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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How to estimate ECL through monte carlo simulation?

I am trying to run a montecarlo simulation based on this model in order to assess whether a loan/company will default or not. Basically the model is $$X_i = \sqrt{1-\rho}Z_i + \sqrt{\rho} Y$$ where $$ ...
StVnilor's user avatar
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The partial derivative of a call option with respect to $t$

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
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mathematics of finance question having to do with table, figuring out the value

The activity date, company, and amount for a credit card bill are shown below. The due date of the bill is July 10. On June 10, there was an unpaid balance of $987.81. Find the finance charge if the ...
Special effects's user avatar
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Future Value/Present Value Question [closed]

You deposit A today; B one year from now; C two years from NOW. Your account value at the end of year 2 is D. Assume compound interest rate. a) Write the Equation of Value (EoV) at “t=2” and solve for ...
Mitchell's user avatar
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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
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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
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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
2 votes
1 answer
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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 ...
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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 ...
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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
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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
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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
101 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 $...
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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
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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
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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
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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
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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 ...
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Optimal weights/Tobin separation

I'm having troubles understanding the following: I understand given, let's say 5 stocks and their historic data, how to compute the efficient frontier of optimal assets. However, when I introduce ...
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How to evaluate $(1-aP)^P=d$ for P where $P>0$ and a & d are known

For context I am working trying to work out a formula based on the payout annuity formulas. With these forums I have been able to derive the expected term for a given payment: Standard payout annuity ...
copperpen's user avatar
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Question about accumulated values and nominal rates of discount

A collection agency pays a doctor $\$5,000$ for invoices that the doctor hasn't been able to collect on. After two years, the collection agency has collected $\$6,000$ on the invoices. At what nominal ...
j.jerrod.taylor's user avatar
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How to implement the following problem in Matrix form? (Linear Algebra and equations on Matrix form)

How to implement the following problem \eqref{Eq. 1} in Matrix form? (Linear Algebra and equations on Matrix form) I am trying to solve the following optimization problem (coming from this another ...
Joako's user avatar
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HJM model forward rate explosion

In Steven Shreve's excellent book, page 436, it says the forward rate $f(t,T)$ of the Heath-Jarrow-Morton (HJM) model explodes as $t\to T-$. The attached screenshot shows the calculation where the HJM ...
Andras Vanyolos's user avatar
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2 answers
172 views

Best books covering SOA Exam Fm

Looking for book recommendations to self learn material for SOA exam FM. (This exam is mostly equations involving compound interest). Particularly something that covers the underlying math intuitively ...
Older Amateur's user avatar
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Ways to solve compounding interest questions?

Ed invests $\$15,000$ into an account that pays an annual rate of $3.5\%$ interest compounded k times per year where $k\in \mathbb{Z}$. At the end of two years the investment is worth $\$16,087$. Find ...
Audric Jardine's user avatar
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How to calculate the accumulated value of a simple interest annuity?

An account is credited interest using 6% simple interest rate from the date of each deposit into > the account. Annual payments of 100 are deposited into the account. Calculate the accumulated ...
j.jerrod.taylor's user avatar
4 votes
1 answer
57 views

Do inequalities in boundary conditions imply inequalities in solutions to a PDE?

I'm working on an unassessed course problem to show the solution $p$ to a PDE satisfies an inequality. I think I can show that of the 2 boundary conditions of the PDE, $p$ is equal to another solution ...
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Calculating the withdrawal amount from a fund as an annuity-immediate incorrectly

I have the following problem. Consider an investment of $5,000 at 6% convertible semiannually. How much can be withdrawn each half-year to use up the fund exactly at the end of 20 years? I can tell ...
j.jerrod.taylor's user avatar
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Show that no arbitrage implies the extension property in $L^p$

Let $(\Omega,\mathcal F,P)$ be a probability space, and let $X:=L^p$ denote the normed space of (equivalence classes) of $p$-integrable real random variables on $(\Omega,\mathcal F,P)$, where $1\leq p&...
Alphie's user avatar
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Expected value of Ornstein-Uhlenbeck process

In the paper "The Impact of Jumps in Volatility and Returns" by Nicholas Polson, Bjorn Eraker, and Michael Johannes (2003), the authors state in footnote 6 on page 1273 that, given an ...
Roberto Palermo's user avatar
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Derivation of $ \pi(\sigma) $

This is my first post on this website so please forgive me for any mistake or inappropriate use. I am taking a Master level Investments course, in which, amongst the rest, we are deriving $$ \pi(\...
Giuseppe Siragusa's user avatar
1 vote
1 answer
59 views

Finding optimal way to pay a set of credit cards

Lets say we have two credit cards. Card $A$ has a balance of \$2000 and card $B$ \$3500. That is, $C_A = 2000, C_B = 3500$. The interest rates are $r_A = 0.20, r_B = 0.25$. What is the optimal way to ...
rudytheduck's user avatar
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Seeking help with the application of Law of Large Numbers and Central Limit Theorem to calculate Investor Risk

I'm a newbie to the forum with zero financial or statistical skills - first time post...seeking some assistance and a solution..thanks in advance! I am trying to create a investor calculator or at ...
Darren's user avatar
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European Option Volga Derivation

This should be a standard exercise involving high-school Calculus, but for some reason, my expression for the European option volga, does not match the one on Wikipedia. I would like to ask if, ...
Quasar's user avatar
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1 answer
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Brownian motion X(t) is with probability 1 a continuous function of t

Here is an excerpt from "An Elementary Introduction to Mathematical Finance" by Sheldon Ross, 3rd edition: I understand this is not meant to be rigorous, but I'm having trouble ...
Bastiza's user avatar
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Estimating Present Bond Value without YTM using yield curve rates

Suppose I have a bond where I know the par value, coupon rate, and maturity date as well as the daily Yield Curve Rates given here How can I go about estimating the ytm needed to determine the present ...
ccj242's user avatar
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1 answer
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Understand a FM question about a bond with varying interest rate. [closed]

Consider a coupon bond with maturity in $2$ years, with a coupon rate of $4.375\%$ (coupons are paid twice a year) and with a face value of $100€$. Let's say this coupon bond has a varying ...
xyz's user avatar
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1 vote
1 answer
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Question about calculating the price of a coupon bond at different times: FM question.

Let's say we are working with a coupon bond with face value of $F = 100€$, maturity of $T = 5$ years and with $10€$ coupons paid anually. Also, consider we're dealing with a continuously compounded ...
xyz's user avatar
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How to interpret the value of money depending on the maturity of a bond? FM question.

Consider the following exercise: Exercise. A financial institution issues bonds with maturities of $13$ weeks, $26$ weeks and $52$ weeks, at zero coupon, and with a discount value $B_1(0) = 98€$, $B_2(...
xyz's user avatar
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Loan Balance Formula for Cumulative Interest Paid?

I'm trying to find a formula to calculate the cumulative interest paid on a loan after x amount of time. I can do this with data science software but it has to amortize every loan for every customer ...
MDYETI's user avatar
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2 votes
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How can Capital Market Line portfolios be efficient when they're not feasible?

My course notes define Suppose now that there are many different investments $A_1,\dots,A_n$ available. We can invest our one unit of currency by investing $t_i$ in $A_i$ for each $1 \leq i \leq n$ ...
mjc's user avatar
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1 vote
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Price sensitivity under Uniswap [closed]

Question. Under the uniswap pricing rule, what determines how much the price of an asset increases when you buy that asset? Note. while this is a mathematical question, answering it requires an ...
afreelunch's user avatar
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How do Brownian/Wiener processes involve randomness?

My financial mathematics course notes have A Brownian motion is a family of random variables $\{B_t|t\geq0\}$ on some probability space $(\Omega,\mathcal{F},P)$ such that: \begin{align} (1) \; & ...
mjc's user avatar
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0 votes
1 answer
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Macaulay Duration?

I have a problem with a question, I don't know if the question is well worded, it reads as follows Having the following information about a loan: Interest rate: 11.5% per annum, compounded monthly. ...
thelast12e1's user avatar
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I am having trouble with applying Leibniz Rule for Differentiation an integral.

I have a function, $V(D) = (1-\tau_E) (1-\tau_C) \int_0^{100} (R-D) dF(R) + (1-\tau_D) \int_D^{100} D dF(R) + (1-\tau_D)(1-b) \int_0^D RdF(R)$ Where R is a random variable $R \in [0,100]$ with ...
closed_form's user avatar
1 vote
1 answer
38 views

Returns of an asset in risk-neutral measure and its PDE

I am a bit confused regarding how an asset returns in a risk-neutral measure (say $\mathbb{Q}$), and subsequently its Black-Scholes-esque PDE. In class, we learned about the approach to take when ...
dismal-audience's user avatar
1 vote
1 answer
47 views

Why is the inequality $c>S-Xe^{-rT}$ (relating European call option price to spot price and strike price) strict?

Define \begin{align} c & \text{ the price of a European call option at time } t = 0, \\ S & \text{ the spot price of a share at time } t=0, \\ X & \text{ the strike price for the option}, \...
mjc's user avatar
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-1 votes
1 answer
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What is the formula for the ratio of the principal payment and interest payment for a loan, given a fixed interest rate and loan term? [closed]

I just saw a youtube video where they mentioned that a closed-form solution exists, bypassing the need for an amortization table. This was framed in the context of having multiple loans to service and ...
Adrian Fletcher's user avatar
1 vote
1 answer
70 views

Is Heston model an affine jump-diffusion?

In Duffie, Pan and Singleton's paper "Transform Analysis and Asset Pricing for Affine Jump-diffusions" (2000) they define affine jump-diffusion (AJD) a process of the following form: $$dX_t=\...
Roberto Palermo's user avatar
2 votes
0 answers
183 views

Are the Black-Scholes equation models still considered accurate predictors for asset pricing? [closed]

Sorry if this equation is not phrase in precise mathematical form. I am open to suggestions to improve the explanation, and I have tried to formulate the problem as precisely as I could. I was talking ...
krishnab's user avatar
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