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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0
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2answers
29 views

Finding the present value of the given cashflow.

A loan is repayable by an annuity certain , which is payable annually in arrear for 16 years and calculated at effective rate of interest $5\%$ pa. The payments at t=1 , t=2 , t=3 , t=4 , . . . . . . ...
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3answers
46 views

Savings question: How long will savings last if I withdraw a certain amount every year?

Let's say that for $30$ years I insert $.20$ dollars every year into a bank account for with interest rate $5\%$. After 30 years, I stop inserting money, and start withdrawing 0.70 dollars every year. ...
0
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0answers
15 views

Proof of the finiteness of integral (in option pricing)

I would like to ask for help with proving the finiteness of the following double integral. $$\int_{0}^{\infty}e^{\alpha+k}\int_{k+\zeta}^{\infty} (e^{-\zeta+x}-e^k)f(x)\ \mbox{d}x\ \mbox{d}k,$$ ...
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0answers
23 views

Annuity question from my textbook

Assuming a pensioner expects to receive an annual pension of $20,000 for the next 5 years from his former employer. What is the present worth of the pension plan? Attempt: I'm solving annuity ...
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0answers
16 views

càdlàg adapted process of finite variation

$X$ is a semimartingale with $X_0=0$. I have to show, that $S_t:=\prod^{}_{s\le t}(1+\Delta X_s)\exp(-\Delta X_s)$ is a càdlàg adapted process of finite variation. Could you please help me?
0
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0answers
9 views

Equivalence between two different representations of exponential Lévy Processes

My questions are: Why do I know that $\frac{Z}{Z_-}$ looks like in the proof? Why $\int \frac{d[Z^c]}{Z_-^2}=[Y^c]$? Why does the part with the sum look like the one below? I only know that ...
2
votes
1answer
37 views

Calculate the VaR

An investor has a portfolio of three positions. The 1-day $95\%$ VaRs for positions 1, 2 and 3 are $\$250$, $\$180$ and $\$480$ respectively. The correlation matrix is given as follows. ...
1
vote
1answer
24 views

Volatility of investment (/w currency hedging)

I´ve been trying to compute a volatility of invesment with currency hedging and I have a question. Let's take this example. We have our money in a fond copying the S&P500 index, which has 16% ...
1
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0answers
43 views

Problems with a Black-Scholes modified equation

I haven't really studied much financial mathematics until about 2 months ago so I'm quite new to this stuff, so I'm sorry if this is a trivial question. At the moment I'm trying to work out what the ...
0
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0answers
27 views

Financial mathematics- finding yield rates for bonds

I'm not sure if its appropriate to post here but oh well QF put me on hold. Joe must pay liabilities of $1,000$ due $6$ months from now and another $1,000$ due one year from now. There are two ...
3
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1answer
1k views

Easy proof of Black-Scholes option pricing formula

I use this Book to read the option pricing in Black-Scholes model in pages 93-99, The proof of the formula given by $$c(s,t)= N(d_1(s,t)- Ke^{-rT}N(d_2(s,t)))$$ where $$d_{1,2}=\frac{\ln(s/K)+(r\pm ...
2
votes
2answers
10k views

How the formula for EMI is derived

I was looking for a formula to calculate EMI (Equated Monthly Installments). I have some fixed known parameters like, Principal Amount, Rate of Interest and No. Of Installments. By googling, I came ...
0
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1answer
642 views

Redington vs full immunization?

I understand that the present values and duration of liabilities and assets are required to be equal to each other under both cases, and furthermore for Redington immunization the convexity must also ...
-2
votes
3answers
72k views

What is the formula for the difference between CI and SI?

if principal, time and rate are given how do i find the difference between Compound interest and Simple Interest? P=12,000 n=1 and a 1/2 yrs. R=10% per year ...
0
votes
1answer
29 views

square-root rule of time

I tried to test the square-root-rule of time for quantiles of a normal distribution. So i created with the statiscal programming language R two variables a<-rnorm(100,mean=2,sd=1) ...
0
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0answers
17 views

Working out payoff of a derivative with random interest rates

For this question, I've worked out the payoffs at N=3 but I'm not able to understand how to calculate the the expectation of the terms inside. If anyone could tell me how to find the expectation of ...
0
votes
1answer
22 views

Trouble understanding the constant before an increasing/decreasing annuity.

I have this question here that I'm having trouble understanding An annuity immediate has semiannual payments of 800,750, 700,..., 350 at $i^{(2)}$ $= .16$ if $a_{\overline10|.08} = A$, find the ...
0
votes
1answer
9 views

Future value given force of interest

Find the future value of a five year annuity ($s_{(n)}$) if $\delta _t=0.02t$ for $0 \le t \le 5$. What I know is $\delta_t= \frac{A'(t)}{A(t)}$ $A(5)=\frac{0.02}{0.02. X5}=0.2$ I am not even ...
1
vote
1answer
32 views

Maximum Likelihood Estimation of Brownian Motion Drift

I'm looking at times series of stock movements over 10 minute windows, and am trying to measure the "trend" of these movements. Method A is to simply calculate $\Delta P$, the difference between the ...
1
vote
1answer
37 views

Outstanding loan balance

A loan of ${$1000}$ is being repaid with annual payments over 10 years. The size of the payment in the first five years is ${$ k}$. It is found that the payments in the last five years are five times ...
1
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1answer
26 views

Finding effective annual rate of interest

The present value of $2x$ paid at the end of $k$ years and the present value of $x$ paid at the end of $2k$ years sum up to $2x$. Show that the annual rate of interest is $(\frac{\sqrt{3}+1}{2})-1$ ...
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0answers
16 views

Solve Black scholes PDE without using any transformation

I know that one of the methods of solving the black scholes PDE given by : $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV ...
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votes
1answer
24 views

Reinvesting and accumulated values

I'm having trouble understanding the solution for this problem Susan invests Z at the end of each year for seven years at an e ffective annual interest rate of 5%. the interest credited at the ...
1
vote
2answers
36 views

Finding the monthly payment for fixed-rate mortgage, but with first month interest free.

I'm trying to calculate the monthly payment of a fixed-rate (annuity) loan, but with the twist that the first month is interest free. I.e., I have a principal $P_0$ - the total sum that I've loaned - ...
0
votes
1answer
15 views

Extimate error, payment rate, continous compounded

Give an estimate of the error, when the payment rate $x_m = r P_0 \frac{(1+r/m)^{mT}}{(1+r/m)^{mT}-1}$ (compounding and repayment m times per year) ist approximated bei $x_{\infty}=\frac{r P_0* ...
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votes
2answers
34 views

Find the price of the bond using its book value

A n year 1000 par-value bond with 8% annual coupons has an annual effective yieled of i, 1+i >0 . The book value of the bond at the end of the third year is 990.92 and the book value of the bond at ...
1
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2answers
68 views

Distribution of a Gaussian variable with a normally distributed mean

Let $X\sim N(0,1)$ and $Y\sim(X,1)$, where $Y-X$ is independent of $X$. Then what is the PDF of $Y$? Specifically, I am interested in computing $P(Y<0\vert X>0)$. For those interested in ...
1
vote
1answer
26 views

Future Value and Present Value of a General Annuity Due

I understand that a general annuity due, the payments are made at the beginning of each payment period, and the compounding period is not equal to the payment period. Then to solve I need to transform ...
0
votes
0answers
16 views

arbitrage and exchange rate: Find x if $(1-t_c)*S_f*(1-t_c)*(\frac{1}{S_f+x})<1$

Okay, so I need help with this math problem. The professor said the answer is $x=.000307785$ I need to find x if: $$(1-t_c)*S_f*(1-t_c)*(\frac{1}{S_f+x})<1$$ $t_c=.02\% $ and $S_f=1.3000$ The ...
-3
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0answers
17 views

Quarterly fee calculation and effective rate using a fee schedule

What formula do I use when I am given the market value for end of Jan, Feb and March and a fee schedule to calculate the quarterly fee and effective rate?
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1answer
25 views

SDE Solution: Hull-White extension of Vasicek model

I am trying to figure out the particular ansatz (if that's all there is) for the solution to the SDE: $ dr_t = [v_t - ar_t]dt + \sigma dW_t, $ where $a$ is constant and $v,t$ are, potentially, ...
0
votes
0answers
14 views

Deciding whether a maximum asset price process is a markov process

I understand how Mn has been drawn. For the second computing part, after computing, I have no idea how to decide if Mn is a markov process I don't understand the solution at all, don't know what ...
1
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0answers
21 views

Finance Algeabra: Converting a Discount Polynomial Function to an Interest Rate Polynomial Function

I have a finance problem that is 99% mathematical. In finance, the price of a bond could be modelled as the discounted value of its future cash flows, so something like: ...
1
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1answer
21 views

Solve for the interest rate while we are not told if it is simple or compound.

You are offered to have a discount of \$20 if you pay cash now for \$1500 due in 120 days. If you pay cash now, at what rate may you consider your money to be earning interest for the next 120 ...
2
votes
1answer
36 views

continuous local martingale brownian motion

$B$ is a one-dimensional Brownian motion and $X_t$ is defined as$\\$ $X_t:=f_{1-t}(B_t)$, $0\le t<1$ and $0$, $1\le t<\infty$ where $f_s(x)=\frac{1}{\sqrt{2\pi s}}e^{-\frac{x^2}{2s}}$. I have to ...
2
votes
0answers
28 views

Integral Representation of Brownian Motion [duplicate]

B is a Brownian motion with values in $\mathbb{R}$. I have to find a process $(F_t)_{t\in[0,T]}$ such that $X=E[X]+\int_0^T F_s dB_s$, for $X=B_T$, $X=\int_0^T B_tdt$, $X=B^2_T$, $X=B^3_T$ and find a ...
1
vote
0answers
12 views

Financial Mathematics--Finding Compounding Period given Annual and Effective Interest Rates

I'm trying to find a compounding period C when given an annual interest rate r and effective annual yield i. I'm working with the following equation: $i=(1+r/C)^C-1$ I'm having trouble re-writing ...
1
vote
1answer
23 views

Loan to be repaid with the interest of the last payment given

A $60$-month loan is too be repaid with level payments of $1000$ at the end of each month. The interest in the last payment is $7.44$. Calculate the total interest paid over the life of the loan. Let ...
0
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1answer
19 views

Repayment of a loan with non level annual payments

A loan of $10,000$ is being repaid with 20 non-level annual payments. The interest rate on the loan is an annual effective rate of$6$% . The loan was originated 4 years ago. Payments of $500$ at the ...
2
votes
0answers
34 views

Asymptotic distribution of zero-drift Geometric Brownian Motion as $t \to \infty$

If we fix the drift at $\mu = 0$, then my geometric brownian motion will have stationary mean, but it seems that the variance will grow without bound. What does the limiting distribution look like for ...
1
vote
0answers
15 views

Calculating benefit of paying off loan ahead of time

So I'm doing a little financial planning, and I'm looking into the worthwhile-ness of paying off my student loans as quickly as possible. Being that I have multiple student loans, I combined them all ...
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0answers
9 views

How to find the swap value?

So the question goes like this: The current term structure of interest rates (3-month, 6-month, 9-month, 12-month maturities, annualized with quarterly compounding) is (1.00%; 1.50%; 2.00%; 2.50%). ...
1
vote
1answer
27 views

present value, continously compounded

Compute the present value of a payment of 10 000 Euro after 3 years, if the continuously compounded interest rate in the first year ist 4%, in the second year 6%, and in the third year 5%. For a ...
0
votes
0answers
25 views

How does this self referencing (circular reference) equation terminate (i.e. not create a paradox?)

I'm working with a financial equation which seems like it should result in a paradox but I'm told doesn't, however I haven't been told why it doesn't. (I don't work in the field I'm a programmer ...
1
vote
1answer
985 views

Comparing annualised volatility from monthly and annual data

I fear there is a very simple answer to this question and its killing me that I can't see it. I am interested in calculating historical volatility: I have monthly index values starting in Jan 2005 ...
0
votes
1answer
20 views

Amortization varying series of payments

Having trouble understanding the solution for this question. A borrower is repaying a loan at 5% effective with payments at the end of each year for 12 years, such that the payment at the end of ...
0
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0answers
29 views

Application of Stochastic Calculus to Interest Rate Model (Ito's Formula)

Above is my question. Now, the setting is of mathematical finance, but the part that I'm stuck on isn't directly related to finance, but stochastic calculus (hence posting on this site). We have the ...
2
votes
6answers
24k views

Most efficient method for converting flat rate interest to APR.

A while ago, a rather sneaky car salesman tried to sell me a car financing deal, advertising an 'incredibly low' annual interest rate of 1.5%. What he later revealed that this was the 'flat rate' ...
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0answers
9 views

Calculate CAGR of an investment Portfolio

I want to calculate the CAGR of investments that have been closed. Example: Investment 1; 2 years duration; CAGR - 10% Investment 2: 3 yrs duration; CAGR - 40% Will the CAGR of the Portfolio be a ...
0
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0answers
16 views

Martingale implies moment generating function exists

Let $X_T = \ln(S_T/S_0)$ where $S_T$ denotes the stock price at time $T$ and $S_0$ is the spot price. There is a well known relationship between the moments of $X_T$ and the characteristic function ...