Questions tagged [mathematical-finance]
For questions about mathematical problems arising from the study of financial markets.
22 questions with no upvoted or accepted answers
12
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0
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1k
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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 ...
- 23.2k
8
votes
0
answers
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 ...
- 2,075
4
votes
0
answers
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 ...
- 51
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, \...
- 39
3
votes
0
answers
171
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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 ...
- 1,835
3
votes
0
answers
518
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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 $...
2
votes
0
answers
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 ...
- 21
2
votes
0
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
2
votes
0
answers
261
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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
2
votes
0
answers
263
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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 $...
- 21
1
vote
0
answers
65
views
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
1
vote
0
answers
71
views
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 ...
- 721
1
vote
0
answers
152
views
Constrained trace optimization with relavance to optimal asset selection
Let $D$ and $Q$ be two real $m\times m$ diagonal matrices given
$$
D=\left(\begin{array}{cccc}
d_1 & 0 & \cdots & 0\\
0 & d_2 & \cdots & 0\\
\vdots & \vdots & \ddots &...
- 43
1
vote
0
answers
328
views
Preservation of variance for log-normal variables under change of measure
Aim: to show that changing a probability measure via the application of a Radon-Nikodym derivative preserves variance of a log-normally distributed random variable (for the case when variance is non-...
- 141
1
vote
0
answers
97
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Applications of Kazamaki Conditions
I'm interested in applications of this theorem by Sekiguchi Kazamaki:
"Continuous Exponential Martingales and BMO" - Theorem 1.12:
Let $M$ be a continuous local martingale and $Z(M):= \exp(M-\frac{1}{...
- 21
1
vote
0
answers
95
views
Non-diagonalizable matrix in a discretized Ornstein-Uhlenbeck process
I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process:
\begin{equation*}
d\mathbf{S}_t = \mathbf{\kappa}(\...
- 11
1
vote
0
answers
68
views
Recovering a Log-Correlated Gaussian Field from a limit-lognormal singular measure
In a paper I (didn't write, but) co-authored, Forecasting Volatility with the Multifractal Random Walk Model, we use explicit formulas that give the law of $(X(t),t>0)$ conditional on $(X(t),t<0)...
- 3,085
1
vote
0
answers
302
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Unique EMM & completeness in the Black-Scholes model
Consider the Black-Scholes model
$$ 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
1
vote
0
answers
114
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
1
vote
0
answers
82
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 ...
- 19
1
vote
0
answers
132
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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 ...
- 21
0
votes
0
answers
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 ...
- 309