Questions having to do with financial mathematics. Please note that for questions in quantitative finance, quant.stackexchange.com is perhaps a better site.

learn more… | top users | synonyms

1
vote
1answer
49 views

Find expected present value of a continuous payment stream

I have a question for the financial part of my course which I am struggling to answer as i am not sure my answer makes sense. Question: Time is counted from the present t = 0 in years. Suppose for ...
0
votes
0answers
60 views

Finding Joint Probability of a Binomial Tree Model given the stock price , then Conditional Probability

Consider a T-period binomial tree model with stock price $S_{t,n} = S_0u^nd^{t-n}$ at each node $(t,n)$ of the binomial tree for every $n = 0,1,...,t$ and every $t = 0,1,...,T$. a) Let $v,t \in ...
0
votes
1answer
43 views

Prove the process is a martingale with respect to the natural filtration

Let $\{M_n\}_{n\ge 0}$ be a symmetric simple random walk. Fix a real $b$. Prove that the process $S_n = e^{bM_n} (\frac{2}{e^b + e^{-b}})^n$, $n = 0,1,2,....$, is a martingale w.r.t. the natural ...
0
votes
1answer
41 views

How to differentiate the standard normal deviation w.r.t. a parameter inside the upper bound

Given that $$N(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{s^2}{2}}\:ds$$ And that $$d=\frac{1}{\sigma\sqrt{\tau}}\ln\left({\frac{S}{e^{-r\tau}K}}\right)+\sigma\sqrt{\tau}$$ How do I take the ...
1
vote
1answer
96 views

How do I calculate interest on short term loan?

I'm trying to work out interest on short term loans - these are loans that extend to months not years, and are typically repaid in monthly chunks, but I also know that some are repayable in weekly ...
1
vote
1answer
62 views

How to differentiate the Black-Scholes formula w.r.t. volatility

The Black-Scholes-Merton formula for determining call option value is given as: $$C(S,K,\sigma,r,\tau)=N(d_1)S-N(d_2)Ke^{-rT}$$ where $N(d_i)$ is the standard normal distribution and ...
1
vote
1answer
37 views

Construct a strategy to profit: Problem involving term structure and interest rates.

I am currently studying about term structure and interest rates such as forward rates, swap rates, etc... The following problem seems like an actual actuarial problem that I might see in the future ...
2
votes
0answers
34 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 ...
0
votes
1answer
39 views

Find the annual yield rate. Exam FM problem.

I'm trying to solve for the following problem and I cannot get the right #. You are given the spot rates at time $t=1,\ 2 \ \text{and} \ 3$ as $s_0(1)=.15,\ s_0(2)=.10,\ \text{and} \ s_0(3)=.05$ ...
1
vote
1answer
26 views

Forward rate example, switching the investment.

I need explanation regarding forward rates for the following specific example. A zero coupon with spot rate $s_0(1)=.08$ and $s_0(2)=.09$ are available. a), Smith borrows $1$ and is obliged ...
1
vote
3answers
85 views

How long an investment will take to compund to a target amount

A man with $\$20,000$ to invest decides to diversify his investments by placing $\$10,000$ in an account that earns $7.2\%$ compounded continuously and $\$10,000$ in an account that earns $8.4\%$ ...
1
vote
2answers
36 views

Calculating monthly compounded interest

To solve the problem How long does it take for an investment to double in value if it is invested at 8% compounded monthly? I figured like this: $$2P = P(1 + 0.08)^t$$ where $P$ is an ...
0
votes
0answers
17 views

Purchasing a unit on fund $X$ calculating the dollar weighted and the time weighted rate of return.

I am currently working on the following problem trying to figure out the rate of return. Fund $X$ has unit values which are $1.0$ on Jan 1 05, $0.8$ on Jul 1 05 and $1.0$ on Jan 1 06. A fund ...
0
votes
1answer
21 views

Dollar weighted method vs. Time weighted method Problem. Exam FM

The following is the problem that I am working on and I am having trouble. On Jan 1 2005, an investment account is worth 100. On Apr 1 2005, the value has increased to 103 and 8 was withdrawn. ...
1
vote
0answers
22 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 ...
0
votes
1answer
27 views

Dollar weighted return. Formula or definition?

I was learning dollar-weighted return and I was a bit puzzled by the following and I would like to have some advice. I understand that it's basically the internal rate return, but using simple ...
2
votes
0answers
46 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 ...
0
votes
0answers
47 views

Calculating a Forward Starting Swap with Forward Equations

I have been trying to resolve this problem for some time but I cannot get the correct answer. The problem is the following one. Compute the initial value of a forward-starting swap that begins at ...
0
votes
1answer
30 views

Loan Interest Discrepancy

Suppose that I have a loan value $x$ and interest rate $r$. The simple interest is then $x\cdot(1+r)$. If I take out a loan compounded annually and paid monthly for $12$ months the amount at the end ...
1
vote
1answer
44 views

Inequality of an expectation (here: perpetual put of an american option)

for a given function $u(x):=\sup_{\tau \in T_{0,\infty}}E[(Ke^{-r\tau}-xe^{\sigma B_{\tau}-(\sigma^{2}\tau)/2})_{+}1_{\tau <\infty}]$ and $x \in [0,\infty)$, K a positive real number, $(B_{t})$ a ...
0
votes
1answer
61 views

Exam FM problem: Financial calculator necessary for finding $i$ from $a_{\overline{n}\rceil i}$? Edited

I am currently studying for the Exam FM for actuaries, and the calculator that I have is a TI 30X IIS, which was very helpful for me during the Exam P. I cam as far as studying bonds, and the ...
0
votes
1answer
20 views

Finding out the minimum yield of a premium bond with a different redemption fee. ($F=100, r^{(2)}=10\%, i^{(2)}=8\%, C=110$)

I am working on a specific problem regarding price of bonds and it is the following. A 10% bond with face amount $F=100$ is callable on any coupon date from $t=15.5$ years after issue up to the ...
1
vote
1answer
104 views

Forward Starting Swaps and Forward Equations

Hi all I have a problem when I have to calculate swaps/swaptions. n=10-period binomial model for the short-rate, ri,j. The lattice parameters are: r0,0=5%, u=1.1, d=0.9 and q=1−q=1/2. 1.Compute the ...
1
vote
1answer
40 views

Why would an investor want the minimum yield?

I am puzzled by a problem related to bonds. When a bond is callable, the purchase price (present value of the bond) can fluctuate and I also understand the difference when the bond is purchased at a ...
1
vote
1answer
76 views

Portfolio VaR with Copula?

Let the portfolio be given by: $$X=X_1+X_2$$ $(X_1,X_2)$ are dependent through a Copula function $C(u_1,u_2)$, such that the joint distribution is given by: $$F(x_1,x_2)=C(F(x_1),F(x_2))$$ What is ...
1
vote
1answer
36 views

Annuity Depreciation Problem from Exam FM

A manufacturer buys a machine for 20, 000. The manufacturer estimates that the machine will last 15 years. It will be depreciated using the constant percentage method with an annual depreciation rate ...
0
votes
2answers
205 views

What is an alternative book to oksendal's stochastic differential equation: An introduction?

What is an alternative book to oksendal's stochastic differential equation: An introduction? But also An alternative that is over 300 pages and at the same level? Some professor refer that book as a ...
1
vote
1answer
66 views

Will this well enough to serve as a prerequisite to oksendal's book?

Will this well enough to serve as a prerequisite to oksendal's stochastic differential equations: an introduction with applications book? I refer to shiryeav's probability, but i guess it still miss ...
1
vote
2answers
27 views

Help for calculating the correct marked price?

A) For this equation I need help calculating the marked price. A retailer wants to make a 22% profit on the sale of a television set. The television set cost the retailer $560. What should the ...
2
votes
1answer
22 views

Total MSRP given monthly payment, downpayment %, and term of autoloan

I want to buy a car. I know the following: - monthly payment - interest rate - # of months of loan - downpayment % How can I calculate the total MSRP I can get for my monthly payment? So for ...
-1
votes
1answer
64 views

What is an elementary yet important application of matrix in finance?

What is an elementary yet important application of matrix in finance? I have difficulty to read anything intermediate/advanced associated with this topics, hopefully I can find something interesting ...
0
votes
1answer
26 views

Are yield rates different from rate of return? (Bonds)

There is a puzzling thing that is bothering me regarding bonds and I would like to have some help. The following is the situation I am dealing with. A 20-yr 8% bond has semi=annual coupons and a ...
3
votes
2answers
66 views

Zero-coupon vs. $10\%$ coupon problem

I am working on Bonds and I am having trouble solving this problem. A zero-coupon bond pays no coupons and only pays a redemption amount at the time the bond matures. Greta can buy a zero-coupon ...
1
vote
1answer
30 views

Time required to make money 6 times of itself

A sum doubles itself in one year at a certain rate of interest, compounded annually. In how many years will a sum become six times itself under the same investment scheme? I got confused in this ...
0
votes
1answer
36 views

Bond prices and how to compare

I have a couple of basic questions regarding bonds that I would like to ask and the following problem is what I used. Find the "price" of the following bonds "redeemable at par". Let $F=100$ be ...
1
vote
0answers
42 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. ...
1
vote
1answer
54 views

Ranking $ d, i, d^{(m)}, i^{(m)}, \delta$

Any actuary or anyone studying mathematics of finance out there? Please help me out. How can I prove or show that $ d< d^{(m)}< \delta< i^{(m)}<i,$ for $m > 1$. Thanks a lot !!!
1
vote
0answers
26 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 ...
1
vote
1answer
62 views

Bonds … whats is it and what is a discount?

I am studying about bonds and I am a bit puzzled. So far my understanding is that bonds are loans issued by the government or a company and it works very similarly to loans. However, I have a ...
0
votes
1answer
61 views

Black-Scholes formula is a monotonic increasing function of the volatility. Proof?

I'm trying to show this statement: that Black-Scholes formula is a monotonic increasing function of the volatility ($\sigma$). I need to proof it from the Black-Scholes formula which is: ...
0
votes
0answers
30 views

Financial mathematics: asion option

I've got a forward starting asian call option: $V_N = max\left( 0,\left(\sum_{j=m+1}^{N}{S_j} - K \right) \right) $ So the payoff is determined by the average stockprice over de latest N-m days and ...
0
votes
1answer
48 views

Derivatives pricing w/ squared and cubed stock prices

I have an assignment in which $S_t$ is a stock price following a geometric Brownian motion. The task is now to show that at time t the risk-neutral price of a derivative on $S_t$ that pays $S_T^3$ at ...
0
votes
0answers
21 views

Shrinkage of variance

I have a sample of N variables, and a sample period of length T. At each time $t$, I have the following information: For each variable i, I have an estimate of variance $\widehat{\sigma^2_{i,t}}$. ...
0
votes
2answers
66 views

Draw the price-yield curves for bonds, B1 and B2 on the same yield-price plane

Consider a bond B1 which matures in 30 years and a bond B2 which matures in 15 years. Both have a facevalue $100 and semi annunal coupon pmt 7%(2). Draw the price yield curves for B1 and B2 on the ...
0
votes
1answer
25 views

How is Excel's `RATE(nper, pmt, pv)` evaluated?

I am trying to come up with the same answer as using the function RATE. I've tried several formulas to no avail. As an example, if I plug in ...
1
vote
0answers
89 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 ...
1
vote
1answer
35 views

What does it mean to be an equivalent repay scheme? (sinking fund vs. amortization)

I am having trouble solving the following problem which seems simple, but I cannot quite get it right. Smith can repay a loan $L=250,000$ in one of two ways 1), 30 annual payments based on ...
0
votes
2answers
42 views

annual and effective annual interest rate

Can anyone help me to find the answer to the question below? I am taking the Engineering Economics exam and need to be sure about the correctness of my answer. Here is the question: Smart Visa, ...
2
votes
0answers
14 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?
0
votes
0answers
13 views

$\frac{L}{a_{\overline{n}\rceil i}}(n-a_{\overline{n}\rceil i})$ vs. $Li \frac{n-1}{2}$ which is larger?

I am having trouble deciding which of the expression is larger. The following is the original problem and I may not have the expression entirely correct, but I am pretty confident. A loan of $L$ ...