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 Please note that for questions in quantitative finance, quant.stackexchange.com is perhaps a better site.
2,031
questions
-1
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12 views
What are some of the best books to learn Mathematics for Quantitative Finance and Machine learning?
I am finding it difficult to understand the math behind quantitative finance and machine learning.
Can you tell what are the math concepts required to master these areas and also suggest some of the ...
0
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0answers
13 views
How to use Itô's lemma to find expressions for mu and sigma?
if $πS = \mu Sππ‘ + \sigma Sππ§$ and
$ππ = \mu_c πππ‘ + π_c πππ§$ , how do i use Ito's lemma to find expressions for $\mu_c c$ and $\sigma_c π$?
Thanks.
0
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0answers
14 views
What kinds of mathematics are used to explain the theories of valuation?
Using economics as example, Marx "explained" to us "philosophically" what happens when capital is "created" from an initial value:
D -> M -> D' (with D' > D)
but even though βdas Kapitalβ is ...
1
vote
0answers
8 views
Non-uniqueness in the $L^1$ martingale representation
Let $\xi \in L^1(P,\mathfrak F_T)$ on some probability space with measure $P$, supporting a Brownian motion, we consider the augmented filtration $\mathfrak F$ associated to $W$, and a time $T>0$. ...
0
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0answers
9 views
What happens to the Black-Scholes pricing formula for a European Call Option when there is a rise in interest rates?
The topic
The Black-Scholes pricing formula for a European Call Option C (S; t) is given by the picture. I'm puzzled about what will happen to the Black-Scholes pricing formula for a European Call ...
0
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0answers
17 views
Acturial Studies: what is the formula for compound discount with simpe discount over the final fractional period?
What is the formula for compound discount with simple discount over the fractional period. I could not find the formula in my book and when I looked online the ones that had the formula where answers ...
0
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0answers
19 views
Black-scholes model for binary options [closed]
Also check if the formula found satisfies Black-Scholes partial differential equation. Thank you!!
enter image description here
0
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1answer
25 views
Can a unit of currency be taken to n power? [duplicate]
I've been building a unit & rate library for a forex trading algorithm and I realized I didn't have an answer to this question:
Can currencies be taken to the Nth power?
Unlike physical units, ...
-1
votes
0answers
15 views
Show that the Value-at-Risk (VaR) at confidence level $c = 95\%$ is $\text{VaR} = 1$. [closed]
Suppose an asset has return $K = 1$ with probability $\frac{1}{2}$ and $K = β1$ with probabilty $\frac{1}{2}$.
Show that the Value-at-Risk (VaR) at confidence level $c = 95\%$ is
$\text{VaR} = 1$.
2
votes
1answer
37 views
Actuarial theory of interest question with effective discount
I have spent several hours trying to solve this problem. $A$ and $B$ both open up new bank accounts at time $0$. The principle for $A$ (the amount deposited at $t=0$) is $100$. The principle of $B$ is ...
0
votes
0answers
28 views
Method of images for Black-Scholes PDE
The paper.
Let $\mathcal{L} = \partial/\partial t + \frac{1}{2}\sigma^2x^2 \partial^2/\partial x^2 + rx \partial/\partial x - r $ be the Black-Scholes operator.
In the paper he mentioned about the "...
3
votes
1answer
44 views
Find Swap Rate given Spot Interest Rate Curve
Question: Spot interest rate curve:
$$\begin{array}{c|c}
t & r_t \\ \hline
0.25 & 1.50\% \\ \hline
0.5 & 1.65\% \\ \hline
0.75 & 1.79\% \\ \hline
1 & 1.92\%
\end{array}$$
I ...
-1
votes
1answer
22 views
Simple finance question [closed]
Is this statement correct? :
Let's say I buy an apple for 1 dollar from a Mexican producer, pack it and export it at a cost of another dollar, and manage to sell it for 4 dollars to the prince of ...
0
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0answers
12 views
Deferred social security break-even with discount factor
Zero Discount
Solving for break-even for social security when not using discounted cash flows is simple.
After full retirement age, an 8% increase is added to the base amount for each year of ...
1
vote
0answers
77 views
+50
Modeling Via ODE System
My Problem
JosΓ© invests $\$2500$ of his $\$5000$ into an account with a $6\%$
annual interest rate, compounded continuously. If he continuously
deposits an additional $3.5\%$ of his annual ...
0
votes
0answers
20 views
Covariance of $X$ and $M$
I was wondering how we compute the covariance of $r$$x$ and $r$$m$. Here is the problem:
We know that:
For stock $X:
E(r_x) = 0.21$, and $Stdev(r_x) = 0.15$.
For stock $Y:
E(r_y) = 0.15$, and $...
0
votes
0answers
24 views
I'm helping out a student with a business calculus question and his textbook does not help him solve this specific problem.
Kate is in the 27% bracket and has 21,000 dollars available for investment during her current tax year. Assume that she remains in the same tax bracket over the next 10 years, and determine the ...
1
vote
1answer
49 views
Proof of Expected value of Brownian Motion
Consider the following exercise:
Let $T_{[-a,a]} = \inf \{t: B_t \notin [-a, a] \}.$ Show that $E[T_{[-a,a]}]$ $=$ $a^{2} \times E[T_{[-1,1]}]$.
Please tell me if this reasoning is correct:
$T_{[-a,...
1
vote
1answer
27 views
Properties of Brownian Motion (Expected value)
Consider the following exercise:
Let $T_{[-a,a]} = \inf \{t: B_t \notin [-a, a] \}.$ Show that $E[T_{[-a,a]}]$ $=$ $a^{2} \times E[T_{[-1,1]}]$.
Please tell me if this reasoning is correct:
$T_{[-a,...
0
votes
1answer
37 views
Itô's formula and Integration by parts
Could someone please help with this question:
By applying the generalized ItΓ΄βs formula to the 2-dimensional process $ {\{(Xt,Yt),t \ge 0 }\}$ with the function $ F(x,y) = xy $, show the integration ...
0
votes
0answers
17 views
Distribution of Itô Formula (Brownian Motion) [duplicate]
can you please help me with this question:
Let $f: [0,T] \to \mathbb R$ be a deterministic function, with $\int_{0}^{t} {f^{2}(s)ds < \infty}$. Prove that $\int_{0}^{t} {f(s)dW_{s}$ has normal ...
0
votes
1answer
22 views
Brownian motion and expectation
I'm having trouble solving the following exercise:
Let $T_{[-a,a]} = \inf \{t: B_t \notin \{-a, a\} \}.$ Show that $E[T_{[-a,a]}]$ $=$ $a^{2} \times E[T_{[-1,1]}]$.
I don't see how I can solve this. ...
1
vote
0answers
24 views
Bank Interest in Terms of Convolution
Example 4 from an old set of MIT notes asks:
Bank interest. On a savings account, a bank pays the continuous
interest rate $r$, meaning that a sum $A_0$ deposited at time $u = 0$ will by
time $...
1
vote
1answer
56 views
Present value of varying annuities
I need help with the following questions. In order to answer these questions, I am only allowed to use two of the tables of values for $\ddot a_n , \bar a_n$ (continuous annuity) and $(I \ddot a)_n$ ...
0
votes
1answer
32 views
Closed form for duration formula
so I was trying to prove this closed form for bond duration formula: D=1+$\frac{1}{r}$ + $\frac{T(r-c)-(1+r)}{c((1+r)^T-1)+r}$ where r- yield to maturity, c-coupon rate,T-time to maturity. I made some ...
0
votes
0answers
18 views
How to find the derivative of a multi-factor brownian motion model of asset prices.
Does anyone know how to find the derivative for a multi-factor geometric brownian motion model ($ \frac { dS_{i}}{S_{i}} $). I have seen solutions for the standard GBM model however I suspect that the ...
1
vote
1answer
23 views
Mean and Variance of Geometric Mean Reverting Process
For the geometric mean-reverting process $dX_{t} = k(\theta - logX_{t})X_{t}dt + \sigma X_{t}dW_{t} $ it is possible to obtain the solution:
$log(X_{t}) = e^{-kt}log(X_{0}) + (\theta - \sigma^2/(2k))(...
1
vote
1answer
43 views
How to solve the Black-Scholes equation using Feyman-Kac with two underlying assets?
If we have the following Black-Sholes equation (PDE) for two independent stocks with the same volatility:
$$f_t +\sum^{d=2}_{i=1}rs_i\frac{\delta F}{\delta s_i} + \sigma^2\frac{1}{2}\sum^{d=2}_{i,j=1}...
2
votes
1answer
61 views
How much bigger is the amount of security papers when comparing strategy X to strategy Y?
I'd like to ask the following question:
Mr. X has a certain capital (10000 Dollars).
He buys one security (from am company, say) now.
The present value of one security paper is 10000 Dollars, but ...
0
votes
0answers
46 views
Equation with matrix exponential integral
For my thesis I came across an equation that involves a matrix exponential within an integral, i.e.
$ \int_0^{\Delta t} e^{-Ks}\Sigma \Sigma'e^{-(K)'s}ds = Q$
Where K is non-symmetric with real ...
0
votes
1answer
10 views
Dividend growth model?
A company whose earnings and dividends are expected to grow at a rate of 3% per year. Next year's dividend is $0.65 per share. The market capitalization rate is 7%.
How do I find the current price ...
0
votes
0answers
10 views
What formula do the majority of banks use for savings accounts?
I apologize if I'm asking this in the wrong forum or if it's already been answered and I missed it. I am a Software Engineer taking a Java course. Our final project it do write a Banking application ...
0
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0answers
10 views
Financial Mathematics Question relating to continuous investment and continuous interest
On 1 January 2014 an investor opens a bank account. The investor plans on depositing an amount of $20,000 continuously per year into the bank account for a total of five years.
From 1 January 2019 ...
0
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0answers
16 views
Risk neutral for Black Scholes
In the Black-Scholes model, consider a forward contract on the asset S, expiring at time T. Let Ft be the corresponding forward price: how does this process evolve in the risk-neutral measure (i.e. ...
1
vote
1answer
39 views
Converge to a Dirac Delta Function [closed]
I was reading something when the author said that "x converges to a dirac delta function". Was wondering if someone could explain what it means. I work in IB and am unsure what this means.
The exact ...
0
votes
1answer
27 views
accumulated value in geometric progression
In a general form, i know how to compute the PV of an annuity that follows a geometric progression.
But how do i compute it when it's the accumulated value?
let's say a payment of 100/ year with ...
0
votes
0answers
29 views
Stock price: Scaled Brownian Motion
Suppose that the share price of a given stock is modeled over time (measured in years) as the stochastic process $\{S(t), t β₯ 0\}$, with $S(t) = 100 + 10B(t)$, and $\{B(t), t β₯ 0\}$ is standard ...
1
vote
1answer
31 views
Layperson's explanation of what it means for something to become more and more like a Gaussian
Question: I was asked by a friend what it means for something to become more and more like a Gaussian and I was unable to come up with a satisfactory answer. Therefore my question is:
How would you ...
0
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0answers
26 views
Black Scholes Model Put-Call Parity alternative proof
I have PDE like this:
$f_t(t,x)=-\frac{1}{2}\sigma^2x^2f_{xx}(t,x)-rxf_x(t,x)+rf(t,x)$
and I have to find final condition $h(x)$ if function $f(t,x)=x-Ke^{-r(T-t)}$ satisfy this equation. So i ...
0
votes
1answer
39 views
accumulated value on the last payment of an annuity due
A $20$-year annuity-due makes payments of $100$ each year for the first $10$ years and then each subsequent payment decreases by $5$ for the next $10$ years. The effective annual interest rate is $9\%...
0
votes
2answers
38 views
Financial Mathematics Probability Question
It was crucial for our no arbitrage computations that there were only two possible values of the stock. Suppose that a stock is now at 100, but in one month may be at 130, 110, or 80 in outcomes that ...
0
votes
1answer
27 views
Determining Pricing Process of Futures Price
So I am studying Paul Glasserman's book on Monte Carlo in finance and I am currently working through Chapter 3 just to brush up on my risk neutral pricing processes (It's been a while). I have a ...
0
votes
1answer
22 views
Cost of the government
I am supposed to solve the following problem.
The bank provides 1,000 dollars loans. Risk-free clients will repay the loan
in full for a year, in addition to paying 4% interest. For a high-risk
...
0
votes
0answers
19 views
Showing CRR formula can be used to price call option: $Ο=15% R=6%$
Suppose that the volatility is 15% and that the annualized
interest rate is 6%. Construct a binomial tree with 15 periods to value a 6-
month instrument. Show that the Cox-Ross-Rubinstein formula ...
0
votes
0answers
26 views
Black Scholes theta at time t
I 'm trying to get the theta of a Call in the classical Black Scholes model.
We have (classical result with usual notations) :
$$C_t = S_tN(d_1) - Ke^{r(T-t)}N(d_2)$$
When deriving according to time,...
1
vote
1answer
31 views
Financial Mathematics Effective rate of interest
If you invest \$1000 , and you get paid \$500 in 5 years, \$1000 in 10 years, and \$1500
in 15 years and then get a final payment of \$2000 in twenty years, what is the effective
annual rate of ...
0
votes
0answers
12 views
Does the American Put have a higher price than payoff
According to shreve the value of a put is equivalent to or greater than the possible payoff, before a stopping time with the condition that its value equals the intrinsic value. First what does it ...
0
votes
1answer
17 views
Compound Interest Question from the textbook *Core Math for Advanced Level by Bostock and Chandler*
Good Day, I came across this question in a Form 6 Math textbook and it stomped me. I know it has to do with constructing the formula for compound interest and continuous interest but I think once I ...
0
votes
2answers
54 views
Modelling continuous compound interest with differential equations (Intuition)
Starting with the equation for continuously compounded interest, we can derive the differential equation. Let $A$ be the amount accumulated, $P$ be the principal amount and $r$ the rate.
\begin{align*...
1
vote
1answer
38 views
Conflicting Loan Amortization Formulae
When running a financial loan payoff simulation, I realized that my calculated monthly payment differed from the calculator offered by popular loan service Sallie Mae.
I found two formulas for loan ...
