Questions tagged [finance]

Questions having to do with financial mathematics. This is not a tag about financing, which is not within the scope of mathematics defined by the help center: http://math.stackexchange.com/help/on-topic Topics may include: option pricing, arbitrage theory, market completeness, and applications of stochastic analysis to finance. Please note that for questions in quantitative finance, quant.stackexchange.com is perhaps a better site.

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Estimating the price of a put using the probability of bankruptcy

If there is a threshold price called $K_1$ for which the only way the stock price could fall below $K_1$ is if the company goes bankrupt over the period. Assuming that were to happen, the stock price ...
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Portfolio value on selling and buying calls

Consider buying a call option with strike $K − δ$, selling two call options with strike $K$ and buying a call option with strike $K + δ$, where $K, δ > 0$ and $K > \delta$, all with maturity $T &...
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Expected value of powers of Brownian Motion

At the moment I am following a uni course on Financial mathematics, the current subject is Brownian Motion. A subject I have now encountered a couple of times which I don't really understand is the ...
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Expectation of Brownian function.

$(W_t)t≥0$ be a standard one-dimensional Brownian motion Let $f : R → R$ be a given function and $0 ≤ s ≤ t$. Write down an expression for $E(f(W_t)|W_s = x)$ in terms of $ϕ(x) = Φ′(x)$ Where $Φ(x)$ ...
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Sampling distribution of GBM Maximum-Likelihood estimator

Given the geometric Brownian diffusion $$ X_t = \mu X_t dt + \sigma X_t d W_t$$ I learnt that its maximum likelihood estimators are the following as this web article suggests $$\hat \mu = \frac{\delta ...
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Find the expected returns of a stock

I would appreciate help with how to get five stocks expected returns as a row matrix in order to solve the task below: Compute the weights of the market portfolio $w_m$.Consider the risk-free rate to ...
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Rice formula on time series

I have a time-series which is a mean-reverting spread with a long-run mean (equilibrium) close to zero. I am looking to use the Rice formula to get the level crossings rate, e.g. how many times the ...
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What does it mean to find the mean vector using simple returns

What does it mean to find the mean vector using simple returns from five selected stocks? I have chosen five stocks and collected their (weekly) data over a 10-year period. And then I have used that ...
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Daily log return of a stock [closed]

How do you calculate the daily log return of a stock? When I have googled this I seem to find two different answers. The first one is by taking the natural log of the ending value divided by the ...
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Question on use of indicator function

For my Financial Mathematics course I have the following exercise (with solutions): I don't really understand the start of the solution of (B). More specifically I do not understand why the $1_{\{X=\...
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2 answers
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How to calculate loss if I give things away for free?

Let's say I sell product A for 100 dollars and it costs me 20 dollars to make A (including all costs associated). So, for each A I sell, I am making a hefty 80-dollar profit. However, if I run a ...
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3 votes
1 answer
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Black-Scholes model with a derivative with payoff $S_{T}^{3}$

Given a Black-Scholes Model and a derivative with payoff $S_{T}^{3}$ at time $T$. Check that the value of that derivative at time t is $V_{t} = g(t, T)S_{t}^{3}$, where $g(t, T)$ has to be determined. ...
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1 vote
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Compounded Annual Growth Rate with Negatives

I was trying to figure out the CAGR of EBIT of certain companies that filed for bankruptcy. Needless to say, most of these show a negative trend in their EBIT values, often starting from a positive in ...
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Stochastic Portfolio Optimization with Recourse

I am given the following problem from a tutorial in my course: (Portfolio Optimization with Recourse). You have £10,000 to invest (without short selling) in a portfolio composed of eight leading ...
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Rearranging Annuity formula

Can someone please show me how this is rearranged/simplified from this: PV = CF[(1-(1+r)^-n)/r] to PV= CF(1/r)[1-(1/((1+r)^n)] with steps, will help me wrap my head around it. Also, how does it go ...
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Conditional expectation implied by expectation of product

For uni I have got the following assignment, and I've gotten stuck at (b). I have proven that E[1_{bi}X] = E[1_{bi}X], but don't know how I would get from this to the required conclusion. Does anyone ...
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A word problem on a sequence of financial values

I need help on a Series and Sequence Assessment Task. PROBLEM DESCRIPTION A business man in Minnesota sets up a prize fund of $\$3000$ with a single investment to provide an annual prize of $\$200$. ...
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Variant on the Monty Hall Problem ($4$ Doors, $2$ Prizes)

I had the following question as part of a Finance course I'm doing. I don't struggle too much with Monty Hall problems in which there are $3$ doors, but this one with $4$ doors ($2$ with Prizes and $2$...
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1 vote
1 answer
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Bernoulli Model: Help calculating variance of asset price?

I'm currently looking at the Bernoulli Model for an asset, with spot price $S_{0}$, which can rise to $uS_0$ with probability $p$ or drop to $dS_0$ with probability $q = 1-p$, over a time of $\delta t$...
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Solve inequality with exponential

I recently need to solve the following equation: $e^{\frac{-A_1^2}{2c^2}}+1>e^{\frac{-A_2^2}{2c^2}}$ where $A_1=ln(K_3/K_1), A_2=ln(K_3/K_2)$ They are actually strike prices (in finance) so $K_1<...
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What are common ways to realistically simulate the stock market using historical market data? [closed]

I am currently using the FinRL library to try to automate Trading using Reinforcement Learning. However, I wanted to understand how FinRL simulates the stock market using historical data. I read here ...
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Black-Scholes Model for options pricing

The Black-Scholes pricing formula for a European call option is given by: $$C(S,t) = SN(d_+) - E\exp(-r(T-t))N(d_-)$$ where $S\ge0$ is the spot price, $t\le T$ is the time,$T\ge0$ is the expiry date, $...
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Price of forward contact

At a market there is an index S, which purchase price at the moment t=0 is 70\$ and selling price is 71\$. This index pays a dividend continuously of 0.7%. At the market is available 2-years forward ...
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Why are these equations describing the variance of residuals equivalent (single-factor model in finance)? [closed]

Residual Equations The above equations describe the residual variance in the single-factor model in finance. I'm struggling to understand why they are equivalent, specifically why the fourth and fifth ...
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Perpetuities with differing payment and compounding periods in practice

I've been studying annuities and perpetuities recently and have tried to solve a simple problem using them. I want to find the lump sum I'd need now so that I could withdraw $\$10$ each month forever. ...
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Proof that existence of specific process implies no numeraire strategies

Consider a discrete-time market with $n$ assets and (possibly negative) prices $(P_t)_{t\geq0}$. A numeraire strategy is a self-financing investment strategy such that the wealth process is almost ...
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Variance of Stock Price - St - given by GBM

I am working on a problem, where I'm interested in computing the variance of the stock price in the next two years. Using the GBM notation for the stock price, I can write St as $ S_t = S_0e^{(\mu - \...
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What is the formula of an APY with an initial borrow fee?

I am working on a platform where lenders can lend an amount of money and there are borrowers who can borrow from that pool. All borrow amounts are subject to a yearly interest rate (4%) compounded on ...
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Mathematically, how do you show that prices follow a lognormal distribution?

In finance, the price of an asset is given by the following formula. r=returns Pt = Price at time "t". P0 = Actual Price. Then. $r=\ln (\frac{P_t}{P_0})$ We assume that prices are Log-...
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2 answers
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How would I re-write a mortgage equation to solve for the APR, knowing the weekly payment?

I'm playing around with Australian house sales and rental data, and I've noticed that over the last couple of years, more than half of all renters are paying more than the entire owners mortgage (in ...
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What is the fastest way to find the product price in combination with an advertising value

I would like to sell many products on Ebay, so it is important for me to find the best price for each item. The lower the price, the better my search ranking (I would like to get on rank 1). So I ...
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Calculate effective rate of the loan - additional fees

I am having following parameters: ...
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0 answers
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Expectation of max function on standard normal variable [closed]

I was asked the following question in a tutorial: $E[\max(ae^{bZ}-c,0)]$, where $Z \sim N(0,1)$ According to the answers, this expectation is equal to $ae^{b^2/2}\Phi(b+\frac{1}{b}\ln(a/c))-c\Phi(\...
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Pricing a call option in the Black Scholes Market - calculation steps

I am working on computing the price of a standard European call option under a Black-Scholes market. Using knowledge of the payoff, I can split the calculation into: $ e^{-rT}(E[S_t] \mathbb{1}_{S_T &...
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Deriving Geometric Brownian Motion solution - dSt multiplication misunderstanding

I am working on the general problem of applying Ito Lemma on function $f(t, W_t)= lnS_t$ - similarly to this question: Deriving Geometric Brownian Motion's solution?. Writing down the equation I ...
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Show that the cash flow of a caplet is equivalent to cash flow of put options

I study Filipovic's book "Term-Structure Models A Graduate Course". I came across exercise 2.7, which wants me to show that the cash flow of the $i^{th}$ caplet at time $T_i$, i.e $$\delta(F(...
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2 votes
1 answer
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Why does interest rates need to be below thee increase of a stock for no arbitrage [closed]

I'm doing a mathematical finance module in discrete time. One of the lemmas given is $0<d<1+r<u$ there is no arbitrage opportunity. Where $d$ is a multiplier for when the stock price goes ...
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1 vote
1 answer
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Discrete time forward rate

I'm reading Jarrow and Turnbull (1997) . They defined p(t,T) as the time t price of a default free zero coupon bond paying a sure dollar at time T where $0\le t \le T$ (in year) . They also defined ...
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1 vote
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Stock price increases twice is a stopping time? [closed]

This question is related to Stochastic Calculus for finance. I don't have a strong foundation on probability theory so I'm struggling to do basic exercises. How to show that the first time stock price ...
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2 votes
1 answer
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Is every càdlàg process locally bounded?

Setting We work on a filtered probability space with finite time horizon $T$ and let $X=(X_t)_{t\in[0,T]}$ be an adapted càdlàg process. Question Is $X$ locally bounded? "Locally bounded" ...
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5 votes
4 answers
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Why does compound interest exist?

Background My understanding is that compound interest arises in the following way: The bank offers its clients some interest rate $r$ on an account with principal $P$ that yields $rP$ after some time $...
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Why Mr.Brown Still get the interest?

here is the example in broverman's book A promissory note is a short-term contract (generally less than one year) which requires the issuer of the note (the borrower) to pay the holder of the note (...
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Derivative pricing and porftofolios

I am struggling a little in understanding a passage in "PDE representations of derivatives with bilateral counterparty risk and funding cost" by Burgard and Kjaer: "As in the usual ...
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How to calculate APY from APR?

I'm trying to figure out how some investments based on compounding interest with bi-weekly contributions compare. One lists the APY, and the other lists the APR. I'm writing a program for comparing ...
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5 votes
1 answer
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Submartingale under equivalent change of probability measure

Setting We work on a filtered probability space with finite time horizon $T$ and let $Q$ be a probability measure equivalent to $P$. Let $\rho_t=E[dQ/dP |\mathcal{F}_t]$, then $\rho$ is a martingale. ...
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2 votes
1 answer
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Is there a math formula to calculate the following math problem? (Days of spending based on income)

I would like to learn how to calculate the following in the most efficient way possible. I guess there is a math function existing that could do that. Right now i am using google sheet functions, but ...
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2 votes
1 answer
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Modelling difference in stock prices after a Brownian motion

Question Stocks $A$ and $B$ open on trading day at the same price. Let $X(t)$ denote the dollar amount by which stock $A$'s price exceeds stock $B$'s price when $100t\%$ of the trading day has elapsed....
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How to compute interest paid during a specified period for an irregular cash flow

I want to know how to compute how much interest was paid during a specified period for an irregular cash flow which varies by amount, frequency, and transaction direction. Furthermore, the specified ...
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2 votes
1 answer
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Calculate maximum affordable property value for given capital, taking stamp duty into account

I want to calculate maximum affordable property value for given capital, taking stamp duty into account Assume the only house-purchasing costs are the house price plus (UK) stamp duty. For any given ...
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2 votes
1 answer
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How to make sense of the formula for the Sum of Geometric Series with finite terms?

this is my first ever question here. My apologies if my formatting is a bit off, still learning. I am now studying the topic: geometric series with finite terms. I am having a very hard time relating ...
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