$ grows to \$240 at $t=2$, what are $Z$ and $B$? Please show work. Also, what would \$1600 invested at $t=6$ grow to at ...