Questions tagged [finance]
Questions related to the various aspects of financial mathematics. Topics include option pricing, arbitrage theory, market completeness and stochastic analysis.
2,546
questions
0
votes
2
answers
27
views
Calculating the exact inverse of a transaction fee with a linear volume-based fee structure
Let's say we have a volume-based fee structure that takes a 5% fee of a transaction from \$0, and a 2% fee at \$1000, with the intermediate amount being a linear interpolation of these two points - ...
- 1
1
vote
0
answers
28
views
On the operational process of fractional and delay Brownian motions (FGBM/GDBM) governing respective market scenarios
I have some knowledge about the fabrication of a stochastic differential equation (SDE) governing asset price ($S(t)$) dynamics (This answer helped me up to some extend).
For instance, the Geometric ...
- 2,283
-4
votes
0
answers
23
views
How to calculate Excel's Yield function for coupons more than 1? [closed]
Below is the actual process for calculating the Bond yield for coupons > 1. Reference : YIELD
If there is more than one coupon period until redemption, YIELD is
calculated through a hundred ...
0
votes
0
answers
9
views
Testing predictability of a predictor of expected weekely returns on the stock market
say I have a T daily observations for the last ten years on a new predictor $x_t$ which I think is a predictor of the expected weekly return on the stock market, $r_{t,t+5} = r_{t+1}+...+r_{t+5}$, ...
- 151
0
votes
2
answers
40
views
What is the formula for cumulative compound interest? [closed]
I would like to start with a principal amount (P) in year 0, then add compound interest (C) to it for year 1, and then add that total value to the starting amount. So for example:
P=1000
C=2.5%
For ...
- 11
0
votes
1
answer
23
views
Partial derivative of a matrix product w.r.t. a vector
What is the partial derivative of
$$
w^T\beta \cdot \Sigma \cdot \beta^Tw
$$
with respect to $w$?
Dimentions:
$w$ is a $N \times 1$ vector ("asset weights")
$\beta$ is a $N \times K$ matrix ...
- 133
-1
votes
0
answers
30
views
Valuing Call Option Expiring Tomorrow given final stock prices
Here is the original question:
At the end of the day, a stock will be 100 with prob. 0.6 or 50 with prob. 0.4. What is the stock trading for right now?
Value an ATM Euro call option expiring at the ...
- 151
0
votes
1
answer
97
views
Martingale property of optimal control
I am trying to solve Exercise 25.4 of Tomas Björk's Arbitrage Theory in Continuous Time.
The exercise goes as follows:
Consider the problem of minimizing
$$
\mathbb{E}\left[\int_0^T F(t, X_t^u, u_t)dt ...
0
votes
0
answers
34
views
How to approximate a function in the H-model
I have been looking to understand the H-model in finance, that is used for stock price valuation. In particular, I wanted to formally derive the final formula:
$$PV=\frac{D}{r-g_2}\left[1+g_2+\frac{H}{...
- 145
0
votes
0
answers
21
views
Looking for an equation to separate Compound interest from a given amount [duplicate]
Maths newbie here.
I am trying to make a formula to discern the monthly payments required to achieve a target amount over a given time with a given compounding interest rate.
The information I have is ...
- 101
0
votes
0
answers
27
views
Properties of an American Derivative Security Process
$$
\newcommand{\cbkt}[1]{\left\{{#1}\right\}}
\newcommand{\rbkt}[1]{\left({#1}\right)}
\newcommand{\sqbkt}[1]{\left[{#1}\right]}
$$
I am self-learning basic stochastic calculus using Shreve's books. ...
- 5,042
0
votes
0
answers
13
views
Does the Inf over t of PDE initial value problem solutions leave any clue about the Initial Value
Assume you have an initial value $\varphi(x)=\vartheta(0,x)$ of a PDE of two variables $t,x$. let the solutions of the PDE be $\vartheta(t,x)$. Let there be a continuous function $\hat{t}:\mathbb{R}_+ ...
1
vote
1
answer
22
views
Calculating the present value of a payment after n days (APY)
I am currently trying to fully understand the APY concept in finance (I know very basic). I would be very greatful, if you could help me with my problem.
The following example is given:
The current 1 ...
- 13
0
votes
0
answers
13
views
Find the spread of an asset swap spread
An Asset Swap Spread contract exchanges the annual defaultable coupons computed on the defaultable term structure $SPS^1$:
$$
SPS^1 = {i^1(0,1) = 0.025; i^1(0,2) = 0.03; i^1(0,3)=0.018}
$$
versus s ...
- 573
-1
votes
0
answers
15
views
Annual Growth Rate for Ratio Values between 0 and 1
I am truly unsure of what is the correct way to solve this issue. I am currently taking ratios for a set of 4 values that have a time range of 15 years (2002, 2007, 2012, & 2017). Therefore, I ...
- 1
1
vote
1
answer
53
views
Differentiating the risk-neutral price of a European call
(Black-Scholes formula) The risk-neutral price of a European call is $$C_t = S_tN(d_1) - e^{r\tau}KN(d_2)$$ where $$d_1 = \frac{log(\frac{S_t}{K}) + (r + \frac{1}{2}\sigma^2)}{\sigma\sqrt{\tau}}$$ and ...
- 77
11
votes
5
answers
2k
views
Why using compound interest formula gives (potentially) wrong answer in this instance
I was doing some catch up exercise on Khan academy and was given this seemingly simple looking problem
Find the compound interest and the total amount after 4 years and 6 months if the interest is ...
- 113
1
vote
0
answers
37
views
Forward price change of variables in Black-Scholes Model
Suppose that $V(S, t)$ satisfies the Black-Scholes PDE:
$$\frac{\partial V}{\partial t} + \frac{1}{2} \sigma ^2 S^2 \frac{\partial ^2V}{\partial S^2} + (r-q)S \frac{\partial V}{\partial S} - rV = 0, \...
- 165
0
votes
1
answer
30
views
Return on investment & Interest rate comparisons
For the question to make sense I need to provide some context on the products available, below is an example of how banks make money for just two products.
We have infrastructure bonds at 12% per/year ...
- 101
0
votes
1
answer
21
views
How to get Effective Rate?
When googling for Effective rates, i'm getting a different result for the formula compared to what we are using in our current company.
Google Result:
Effective Annual Rate(EAR) = ((1+i/n)^n)-1
where ...
1
vote
0
answers
23
views
Solving the max return problem of Markowtiz
I am trying to solve the following Lagrangian but I am having a hard time to find the solution. My end goal is to obtain a formula for x with a closed form solution for the lambda's
$$
L(x, \lambda_1, ...
- 119
0
votes
0
answers
25
views
Volatility matrix understanding
I have problem to understand volatility matrix on page 9 section 2.3 The Yield-Adjustment Term in the following article: Volatility matrix in Nelson-Siegel. It is volatility matrix in AFNS model. Why ...
- 149
2
votes
0
answers
31
views
On the existence and uniqueness of solution for a regime-switching Black-Scholes problem (coupled parabolic problems)
I am currently working on a regime-switching Black-Scholes model and am having trouble determining the existence and uniqueness of a solution for the problem. Specifically, I am interested in finding ...
- 701
0
votes
0
answers
20
views
How to get expressions for interest and principal payments over amortization schedule?
Say I have the following loan:
principal: 375000$
yearlyRate = 0.055 (5.5%)
years = 30
I already know how to calculate the amortized monthly payment:
$$monthlyRate = \dfrac{0.055}{12}$$
$$r = \...
0
votes
0
answers
34
views
Determine the value of the monthly payment if the sum of $5000$ is paid in the last year
The company wishes to sell its car for $23000$ usd cash. A client who wants the car offers to make semi-annual payments with a rate of $11.5\%$ with semi-annual capitalization, the first at the end of ...
- 100
0
votes
0
answers
22
views
Simpler approximation of Skew-Normal Distribution using Normal distribution pdf
While looking for a simple approximation of the Skew-Normal distribution, I found this answer. This says that the sum of a gaussian and its first derivative multiplied by a constant can approximate a ...
- 36
0
votes
1
answer
65
views
On the time value asymptotic behaviour of call option in the generalized Black-Scholes model
In the context of generalised Black-Scholes models,
$$\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2(S, t) S^2 \frac{\partial^2 V}{\partial S^2}+(r(t)-D(t)) S \frac{\partial V}{\partial S}-r(t) V=...
- 2,283
0
votes
0
answers
19
views
A put option and a call option with identical exercise price are both marketable or neither is.
I am doing Exercise 1.13 in Introduction to Mathematical Finance: Discrete Time Models by Pliska.
Exercise 1.13 Suppose the interest rate $r$ is a scalar, and let $c$ and $p$ denote the prices of a ...
- 615
0
votes
2
answers
51
views
How to do this financial math (Exam FM) question from first principles (summation)
A perpetuity costs 77.1 and makes end-of-year payments. The perpetuity pays 1 at the end
of year 2, 2 at the end of year 3, ...., n at the end of year (n+1). After year (n+1),
the payments remain ...
- 188
-1
votes
1
answer
22
views
Financial Calculation with compound interest: Determine how much a person will need to have saved in year 0 of retirement based on monthly withdrawals [closed]
I am trying to come up with a formula to determine how much a person will need in retirement. So every year, they will take their inflation-adjusted withdrawals out of their retirement savings, and ...
- 101
1
vote
1
answer
27
views
why can we add accumulated values in example shown below?
Can you explain to me why we're able to add the accumulated amount of 1000 invested for 5 years to the additional accumulated value of 1000 they invested after 3 years?
1
vote
0
answers
22
views
Price a SWAP contract in no arbitrage market
We have a single period, arbitrage free market with riskless rate of return $r$. A swap contract operates in the following way:
At $ t = 0$ the buyer pays the seller amount $q$.
The seller agrees to ...
- 449
0
votes
0
answers
28
views
How is the Dybvig (1981) asset pricing kernel in return space derived?
Coval and Shumway ("Expected Option Returns", Journal of Finance, 2001) cite to a working paper by Dybvig (1981), which does not appear to be available online. Dybvig shows that, under Black-...
- 1
0
votes
1
answer
37
views
Are this equations somehow equivalent? (newbie mistakes...)
When reading the next question about Differential Equations and their application in Finance (in this scenario they are modelling Annuities; Differential Equation)
The Differential equation is:
$$\...
1
vote
1
answer
61
views
Calculate risk-neutral probability
Consider a two-step binomial model in which at each step the share price either doubles, with probability $p ∈ (0, 1)$, or halves, with probability $1 − p ∈ (0, 1)$. Initially the price is $S_0 = 4$. ...
- 155
1
vote
1
answer
37
views
Expectation of stock price that follows a stochastic differential equation
Let the price $S_t$ of an asset satisfy $dS_t = \alpha (\mu - \ln{S_t})S_tdt + \sigma S_t dW_t$, where $W_t$ is a Brownian motion.
I managed to show that $x_T = x_te^{-b(T-t)} + \frac{a}{b}(1 - e^{-b(...
- 165
2
votes
3
answers
62
views
Is My Textbook Incorrect On Explaining How To Solve Annual Interest Compounded Monthly?
I doubt that the textbook solution is correct. If I have $\$100$ and put it into a bank with annual interest compounded monthly of $6\%$, how much money $y$ would I have after $t$ years?
The equation ...
- 21
2
votes
1
answer
62
views
Proof that $VaR_c(L)=(\Phi^{-1}(\frac{c+1}2))^2$
The loss $L$ has the $\lambda_1^2$ distribution, i.e. the distribution
of the random variable $X^2$, where $X$ has a standard normal
distribution. Proof that $VaR_c(L)=(\Phi^{-1}(\frac{c+1}2))^2$, ...
- 327
1
vote
1
answer
63
views
Understanding the definition of arbitrage opportunities (one period model)
The setting is the one period model of a financial market with $d+1$ assets, where the $0$-th asset is a riskless bond:
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $\overline\...
- 2,614
0
votes
1
answer
29
views
Risk free interest and integral setup
A very simple question im wondering why we would use an integral for I in the line after the graph?
1
vote
1
answer
55
views
Is it possible to simplify this function in a way that the summation symbol is removed?
$$
f(t)=\sum_{t_0=0}^{t} ab^{t_0}
$$
In which $a$ and $b$ are constants, and $t$ is variable.
The purpose of the function is to describe the performance of a fixed income investment with monthly ...
- 31
0
votes
2
answers
74
views
What's the math formula that is used to calculate maximum mortgage amount like in this calculator? [closed]
What's the math formula that is used to calculate maximum mortgage amount like in this calculator?
Visual Reference: Take a look at these images to see which tool I'm specifically referring to:
...
0
votes
1
answer
45
views
Taylor Expansion for the Return averaged over k periods? [closed]
this is my first question here. I need help to understand the Taylor Expansion which gives the (2.2.5) equation (see the pictures). Thanks
(pictures from: Schmidt - Quantitative Finance for Physicists....
- 11
1
vote
0
answers
31
views
Sufficient condition for no arbitrage in a discrete time market
Assume there is a multi period discrete market i.e. a finite Probability space, finite time periods $\lbrace 0,1\ldots, n\rbrace$, a stochastic process $(S(t))_{t\in\lbrace 0,1,\ldots,n\rbrace}\...
1
vote
1
answer
42
views
Regression relation to casual relationship
If the correlation coefficient of two variables is 0, can there still be a causal effect between them? And can the causal relationship between these two variables be studied by regression analysis?
- 69
0
votes
0
answers
29
views
Many questions about a loss distribution
Long question, but mostly set up:
I'm trying to understand the capital requirements formulas of Basel III. They come from the Merton-Vasicek model for the value of a company's assets. Assume we have $...
- 2,920
3
votes
2
answers
73
views
Betting on the outcome of a semidecidable computation
If my neighbor Colin insists that
Bruce Willis will be the winner of the 2024 U.S. presidential election
we can bet on it, and (barring some very strange outcomes) there is some well-defined future ...
- 63.5k
-1
votes
1
answer
64
views
Finding the Rate of Decrease to End Up at Goal Number [closed]
I'm trying to find the Rate of decrease at which 1,262 is multiplied by every day (16 times in total) and added to the Base number of 17,584 in total 16 times to get to the goal number of 24,000. I've ...
- 1
0
votes
0
answers
20
views
Definition of transition density
Sorry if this is a basic question but I have a hard time grasping the definition of transition density in a financial setting.
From what I understand, transition density is the expected (future) value ...
2
votes
1
answer
25
views
Expected present value and expected future value
Suppose that the interest rate $r$ is a random variable.
Given a future value $FV$, the expected present value is $\mathbb E (\frac{FV}{1+r})$.
Given a present value $PV$, the expected future value is ...
- 385