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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245 views

In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ ...
3
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50 views

Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta ...
3
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0answers
119 views

What is the name of this symbol ( ┐) and what does it mean

Sorry bout the dumb question, it's just that I'm taking a mathematical finances class and the teacher started using this symbol today but I've never seem it before, was trying to google it but don't ...
3
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0answers
38 views

Stochastic control with stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ and smooth functions $u,F: [0, +\infty) \rightarrow \mathbb{R}$, how can we optimise the ...
3
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0answers
97 views

How to get interest in the mathematics of tax

In a similar vein to my previous thread, I will also be teaching about the mathematics behind taxation - to a lot of people, this is very mundane - but that is not true of everyone. The practicality ...
3
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0answers
168 views

Algorithm/Formula to compute adding and/or removing compound and/or non-compound percentages from a value?

I will first start with a scenario, I have to apply some adjustments to a particular value. These adjustments are either compound or non-compounded and they can either be added or subtracted to the ...
2
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0answers
29 views

Linear combination of Geometric Brownian Motions

Let $X_t= e^{\left(\mu-\sigma^2/2 \right)t+\sigma W_t}$ be a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. I am trying to derive an analytical solution to $$\mathbb{E}\left[ ...
2
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0answers
14 views

Mean and variance regime-switching model

Suppose we have the following model for stock price: $$ X_{t}=X_{0}\exp\left(\int_{0}^{t}(r-\frac{1}{2}\sigma_{\epsilon(s)}^2)ds+\int_{0}^{t} \sigma_{\epsilon(s)}dW_{s}\right) $$ This follows a normal ...
2
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0answers
32 views

Term Structure and short rates

If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t $$ and want to find the short/spot rate $r_t$, is this simply: $$f(t,t) = f(0,t) + ...
2
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41 views

Reformulate this PDE in different notation

I would like to rewrite this general PDE \begin{equation} \alpha\partial_tu+\beta\partial_xu+\gamma\partial_{xx}u+\delta u=\varepsilon \end{equation} in this form $$c\left(x,t,u,\frac{\partial ...
2
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0answers
62 views

Simplifying $\sum_{t=1}^{n}t^2v^t$ using actuarial notation.

In financial mathematics involving immunization, I encounter situations where I am trying to calculate $$(A) \quad v+4v^2+9v^3+ \cdots +n^2v^n=\sum_{t=1}^{n}t^2v^t $$ where $v$ is the present value ...
2
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36 views

Are these two option valuation formulas equivalent? Why?

I have been reading a finance paper that claims that the following function, which is a value for a financial derivative (1): $$V(s,t)=E_{Q} \left[\zeta\big(S(T)\big)e^{-\int_t^T r_F(\nu) ...
2
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0answers
46 views

Macaulay duration for a coupon bond. Proof

I am working on showing the following. There is a coupon bond redeemable at par with annual coupon rate $r$ per year. The yield to maturity is $i$. The total number of coupons is $n$. Show ...
2
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0answers
38 views

What is this sort of optimisation called?

I am reading a book in mathematical finance. There is something about constrained optimisation. They have specialised it for the financial market, but I am wondering what the general name for this ...
2
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0answers
67 views

Pricing/Valuation of American Options

Hi i'm a litte bit confused by the pricing valuation of American options. For simple Assumtions on the Blacksholes Model and no dividends, and constant rates else one can show, that for a given ...
2
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0answers
20 views

How to calculate present value with changes interest

How to calculate present value with period of 5 years and 6 months? Besides that, there is interest changes and compounded differently. Is there any formula?
2
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0answers
177 views

Engineering Economics Cash Flow Diagram

I have the following question and solution below. What I don't understand is why is the 100,000 seen as savings/revenue when clearly it is coming out of pocket? Additionally, the monthly loan payment ...
2
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0answers
277 views

What does it mean to “pass to the limit” in mathematics?

I've been reading a finance paper and stumbled upon this phrase. What does passing to the limit mean in this context (or overall in mathematics)? Here is an excerpt from the paper: It is ...
2
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0answers
45 views

Reinvesting the interest (generalized version)

If I deposit \$1 at $t=0$ into an account which credits interest at the end of each year at a force of interest $\delta_t$ (assume it's integrable.) Then, if I reinvest the interest at an annual ...
2
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0answers
40 views

Stochastic control, numerical, need expectations given coupled SDEs

I'm looking at a trio of processes which arises in a stochastic control situation. I have a process $(V_t)$ which I may control, and $(V_t)$ influences a diffusive stock price process $(S_t)$. The ...
2
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92 views

Doob Decomposition of American Option

I am trying to figure out the Doob decomposition of an American put option in a discrete time binomial model. I know how to price the American put, but I'm having trouble expressing it as the sum of ...
2
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0answers
453 views

Analogue of Leibniz Rule for Stochastic Integrals

Suppose $$f(t,u)=f(0,u)+\int_0^t{\mu (w,u)dw}+\int_0^t{\sigma(w,u)dB_w}$$, where $B_w$ is a standard Brownian motion. I would like to calculus the drift and diffusion of $Y_t=-\int_t^s{f(t,u)du}$ ...
2
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0answers
29 views

Standard practice for identifying outlying spend amounts

Hope this is mathematical enough to qualify as a question - I'm no mathematician! I have a set of individuals travel & entertainment credit card spend, and I'd like to highlight any outliers that ...
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0answers
22 views

Annunity calculation with and without tax

I'm doing a annunity calculation: payment = 331880*( 0,002458333 /( 1-(1+0,002458333)^-84) ) This will return me the payment per. month of the loan ...
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0answers
26 views

Looking for a Formula for ROI, couldn't get an answer in Finance

This is honestly a pretty simple problem, but for whatever reason I am not able to pull it all together. I was talking theoretically with a friend and neither of us can nail down the maths so I coming ...
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0answers
64 views

A question of odds

Consider an experiment with four possible outcomes, and suppose that the quoted odds for the first three of these outcomes are as follows. What must be the odds against outcome 4 if ...
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0answers
35 views

Compound Interest Calculation (Years + Months)

My question is with regards to the calculation of "Compound Interest". I have the formula below where I would get an answer to the total value of the investment over a period of "years". $A$ = ...
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0answers
24 views

Estimating compound growth

I have a compound interest function with the following parameters: Value at time 0 = 13.8 Interest rate = 0.05 time interval = 10 I need to check quickly, (without a calculator, only pen and paper) ...
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0answers
34 views

why hull white model has normal distribution?

consider hull white model $dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$ when we solve the SDE above we have $r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha ...
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0answers
52 views

Optimal Insurance Coverage - risk neutral and risk loving consumers

I'm struggling with understanding a problem in finance: We have 2 states: $S_1$: bad state $Y-K = 2000$; probability $\pi$ = 10% $S_2$: normal state $Y = 5000$; probability $1-\pi$ = 90 ...
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0answers
30 views

Common term for “present value” and “future value”

In the past, I have always used the term "present value" for the value of a payment made at some point in time $t$ from the perspective of some other valuation point in time $T$. I did not distinguish ...
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0answers
46 views

Is this a self-financing portfolio?

I have $S_t = 10 + B_t$, $\beta_t = 1$, $a_t = 2B_t$, $b_t = -t - B_t^2 - 20B_t$ Then the value, $V = a_t S_t + b_t \beta_t$ Is this a self financing portfolio? Note, $B_t$ is brownian motion I am ...
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0answers
33 views

How to use the BA II Plus financial calculator to solve for IRR and NPV?

I've calculated the answers manually but would like to learn how to do so on the financial calculator to save time on the test and minimize errors. How to do this? Problem: You have been offered a ...
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0answers
24 views

Find the highest price which an investor can pay and still be certain of a yield of:

I'm having trouble understanding this example in Kellison's Theory of interest: Consider a 100 par value 4% bond with semiannual coupons callable at 109 on any coupon date starting 5 years after the ...
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0answers
16 views

How to analytically find these rounding issues

Let's say we have a fixed yearly amount that we have to divide equally among an amount of days. For instance for $1,600 we may have: ...
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0answers
41 views

Negative option value

I have an exercise where I need to replicate the following graph: with my own parameters. To do this I use: $\begin{align*} \text{Call option value} =SN(d_1)-Ee^{-r(T-t)}N(d_2) \end{align*}$ ...
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0answers
21 views

Financial Mathematics Question - How to approach?

I know the answer, but I'm not sure how to 'approach' the question the right way. The question is "Katarina would like to buy a house in 4.5 years time and requires a deposit of $40000. What ...
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0answers
12 views

Convergence under rank correlation

I have a following setup: Let $c\in{\Bbb R}$, $R^2\in [0,1]$ and $\Psi,\varepsilon_1,\varepsilon_2,\ldots$ independent random variables on a probability space $(\Omega,{\cal A},{\Bbb P})$. Define the ...
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0answers
20 views

Is there a goodness-of-fit chi-square test for muli-factor SDE models?

I read in the book 'Modeling with Itô Stochastic Differential Equations' by Edward Allen about a chi-square test for SDE models. In section 5.5 this test is explained for a one-factor model. Can this ...
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0answers
25 views

Calculate year for a provided yield

\$146.25 will yeild \$46.25 at 7.5% per annum. How to get the number of years? Answer is 6 but how do you get it? What is the formula?
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0answers
27 views

Finite expectation of bank account with CIR interest rate model

The CIR interest rate model is $$dr_t=(\theta-ar_t)\,dt+\sigma\sqrt{r_t}\,dW_t\;.$$ The money account with this interest rate is $$e^{\int_0^tr_s\,ds}\;.$$ It is known that ...
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0answers
45 views

Help with integrating stochastic calculus expression from yield curve model

I am very rusty on stochastic calculus, and I am having trouble integrating the following simple term from a yield curve model: $$z(t)=\int_0^t\exp(-k(t-s))dW(s)$$ Any suggestions appreciated. ...
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0answers
28 views

Force of interest for simple interest

I am struggling to work out what the force of interest for simple interest is when using differential equations. I know that it is $\delta=\frac{r}{1+rt}$ where $r$ is the interest rate, but when I ...
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0answers
90 views

Help writting financial distribution formula

I need help writing a function to calculate the financial contribution of a product subscription into a given month. Not so straight forward however, since it has to consider months with fixed length ...
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0answers
165 views

Calculating the interest rate for an annuity (Exam FM)

I have been searching for a way to solve for the interest rate given the monthly payments of a loan. I would like to set up a problem as the following. $X$=monthly payment , $i$=effective ...
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0answers
40 views

Finding drop payment with varying interest rates..

Person A is accumulating a 10,000 fund by depositing 100 at the end of each month starting September 1,2002. If the nominal interest rate on the fund is 12% convertible monthly until May 1,2005, and ...
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0answers
38 views

Black and Scholes solution

Consider the Black and Scholes equation. What is the (financial) interpretation of the solutions S and $e^{rt}$? They both satisfy the equation.
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0answers
31 views

Finding the annual withdrawal, given initial and final amounts, and interest rate

I am working on the following problem and I keep getting a different answer. The principal $P=10,000$. The annual interest rate is $i=4\%$. The money is deposited at time 0 and the interest is ...
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0answers
45 views

Locate proof of Second Fundamental Theorem of Asset Pricing

Where can I find a $\textbf{rigorous}$ proof of the Second Fundamental Theorem of Asset Pricing. That is, A market is complete if and only if it has a unique risk neutral measure. Please do not ...
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0answers
55 views

Intuition behind American option pricing

The price of an American option is given by $V_n = \max\{G_n, \frac{1}{1 + r}(pV_{n +1}(H) + qV_{n + 1}(T)\}$, where $p$, $q$ are the risk neutral probabilities. I have two questions. How can ...