Questions having to do with financial mathematics. Please note that for questions in quantitative finance, quant.stackexchange.com is perhaps a better site.

learn more… | top users | synonyms

5
votes
0answers
409 views

In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ ...
4
votes
0answers
68 views

Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta )S{{P}...
4
votes
0answers
52 views

Stochastic control with stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ and smooth functions $u,F: [0, +\infty) \rightarrow \mathbb{R}$, how can we optimise the ...
3
votes
0answers
31 views

Stock Price Dynamics correlated with Bond market returns

I am currently working on to derive the following form of the stock price dynamics: $$dS_t = S_t[(r_t + \psi\sigma_S)dt + \rho \sigma_S dz_{1t} + \sqrt{1-\rho^2}\sigma_S dz_{2t}$$ where the ...
3
votes
0answers
278 views

What is the name of this symbol ( ┐) and what does it mean

Sorry bout the dumb question, it's just that I'm taking a mathematical finances class and the teacher started using this symbol today but I've never seem it before, was trying to google it but don't ...
3
votes
0answers
238 views

Math for Future Value of Growing Annuity

Am I working this out correctly? I need to verify that my code is correct... $$1000 \cdot \left(\frac{(1 + 0.1 / 12)^{40 * 12} - (1 + 0.06 / 12)^{40 * 12}}{(0.1 / 12) - (0.06 / 12)}\right)$$ ...
3
votes
0answers
99 views

How to get interest in the mathematics of tax

In a similar vein to my previous thread, I will also be teaching about the mathematics behind taxation - to a lot of people, this is very mundane - but that is not true of everyone. The practicality ...
3
votes
0answers
111 views

Doob Decomposition of American Option

I am trying to figure out the Doob decomposition of an American put option in a discrete time binomial model. I know how to price the American put, but I'm having trouble expressing it as the sum of ...
3
votes
0answers
171 views

Algorithm/Formula to compute adding and/or removing compound and/or non-compound percentages from a value?

I will first start with a scenario, I have to apply some adjustments to a particular value. These adjustments are either compound or non-compounded and they can either be added or subtracted to the ...
2
votes
0answers
45 views

Problems with a Black-Scholes modified equation

I haven't really studied much financial mathematics until about 2 months ago so I'm quite new to this stuff, so I'm sorry if this is a trivial question. At the moment I'm trying to work out what the ...
2
votes
0answers
35 views

Asymptotic distribution of zero-drift Geometric Brownian Motion as $t \to \infty$

If we fix the drift at $\mu = 0$, then my geometric brownian motion will have stationary mean, but it seems that the variance will grow without bound. What does the limiting distribution look like for ...
2
votes
0answers
34 views

Using reflection principle to find probabilities

I'm not able to answer these questions because firstly I don't understand the reflection principle properly. Secondly if someone could provide a visual explanation as to how this process works then ...
2
votes
0answers
24 views

Under which conditions on $\sigma_1, \sigma_2$ and $\rho_{12}$ the minimum variance portfolio involves no short selling?

If $\rho_{12} \lt 1$ or $\sigma_1 \ne \sigma_2$ then $\sigma_{V}^2$ representing the variance of the portfolio with weights $(w_1, w_2)=(s, 1-s)$ as a function of $s$ attains its minimum value at $$...
2
votes
0answers
41 views

Mean-variance portfolio probelm

So the question asks: Consider three uncorrelated stocks in the market. Each stock has variance 1. The expected returns are given by $2, 3 $ and $ 5$ respectively. Find the optimal mean-variance ...
2
votes
0answers
32 views

Proof that no futures trading system always wins

Hopefully someone here has some knowledge in both finance and maths. I am pondering on the existence/impossibility of a trading system (or algorithm) that ALWAYS ends up winning money, no matter how ...
2
votes
0answers
25 views

optimal derivative position through optimization

So I have the following optimization problem: min. $-E^Q[u(h(x))]$ s.t $\int h(x)q(x)dx \leq \frac{V_0}{B_0}$ Where $Q$ is the subjective probability which then gives: $E^Q[u(h(x))]=\int u(h(x))p(...
2
votes
0answers
90 views

Linear combination of Geometric Brownian Motions

Let $X_t= e^{\left(\mu-\sigma^2/2 \right)t+\sigma W_t}$ be a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. I am trying to derive an analytical solution to $$\mathbb{E}\left[ \...
2
votes
0answers
23 views

Mean and variance regime-switching model

Suppose we have the following model for stock price: $$ X_{t}=X_{0}\exp\left(\int_{0}^{t}(r-\frac{1}{2}\sigma_{\epsilon(s)}^2)ds+\int_{0}^{t} \sigma_{\epsilon(s)}dW_{s}\right) $$ This follows a normal ...
2
votes
0answers
34 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) + \sigma^2t(t-\...
2
votes
0answers
48 views

Reformulate this PDE in different notation

I would like to rewrite this general PDE \begin{equation} \alpha\partial_tu+\beta\partial_xu+\gamma\partial_{xx}u+\delta u=\varepsilon \end{equation} in this form $$c\left(x,t,u,\frac{\partial u}{\...
2
votes
0answers
66 views

Simplifying $\sum_{t=1}^{n}t^2v^t$ using actuarial notation.

In financial mathematics involving immunization, I encounter situations where I am trying to calculate $$(A) \quad v+4v^2+9v^3+ \cdots +n^2v^n=\sum_{t=1}^{n}t^2v^t $$ where $v$ is the present value ...
2
votes
0answers
49 views

Are these two option valuation formulas equivalent? Why?

I have been reading a finance paper that claims that the following function, which is a value for a financial derivative (1): $$V(s,t)=E_{Q} \left[\zeta\big(S(T)\big)e^{-\int_t^T r_F(\nu) d\nu}\\+\...
2
votes
0answers
41 views

What is this sort of optimisation called?

I am reading a book in mathematical finance. There is something about constrained optimisation. They have specialised it for the financial market, but I am wondering what the general name for this ...
2
votes
0answers
30 views

How to calculate present value with changes interest

How to calculate present value with period of 5 years and 6 months? Besides that, there is interest changes and compounded differently. Is there any formula?
2
votes
0answers
516 views

What do two number on top of each other in square brackets mean?

Im currently going through "Universal Portfolios with Side Information" by Cover and Ordentlich [96]. Near the end of the paper, they provide a formula for calculating weights of a Universal Portfolio ...
2
votes
0answers
286 views

Engineering Economics Cash Flow Diagram

I have the following question and solution below. What I don't understand is why is the 100,000 seen as savings/revenue when clearly it is coming out of pocket? Additionally, the monthly loan payment ...
2
votes
0answers
469 views

What does it mean to “pass to the limit” in mathematics?

I've been reading a finance paper and stumbled upon this phrase. What does passing to the limit mean in this context (or overall in mathematics)? Here is an excerpt from the paper: It is ...
2
votes
0answers
48 views

Reinvesting the interest (generalized version)

If I deposit \$1 at $t=0$ into an account which credits interest at the end of each year at a force of interest $\delta_t$ (assume it's integrable.) Then, if I reinvest the interest at an annual ...
2
votes
0answers
44 views

Stochastic control, numerical, need expectations given coupled SDEs

I'm looking at a trio of processes which arises in a stochastic control situation. I have a process $(V_t)$ which I may control, and $(V_t)$ influences a diffusive stock price process $(S_t)$. The ...
2
votes
0answers
575 views

Analogue of Leibniz Rule for Stochastic Integrals

Suppose $$f(t,u)=f(0,u)+\int_0^t{\mu (w,u)dw}+\int_0^t{\sigma(w,u)dB_w}$$, where $B_w$ is a standard Brownian motion. I would like to calculus the drift and diffusion of $Y_t=-\int_t^s{f(t,u)du}$ (...
2
votes
0answers
30 views

Standard practice for identifying outlying spend amounts

Hope this is mathematical enough to qualify as a question - I'm no mathematician! I have a set of individuals travel & entertainment credit card spend, and I'd like to highlight any outliers that ...
1
vote
0answers
11 views

Why is a risk neutral measure unique in a discrete time market with continuous states?

Why is the radon nikodym derivative unique in a discrete time market with continuous states? By radon nikodym derivative, I meant the derivative with respective to the risk neutral measure and the ...
1
vote
0answers
23 views

Conditional expectation and set times random variable??

On page 62, what in the world is the meaning of equation (5.2)? $\mathcal{F}_t$ is a $\sigma$-algebra, so $Z_t \in \mathcal{F}_t$ is a set. $X_u$ is a random variable, so what is $Z_t X_u$?
1
vote
0answers
31 views

implied volatility

I have a question about calculating the implied vol. Assuming the implied vol that a option will expire in 1 day is $\sigma_1$, and the implied vol that the option will expire in 2 days is $\sigma_2$. ...
1
vote
0answers
27 views

subaddivity of VaR

It is known that the VaR (Value at risk) doesn't fulfill subadditivity, i.e. $VaR(X)+VaR(Y) \le VaR(X+Y)$. But for elliptical distributions subadditivity is true. Questions: (1) Which ...
1
vote
0answers
36 views

Options on Futures Black-Sholes

I am taking the Financial Risk Management course, and the topic now is "Variations on the Black-Scholes Model". I am following Paul Wilmott's "The Mathematics of Financial Derivatives: A Student ...
1
vote
0answers
26 views

Proof of the finiteness of integral (in option pricing)

I would like to ask for help with proving the finiteness of the following double integral. $$\int_{0}^{\infty}e^{\alpha+k}\int_{k+\zeta}^{\infty} (e^{-\zeta+x}-e^k)f(x)\ \mbox{d}x\ \mbox{d}k,$$ ...
1
vote
0answers
25 views

Finance Algeabra: Converting a Discount Polynomial Function to an Interest Rate Polynomial Function

I have a finance problem that is 99% mathematical. In finance, the price of a bond could be modelled as the discounted value of its future cash flows, so something like: ...
1
vote
0answers
13 views

Financial Mathematics--Finding Compounding Period given Annual and Effective Interest Rates

I'm trying to find a compounding period C when given an annual interest rate r and effective annual yield i. I'm working with the following equation: $i=(1+r/C)^C-1$ I'm having trouble re-writing ...
1
vote
0answers
17 views

Calculating benefit of paying off loan ahead of time

So I'm doing a little financial planning, and I'm looking into the worthwhile-ness of paying off my student loans as quickly as possible. Being that I have multiple student loans, I combined them all ...
1
vote
0answers
34 views

deriving mean-variance form

I have a function $$W(x_1,x_2,...,x_n)=A\sum_{i=1}^{n}x_i^2+\sum_{i\ne j}^{n}x_ix_j,$$ where $A \in[0,1)$. And in a paper I am reading, $W$ can be written in the "mean-variance" form:$$n(n+A-1)[\mu^2(...
1
vote
0answers
19 views

How to tell whether put or call is more expensive using put-call parity?

Using the following example can someone explain to me why the put is more expensive in part a and why the call is more expensive in part b? $$C_E-P_E=S(0)-X \cdot \frac{1}{1+R}$$ where $X$ is the ...
1
vote
0answers
46 views

Relations between Call and Put

I am trying to solve a question in finance but I am pretty much stuck and would need your help :) Suppose you know the following information about a market: Future is at 66 70 strike straddle is ...
1
vote
0answers
22 views

Calculating ROI of Feature

I have the following figures: (x) Quote for new software feature: $10,000 (y) Employee Rate: $15.00/hour (z) Hours per Day to perform task manually: 2 (n) Number of working days in a year: 251 (a)...
1
vote
0answers
28 views

Finding accumulated profit (Finance).

We need to find the accumulated profit at $t=5$ ( in years ) for the following project given that it is financed by a loan subject to interest $6.25\%$. Project : Initial outlay of $\$ 100,000$ , ...
1
vote
0answers
72 views

Exact match of assets and liabilities.

Liabilities of $1$ each are due at then ends of periods $1$ and $2$. There are three securities available to produce asset income to cover these liabilities, as follows: (i) A bond due at the end of ...
1
vote
0answers
19 views

Replication strategy of European call option

So the question asks: L et $S(0) = 120$ dollars, $u = 0.2$, $d = −0.1$ and $r = 0.1$. Consider a call option with strike price $X = 120$ dollars and exercise time $T = 2$. Find the option price and ...
1
vote
0answers
51 views

Pricing Function is convex

I am now reading Alternative Characterization of American Put Options by Carr et all (available at http://www.math.nyu.edu/research/carrp/papers/pdf/amerput7.pdf). There is a theorem called 'Main ...
1
vote
0answers
21 views

Intuition behind a market uncertainty represented by a filtered complete probability space?

What is the intuition behind a market uncertainty represented by a filtered complete probability space $(\Omega, F, P, {F_t})$, on which an m-dimensional standard Brownian Motion $W(t) = (W_1 (t), W_2 ...
1
vote
0answers
38 views

what am I doing wrong when finding the weights of the market portfolio?

I need to find the weights of the market portfolio with three risky securities given the following information: $\mu_1=0.08$ $\sigma_{1}^{2}=0.0255$ $c_{12}=0.00225$ $\mu_2=0.1$ $\sigma_{2}^{2}=0....