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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Linear functional and Riesz' Rep theorem

On page 59 in these Finance notes, a positive linear functional is defined, and then Riesz' representation theorem is used (the scalar product is defined on bottom part of page 56). I don't ...
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2answers
63 views

help for an integral

I need help calculating this integral: $$\int_0^x \frac{2(e^{\gamma u}-1)}{(\gamma+\kappa)(e^{\gamma u}-1)+2\gamma} du$$ I tried with the integration by parts but the situation seems to get ...
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0answers
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Need to find how much more a monthly payments is? [on hold]

$$P=A(r)\frac{(1 + r)^{n}}{(1 + r)^{n-1}}$$ Using the given formula for a loan of $7,000 at 7% annual interest, how much more is the monthly payment for a 12-month term than for an 18-month term? 
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2answers
26 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 ...
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0answers
20 views

implied volatility

I have a question about calculating the implied vol. Assuming the implied vol that a option will expire in 1 day is $\sigma_1$, and the implied vol that the option will expire in 2 days is $\sigma_2$. ...
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1answer
16 views

Transpose Present Value of an Ordinary Annuity Formula for Interest Rate

I'm having trouble transposing the formula for Present Value of an Ordinary Annuity in order to find the interest rate. The formula is: Where P=Present Value of an Ordinary Annuity PMT=Payment ...
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0answers
16 views

Financial Math (differentiate) [on hold]

So I was given an equation in financial math $R(φ)=E^p[(\bar X-\bar V_0(φ)-\bar G)^2]=E^p[(\bar X-\bar V_1(φ))^2]$ where $X$ is contingent claim How do I differentiate $R(φ)$ with respect to $φ^0$ ...
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1answer
29 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
32 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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2answers
28k views

Formula for calculating the total interest payable over the life of a loan

wondering if someone can help a non-mathematician out. I am looking for the formula for calculating the total interest payable over the life of a loan. Given that we know: ...
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1answer
44 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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1answer
1k 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 ...
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1answer
25 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
13 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
39 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
11 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 ...
2
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1answer
31 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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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 $1$ ...
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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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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
69 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
26 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
111 views

Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator ...
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1answer
43 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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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
21 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 ...
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2answers
8k views

What's the math formula that is used to calculate the monthly payment in this mortgage calculator?

What's the math formula that is used to calculate the monthly payment in this mortgage calculator? I would like to know this math formula so that I can plug in the following values ...
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1answer
61 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
23 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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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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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 ...
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1answer
37 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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2answers
116 views

Can someone explain what a portfolio is in financial math?

I took mathematical probability last semester and now I am taking financial mathematics, but only probability was a pre requisite for financial math (no finance classes were required). These types of ...
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2answers
817 views

compound interest with geometric series

Were studying geometric sequences in maths and this came up as one of the questions: A mortgage is taken out for 150000 and is repaid annually with 20000 installments. Interest is charged on the ...
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2answers
33 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
49 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
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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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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
49 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. ...
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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% ...
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0answers
45 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
28 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
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 ...
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2answers
11k 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 ...
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1answer
682 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 ...
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3answers
75k 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 ...
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1answer
32 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) ...
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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
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 ...
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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 ...