Questions tagged [finance]

Questions having to do with financial mathematics. This is not a tag about financing, which is not within the scope of mathematics defined by the help center: http://math.stackexchange.com/help/on-topic Please note that for questions in quantitative finance, quant.stackexchange.com is perhaps a better site.

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20 views

Analytical expression for the optimal trading fraction when returns are skew t distributed

Thorp's paper here details the derivation of a continuous approximation of the Kelly formula when returns are normally distributed. Let X be a random variable with $P(X = m + s) = P(X = m − s) = 0.5$. ...
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1answer
36 views

How to get the weight between two stocks in a portfolio?

So in this exercise I need to get the weight of proportion between two stocks, the problem says the following: An investor has a certain amount of dollars to invest into two stocks (...
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24 views

Formulas for determining maximized efficiency when allocating resources [closed]

Update - this assumes fragmented shares are not available as an option. Given an investor had a random finite amount of capital to invest and a series of potential investments with fixed returns, how ...
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23 views

Interest is increasing for remaining terms [closed]

Any one can help me with this problem? i) A loan is repayable over 20 years by level installments of $£1,000$ per annum made annually in arrear. Interest is charged at the rate of $5$% per annum ...
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14 views

Pricing of $(S(T_0)-S(T))^+$

Problem: Consider a new derivative that at time $T$ pays $Y =(S(T_0) − S(T))^+$ where $0 < T_0 < T$ is a fixed date. (i) Show that the arbitrage-free of Y at time $t = T_0$ is given by $\pi_{...
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1answer
21 views

How to restructure Compound interest formula (with regular contributions) to solve for the periodic payment amount

I need help restructuring this formula to solve for the payment PMT rather than the Total: $$ \text{Total} = \text{Compound ...
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24 views

Calculating profit percentage and profit absolute value

The question is about calculating profit percentage and profit abs which is related to crypto trading. I'm using Binance as a crypto trading platform and the following examples are based on TRX/USDT. ...
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2answers
49 views

Financial math- Calculating Perpetuity problem

A investor is hesitating between two projects. The first will yield steady returns of $X$ every $6$ months for the first $10$ years and $X$ every year after. The second will return $500$ per month for ...
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43 views

Loan and Annuity problem - finding X

I am new in finance math, and I am working on this problem. I was wondering if someone could help me solve it. A loan of $10,000$ is to be repaid with annual payments, at the end of each year, for the ...
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1answer
44 views

What is the difference between effective, real and nominal interest rate?

I have been somewhat confused in financial math when to change the rates and the difference between effective, real and nominal rates. I still am confused after searching on Google and asking the ...
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68 views

Having deposited $\$5,000$ at $9\%$ annual interest, for how many years can you withdraw $\$500$? What's a good way to iterate this?

NOTES: I hope the tags are appropriate, this is a homework problem EDIT (another note): if it makes a difference, this question may involve calculus since it's in my calculus homework... I doubt it ...
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1answer
35 views

differentiating covariance

$$\sum_{i=1}^n x_{i}^2C_{ii} + \sum_{i=1}^n \sum_{j=1 \atop{j\neq i}}^n x_{i}x_{j}C_{ij}$$ I am trying to differentiate the above expression with respect to $x_i$. I did partial differentiation to get ...
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1answer
18 views

Is there a formula of calculating the significance of an increase of a value relative to its size?

This is an idea I had about volume in stock trading (amount of shares traded over a given time interval) If a stock volume at some point is 5 and the next time unit it's 10, that represents a 100% ...
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25 views

Force of interest piecewise function to find nominal yearly interest

I have been given a question that asks to calculate the nominal yearly interest rate from the force of interest of the following piecewise function: I initially approached this question by doing the ...
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1answer
54 views

Connection between semimartingales in Itô calculus and measures in integration theory

I'm currently learning about stochastic calculus for the first time. I have taken a first course in Real Analysis covering Lebesgue integration. According to Wikipedia, we write out the Itô stochastic ...
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30 views

Price of dividend paying stock

Consider a dividend paying stock $S$ and suppose that its value just before a dividend payment of $D >0$ at time $t$ is denoted by $S(t−)$. The price after the payment should be $S(t)=S(t-)-D$? How ...
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21 views

Is there a formula to calculate future value with compound interest where only part of the interest of each period is applied?

Based on FV=C0*(1+r)^n Where C0 = Cash flow at the initial point (Present value) r = Rate of return n = number of periods Is there a way to adapt this formula to (...
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29 views

Question on two equivalent densities

I have two integrals $I_1$ and $I_2$ that are almost similar : $$I_1=\int_K^{+\infty}(x-K)f_1(x)dx$$ $$I_2=\int_K^{+\infty}(x-K)f_2(x)dx$$ with$f_1$ and $f_2$ being two equivalent densities (so they ...
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How do I convert from a semi-annual interest rate to montly?

On a practice exam I am given a $.06$ compounded semi-annually and need it converted to monthly. I assumed it would be $.005$ because it has annual percentage of $.06/12$ but my answer is slightly off ...
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1answer
40 views

How can an annuity pay you so much money that, even if you live to 100, you can outlive your assets? [closed]

Kindly see the emboldened sentence below. The last time I took math was when I was 16.! I Googled annuity and found mathematical formulas. But how do I use them to prove the emboldened sentence below? ...
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What's “the incremental fee for active management as a percentage of the incremental risk-adjusted returns above the market index”?

Kindly see the emboldened sentence below. It contains too many technical lingo. Can you unpack and simplify it please?       But even this recalculation substantially understates the real cost of ...
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1answer
30 views

How to price a contract that is denominated in another currency using the Martingale Approach to the Black and Scholes theory?

I am taking a course in asset pricing and I have the following problem at hand: Suppose that the level of the UK FTSE100 index (in British pounds) evolves according to $$\frac{\mathrm{d}S_t}{S_t}=\...
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1answer
57 views

Why does calculating the return on a portfolio differ when calculated at a stock level rather than at a portfolio level?

I have a portfolio of three assets A, B and C, each with a beginning value of 100. I trade the portfolio over two days. On day 1, asset A returns 50, B returns 20, and C returns -10. On day two, A ...
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1answer
68 views

Show that $Y_t \int_0^t \frac{1}{Y_s} ds$ and $\int_0^t Y_s ds$ are equal in distribution

Let $B=(B_t)_{t \geq0}$ be a standard Brownian motion and let the two stochastic processes $X=(X_t)_{t \geq0}$ and $Y=(Y_t)_{t \geq0}$ solve the following two stochastic differential equations $$dX_t =...
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106 views

Forward Price of a Stock with and without dividend

I have some questions about the forward price calculation. These two examples both use special compounding from t to T and no need to use continuous compounding. I hope to understand this concept. ...
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1answer
42 views

Show that $Z_t = \sqrt{\sqrt{2t}e^{-\sqrt{2t}}} B_{e^{\sqrt{2t}}-1}$ solves $dZ_t = f(t)Z_tdt +dM_t$ and find $f(t)$.

Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion started at $0$ and define $M=(M_t)_{t\geq0}$ by $$M_t = \int_0^{e^{\sqrt{2t}}-1} \sqrt{\frac{\log{(1+s)}}{1+s}} dB_s$$ so that M is also a standard ...
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76 views

Apply Ito's formula, prove a martingale, and calculate $\mathbb{E}(\tau)$

Let $B = (B_t)_{t≥0}$ be a standard Brownian motion started at zero, and let $X = (X_t)_{t≥0}$ be a non-negative stochastic process solving $dX_t = 7dt+ 2\sqrt{X_t}dB_t$ with $X_0 = 0$, and let $F(t,x)...
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1answer
52 views

Let $B = (B_t)_{t \geq 0}$ be a standard brownian motion and $I_t = \inf_{0 \leq s \leq t} B_s$, show $F(t, B_t, I_t)$ is a martingale.

Let $B = (B_t)_{t \geq 0}$ be a standard brownian motion and $I_t = \inf_{0 \leq s \leq t} B_s$, show that $F(t, B_t, I_t) = (B_t−I_t)^6−15t(B_t−I_t)^4+45t^2(B_t−I_t)^2−15t^3$ is a martingale. I'm ...
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38 views

Apply Ito’s formula to $F(X_t,Y_t,Z_t)$, where $Z_t = X_t Y_t$.

Let $(X)_{t \geq 0}$ and $(Y)_{t \geq 0}$ be continuous semimartingales with values in $\mathbb{R}$. Apply Ito’s formula to $F(X_t,Y_t,Z_t)$, where $Z_t = X_t Y_t$ and $F$ is a $C^2$ function. ...
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18 views

Valuing Bonds With Continuous Coupon Yields

How do I find the value of bonds with continuous coupon yields and interest rates that are both a function of time?
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69 views

A good textbook to teach Financial Mathematics

I recently was told that I will need to teach the University course Financial Mathematics 2, as a replacement of retired person. This is urgent, and I do not have time to write good lecture notes. So, ...
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1answer
45 views

Payoff from an option contract

In period 1 the consumer of type $\theta$ selects an option contract consisting of an up-front fee, $B>0$, and exercise price, $\bar{R}$. The consumer pays $B$ at the end of the first period. In ...
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12 views

Interest Expense Optimization (Linear Programming)

So I have a problem I need to solve and no idea how to approach it. Its a verbal problem without any specific numbers given except for those below. So it is up to me to determine how to structure the ...
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1answer
25 views

Geometric expectation & perpetuity discount rate

Two observations: If $X \sim \text{Geom}(p)$ then $E(X) = \frac{1}{p}$ (using the "first success" definition, i.e., X is supported on the set $\{1, 2, 3, ...\}$). The present value of a ...
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1answer
11 views

Revenue Equivalence in Auction Theory: how does an English auction generate the same revenue as First Price Sealed Bid?

The theroem: The revenue equivalence theorem states that, if all bidders are risk-neutral bidder and have independent private value for the auctioned items, then all four of the standard single unit ...
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16 views

Interpreting candlestick charts for edge case in stock market

Per my understanding: The close price is the last price traded during the period of the candle formation. If the close price is below the open price the candle will turn red as a default in most ...
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1answer
30 views

(Wikipedia Bug?) Equilibrium in natural ordering of Auction prices

According to wikipedia: Natural ordering Order the buyers in decreasing order of their bid: b1≥b2≥...≥bn. Order the sellers in increasing order of their bid: s1≤s2≤...≤sn. Let k be the largest index ...
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1answer
35 views

Vickrey vs English Auction Equivalence

Wikipedia states: When the auction involves a single item for sale and each participant has as an independent private value for the item auctioned, the expected payment and expected revenues of an ...
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1answer
23 views

Equilibrium Price in a Double Auction

I am trying to understand how to find the equilibrium price in a double auction. From wikipedia: Natural ordering Order the buyers in decreasing order of their bid: b1≥b2≥...≥bn. Order the sellers ...
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1answer
40 views

How do I calculate marginal cost?

Link to excel Marginal calculation I want to know if I did the calculations for the terms mentioned in the question correctly. I am trying to see how much the organization saves in costs if the ...
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19 views

if $h^T D \psi = 0 \forall h \in \mathbb{R}^N$, then what can we say about $\psi$ and $D$?

$D$ is a $N \times M$ matrix. $h \in \mathbb{R}^N$ and $\psi \in \mathbb{R}^M$ Would this mean that $\psi$ is in the Nullspace of $D^T$. because irrespective of h, $D$ is transforming these to $0$; ...
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57 views

Derivation of the Black–Scholes equation

I am studying about Derivation of the Black–Scholes formula. I have two questions. First one: please check my writing if it's correct, and the second one: Is there another method to obtain the Black–...
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14 views

Construct a zero coupon bond

Suppose a 3% 10-year bond is trading at 89 and a 7% 10-year bond is trading at 97. Then (assuming no arbitrage) the price of a 10-year zero-coupon bond would be: The answer should be 83. How using ...
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1answer
25 views

find the interest rate in the form of compound interest with recurring contribution

I have the following formula to find the final value of compound interest with recurring contribution: ...
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1answer
47 views

Maximizing compound interest with fee per compound

Given the following: initial principle $P$, interest rate $r$, flat fee whenever interest is applied $x$, period between applying interest $p$, total time period $t$ How do I calculate the final ...
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42 views

Does “Every derivative security can be hedged” imply that the “filration is generated by the underlying Brownian motions”?

The market model is as follows: $$dS_i(t)=S_i(t)\alpha_i(t)dt+\sum_{i=0}^{n-1}S_i(t)\sigma_{i,j}(t)dW_j(t) \ \ \forall \ i = 0,...,n-1 $$ Where each of $\alpha_i,\sigma_{i,j}$ are adapted to the ...
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1answer
47 views

Showing whether $\tau=\inf\{t:X_t=\sup_{0\leq t\leq T}X_t\}$ is a stopping time or not

As the title suggests, I am not sure how to show if $\tau=\inf\{t:X_t=\sup_{0\leq t\leq T}X_t\}$ is a stopping time or not, where $\{X_t\}$ is an adapted process. Intuitively, I see that it is not a ...
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32 views

Why are these two definitions of a trading strategies as discrete processes equivalent?

This is my first time asking a question here, so please tell me if there is something I should ask differently. We are in the setting of $(\mathbb R^n, \mathcal F, \mathbb Q)$ being a probability ...
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1answer
22 views

Find an expression for $n$ in terms of $x$, $y$, and $z$.

At a certain rate of compound interest, $100$ will increase to $200$ in $x$ years, $200$ will increase to $300$ in $y$ years, and $300$ will increase to $1,500$ in $z$ years. If $600$ will increase to ...
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1answer
61 views

Interpretation of Value at Risk

Let $X$ be a Loss random variable (Positive values of X represents Losses) and let $p \in (0,1)$. I know that the Value at Risk at level $p$ of $X$ is defined as: $$VaR_p(X) = inf{\{x \in \mathbb{R} : ...

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