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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2answers
30 views

Why does $\operatorname{var}\sum_{i=1}^n w_ir_i=w^\top\Sigma w$ hold?

Let $r_1,\ldots, r_n$ be real-valued random variables, $\Sigma$ be the covariance matrix, $\mu_i=E(r_i)$, and let $w=(w_1,\ldots,w_n)^\top$ be an arbitrary vector in $\Bbb{R}^n$. Why does $$\mathrm{...
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1answer
32 views

Question on share valuation involving rate of return and dividend

BLC industries is expected to pay a dividend of $1.50$ and the dividend is expected to grow at a constant rate of $7$%. This stock is $15$% less risky than the market as a whole. The risk-free rate ...
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1answer
38 views

How to find the standard deviation from the given information and what is $B(0)$ equal to?

Assume that the risk free rate is $0$ and that the stock price is given by the equation $S(t)=6e^{2t+2B(t)}$ where $B(t)$ is the standard Brownian motion. Determine the price at time $0$ of the ...
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1answer
48 views

Nominal rate of interest for interest reinvested.

Sally lends $10000$ to Tim. Tim agrees to pay back the loan over $5$ years with monthly payments at the end of each month. Sally can reinvest the the monthly payments from Tim in a savings account ...
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1answer
14 views

Interest re-invested given Force of interest

Jason deposits $3960$ into a bank account at $t=0$. The bank credits interest at the end of each year at a force of interest $\delta_t=\frac{1}{8+t}$ Interest can be reinvested at an annual ...
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1answer
40 views

Re-investment of interest

Thomas invests X into Fund $1$ at the beginning of each year for $10$ years. Fund $1$ pays interest annually into Fund $2$. Fund $1$ earns $7$% annually while Fund $2$ earns $6$% annually. After $10$ ...
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1answer
13 views

Net present value given cash flows for 2 different projects

Project P requires an investment of $4000$ at time $0$. The investment pays $2000$ at time $1$ and $4000$ at time $2$. Project Q requires an investment of X at time 2. The investment pays $2000$ at ...
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3answers
18 views

question on force of interest with investment at 2 different times

You invested $500$ on Jan $1$ $2012$. To save for this amount, you invest $x$ on Jan $1$ $2008$ and $2x$ on July $1$ $2008$. The force of interest is $\delta_t=0.02t$ where $t$ is $0$ on Jan $1$ $...
2
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1answer
33 views

Buyer's price in terms of risk-neutral measures

Let us consider a finite arbitrage-free market model $(B,S)$, where $B$ is a bank account and $S$ is a share. Let $X$ be a claim. We define a buyer's price of $X$ as follows:$$\Pi^b_0(X)=\sup \lbrace ...
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0answers
13 views

Help optimizing payments on 3 loans - planning to use AMPL program but I have math problems first.

So the basis is that I have 3 loans with different interest rates and different principal amounts as well as different minimum monthly payments and different amortization (is that the right word? Time ...
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2answers
80 views

If A can either increase by 100% or decrease by 50% with equal probability, what will be the arithmetic mean return over n periods?

This is more of a finance related question but deals with some discrete probability and or combinations. The question goes like this. If you buy stock A, and it has a 50% chance of going up 100% in ...
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0answers
27 views

subaddivity of VaR

It is known that the VaR (Value at risk) doesn't fulfill subadditivity, i.e. $VaR(X)+VaR(Y) \le VaR(X+Y)$. But for elliptical distributions subadditivity is true. Questions: (1) Which ...
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1answer
56 views

Differentiating an Option Payoff

Okay this is probably going to be an extremely easy/straightforward question but I thought I should post it here just to double check. Suppose I have a payoff $\Phi = (S_{T}-K)^{+}$. Now let's say I ...
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1answer
52 views

What is the difference among three kinds of continuous income stream?

In the chapter of our book , we discuss "Tolal value of continuous income stream:$\int_a^bR(t)dt$" "Future value of continuous income stream:$\int_a^bR(t)e^{r(b-t)}dt$" "Present value of continuous ...
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0answers
12 views

Replicant portfolio with commissions (Jarrow rudd)

I have created a Jarrow Rudd three for a call option that I know how to replicate with a portfolio. A replicating portfolio of a option works this way: At time 0 we form a replicating portfolio ...
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1answer
68 views

Forward price in Black Scholes Model

Recall that a forward contract on $S_T$ contracted at time $t$, with time of delivery $T$, and with forward price $f(t; T, S_T)$ can be seen as a contingent T-claim $X$ with payoff: $$ X = S_T - f(t; ...
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1answer
28 views

Which Option is more expensive?

Consider two European put options, written on the same asset, with the same maturity, but different strike prices: K1< K2 Which option is more expensive? Then Answer the same question, but using ...
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0answers
13 views

Esscher Transformation structure preserving

$X$ is a Lévy Process with the characteristic triplet $(\gamma, \sigma^2,\nu)$ and it exists a $\nu$ such that $E[exp(\nu X_1)]<\infty$. I would like to know if the Esscher transform $Q^\nu$ is ...
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0answers
36 views

Options on Futures Black-Sholes

I am taking the Financial Risk Management course, and the topic now is "Variations on the Black-Scholes Model". I am following Paul Wilmott's "The Mathematics of Financial Derivatives: A Student ...
1
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1answer
74 views

Is there an interpretation of the hyper skewness?

Let $X$ be a random variable. The standardized $n$th moment of $X$ is defined as $$\frac{E[(X-\mathbb{E}[X])^n]}{\mbox{Var}[X]^{n/2}}. $$ Special cases are the skewness ($k=3$) and the kurtosis $k=4$. ...
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0answers
27 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
31 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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1answer
23 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 $f(x)*\...
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1answer
32 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?
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1answer
30 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% ...
2
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0answers
46 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 ...
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0answers
30 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 ...
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1answer
42 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) b<-...
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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 ...
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1answer
23 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 ...
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1answer
10 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 ...
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1answer
50 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 ...
0
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2answers
34 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 , . . . . . . ...
2
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3answers
52 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. ...
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0answers
22 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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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
27 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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1answer
26 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 ...
0
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1answer
16 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* e^{rt}}...
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2answers
37 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
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2answers
39 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 ...
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1answer
44 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 ...
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0answers
18 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 ...
1
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1answer
49 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, time-...
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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 the ...
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0answers
28 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: ...
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1answer
22 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 days? ...
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0answers
13 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 ...
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1answer
24 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 ...
2
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1answer
51 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 ...