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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Show that formula for European call option satisfies Black-Scholes equation and boundary conditions.

Can anyone please help me out with this question? Show that the formula for the European call option satisfies the Black Scholes equation, and that it satisfies the appropriate initial and boundary ...
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1answer
19 views

Replication in the multi-period binomial model.

For the solution: Just wanted to ask for $V_{3}(HTH)$ I get $S_{1}(H)$ and $S_{3}(HTH)$ to give me the same maximum value of $16$ so would it also be right if I used $S_{1}(H)$ nstead of $S_{3}(HTH)$?...
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2answers
42 views

Compound Interest Formula adding annual contributions

I'd like to know the compound interest formula for the following scenario: P = Initial Amount i = yearly interest rate A = yearly contribution or deposit added. n = the deposits will be made for 10 ...
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0answers
51 views

Pricing Function is convex

I am now reading Alternative Characterization of American Put Options by Carr et all (available at http://www.math.nyu.edu/research/carrp/papers/pdf/amerput7.pdf). There is a theorem called 'Main ...
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1answer
33 views

Potential arbitrage profit and proof

So the question asks: Consider 4 following European call and put options with the same maturity time: Call option with strike price $100$ sell for $45$ Call option with strike price $110$ sell for $...
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2answers
27 views

Annuities and interest conversion

I am having trouble understanding how to find the equivalent rate of interest per payment periods for annuities. For example, for this question: Find the accumulated value at the end of four years ...
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1answer
20 views

IRR for Incremental Investment Doubt

I had a doubt in the following question's solution: Aren't the signs in the equation wrong? Shouldn't it be $-(6,000,000 - 400,000) + 15,000,000(P/F, i^*,15) - 400,000(P/A,i^*,14) = 0$ (using ...
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1answer
43 views

Derive an expression for the value of the asset as a function of time, V(t), t>=0

An investor deposits USD 300 in a bank account at time 0, reinvests all interest payments and also additionally continuously invests USD 300 per annum, until the total value of the deposits reaches ...
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2answers
24 views

Nominal annual interest

A bank offers the following certificates of deposit: $$ \begin{array}{c|lcr} \text{Term in years} & \text{Nominal annual interest rate(convertible semi-annually)} \\ \hline 1 & 0.05 \\ 2 &...
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1answer
20 views

Annuity and future equivalent values

I need some clarification on the formulas to use for these questions. Q1. If $\$30,000$ is deposited now into a savings account that earns $7\%$ per year, what uniform annual amount could be ...
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0answers
21 views

Intuition behind a market uncertainty represented by a filtered complete probability space?

What is the intuition behind a market uncertainty represented by a filtered complete probability space $(\Omega, F, P, {F_t})$, on which an m-dimensional standard Brownian Motion $W(t) = (W_1 (t), W_2 ...
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1answer
41 views

Financial Mathematics problem.

Consider a property developer who buys a property at time $0$ for $\$90,000$. He also spends $\$10,000$ at time $0$ to buy some materials he will use to develop the property. Ignoring Inflation , the ...
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1answer
21 views

Ordinary Annuity future value

I have a problem to calculate future value of this problem: "The parents of a newborn baby set up an account to cover the cost of college they deposited 1,500 every birthday in an account that pays 8%...
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0answers
36 views

Find the value of forward contract

The current spot gold price is $\$1788$ per ounce. The storage cost is $\$24$ per ounce per year, to be paid quarterly at the beginning of each quarter. Suppose the current $3$-month and $6$ month ...
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1answer
19 views

Need to find annual payment in 2 halfs of the payment period [closed]

A loan of $4000 is to be repaid over a period of 8 years. During the first years, exactly half of the loan principal is to be repaid (along with accumulated compound interest) by a uniform series of ...
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1answer
46 views

Finding implied interest rate from swap exchange

Suppose the current term structure of interest rates (0.5 year, 1 year, 1.5 year, 2 year maturities annualized with semi annual compounding) is (5.00%, 6.00%, 7.00%, 8.00%). A 2-year vanilla interest ...
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1answer
26 views

loan repayment involving added amounts

A bank charges $5\%$ interest p.a on loan. At the end of the year, the interest is added and then a fixed amount $R$ is paid off. If the amount borrowed is $1000$, show that the amount owed at the ...
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1answer
40 views

Financial mathematics - earning compound interest

This I assume is a very simple question but can't really wrap my head around it. So the question is: If 100 dollars is deposited at time t = 0 into an account earning 10% interest and $20 is ...
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1answer
33 views

how to prove that there is no arbitrage if the following inequality is satisfied?

The continuously compounded interest rate is $r$. The current price of the underlying asset is $S(0)$, and the forward price with delivery time in one year is $F(0,1)$. Short selling of the stock ...
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2answers
64 views

Simply this formula

Is there a way to write $\sum_{n=a}^b (Q+P(n-1)) v^n$ in terms of $\sum_{n=a}^b v^n$? So far I've gotten: (Q-P)$\sum_{n=a}^b v^n$ + P$\sum_{n=a}^b nv^n$. *I know the last term can be rewritten as $...
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1answer
33 views

Future value and simple interest

On a loan of $3,000 at an interest rate of 12% per year when half of the loan principal is repaid as a lump sum at the end of four years and the other half is repaid in one lump sum amount at the end ...
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0answers
38 views

what am I doing wrong when finding the weights of the market portfolio?

I need to find the weights of the market portfolio with three risky securities given the following information: $\mu_1=0.08$ $\sigma_{1}^{2}=0.0255$ $c_{12}=0.00225$ $\mu_2=0.1$ $\sigma_{2}^{2}=0....
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1answer
33 views

How do I calculate the delivery price for a forward contract?

How do I calculate the delivery price for a forward contract? In the time interval $[0,t]$ the interest rate is $r_1$ and in the time interval $[t,T]$ the interest rate is $r_2$. Determine the ...
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1answer
25 views

Sum of Present value [closed]

The sum of the present value of 1 paid the end of n periods and 1 paid at the end of 2n periods is 1. Find $(1+i)^{2n}$. Present value for n periods is given as $(1+i)^{n}$ and that of 2n follows the ...
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1answer
15 views

Compounded discount rate

The total amount of a loan to which interest has been added is $20000$. The term of the loan was for $4.5$ years. If the annual rate of interest was $6\%$ and interest was compounded annually, what ...
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2answers
17 views

Present value and discount function

What deposit made today will provide for a payment of $ \$1000$ in $1$ year and $ \$2000$ in $3$ years, if the effective rate of interest is $0.075$. Answer is $\$2540.15$ I have calculated $d=0.069$ ...
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2answers
56 views

Compound interest with penalty in withdrawal amount

Carl puts $10000$ into a bank account that pays an annual effective interest rate of $0.04$ for $10$ years. If a withdrawal is made during the first five and a half years, a penalty of $5\%$ of the ...
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1answer
23 views

Expectation equals to Black-Scholes Equation

Let $S_t$ be ageometric brownian motion with parameters $\sigma$ and $r$ and fix $T,K\in (0,\infty)$. How can I show that: \begin{align} \mathbb{E}[e^{-rT}max\{(S_T-K),0\}] & = x\Phi(d_+(T-t,x))...
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1answer
34 views

How do you calculate the breakeven amount for two credit cards with different fees and rebates?

Credit Card (CC) M offers a rebate of cash back of $a$ dollars, with NO annual fee. CC S offers a rebate of cash back of $b$ dollars, but charges an annual fee of $f$. How much must I spend ...
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1answer
36 views

Empirical Formula for Financial problem

I have a financial problem, which is strictly related to math of course. The problem states that on the last year the steel market price was about $450$ \$, and a company, that sells steel, used to ...
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0answers
24 views

Under which conditions on $\sigma_1, \sigma_2$ and $\rho_{12}$ the minimum variance portfolio involves no short selling?

If $\rho_{12} \lt 1$ or $\sigma_1 \ne \sigma_2$ then $\sigma_{V}^2$ representing the variance of the portfolio with weights $(w_1, w_2)=(s, 1-s)$ as a function of $s$ attains its minimum value at $$...
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2answers
63 views

Does the term “selling price” mean the “cost price” or the “sale price” of a product/commodity?

I have been told that the idiom "selling price" is the same as the cost price of an item, that is the amount which a seller pays to, e.g. a wholesale merchant. The seller later sells the commodity at ...
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5answers
39 views

How to calculate the price of a product without the sales tax, if we know the price including the tax and the rate of the tax?

The question is The price of a mobile phone is $8800 inclusive of a 10% GST (General Sales Tax). What is the original price of the mobile phone? This is how I approached it: The Sale Price <...
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2answers
34 views

Determine the weights in the minimum variance portfolio.

Consider three securities with the following expected returns, standard deviations of returns, and correlations between returns: $$ \begin{matrix} \mu_1=0.20, & \sigma_1=0.31, & \rho_{12}=\...
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0answers
24 views

Describe a process mathematically

I am simply wondering how to explain this process mathematically: Lets say that we have a set $A_j$ of sets $B_i$ such that $A_j$ is the set off all $B_i$ where $i\leq j$ Now lets say that set $...
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1answer
34 views

Annuity payments $2n = n$

I'm having trouble with this question: For an annuity of $3n$ payments of equal amount at periodic interest rate $i$, it is found that one period before the first payment the present value of the ...
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0answers
12 views

Is complete market or not if appreciation rate is random?

Consider the stock price process satisfies the following SDE: $dS_t=\mu_t S_tdt + \sigma S_t dW_t , S_0=s $ and the appreciation rate process $\mu_t$ satisfies the following SDE: $d\mu_t=(a-\mu_t)...
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0answers
41 views

Mean-variance portfolio probelm

So the question asks: Consider three uncorrelated stocks in the market. Each stock has variance 1. The expected returns are given by $2, 3 $ and $ 5$ respectively. Find the optimal mean-variance ...
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1answer
48 views

Minimum variance portfolio problem

So the question asks: There are N (N > 1) stocks with the same variance $σ^2$ and the same pairwise correlation coefficient γ (i.e. $c_ij$ = γ for all i = j. γ is a given constant such that 0 ≤ γ < ...
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1answer
23 views

Compute the risk measured by the standard deviations $\sigma K_1, \sigma K_2, \sigma K_3$, does this have to do with weights?

Compute the risk measured by the standard deviations $\sigma K_1, \sigma K_2, \sigma K_3$ for each of the investment projects, where the returns $K_1, K_2$, and $K_3$ depend on the market scenario: $$...
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1answer
16 views

Calculating retirement amount and time

I am asking for three inputs from the user: How much money do you want to retire?(Savings goal) Initial Investment Annual contribution Interest rate And I want to calculate the amount you end up ...
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0answers
23 views

Incompatibility of Libor and Swap Market Models

I'm trying to see why the Libor and Swap Market Models are incompatible with each other. I have found numerous sources such as the book of Brigo and Mercurio but my main source is a paper of ...
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1answer
70 views

Proportion of Stock investors owning stocks in companies

Suppose stocks of companies A, B, and C are popular among investors. Suppose $18$% of the investors own A stocks, $49$% own B stocks, $32$% own C stocks, $5$% own all three stocks, $8$% own A&B ...
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1answer
39 views

The three year bond has face value $100, and pays \$5 coupons annually, the last one at maturity. Assume that the continuously compounding rate is 7%.

The three year bond has face value USD 100, and pays USD 5 coupons annually, the last one at maturity. Assume that the continuously compounding rate is 7%. (a) Find the price of this bond. (b) ...
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1answer
34 views

Is this the correct continous compounding equivalent to daily compunding of 90%?

Find the rate of continuous compounding equivalent to daily compounding of 90%, if we assume that a year has 365 days. P=principle daily = $P(1+(\frac{0.9}{365}))^{365}$ continuous = $Pe^x$ $$(1+(\...
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2answers
32 views

why does this answer on paying a mortgage two years earlier make sense?

Suppose that you took a mortgage of 100000 on a house to be paid back in full by 10 equal annual installments, each consisting of the interest due on the outstanding balance plus a repayment of a part ...
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1answer
28 views

How to find arbitrage for forward rate of exchange?

Assume that $S(0)$ is the current rate of exchange for foreign currency. Assume that and $K_h$ and $K_f$ are rates of return on home and foreign currency if it is invested over a period $T$. (A) ...
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1answer
26 views

can someone explain how bonds work and how to calculate the value of a bond?

I am taking financial mathematics and in class I learned a formula to calculate the value of a bond, $B=\frac{F}{1+r}$ if one payment or if coumpund $B=\frac{F}{(1+r)^n}$ where $B$ is the value of the ...
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1answer
30 views

Models for Financial Economics: Determining The Amount to Lend in Synthetic Assets

You are given: (i) The price of a stock is 43.00$ (ii) The continuously compounded risk-free interest rate is 5% (iii) The stock pays a dividend of 1 three months from now (iv) A 3-month European ...
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1answer
39 views

Question About Representation of Brownian Motion

In Stochastic Calculus with Financial Applications by J. Michael Steele he makes a specific representation of a Brownian Motion using wavelets. At one point he calculates the covariance, and he uses ...