Questions tagged [mathematical-finance]

For questions about mathematical problems arising from the study of financial markets.

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Relationship between an optimizer and a mode/mean of pdf

Let $X$ be a random variable, uniformly distributed over a support $S$. Let $f(X;\theta)$ be a function of $X$, parameterized by $\theta$. I am hoping to think of a relationship between two quantities:...
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How to derive the HJM drift condition?

I'm trying to derive the Heath Jarrow Morton drift condition (from Björk, page 298) and this equation is the part that I'm not able to derive: $$ A(t,T) + \frac{1}{2} ||S(t,T)||^2 = \sum_{i=0}^d S_i(t,...
2 votes
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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 ...
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7 votes
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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 ...
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1 vote
0 answers
191 views

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-...
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1 answer
116 views

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 ...
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0 votes
1 answer
137 views

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 ...
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186 views

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 ...
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2 votes
1 answer
346 views

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 ...
1 vote
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69 views

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}{...
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1 answer
259 views

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 ...
1 vote
0 answers
77 views

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}(\...
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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)...
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1 vote
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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. ...
3 votes
1 answer
210 views

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 ...
3 votes
2 answers
287 views

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 \...
1 vote
0 answers
82 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}))...
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1 answer
112 views

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 $C$ ...
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3 votes
2 answers
893 views

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 vote
1 answer
722 views

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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2 votes
3 answers
341 views

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 ...
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1 vote
1 answer
549 views

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 ...
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1 answer
381 views

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....
3 votes
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219 views

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, \...
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2 votes
1 answer
399 views

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 ...
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1 vote
2 answers
121 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 $...
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2 votes
0 answers
230 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 ...
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25 votes
2 answers
2k views

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 ...
1 vote
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76 views

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 ...
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4 votes
1 answer
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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 ...
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2 votes
1 answer
895 views

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 ...
3 votes
0 answers
168 views

compactness of a probability set

I have a question about the compactness of a set of martingale measures. Let $\Omega=\mathcal{C}[0,1]$ be the space of continuous functions on $[0,1]$ and $\mathcal{M}_{\Omega}$ be the family of ...
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16 votes
2 answers
1k views

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 ...
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1 answer
146 views

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 $...
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2 votes
0 answers
247 views

A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$

Hi to everyone, The ingredients of my problem are the following: I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on $...
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5 votes
3 answers
1k views

One can earn nothing on the Brownian motion, true ?

Consider any discrete time stochastic process $p(n)$ (price) with independent increments $\xi_k$ and $E(\xi_k)=0$. E.g. Brownian motion (i.e. $\xi_k = N(0,1)$). Consider some "trading strategy" ...
7 votes
3 answers
1k views

How much one can earn on a white noise ?

Consider the simplfied math. model for asset price (it is nevertheless quite practical for specific situations see "PS" part below) assume price "p(n)" at moment "n" is equal to N(0,1) - i.i.d - ...
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1 answer
476 views

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 ...
12 votes
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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 ...
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4 votes
1 answer
378 views

Trajectorial version of Doob's $L^2$ inequality

In the paper http://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0154.pdf you can find a trajectorial version of Doob's inequality. It is given by: $$\bar{s}^2_T+4\sum_{k=0}^{T-1}\bar{s_k}(...
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1 vote
1 answer
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Solving an Ornstein-Uhlenbeck-like SDE $y(t,T)=H_t + \mathbb{E}[\int_t^T y(s-,T)dX_s|\mathcal{F}_t]$

I have asked a similar question involving some finance background some time ago here math.stackexchange, however no really good answer came up. I was able to find a solution at least for a special ...
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1 vote
2 answers
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market completion in stochastic volatility model

Hi all, Consider a stochastic volatility model. As there are two sources of risk and one asset only, this is an imcomplete market. One can complete the market by considering a derivative V1 used to ...
1 vote
0 answers
128 views

stochastic volatility valuation equation

I'm trying to derive the valuation equation under a general stochastic volatility model. What one can read in the litterature is the following reasonning: One consider a replicating self-financing ...
5 votes
1 answer
275 views

Arbitrage free price of a derivative when the price is collected over the lifetime of the derivative [closed]

Let $X_t$ be an american style financial derivative with random exercise time $T$ where $t$ and $T$ belongs to some finite set $A$. Buying this derivative requires the buyer to pay $p_t$ up to time $T$...
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12 votes
3 answers
1k views

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^\...
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10 votes
10 answers
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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 ...
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3 votes
0 answers
505 views

Laplace transform of a stopping time for stochastic volatility models

Let $V_t$ be a solution of the SDE $$dV_t=V_t(rdt+\sigma_t dW_t) $$ where $\sigma_t$ satisfies some other SDE $$d\sigma_t=\alpha(t,\sigma_t)dt+\beta(t,\sigma_t)dW^{\\ \prime}_t $$ and $W_t$ and $...
19 votes
10 answers
3k views

Expected value as decision criterion in the context of rare events

I have often seen discussions of what actions to take in the context of rare events in terms of expected value. For example, if a lottery has a 1 in 100 million chance of winning, and delivers a ...
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5 votes
1 answer
439 views

Stieltjes integrals of predictable processes

I am looking for a direct proof of the fact that, roughly speaking, if $S=S_0+A+M$ is an $L^2$ semimartingale, and $M$ (the martingale part) has the martingale representation property, then for any ...
3 votes
2 answers
826 views

Characteristic operator

Let $X_t\in\mathbb{R}$ be an Ito diffusion process given by $$ dX_t=a(b-X_t)dt+\sigma dW_t$$, then the characteristic operator of $X_t$ is given by $$L=a(b-x)\frac{\partial}{\partial x}+\frac{\sigma^...