Questions tagged [mathematical-finance]
For questions about mathematical problems arising from the study of financial markets.
71 questions
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Reconstruction of law of diffusion process from call option values
Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the
$$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$
Then, ...
- 128k
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198
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Pricing zero coupon bonds through PDE
I'm currently studying Paul Wilmott on quantitative finance and saw an interesting idea for an interest rate model that went unexplored in the book.
The idea is to model the market price of risk as a ...
- 6,155
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65
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Upsampling parameters in the Takahashi-Alexander model
Let me start by begging your forebearance; this question might at first glance appear to belong more on a forum for economics, but I hope by the end to convince you that there is mathematical content ...
- 721
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8
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Reference for elementary and "cool" statistics or financial math
I signed up for a Math Mentorship Program (for high school students) this term, but one of the students assigned to me is more interested in Statistics and Finance - something that would help him to ...
- 12.9k
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3
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Are symplectic methods used in (classical) Economics?
The tl;dr question is this: are economists using coordinate-free formulations in studying theory?
Borrowing from classical mechanics, the framework I have in mind for classical economics--involving ...
- 121
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1
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81
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Stochastic Geometric Progression [closed]
Let $\mu_1, \mu_2, \ldots, \mu_n, \ldots \in \mathbb{R}$, let $\sigma_1, \sigma_2,
\ldots \in [0, \infty)$ be sequences of numbers.
Let $z_1, z_2, \ldots, z_n, \ldots$ be independent random variables ...
- 258
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10
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Is there any straightforward way to substitute for Gaussian/Brownian assumptions in financial mathematics?
A huge amount of financial mathematics assumes Gaussian distributions of risks and Brownian movement of prices. What efforts have there been to replace these with heavy-tailed distributions? For ...
- 4,713
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71
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Reference request: finding entries that prevent matrix from being correlation matrix
I am currently doing some research with a quantitative finance firm and my supervisor has raised an interesting question that shows up a lot with their clients: quite often, clients will want to do ...
- 869
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437
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Elliptic PDEs in Finance
In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
- 6,367
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152
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Constrained trace optimization with relavance to optimal asset selection
Let $D$ and $Q$ be two real $m\times m$ diagonal matrices given
$$
D=\left(\begin{array}{cccc}
d_1 & 0 & \cdots & 0\\
0 & d_2 & \cdots & 0\\
\vdots & \vdots & \ddots &...
2
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59
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How to determine speed (rate) in large deviation principle for geometric Brownian motion
By reading Asymptotics for volatility derivatives in multi-factor rough
volatility models by Lacombe, Muguruza and Stone, I am not familiar with the way they deduce the speed (or rate) when showing ...
- 63.9k
3
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1
answer
159
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Are there any known results on the probability distributions of perpetuities with power law discount rates?
Currently I am working on studying stochastic integrals of the form: $$Z_\infty = \int_0^\infty e^{-f(t)}\mathop{d}S_t$$
where $S_t$ is a Compound-Poisson process with Exponentially-distributed ...
- 5,474
1
vote
1
answer
1k
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The stock market polytope: explanation?
Ovidiu Racorean.
"Crossing stocks and the positive Grassmannian I: The Geometry behind Stock
Market."
(arXiv Abstract link)
Anyone care to offer a summary of what's going on here?
(The ...
- 63.9k
1
vote
3
answers
1k
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Matching dynamic trading strategies with derivatives
The famous Black-Scholes framework is usually derived using a hedging approach where a self-financing portfolio is constructed and the resulting stochastic differential equation is being solved under ...
- 63.9k
2
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answers
79
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Convex optimization over compact sets defined as Aumann set-valued integrals
Let $(X,P)$ be a probability measure space. Let $K$ be a convex compact subset of $\mathbb R^d$ and let $F:X \to 2^{K}$ be a set-valued map. Assume that $F$ is:
closed (i.e $F(x)$ is closed for ...
- 6,853
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304
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"Meritocratic" pyramid schemes
There have been a couple of times in my life when people from multi-level marketing organizations attempted to recruit me. I listened to what they had to say, and both times I did not get involved ...
- 63.9k
12
votes
3
answers
2k
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Compactness of the set of densities of equivalent martingale measures
Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal P^\...
- 2,821
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328
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Preservation of variance for log-normal variables under change of measure
Aim: to show that changing a probability measure via the application of a Radon-Nikodym derivative preserves variance of a log-normally distributed random variable (for the case when variance is non-...
- 141
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1
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151
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Construction of a probability measure from a sequence of probability measures
Summary
I would like to pass from a sequence of probability measures whose "limit" satisfies a desired property to a new probability measure that satisfies this property.
Details
We work on ...
CommunityBot
- 1
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2
answers
3k
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Ito's lemma in differential form
Basically you'll find two versions of ito's lemma in the literature: an integral and a differential form. The integral form is based on an Riemann-Stieltjes-integral approach, the differential form is ...
- 103
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1
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357
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Integral over a Markov process
I have the following questions:
Let $Z$ be a continuous one-dimensional Markov process on some probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{F}_t = \sigma(Z_s,s \leq t)$. Then show ...
-3
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3
answers
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Fuzzy Logic in Finance
Has fuzzy logic been commercially applied in finance fields and has it been successful ?
I have got knowledge that it has been applied in Algorithmic trading and operational risk, but I want to know ...
CommunityBot
- 1
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340
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Why are financial markets modeled by càdlàg processes?
When opening a book or reading an article on mathematical finance, financial markets (e.g. stock prices) are always modeled by càdlàg semimartingales. I was wondering why it is that these processes ...
- 3,065
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2
answers
2k
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On mathematical aspects of the most recent Nobel Prize in economics winners' work
Can somebody briefly introduce the mathematical aspects, in particular, those related to mathematical finance, of the three economists who were just awarded this year's Nobel Memorial Prize in ...
3
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1
answer
392
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A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to $L_{n}$
Edit: According to the comment by @LSpice we realise the existing link to the main motivation of the question is not available. Then we search for the paper we found the following version:
https://www....
- 356
2
votes
1
answer
495
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Stochastic integral with respect to a random field
I came across a generalized Black-Scholes equation formulation in this paper.
Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...
- 10.3k
12
votes
6
answers
5k
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Discrete version of Ito's lemma
Could anyone give me some references where I could find
(a) discrete version(s) of Ito's lemma
(b) a proof how it converges to the continuous form in the limit
(c) its usage within stochastic ...
- 5,405
1
vote
0
answers
97
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Applications of Kazamaki Conditions
I'm interested in applications of this theorem by Sekiguchi Kazamaki:
"Continuous Exponential Martingales and BMO" - Theorem 1.12:
Let $M$ be a continuous local martingale and $Z(M):= \exp(M-\frac{1}{...
- 21
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2
answers
380
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Large deviation bound for O-U process
Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X_t = -\alpha X_t dt + \sigma dB_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max_{t\le T} |X_t| \ge z
\...
- 325
-4
votes
1
answer
303
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Reference request in optimal stopping [closed]
I am given the following task. Distributed over a trading day, I am supposed to buy a certain quantity of a good. The price of this good changes during the day. The goal is to buy the required ...
6
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1
answer
5k
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Transformation of the Black-Scholes PDE into the diffusion equation - shift of coordinate system
The aim of transforming the Black-Scholes PDE is of course to find a form where an relatively easy solution exists. Most of the steps seem to be straightforward - please use this reference:
https://...
- 111
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3
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379
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Compute inverse series for implicit equation $b=-\log(1-e^{-x})/x$
In financial mathematics, the inverse series of: $$b(x) = -\frac{\log(1-e^{-x})}{x}$$ is needed in order to perform fast calculation on swaptions for G2++ calibration model. (see this post for ...
- 63.9k
1
vote
0
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95
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Non-diagonalizable matrix in a discretized Ornstein-Uhlenbeck process
I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process:
\begin{equation*}
d\mathbf{S}_t = \mathbf{\kappa}(\...
- 11
1
vote
0
answers
68
views
Recovering a Log-Correlated Gaussian Field from a limit-lognormal singular measure
In a paper I (didn't write, but) co-authored, Forecasting Volatility with the Multifractal Random Walk Model, we use explicit formulas that give the law of $(X(t),t>0)$ conditional on $(X(t),t<0)...
- 3,085
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0
answers
302
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Unique EMM & completeness in the Black-Scholes model
Consider the Black-Scholes model
$$ dS(t) = \mu(t) S(t) dt + \sigma(t) S(t) dW^{\mathbb{P}}(t) $$
$$ dB(t) = r(t) B(t) dt$$
Steele shows now in "Stochastic Calculus & Financial Applications" (Ch. ...
- 203
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1
answer
502
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Mathematical properties of financial prices
Prices of financial assets (stock-market prices or currency exchange rates) obviously resemble trajectories of stochastic processes.
What is known about their mathematical properties ?
I know ...
CommunityBot
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3
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1
answer
214
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Inverting the cumulative probability function to find roots of stochastic function
Given a function:
$$f[x]=a\, \Phi \left[-x+\sigma \sqrt{\tau}\right]-\left(b+c\, e^{-d \tau}\right)\Phi \left[-x\right]$$
where $\Phi$ is the cumulative density function of the standard normal ...
- 188k
0
votes
1
answer
161
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Ratios of random variables with weak moment condition
Let $X_n$ be a sequence of iid positive random variables. Assume that $X_n$ has finite $\alpha$th moment for some value $\alpha \in (0,1)$, but infinite first moment. Assume also that the reciprocal $...
CommunityBot
- 1
1
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0
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views
Extending risk neutral measure to insurance/mortality filtration
In insurance mathematics, one often models the underlying of an insurance policy with a Black Scholes model on a filtered probability space $(\Omega,\mathbb{Q},\mathcal{F},\mathbb{F}=(\mathcal{F}_{t}))...
- 203
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1
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120
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Is it possible to solve $P = Cny^{-1}(1-1/(1+y/n)^{nT}) + M/(1+y/n)^{nT}$ for $y$? [closed]
The equation
$$
P = \frac{Cn}{y}\left(1-\frac{1}{(1+\frac{y}{n})^{nT}}\right)+\frac{M}{(1+\frac{y}{n})^{nT}}
$$
represents the present value (price $P$) of a government bond which pays $C$ ...
- 117
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votes
2
answers
1k
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Is the "hybrid" Black-Scholes Hull-White model arbitrage free?
Given a "hybrid" Black-Scholes Hull White (BSHW) model. That is, the stock price is modelled by a Black Scholes SDE:
\begin{equation} dS(t) = \mu(t)S(t)dt + \sigma_{S}(t)S(t)dW^{\mathbb{P}}_{S}(t)
\...
- 1,533
1
vote
1
answer
770
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Beginning books on stochastic calculus and finance [closed]
my background is mathematics i would like to do research in financial mathematics. So I read some part of wilmott's book but it required stochastic calculus. I did not understand that book. So which ...
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American put option pricing by "binomial trees"
I'm teaching a financial mathematics course and have found a fascinating (to me) numerical phenomenon and wonder if anyone has studied it, or knows anything similar.
I'll try and give a description ...
CommunityBot
- 1
1
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1
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679
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Taylor Series expansion for an implicitely defined family of functions
Can we find a Taylor Series expansion for $y(x)$ implicitly defined by:
$$\sum _{i=1}^nA_ie^{a_ix+b_iy} = 1 ?$$
In financial mathematics, the two-additive-factors Model G2++ is commonly used for ...
- 181
3
votes
0
answers
234
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European call option pricing under mean reverting stock return
Consider the stock price process satisfies the following SDE:
$dS_t=\mu_t S_tdt + \sigma S_t dW_t , S_0=s $
and the mean return $\mu_t$ satisfies the following SDE:
$d\mu_t=(a-\mu_t)dt +dB_t, \...
- 410
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1
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461
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Is it safe to work on a Cadlag modification of a Feller process?
Let $f$ be a continuous bounded function.
$X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write
$$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...
- 29.7k
1
vote
2
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134
views
Is zero a regular point for a drifted $\alpha$-stable process?
We consider 1-d process of the form $Y_{t} = bt + M_{t}^{\alpha}$,
where $M_{t}^{\alpha}$ is $\alpha$-stable process for some $\alpha
\in (0,2)$ with its levy symbol $\eta(u) = - |u|^{\alpha}.$,
and $...
- 126
2
votes
0
answers
261
views
Asymptotics of Variable Drift Ornstein–Uhlenbeck Process
The Ornstein–Uhlenbeck process is defined as the stochastic process that solves the following SDE:
$dx_t = \theta (\mu-x_t)\,dt + \sigma\, dW_t$
where $\theta>0$, $\mu$ and $\sigma>0$ are ...
- 3,626
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0
answers
82
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The Stratonovich formulation of the Double Mean Reverting Model
I am writing my Bachelor's Thesis on the fast Ninomiya-Victoir calibration of the Double Mean Reverting model and have a question to its Stratonovich formulation. I am new to mathoverflow and a novice ...
- 8,086
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1
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Algebraic Number Theory in Financial Mathematics
I am currently doing my masters studies in financial mathematics. However, I have had a good background in number theory and I don't feel like leaving it just like that. I am thus inquiring on any ...
- 16.5k