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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Delta Hedging a call option

I'm revising for a class test and I'm using an old past paper but I can't solve this problem and there isn't an example in my lecture notes! Consider a four-month European call option C with a strike ...
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-3 votes
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What are some examples/applications of Functions (Surjective, Bijective, Injective) in Economics? [closed]

Need help in this assignment Topic 'How Functions are used in Economics' and give and explain examples. I have searched a lot on google, but I only find thing related to profit, loss, revenue etc ...
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1 answer
25 views

Deriving the Nelson-Siegel model

The Nelson-Siegel (1987) model states that the instantaneous forward rate at maturity, $r(m)$ is given as: $$\beta_0 + \beta_1 e^{(-m/\tau)} + \beta_2\left(\frac{m}{\tau}e^{(-m/\tau)}\right) \tag{1} $$...
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Double infinite product $\prod_\limits{0<i<j<\infty} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\right\}$

Question Compute the products: $\prod_\limits{0<i<j<\infty} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\right\}$ $\prod_\limits{0<i<j<2020} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\...
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finance mathematics

A factory contributes £1 million into a small city. It is estimated that 75% of that money is respent back into the community. Economists assume that the money is re-spent again and again at a rate of ...
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38 views

Inflection point of Black & Scholes formula for implied volatility using Newton - Raphson Algorithm

We have the formula for pricing the call type of an option under the Black Scholes formula (European type) : $$C(S_t,t;K,T)= S_t \cdot e^{-D(T-t)} \cdot\Phi(d_+) - K\cdot e^{-r(T-t)} \cdot\Phi(d_-)$$ ...
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Investment Formula for Initial and Final Investment and Periodic Contribution.

I am able to re-arrange the following formula to calculate Initial Investment, final investment and Periodic Contribution. But how do I manipulate it to calculate values at the beginning of periodic ...
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22 views

How to prove combination by mathematical induction? [closed]

Prove using previous calculations and mathematical induction that n!/{(n-j)!*j!} I'm not able to solve B3. I don't know where to start the induction base. These are ...
1 vote
1 answer
51 views

Time Value of Money

Santana is planning for his pension savings: He plans to work for $20$ years and then retire. He expects to receive an annuity income of $\$\, 20000$ at the start of each year of his retirement for $...
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1 answer
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How to derive the following in Ho Lee Model?

I am trying to understand the proof of the zero bond price $Z(t)$ of the Ho-Lee model which is the unique solution of the following SDE: $$ dZ(t) = -Z(t) [ \sigma(T-t)dW(t) + [ \int_t^T \alpha(t,u)du -...
2 votes
1 answer
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How to solve SDE: $ dX_t = \bigg(\sqrt{1+X_t^2}+\frac{1}{2}X_t \bigg)dt + \sqrt{1+X_t^2} dW_t$

Problem: I would like to find a solution to the following SDE: $$ dX_t = \bigg(\sqrt{1+X_t^2}+\frac{1}{2}X_t \bigg)dt + \sqrt{1+X_t^2} dW_t$$ Calculations: by Ito's lemma: $$df(t,X_t) = \Bigg(f_t(t,...
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Why do we need the AEP algorithm? The discrete case

I recently found a paper describing the AEP algorithm, an algorithm to compute $$P(X_1 + X_2 + \dots + X_d \leq s)$$ , given the joint distribution of $(X_1, \dots, X_d)$. O.k., why do we need this ...
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22 views

Value at Risk for Pareto Distribution

i want to calculate the value at risk for pareto disttribution and while i was searching in the web i came across to this one here. Trying to do it by myself i did : \begin{equation} \label{eq1} \...
1 vote
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27 views

Prove that the stochastic process $s_t$ follows a normal distribution where the mean and the variance are functions of time in each case.

The two basic models of finance are the following: $\textbf{The Samuelson SDE (aka Black - Scholes - Merton model):}$ Suppose that $Z=\left(Z_t, t\in\mathbb{R}^{+}\right)$ is a Wiener process (aka ...
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25 views

Brownian Motion with replicating portfolio

Consider Black's model, where price dynamics of futures contract $f_t=f_s(t,T)$ with maturity date T=3, is describted by stochastic differential equation: $$ df_t=\mu f_tdt+\sigma f_tdB_t, $$ where $...
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why does maximized utility of consumption(merton problem) exist?

Agent controls his proportion of wealth invested in the stock $ \alpha_t $ and his consumption rate $c_t$. Dynamic of wealth:$ dX_t=X_t[(\alpha_t(u-r)+r)dt+\alpha_t \sigma_t dW_t]-c_t dt $ value ...
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11 views

Equivalent definition of Markov process from Shreve

In the book "Stochastic calculus for finance-II" http://home.ustc.edu.cn/~matheming/Steven%20E.%20Shreve-Stochastic%20Calculus%20for%20Finance%20II.pdf , the author mentions two definitions ...
1 vote
1 answer
37 views

Price of an option that pays $1$ when the stock hits $\$H$ for the first time

I am trying to understand the no arbitrage argument for to determine the price of an option that pays $1$ when the stock hits $\$H$ for the first time. The current price of the stock is $\$1$. The ...
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42 views

How to solve this integral about the normal distribution

This question comes from the Merton-Vasicek model for credit risk. The distribution of the total losses $L$ a bank can suffer in a credit risk event has the following cumulative distribution function $...
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1 vote
1 answer
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Is the weighted average interest rate (aka WACC) strictly less than or equal to the combined value of the interest on those investment kept seperate?

When combining multiple loans with separate interest rates into one interest rate, is it possible to reason about if the resulting interest rate will always be less than that of the combined ...
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1 answer
27 views

Find out if a variable is a martingale (discrete time)

Let X be a stochastic variable on a finite, filtered probability space. If $$ M_1(t) = E(X| \mathcal{F}_t) $$ Is $M_1(t)$ a martingale? I have a lot of trouble checking if variables are martingales or ...
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16 views

Multi-variable chain rule where one variable is a function of others

I’m trying to understand the derivation of the Black-Scholes equation for futures price. Here, we have a PDE for V(S,t) in S and t, and want to substitute S for F(S,t), as presented in this question: ...
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46 views

Evaluation of a path integral for Call option on Zero Coupon Bond

I am given the following identities $$ Z[J,t_1,t_2]=\int D W e^{\int_{t_1}^{t_2}dtJ(t)W(t)}e^{S}=e^{\frac{1}{2}\int_{t_1}^{t_2}dtJ(t)^2} $$ $$ \int_t^Tdx\alpha(t,x)=\frac{1}{2}\left[\int_t^Tdx\sigma(t,...
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1 vote
1 answer
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Basic concepts of Black Scholes formula

The Black Scholes formula gives the formula for European calls, for stock with no dividends, $$ c = S N(d_1) - K e^{- r (T-t)}N(d_2) $$ with $S$ the price of stock at time $t$, $T$ is the maturity ...
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0 votes
1 answer
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Given a sum, how to find a set of numbers that grow by a percent of the preceding number that total to the sum.

This is used for calculating monthly costs that increase by a certain percentage month-over-month when given a forecasted annual budget. An annual increase of 15% annually or 1.28% monthly (equalling ...
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24 views

When finding compound interest, the term "annual interest rate" seems misleading?

The equation for compound interest can be derived by considering the recurrence relation $a_{n+1}=a_n+sa_n=(1+r)a_n$ where $s>0$ is the growth rate, $a_n$ is the current balance, and $a_{n+1}$ is ...
0 votes
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16 views

Upper hedging price no-short-selling

Let $q,\sigma>0$ and $r$ constants and a European call-option $Y=(P(T)-q)^+$, under no ho-short-selling constraint $K=[0,\infty)$ with $\tilde{K}=\{x\in\mathbb{R}|x\pi\geq0, \pi\in K\}$. I would ...
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2 votes
2 answers
76 views

Discrepancy between two expected profit formulas

I have two formulas for calculating the expected ratio of the account value after a trade to before it (we'll call this expected account ratio or EAR) that both make intuitive sense to me, yet they ...
0 votes
0 answers
45 views

Why is this not correct way of calculating real value ( stupid question)?

inflation $= 3\%$ per year, amount of time $= 5$ years, nominal value = $\$5070.6$, real value = ? real value* $1.03^5 = \$5070.6 $, real value = 4373.94, this is correct answer why not : $5070....
1 vote
0 answers
40 views

How to calculate years to gain 5 million net worth?

Givens: 20% annual growth 30k payments/year FV of Annuity$=P[\frac{(1+r)^n-1}{r}]$ $P=$ Periodic payment $r=$ Rate per period $n=$ Number of periods $5000 = 30\cdot((1+0.2)^n-1)/0.2$ https://www....
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98 views

Pathwise differentiation under integrated measurement (stochastic volatility) models.

Setup Consider the (general) stochastic volatility model \begin{align} \mathrm{d}X_t &= \mu^X(X_t)\mathrm{d}t + \sigma^X(X_t)\mathrm{d}W^X_t \\ \mathrm{d}Y_t &= \mu^Y(X_t)\mathrm{d}t + ...
1 vote
0 answers
55 views

Liam opens a bank account and makes deposits at a continuous rate

Liam opens a bank account with an initial balance of $2000$ dollars. Let $b(t)$ be the balance in the account at time $t$. Thus $b(0)=2000$. The bank is paying interest at a continuous rate of $3\%$ ...
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0 votes
1 answer
39 views

Most Efficient Way Saving Money (Compound Interest)

If I have four children and I want to ensure each have £8,500 come there 18th birthday, assuming that I put money in each month, and gain 10% interest per year (taking into account compound interest). ...
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1 answer
43 views

Risk free profit with a put option

From Terence Tao Blog the price of a put option at time $t_0$ cannot exceed the strike price P at time $t_1$. The reason is that otherwise there would be an arbitrage opportunity. Everyone in the ...
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0 votes
1 answer
40 views

Annuity future value, getting extremely low value

The question states: Relative to your current age, the present value at age 67 found in question #1 becomes the future value you need to obtain given your current age now, How much money do you need ...
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How much loan you owed after 20 years

Statement: Paying down a mortgage: you took a $\$200000$ home mortgage at an annual interest rate of 3%.suppose that the loan is amortized over a period of 30 years and let P(t) denote the amount of ...
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33 views

*Simple* nominal interest rates

I have just finished learning nominal interest rates in my actuarial math class and I am curious on why the nominal rate is always tied with a compounding interest model. I tried to do some proving on ...
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1 vote
1 answer
83 views

Whose statement about Brownian Motion correct?

My classmate and I are disagreeing about a point about $X(t)$, a BM. His statement is that $X(t)$ is normal, my statement is that only increments of $X(t)$ are normal (thus (thus $X(t) - X(0)$ is ...
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1 vote
1 answer
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Nash equilibrium in insurance pricing

The insurance market is considered to be a competitive market, so in order to study competition to determine a competitive premium, game theory seems to be a useful tool for studying those situations. ...
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0 votes
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21 views

Calculating APY from N percent earned over X days

What equation could I use to calculate the APY where percent interest and number of days are variable? For instance, suppose I invested $100 and earned 10% interest in 5 days.
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0 answers
26 views

Turnbull and Wakeman Asian Option Price Derivation

I'm attempting to derive the Turnbull and Wakeman (1991) asian option pricing algorithm. I am currently stuck at deriving the cumulants. First, we let: $$A(t)=\frac{S[T-n+1]+S[T-n+1]R_{(T-n+1)+1}+S[T-...
0 votes
1 answer
68 views

Expected daily return given binary outcome

Every day a trader either makes $50$% with probability $0.6$ or loses $50$% with probability $0.4$. The average return per day is: $1-\exp (0.6 \ln 1.5+0.4 \ln 0.5)=-3.34 \%$ How is this average ...
3 votes
1 answer
56 views

Integration of Dirac and Heaviside Functions - Quant Finance

Need some help in understanding the author's simplification here (from Emanuel Derman's The Volality Smile). The equation is set to $V(S, t)$, which represents the value of a derivative security at ...
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0 answers
37 views

What is the total cost of 55k units?

Kaplan Ltd. is one of the premier industries that manufactures fan belts for automobiles. If a belt is not sold in the year of production, it becomes defective and has to be scrapped. The following ...
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0 answers
46 views

Stock price change, quantitatively

People say that when you sell a stock its price goes down, and when you buy it the price goes up. This is clear, qualitatively, but I never really understood how much the price changes. Could you ...
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0 votes
1 answer
24 views

Correct calculation of assets profit and loss

I would like to make some profit and loss calculations based on the market prices. Scenario 1 (profit): Given a crypto is worth $1500. I've made a purchase for <...
2 votes
2 answers
61 views

Using $(I^{(m)}a)^{(m)}_{\bar{n}|}$ to solve for the present value of an annuity where payments increase monthly

I've seen this answer and I understand the methodology, but I am wondering why my original solution using a different method did not work. This is the sample problem in my study guide for Exam FM: ...
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2 votes
1 answer
71 views

Prove that $V(t)=e^{-r(T-t)} \mathbb E\left[S_{t}\right]$ satisfies the Black–Scholes PDE

Let us consider the geometric Brownian motion: $$ d S_{t}=\mu S_{t} d t +\sigma S_{t} d B_{t} $$ where $\mu$ is the drift, $\sigma \in \mathbb{R}^{+}$ is the volatility and $B_{t}$ is the Wiener ...
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2 votes
1 answer
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Cauchy–Schwarz-like inequality justifying the maximum of Sharpe ratio.

Let ${\bf \Delta}=(\delta_1,\,\ldots,\,\delta_n)\in\mathbb{R}^n\setminus(0,\,\ldots,\,0)$, ${\bf 1}=(1,\,\ldots,\,1)_{1\times n}$ and $\bf 1\Delta$$^T=0$, where $T$ denotes transpose. Let ${\bf \mu}=(\...
2 votes
1 answer
43 views

Current value of option

risk free rate=$r$ volatility of stock price=$\sigma$ continuous dividend rate=$q$ $a>0,K>0$ If your stock price S becomes below $K$ at maturity T, the option A pays you $aS_T$. Otherwise, this ...
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