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Questions tagged [finance]

Questions related to the various aspects of financial mathematics. Topics include option pricing, arbitrage theory, market completeness and stochastic analysis.

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Finding duration of given payment

Question :A corporation bond with annual coupon rate 7,5% will mature at 30 june 2025.Find duration of the bond on 31 december 2023 given that the annual interest rate is 5,5%.Assume the par value is ...
user1259172's user avatar
1 vote
0 answers
19 views

How to Compute the Derivative of Maximum Drawdown (MDD) with Respect to Portfolio Weights

I am working on a financial model and need to calculate the derivative of Maximum Drawdown (MDD) with respect to the portfolio weight vector (assume we fix it at the beginning and keep unchanged). The ...
ElonMuskofBadIdeas's user avatar
1 vote
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Is there a notion of approximation of continuous-time Markov processes by finite-valued Markov processes?

Recall that in practice, to simulate a Brownian motion on $[0,1]$, we usually use the interpolated process $X^n=(X^n_t)_{t\in[0,1]}$ between the jumps of a random walk $(S_k)_{k=1,...,n}$ with $n$-...
Jeffrey Jao's user avatar
1 vote
1 answer
56 views

How to solve for monthly interest rate given principal, number of payments, and total payment

The problem: A lottery winner is given two payment options: Receive 131 million dollars in 25 yearly installments of equal size, the first payable immediately, or receive a single immediate payment of ...
ERROR 404's user avatar
3 votes
1 answer
43 views

Analytical function from summation representing Dollar Cost Averaging

I would like to understand what is the math necessary to go from a summation representing the Dollar Cost Averaging (DCA) to the analytical function. In DCA, a certain capital $C_0$ is invested ...
Marco Cappelletti's user avatar
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33 views

No Arbitrage iff no generalized Arbitrage

Let’s consider a market in finite discrete time with trading dates $0,1,\dots,T$ probability space $(\Omega, \mathcal{F}, \mathbb{P})$, filtration $\{\mathcal{F_t}\}_{t \in \{0,1, \dots, T\}}$, $N$ ...
Henry T.'s user avatar
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1 answer
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Should I consider the price for "option pricing" problem?

I'm trying to solve the following problem from "Probability and Statistics" book by Morris H. DeGroot and Mark J. Schervish. Suppose that common stock in the up-and-coming company A is ...
Claptar's user avatar
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How to formally justify fudge factor in this difference equation solution?

In Exercise $11$ from Section $3.3$ of Differential Equations With Boundary Value Problems by Polking, Boggess, and Arnold, we first develop the difference equation $P[n + 1] = (1 + \frac{I}{m})P[n],\ ...
user10478's user avatar
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2 answers
54 views

What formula would I use to calculate total income over $x$ years with a fixed salary increase rate.

I'm wanting to calculate how much total money I would have earned over x amount of years, but while accounting for increase in pay at the end of each year (or maybe even continuous?) Lets say for ...
Apples71's user avatar
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Financial based equation for Cash Flow analysis on Annuity ( Sequences and Series Analysis)

I have a particular Annuity type of question. I am familiar with all the Cash Flow type of equations such as Present Value, Future Value for compound interest, and for Annuities and Mortgages etc. ...
Palu's user avatar
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For $A_t= B_t^l Y_t \mathbb{E}(\frac{A_T}{B_T^l Y_T} \mid \mathcal{F}_t)$, find the values of $l$ to replicate $A_t$ by a self financing portfolio $X$

Background: In attempting to resolve the below problem, I have arrived at an answer that appears to counter intuition (and therefore, I suspect that it is wrong). I would appreciate assistance in ...
FD_bfa's user avatar
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1 vote
3 answers
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Why does total salary differ over 5 years if the average salary increase averages to the same?

I'm considering 2 general scenarios of both having an average salary increase of 6% over 5 years. But in the first scenario, the person ends up with a lower total salary compared to the second ...
sheavictoria's user avatar
0 votes
1 answer
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Analyzing Expected Profit in a Symmetric Random Walk with Trading Actions

Problem Formalization: I am examining a problem where a stock price $X_t$ follows a symmetric random walk starting at 10, and increments or decrements by 1 unit at each step with equal likelihood. The ...
XiaoBanni's user avatar
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1 answer
24 views

Proof that annual nominal interest rate, convertible $m$ times in a year, is a decreasing function

I'm reading the following notes on financial mathematics (they are in italian). In here the annual nominal interest rate, convertible $m$ times in a year, is defined as (page 32) $$j(m)=m i_{1/m}$$ it ...
robertspierre's user avatar
4 votes
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Question regarding a value in a one-period model

There is a script at my university (can't post it for copyright reasons) for a course on discrete time financial mathematics. I decided to give it a try and found this problem: Consider a financial ...
ryen.xain's user avatar
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simulation with transition probability

I would like to simulate a CIR process defined by a probability transition enter image description here . However, I'm stuck. For example, how should I conduct the first simulation? I have experience ...
Ncr's user avatar
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1 answer
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Confusion about price at maturity of a call option allowing for arbitrage

I have been reading the book Tomas Bjork's Arbitrage Theory in Continuous Time and could not understand how there could be arbitrage if the price of a contingent claim is not $X$. To give some ...
KMR's user avatar
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1 vote
1 answer
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Trading model - apply fee at specific time point

I am currently developing an energy trading model, where I look a few hours ahead of the current time. This model is runned for several time points (but discretized into hours), namely for $\tau \in T=...
osi41's user avatar
  • 99
4 votes
1 answer
109 views

Showing a basic market admits no arbitrage

Setting We work in $\left(\Omega, \mathcal{F},\left(\mathcal{F}_t\right)_{t=0}^1, \mathbb{P}\right)$. Let $d=1, T=1$ and assume the discounted price equals the non-discounted price. Take $S_0^1 \in \...
portero's user avatar
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1 vote
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ARCH-Vasicek model closed-form solution

I understand how we can obtain the solution of Vasicek model $dr_t=\alpha(\mu-r_t)dt+\sigma dW_t$: $$ r_t=r_0e^{-\alpha t}+\mu(1-e^{-\alpha t})+\sigma\int_0^te^{-\alpha(t-s)dW_{s}} $$ This easily ...
KiNest's user avatar
  • 11
2 votes
0 answers
60 views

Deriving the CAPM pricing kernel from the general SDF and consumption-based kernel

I'm reading the paper "Quality minus junk" by Asness et al. (2019) and trying to understand the pricing kernel definition they provide on page 6. The authors present the following pricing ...
Newbie's user avatar
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Boundary hitting and arbitrage in Reflecting Brownian Motion approximation

I am facing the following quantitative finance problem. Suppose that $z$ follows a Brownian Motion centered at $0$. There are two boundaries that define a no-arbitrage region: $[-b,b]$. Trades take ...
mkb90's user avatar
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1 vote
0 answers
38 views

Hedging a long position, multiple periods (Steven E. Shreve, Stochastic Calculus for Finance I)

I have attempted to answer this question, but I'm unsure if I'm on the right track. I've started with setting the value of the portfolio at time 3 to the desired value: $$ X_3(HHH) = X_3(HHT) = X_3(...
Alireza Azimi's user avatar
0 votes
1 answer
29 views

A concrete example with Arrow-Pratt coefficient of absolute risk aversion

Let $u_1$ and $g$ be increasing strictly concave functions from $\mathbb{R}$ to $\mathbb{R}$. Let $u_2:=g\circ u_1$. If we regard $u_1$ and $u_2$ as utility functions of two players, this is saying ...
No-one's user avatar
  • 667
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0 answers
50 views

While we are optimising the Sortino ratio, whether this objective function is convex, concave, or neither.

I am going to formulate an optimization problem for finding a feasible portfolio with the largest value of Sortino ratio. It will defined as follows: \begin{align*} \max_{x} \quad & STR(x) = \frac{...
asfsafasf's user avatar
3 votes
1 answer
72 views

Characteristic function of a random variable by Fourier transform

this is character function in probability theory $$\phi(u)=\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}ux}f(x)\mathrm{d}x$$ Let an asset price $S_t$ (e.g. a stock) be modeled with a Geometric ...
Yehui He's user avatar
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1 answer
28 views

Find conditional probability that stock return will exceed some threshold value

Suppose we have some financial data, e.g., stock return time series. The theoretical distribution is unknown, while we can construct the empirical distribution through historical data. The problem is ...
Sane's user avatar
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Find the trivial interval for fair prices

Suppose there is a risk-free asset with $ r^0 = 5 $ and $2$ risky assets with $E r^1 = 8, \sigma^1 = 5, E r^2 = 20, \sigma^2 = 20, \text{corr}(r^1, r^2) = 0 $. What is the maximum average profit and ...
WOL - THE WORLD OF LESSONS's user avatar
0 votes
1 answer
62 views

derivate discounted payoff

Let $S(t)$ be the stock price at time $t\geq 0$ with $S(0)=s_0$ and let $\Pi(S_t)=\max\{K-S_t,0\}=(K-S_t)_+$ the payoff of an american put with strike price $K$. How can I calculate the derivate $\...
Robert's user avatar
  • 386
3 votes
1 answer
80 views

Obtaining a tighter lower bound on an American call with two discrete dividends

Question Suppose a stock pays 2 discrete dividends $d_1, d_2$ at times $t_1, t_2$ respectively, where $ t < t_1 < t_2 < T.$ Assume the risk-free rate, $r$, is a positive constant. Given that ...
Hmmmmm's user avatar
  • 333
1 vote
1 answer
99 views

Seeking clarity on the concept of equivalent martingale measures

Question Consider a one-step trinomial tree, where there are two traded assets, a bond with risk-free rate, $r$, a stock with initial price, $S_0$, and terminal price $$S_T = \begin{cases} S_0u,& ...
Hmmmmm's user avatar
  • 333
0 votes
1 answer
67 views

Continuous discounted cash flow generation

Problem statement The problem is to find the limit of the following sequence: \begin{equation} D(r) = \lim_{n \to \infty} \frac{\sum_{i=1}^{n} \frac{1}{(1+\frac{r}{n})^i}}{n}. \end{equation} ...
petr jilek's user avatar
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0 answers
49 views

How to estimate interest rate cut from futures contracts?

I want to calculate the expected rate cut/change given futures contracts on the basic interest rate, knowing when the rate cut occurs (resembling the FED's FOMC). Consider compound rates and a 252 ...
ffsffs's user avatar
  • 1
1 vote
1 answer
63 views

Conditional distribution of Brownian motion given the passage time?

I am stuck with this question. Suppose a stock price follow Brownian motion starting at price $p$. Denote by $\tau_r$ the passage time reaching level $r$ with $r<p$. What would be the distribution ...
FARRAF's user avatar
  • 197
2 votes
1 answer
88 views

Heaviside under Geometric Brownian Motion

I'm new to using Geometric Brownian Motion, so I'm not sure if what I've done is correct. Be the Geometric Brownian Motion $dS_t = \mu S_tdt + \sigma S_t dW_t$, $H$ a Heaviside, and $p_r, r_k$ ...
Luca Herrtti's user avatar
-1 votes
1 answer
94 views

The partial derivative of a call option with respect to $t$ [closed]

In Black-Scholes related computations, why do we not treat the stock price $S$ as a function of $t$ when taking partial derivatives with respect to $t$? For example, if $$c(t,T)=SN(d_1)-Ke^{-r(T-t)}N(...
Not_a_topologist's user avatar
1 vote
0 answers
46 views

Time Weighted Decay

I've been tasked to calculate/forecast the weighted exposure of a financial product. I work with bunker prices and we have access to bunker future prices everyday. They look similar to this in an ...
zacchhh's user avatar
  • 11
0 votes
0 answers
19 views

How to calculate holding period return of a long-short strategy?

I have daily close prices of two stocks, A and B. Suppose that we long stock A and short stock B. Assume that we do the long-short every day and hold that portfolio for some days. How to calculate ...
user546106's user avatar
-1 votes
1 answer
28 views

Finding the effective annual rate of interest (no constant)

A deposit of 10,000 is done. During first year, the bank credits an annual effective interest rate of $\text {i}$. During the second year, the bank credits an annual effective interest rate of $\text {...
Roma_Rayado's user avatar
3 votes
1 answer
147 views

Why are there different definitions of admissibility in the literature, and why do we need admissibility?

Wikipedia essentially defines an admissible trading strategy as a stochastic process $H = (H_t)_{t\geq 0}$ such that the associated value process $\int H(u) d S(u)$ is lower bounded. As I understand ...
xy z's user avatar
  • 135
0 votes
0 answers
111 views

Is this the correct proof of Proposition 10.23 in Björk's ''Arbitrage theory in continuous time''?

I want to prove Proposition 10.23 from Tomas Björk's ''Arbitrage theory in continuous time'' in the snippet below. My attempt: For simplicity, assume everything is one-dimensional, with one risky ...
xy z's user avatar
  • 135
0 votes
1 answer
50 views

How do I approach this proof to show that P (T, K) is non-decreasing function of T without making any model assumption

Let P (T, K) be the price of a Put option with maturity T and strike K, and assume that the interest rate is zero, i.e., r = 0. By no-arbitrage pricing rule, show that P (T, K) is non-decreasing ...
Statnerd's user avatar
0 votes
1 answer
40 views

How to compute a contract price from an original price by adjusting the taxable total while accounting for the tax being part of the contract price

I'm not sure how to best describe this in the title, so I will do my best here: Let's say we have an order that prices out to $48,967.00 - but we have a few categories that make this up: taxes, ...
Brandon Ragland's user avatar
1 vote
0 answers
78 views

How to self-learn probabilities [closed]

Bit of background: I’m 27, graduated 4 years ago with a bachelors in Computer Science in which I did well. Since I graduated, I’ve been working as an algo trader for a bank. I’d like to start applying ...
IGottaLearnMath's user avatar
3 votes
1 answer
107 views

Functional Analytical definition of no arbitrage

Let $ {(S_t)}_{t\in[0,+\infty[} $ be a semimartingale and ${(x_t)}_{t \in[0,+\infty[}$ an admissible strategy. We denote by $(x.S)_{+\infty}=\lim \int_{0}^{t} x_u dS_u$ if such limit exists, and by $...
Pedro Gomes's user avatar
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0 votes
0 answers
33 views

Approximating the half-life of a shock to a system?

I found the following statement in here regarding the effect of twice lagged differences of CO2 ($\Delta C$) in the atmosphere on the once lagged values, i.e. $$\Delta C_{\text{ @ }t=-1}= 0.83 \times ...
JAP's user avatar
  • 609
1 vote
1 answer
51 views

Interpretation of Value at Risk and its relationship with the upper percentile

I found this definition of VaR (Value at Risk) in a paper: VaR is defined as the “possible maximum loss over a given holding period within a fixed confidence level”. That is, mathematically, VaR at ...
Kolmogorovwannabe's user avatar
0 votes
0 answers
10 views

Dummy variable with multiple criteria - how to

Dummy variable is simple when it is true or false. What happens if the 'true' has multiple criteria that needs to be met? For example, there are 4 criteria in total. And 3 out of 4 must be true for ...
MLux's user avatar
  • 1
2 votes
2 answers
134 views

An exercise about replicable random variables

The following question is an exercise which I have in my course for Financial Mathematics: Let $h:[0,\infty) \to [0,\infty)$ be twice differentiable with $h'' \geq 0$. Establish, with the help of ...
user1265841's user avatar
2 votes
0 answers
29 views

Characterization of Optimal Payoff (under Expected Utility) via Gateaux-Derivative/Fréchet Derivative

Background: Let $(\Omega, \mathcal{F}, \mathbb{P})$ model a financial market and $T>0$. Denote by $(S_t)_{t\in[0,T]}$ the price process of the risky asset in the financial market. Assume that the ...
MWilk's user avatar
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