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.
2,504
questions
-2
votes
0
answers
9
views
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 ...
- 9
-3
votes
0
answers
30
views
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 ...
0
votes
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:
$$\beta_0 + \beta_1 e^{(-m/\tau)} + \beta_2\left(\frac{m}{\tau}e^{(-m/\tau)}\right) \tag{1}
$$...
- 145
0
votes
0
answers
36
views
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}}\...
- 1
0
votes
0
answers
34
views
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 ...
0
votes
0
answers
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) :
$$C(S_t,t;K,T)= S_t \cdot e^{-D(T-t)} \cdot\Phi(d_+) - K\cdot e^{-r(T-t)} \cdot\Phi(d_-)$$
...
0
votes
0
answers
11
views
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 ...
0
votes
0
answers
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 $...
- 21
0
votes
1
answer
71
views
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:
$$ dZ(t) = -Z(t) [ \sigma(T-t)dW(t) + [ \int_t^T \alpha(t,u)du -...
- 145
2
votes
1
answer
45
views
How to solve SDE: $ dX_t = \bigg(\sqrt{1+X_t^2}+\frac{1}{2}X_t \bigg)dt + \sqrt{1+X_t^2} dW_t$
Problem:
I would like to find a solution to the following SDE:
$$ 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,...
- 85
0
votes
0
answers
12
views
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 ...
- 699
0
votes
0
answers
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
0
answers
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 ...
- 51
0
votes
0
answers
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 $...
- 13
0
votes
0
answers
15
views
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 ...
0
votes
0
answers
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 ...
- 87
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 ...
- 2,859
0
votes
0
answers
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
$...
- 2,848
1
vote
1
answer
77
views
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 ...
- 113
0
votes
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 ...
- 230
0
votes
0
answers
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: ...
- 11
0
votes
0
answers
46
views
Evaluation of a path integral for Call option on Zero Coupon Bond
I am given the following identities
$$
Z[J,t_1,t_2]=\int D W e^{\int_{t_1}^{t_2}dtJ(t)W(t)}e^{S}=e^{\frac{1}{2}\int_{t_1}^{t_2}dtJ(t)^2}
$$
$$
\int_t^Tdx\alpha(t,x)=\frac{1}{2}\left[\int_t^Tdx\sigma(t,...
- 195
1
vote
1
answer
43
views
Basic concepts of Black Scholes formula
The Black Scholes formula gives the formula for European calls, for stock with no dividends,
$$
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 ...
- 2,859
0
votes
1
answer
44
views
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 ...
0
votes
0
answers
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 ...
- 646
0
votes
0
answers
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 ...
- 167
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....
- 11
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....
- 127
0
votes
0
answers
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\%$ ...
- 1,721
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). ...
- 123
0
votes
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 ...
user1095117
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 ...
- 169
0
votes
0
answers
58
views
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 ...
- 115
0
votes
0
answers
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 ...
- 75
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 ...
- 333
1
vote
1
answer
55
views
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.
...
- 11
0
votes
0
answers
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.
- 133
0
votes
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-...
- 11
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 ...
- 503
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 ...
- 59
0
votes
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 ...
- 160
0
votes
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 ...
- 11
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 <...
- 103
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:
...
- 23
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 ...
- 7,498
2
votes
1
answer
93
views
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 ...
- 159