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

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5
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398 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
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0answers
65 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 ...
4
votes
0answers
51 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
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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
272 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
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0answers
232 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
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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
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0answers
108 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
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0answers
170 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
23 views

Why could we observe trends in (pseudo-)random graphs?

In finance, markets are evolving from the interactions of people, and thus pure deterministic models are unlikely to provide accurate representations of the data, so they are stochastic by nature. ...
2
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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
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34 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
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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 ...
2
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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
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0answers
40 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
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0answers
31 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
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0answers
24 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 ...
2
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0answers
84 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
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0answers
22 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) + ...
2
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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 ...
2
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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
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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) ...
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
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0answers
27 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
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0answers
282 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
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0answers
457 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
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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
563 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
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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
21 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 $Z_t X_u$?
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0answers
23 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 $\sigma_2$. ...
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0answers
26 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 ...
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0answers
33 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 ...
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0answers
25 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. $$\int_{0}^{\infty}e^{\alpha+k}\int_{k+\zeta}^{\infty} (e^{-\zeta+x}-e^k)f(x)\ \mbox{d}x\ \mbox{d}k,$$ ...
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0answers
23 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: ...
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0answers
12 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 ...
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0answers
16 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 ...
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0answers
33 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" ...
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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 ...
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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 ...
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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 ...
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0answers
27 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 $\$ 100,000$ , ...
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0answers
65 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 ...
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0answers
17 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 ...
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0answers
50 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 ...
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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 ...
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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: $\mu_1=0.08$ $\sigma_{1}^{2}=0.0255$ $c_{12}=0.00225$ $\mu_2=0.1$ ...
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0answers
24 views

Describe a process mathematically

I am simply wondering how to explain this process mathematically: Lets say that we have a set $A_j$ of sets $B_i$ such that $A_j$ is the set off all $B_i$ where $i\leq j$ Now lets say that set ...