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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Financial mathematics problem. ( Deferred annuities ).

We need to calculate present value(value at t=0) of the payments of amount $1$ made at $t = m+1 , m+2 , ...... , m+n$ and no payments are made between $t=0$ to $t=m$ , effective rate of interest is ...
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0answers
28 views

Mathematical proof solving an accounting issue

For whom it may concern, Consider the following situation, a parent company "P" holds two subsidiaries (i) A1 which in turn has a subsidiary, A2, and a sub-subsidiary A3 (lets call these the A-chain) ...
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2answers
55 views

What does it mean when a letter has both superscript and subscript?

I have a formula for Bond Valuation of a Level Coupon Bond, but I don't understand the notation. It looks like: It's the bottom formula in the image below, starting with PV = What does it mean ...
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1answer
35 views

self financing strategy

how could one prove the following proposition from stochastic calculus applied to finance? Proposition : Let $\Phi$ a trading strategy. Then, $\Phi$ is self financing if and only if ...
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0answers
27 views

Conditional expectation in continous markov chains

I am trying to understand the double integral in calculating the conditional expectation. In calculating $V_i$, the second and third equalities are due to the law of total probability. I have the ...
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28 views

Mathematics of finance - reference - Exercise, Problem books for High schools [duplicate]

Please, could you give me any examples of books (Problem books with exercises) of Mathematics of finance for High schools? Thanks for any advice.
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2answers
30 views

Total Present Value of Multiple Cash Flows

I understand how to calculate the total accumulated and present values of multiple cash flows over n years, but I don't quite understand how this works when m of those n years aren't included. For ...
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1answer
45 views

Discount factor problem.

We need to state whether the given statement is true or false : $$ v(t_2) = v(t_1)v(t_2 - t_1)$$ where $v(t)$ is the discount factor. I found it ti be true as $v(t) = (1 - d)^{t}$ where $d$ is the ...
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1answer
26 views

Compounding cash, at a gambling game with certain payout and your certain win rate.

Say there is a gambling game that pays $p=.70$ of bet per win and your win rate is $w = .70$. What is an expression for your account balance $P$, given a starting balance of $P_0$ if you're betting ...
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1answer
71 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 $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...
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2answers
30 views

Where the constant comes from in the Compound Interest formula?

I want to understand where the 1 constant comes from in the Compound Interest formula. I'm a programmer, I can find a logical way to calculate it using a programming language, this is a way I can ...
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1answer
50 views

A 39-year annuity-immediate will pay 13 in each of the first 3 years…

A 39-year annuity-immediate will pay 13 in each of the first 3 years, 12 in each of the next 3 years, etc., until payments of 1 are made in each of the last 3 years. The present value of the payments ...
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0answers
27 views

What are the differences between large deviations theory & extreme value theory?

I need to study both for my Master's thesis in finance. (Probably, I'll have to apply them on the Value at Risk and Conditional Value at Risk estimation, so, on quantile estimation, loosely speaking; ...
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1answer
84 views

How is APR calculated?

I would be extremely grateful if someone could help with this problem. I'm planning a series of lessons on basic finance and wanted to brush up. I will not have to teach this but I like to keep a ...
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0answers
44 views

AR(1) process with exponential noise.

For the AR(1) process defined by $Z_t = aZ_{t-1} + \epsilon_t$, $\epsilon_t \sim Exp(\lambda)$, $a \in (0,1),\lambda >0$, compute $P(Z_t|Z_{t-1})$. I was only able to compute $E(Z_t|Z_{t-1}) = ...
3
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1answer
64 views

How does filtration model information?

Lets say you have a probability space $(\Omega, \mathcal{F},P)$ And a stochastic process on this space $\{X_t, t \in T\}.$ Assume that our process takes vaslues in $\mathbb{R}$. T is a totally ordered ...
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0answers
31 views

Proof that no futures trading system always wins

Hopefully someone here has some knowledge in both finance and maths. I am pondering on the existence/impossibility of a trading system (or algorithm) that ALWAYS ends up winning money, no matter how ...
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2answers
39 views

Calculating asset returns by portfolio weights

I have a securities portfolio (Fixed Income) and corresponding information regarding the single issues weights (in %) in the portfolio, sector identifiers (i.e. Industrial, Retail) and Total Return ...
2
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0answers
24 views

optimal derivative position through optimization

So I have the following optimization problem: min. $-E^Q[u(h(x))]$ s.t $\int h(x)q(x)dx \leq \frac{V_0}{B_0}$ Where $Q$ is the subjective probability which then gives: $E^Q[u(h(x))]=\int ...
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1answer
28 views

help with $\nabla$ and Lagrangian in optimization / portfolio theory?

So $\nabla$ as I know it from calculus means gradient. We have $\min \ \ \frac{1}{2}w^T\Sigma w$ $s.t. \ \ \ \ \ w^T1 = V_0, \ \ V_0 = 100$ where $w$ is weights in vector, $\Sigma$ is the ...
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0answers
18 views

Affect of projection matrix on the covariance matrix

Suppose I have a projection matrix $P$ which transforms a random vector $u$ to a new random vector $w $ with a mean of zero. such that the constraint applies $Xw = 0$ $P = I - ...
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1answer
27 views

Correlation and Covariance

The book I'm reading gives this as an example for lognormal variables. Starting at some fixed time, let $S(n)$ denote the price of a security at the end of $n$ additional weeks, $n \ge 1$. A popular ...
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1answer
60 views

An algebraic manipulation I can't get

I'm working with present value models. I don't get this passage: $$ R_t = \theta_t + \frac{1}{h}\sum_{j=0}^{h-1}\mathbb{E_t}r_{t+j} $$ Where $\mathbb{E_t}$ denotes the expectation conditioned on $t$, ...
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2answers
65 views

Financial Mathematics problem

$i^{(p)}$ is the nominal interest converted p-thly i.e the total interest per unit of time paid on a loan of amount 1 at time 0 where interest paid in p equal installments at the end of each p-th ...
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0answers
34 views

Feynman-Kac for dividend stream

Suppose an asset value process $V_t$ that solves the PDE $$dV_t=\mu V_tdt+\sigma V_tdW_t \text{ with }\mu\in\mathbb{R},\sigma>0, W \text{ Brownian Motion}.$$ I want to price a dividend stream ...
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1answer
24 views

What was the Shopkeepers Loss

Here is the problem : A lady buys goods worth 200 from a shop where the goods are sold non-profit.The lady gives the shopkeeper a 1000 rupee note . The shopkeeper gets the change from the next ...
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0answers
31 views

Proof of “Law of one price” multi period market

I'm struggling with the proof of the LOP. The task is the following: There are two self financing strategies $\psi$ and $\theta$ in a multi period Market $(S^0,S^1,...,S^d)$ and $V_T^\psi$ = ...
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0answers
42 views

Theory of Interest example on annuities (need some clarification)

I'm going through Theory of Interest by Kellison in preparation for actuary exam FM. I'm wading through chapter 3 on annuities right now and I'm completely confused on how this example from the ...
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0answers
29 views

Ito rule for a given ratio and exponential

Helo, I have trouble performing the following differentiation following Ito calculus $$d(e^Z/B)$$ Given that $Z_t$ is a logarithm of a certain process and follows $$dZ=mu_zdt+sigma_zdW$$ $$dB=rBdt$$ ...
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2answers
35 views

Max price of a share

Company is planning to pay a dividend of 5\$ per share (dividend for previous year). Investor that wants to buy a shares of this company assumes that dividend will be stable (Thus will not change in ...
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1answer
33 views

Compound Interest : Clarification Of The Given Solution To This Problem.

I was solving some problems on CI from my textbook, there's a problem whose solution is given in my book but I can't understand it. Here is the problem - A man borrowed a sum of money and agrees ...
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0answers
31 views

Black Scholes partial differential equation; Derivation

I have an exam tomorrow and the issue is, my notes just really briefly mentions it. It doesn't even take a full 2 pages to mention the partial differential equation. I haven't even seen it in ...
2
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2answers
41 views

Annuity and Loan Repayment Question. Show the amount of Loan.

A loan was taken out on 1 September 1998 and was repayable by the following scheme: The first repayment was made on 1 July 1999 and was £1000. Thereafter, repayments were made on 1 November 1999, 1 ...
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2answers
31 views

Nominal Rates/Effective rate computation, confusion.

Given a nominal rate of 6% per annum. Change it to an effective rate per month. What I do is: $$(1+\frac {0.06}{12})^{12}=(1+i)^{12}$$ where $i$ is the effective interest rate per month. Now what ...
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1answer
72 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 ...
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1answer
53 views

How to price a supershare option; expected value of a payoff function?

I thought I'd be able to do this but evidently not. Let $S_t=S_0e^{(r-\frac{\sigma^2}{2})t+\sigma W_t}$ for all $t$. $W_t$ is a standard brownian motion. We have the following function for payoff ...
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1answer
157 views

Confused with “nominal” “convertible” rates; How to calculate a rate for an $n^{th}$ of a year?

The terms and how they're calculated is very unclear to me. My understanding of "nominal" is that this is a rate which isn't in unit time. i.e. $5\%$ per annum "is" in unit time (year) but $5\%$ ...
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0answers
23 views

Log-normally distributed variable; finding the distribution, mean and variable

The question here might be "when do you know it's best to use moment (generating) functions, when do you know you should integrate"? Basically, sometimes, we use "moment functions" and compare the ...
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2answers
50 views

Loan calculation; what went wrong?

I am attempting a question given as follows A loan is payable over 20 years by level installments of $\$1000$ per annum made annually in arrear. Interest is charges at $5\%$ per annum effective ...
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2answers
44 views

Annuity calculation; what is wrong with my calculation to the following question?

First off, I will be honest that I am rather confused not to much with the concepts but more of the language used in questions in finance. So I must acknowledge the possibility that I have ...
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1answer
52 views

Application of Integration on Investments

A small business expects an income stream of $\$300$ per month for a period of $9$ years. The income will be invested at an annual interest rate of $17\%$, compounded continuously. How much interest ...
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1answer
23 views

annuity certain, financial mathematics.

A 20 year annuity certain provides payments of 200 at time 1 year, 180 at time 2 years , 160 at time 3 years, and so on until the payments have been reduced to $ 60. Payments then continue at 60 ...
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1answer
71 views

Actuarial : “ Amortization - mortage”

What is the monthly payment for a $800,000 mortgage for the first 119 payments that is due in 10 years, has a 25 year amortization, at 5% interest? What is the amount of the 120th payment? I ...
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1answer
28 views

Present Value of an annuity payable with $n^2$ at $t=n$

Find the Present Value of an annuity payable with $n^2$ at $t=n$ , $t\in [0,n]$ What I have is: PV=present value $PV=1u+2^2u^2+3^2u^3+\cdots+n^2u^n$ I don't seem to know how to simplify it to: ...
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1answer
28 views

Have I understood the question properly? Annuities in Actuarial math

I am wondering if I have interpreted the language correctly in the following question The force of interest at time $t$ is given by $\delta(t) = 0.05-0.005t$ for $\leq t < 5$ and ...
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2answers
34 views

Investment in a treasury bill

You invested 968 710 in a treasury bill with the face value of 1 000 000 with 91 days left till maturity. After 60 days you have the option to sell it for 989 250. Which option is more profitable? My ...
1
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0answers
41 views

Black Scholes derivation; How and Why

A 15 mark past paper question essentially ask s me to derive the Black Scholes formula for pricing options. Let $S_t=S_0e^{(r-\frac{\sigma^2}{2})t+\sigma B_t}$ where $B_t$ is a standard Brownian ...
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1answer
15 views

Does anyone know about Forced rate of interest and present value $v(t)$? Because it's confusing!

my notes doesn't explain these concepts well and I am very stuck at this example solution. I don't get where the numbers come from at all. Suppose that the forced rate of interest is $\delta(t)$ ...
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1answer
18 views

How to determine loan payment and total price of a loan

Let's say, that we have borrowed a $100,000 for a 5 years, with 6% p.a. interest rate. How can one determine the value of a loan payment, if we are making payments every quarter and at the beginning ...
2
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1answer
43 views

How to integrate the following geometric brownian motion in Black-Scholes framework

As my previous questions make it obvious, I am very new to this field of mathematics and wondering if I am doing things right in the following question. Let $T \in (0, \infty)$ and consider a ...