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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8 views

Lemme itô and Martingale

I want to to find values of $a$, $b$ such that the process: $$e^{W_{t}^2+at+b\int_\limits{0}^{t}W_{s}^2\,ds}$$ be a martingale Could you please help me do that Thank you
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0answers
19 views

Estimating compound growth

I have a compound interest function with the following parameters: Value at time 0 = 13.8 Interest rate = 0.05 time interval = 10 I need to check quickly, (without a calculator, only pen and paper) ...
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0answers
24 views

Call spread derivative [on hold]

delete this tbh lads, this is not the site for finance questions apparently Using the notation $V(E)$ to mean the value of a European call option with strike $E$, what can you say about ...
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2answers
21 views

Pricing call options with binomial trees (proof) [on hold]

I need assistance in proving that the following line: $$f = S_0\left(\frac{f_u - f_d}{S_0u - S_0d}\right)\left(1 - ue^{-rT}\right) + f_ue^{-rT}$$ Equals this line: $$f = \frac{f_u\left(1 - ...
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3answers
45 views

I'm trying to reverse engineer a formula to find answers without trial and error.

As an example, I need to pay €100 to this business, and there are 2 separate fees I need to also pay, the fee for the payment processor and the VAT (Value Added Tax). I know that the fee for the ...
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0answers
18 views

Solve the following stochastic differential equation dX(t) = (1 + (X(t))^2 ) X(t)dt + dZ(t) [closed]

Please solve this stochastic differential equation with steps dX(t) = (1 + (X(t))^2)X(t)dt + dZ(t)
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1answer
17 views

What is the difference between simple interest and simple discounting?

I have been given the following statements: "Simple interest: $C$ now $\equiv (1+in)C$ in $n$ years; $C$ in $n$ years $\equiv \frac{C}{1+in}$ now. Simple discounting: $C$ in $n$ years $\equiv ...
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0answers
49 views

Black-Scholes model [closed]

You take a short position in one European put option contract, with strike price 100 and maturity six months, on a stock that is trading at 100. The annual volatility of the stock is constant and ...
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0answers
21 views

Black-Scholes Merton model [closed]

Consider a Black-Scholes-Merton model with $r=0.1$, $T=0.5$ years, $S(0)=100$. Suppose the Black-Scholes price of the digital option that pays one dollar if $S(T)≥100$ and $0$ otherwise, is equal to ...
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0answers
15 views

why hull white model has normal distribution?

consider hull white model $dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$ when we solve the SDE above we have $r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha ...
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0answers
22 views

Risk-neutral (i.e. martingale) measure if density is given for a single random variable (i.e. asset)

Let $(\Omega,\mathcal F, P)$ be a probability space. And let $S : \Omega \to \mathbb R$ be a random variable, called an asset, also we are given $\pi > 0$ called a price and some $r \ge 0$ called ...
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1answer
23 views

Maximum value for a dependent variable in a marginal effect model

I am unsure as to whether my calculations are correct. Currently, the model given is: The first question was to derive an equation for the marginal effects of EDU on In(Wage). I obtained the ...
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1answer
24 views

Ratio vs Difference in terms of growth

Suppose we have the following data: start date end date quantity 01/05/2014 07/05/2015 5 07/06/2015 02/06/2016 8 What ...
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0answers
19 views

Automatic differentiation for finance

we're estimating sensitivities with automatic differentiation. What we have read about it the adjoint (reverse) should perform more efficiently than the forward mode when there are more input ...
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1answer
20 views

How would I solve for a rate that compounds m times per annum?

Please excuse me, this is my first time using the site and I have absolutely no idea what I'm doing with the notation. Anyways, I am attempting to prove that: $$R_m = ...
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0answers
43 views

Optimal Insurance Coverage - risk neutral and risk loving consumers

I'm struggling with understanding a problem in finance: We have 2 states: $S_1$: bad state $Y-K = 2000$; probability $\pi$ = 10% $S_2$: normal state $Y = 5000$; probability $1-\pi$ = 90 ...
2
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0answers
28 views

Term Structure and short rates

If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t $$ and want to find the short/spot rate $r_t$, is this simply: $$f(t,t) = f(0,t) + ...
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1answer
26 views

No arbitrage iff there EMM $P^*$ theorem [closed]

The definition of an arbitrage I was given: "An arbitrage strategy is an admissible strategy with zero initial value and positive probability of a positive final value." I think that an initial ...
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0answers
12 views

Interpretation of the Snell Envelope wrt European and American options.

In lectures we were taught that the Snell envelope U of a process Z, passes from European to American option prices. I thought that price processes are supposed to be martingales, but the Snell ...
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1answer
35 views

Promissory note example Financial Math

Brenda owes Cathy $\$8500$ and has signed a promissory note to repay the debt in 15 months from the signing date. The note was signed on December 6, 2009, and the maturity value of the note is ...
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1answer
45 views

Ornstein–Uhlenbeck SDE.

I am trying to understand the solution to the following exercise, however it is kind of poorly written. Can someone please explain it to me? For $V = (V_t)$ the solution to the Ornstein-Uhlenbeck SDE ...
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2answers
38 views

Brownian motion and covariance

Show that for $B = (B_t)$ Brownian motion, its covariance is $cov(B_s, B_t) = min(s, t)$. The solution I was given was: For $s ≤ t$, $B_t = B_s + (B_t − B_s)$, $B_sB_t = B_s^2 + Bs(Bt − Bs)$ ...
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0answers
7 views

Question on population related to simple and compound interests [closed]

The annual birth and death rates per 1000 are 39.4 and 19.4 respectively. The number of years in which the population will be doubled assuming there is no immigration or emigration is?
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0answers
26 views

Common term for “present value” and “future value”

In the past, I have always used the term "present value" for the value of a payment made at some point in time $t$ from the perspective of some other valuation point in time $T$. I did not distinguish ...
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1answer
24 views

Martingale Properties

Here is a proof of a property of a martingale $X$ relative to the filtration $(F_{n})$: $n\gt m,\\$ $\\ \\ E[X_n|F_m]=E[E[X_n|F_{n-1}]|F_m]=E[X_{n-1}|F_m]=...=E[X_m|F_m]=X_m$ In the definition of a ...
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1answer
23 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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1answer
13 views

Ito's formula for this stochastic differential - please explain this step?

Referring to those two lines, can someone please explain how those results were obtained? My understanding is, the following formula is being referenced: $$dV_t = dV(S_t,t) = \frac{\partial ...
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0answers
40 views

Is this a self-financing portfolio?

I have $S_t = 10 + B_t$, $\beta_t = 1$, $a_t = 2B_t$, $b_t = -t - B_t^2 - 20B_t$ Then the value, $V = a_t S_t + b_t \beta_t$ Is this a self financing portfolio? Note, $B_t$ is brownian motion I am ...
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1answer
46 views

Compounded Interest with Exponentially Increasing Periodic Payments

Given the formula $$v_a = p\left(\frac{\left(1+\frac{r}{n}\right) ^{nt}-1}{\frac{r}{n}}\right)$$ for the value $v_a$ of an account growing at a periodic rate $r$ with a regular deposit $p$ compounded ...
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1answer
54 views

Single factor model question, related to the benefits of diversifying one's portfolio.

The question: Suppose in a single period investment problem we may divide our wealth between n assets and that the return on the ith security is given by $r_i = \alpha + \beta_i\theta + \epsilon_i,$ ...
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0answers
11 views

Request for recommendation: Transition textbook for graduate course in mathematical finance or classical math reference book

I am looking for a well-written, theoretically rigorous textbook that contains all the mathematics necessary to transition smoothly to a graduate course in mathematical finance. I am graduating with ...
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1answer
41 views

Prove $\sum \frac{t}{(1+y)^t }= \frac{y+1}{y^2}$

I see on Wolfram Alpha that $\sum \frac{t}{(1+y)^t} = \frac{y+1}{y^2}$ when t goes to infinity. I cannot, however, proove it myself. What theory is used and how do I start the proof?
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1answer
53 views

Regarding “Two Singular Diffusion Problems” by William Feller

I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his ...
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0answers
19 views

How to use the BA II Plus financial calculator to solve for IRR and NPV?

I've calculated the answers manually but would like to learn how to do so on the financial calculator to save time on the test and minimize errors. How to do this? Problem: You have been offered a ...
2
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2answers
36 views

To mark up in retail by $20$%, do I add $0.20$ times the original cost, or divide by $0.80$?

Why is it that when I take a cost of say $\$15.60$ and want to mark the item up at retail 20% that I'm being told two different ways with two different answers? The first way (my way) would be to ...
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1answer
49 views

Recurrence relation for a mortgage

Find a recurrence relation for the amount of money outstanding on a \$40,000 mortgage after n years. The interest rate on the mortgage is 10% and the yearly payment is \$2,000( the yearly payment is ...
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0answers
20 views

Isolate n in Compound Interest Formula

How would one isolate n in this comound interest formula? Another forum says that it is impossible and can only be estimated with a series but I'm wondering if anyone can confirm that or knows how. ...
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1answer
14 views

Equivalent Interest Rates

Suppose that the APR on a certain product $x$ whose dollar value is $x_1$ is 5%. Now suppose we subtract some amount $x_2$ from $x_1$ where $x_2 < x_1$. Call this new amount $y$. How does one ...
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0answers
23 views

Find the highest price which an investor can pay and still be certain of a yield of:

I'm having trouble understanding this example in Kellison's Theory of interest: Consider a 100 par value 4% bond with semiannual coupons callable at 109 on any coupon date starting 5 years after the ...
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1answer
40 views

Is there any interpretation to the imaginary component obtained when computing the geometric mean of a series of negative returns?

When computing returns in finance geometric means are used because the return time series of a financial asset is a geometric series: $\mu_r = \sqrt[T]{\prod_{t=1}^T r_t}$ where the return is computed ...
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1answer
31 views

Properties of brownian motion

I was doing some revision and had an admittedly elementary question. My lecture notes say, the following are properties of Brownian Motion {$B_t$} (Normal or Gaussian increments) For all $s < t, ...
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1answer
25 views

Savings account interest rate

Just a brief question regarding bank interest rates, my apologies if this is a duplicate, I did a quick search but came up with no results relating to my question, surprisingly. Also, please excuse ...
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1answer
38 views

Method for finding a arbitrage opportunity when market price of call is incorrect

The solution of the Black-scholes equation is the price of a European call. And the option price assumes the underlying stock is a geometric Brownian motion with volatility $\sigma_{1}>0$. ...
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1answer
40 views

Continuous Annuity Question

I need to calculate the present value of a level continuous annuity which pays $1000/mo. for 10 years. The force of interest is 5/(3+2t). I tried taking the integral of e^(integral of force of ...
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2answers
46 views

How to find the actual doubling time with the rule of 72.

I have a programming assignment in C# from my professor that involves the Rule of 72. He clearly says that in order to find the amount of time in years it will take for an amount to double, you have ...
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3answers
39 views

Finding the probability of loss from standard deviation in normal distribution

I am unsure how to approach the following question. The returns from a project are normally distributed with a mean of \$220,000 and a standard deviation of \$160,000. If the project loses more than ...
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1answer
40 views

How to calculate inverse of Variance Gamma call price formula using Newton-Raphson search

The Variance Gamma call price formula is given by: $$C(0)= \int\gamma(R) e^{-rT} \int f\left(S(0) e^{\theta R+\omega T+\frac12 \sigma^2 R} e^{rT-\frac12 \sigma^2 R+\sqrt{T}\sqrt{R/T} \sigma ...
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1answer
35 views

Factorals with exponents. Is their a way?

I know of multiplication factorials with the 4! = 4*3*2*1 and I know of the addition with the nth triangle. I am busy deriving my own equation for something, and i am getting stuck on how to furthur ...
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0answers
29 views

Optimization of stochastic differential equations

Is there a way to optimize or maximize a set of differential equations. such that each equation is represented by a time series S_((t+1),μ) = μ*(S_(t+1)-S_t) + S_t and μ = 2/(i+1), i=1,...,n. Then I ...
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0answers
28 views

Gaussian distribution finite population with unknown cardinality

I have taken a sample population of a population with unknown size. The sample size is 54 trades. The sample mean is 2.1% (1.021) return per trade. The sample standard deviation is 0.01. 100% of ...