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Questions tagged [mathematics]

Used for question on application of mathematics in finance - from interest calculation to mathematical description of random processes.

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45 votes
8 answers
17k views

Recommendations for books to understand the math in quantitative finance papers?

Can anyone recommend books that explain the math used in quantitative finance academic papers?
32 votes
5 answers
8k views

Random matrix theory (RMT) in finance

The new kid on the block in finance seems to be random matrix theory. Although RMT as a theory is not so new (about 50 years) and was first used in quantum mechanics it being used in finance is a ...
vonjd's user avatar
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4 votes
2 answers
2k views

Understanding the solution of this integral

The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price): $$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\...
emcor's user avatar
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144 votes
15 answers
176k views

How can I go about applying machine learning algorithms to stock markets?

I am not very sure, if this question fits in here. I have recently begun, reading and learning about machine learning. Can someone throw some light onto how to go about it or rather can anyone share ...
zubinmehta's user avatar
  • 1,551
2 votes
1 answer
187 views

Value of a perfect hedge

Background Information: The price of a portfolio at time $t$ ($t = 0 ,1$) is $$V_t(\pi) = \phi S_t + \psi B_t$$ The portfolio $\pi$ is a perfect hedge for the claim $X$ if $V_1(\pi) = X$ a.s. as ...
Wolfy's user avatar
  • 728
2 votes
1 answer
898 views

Black-Scholes evaluating the squared of the stock price

Consider a Black-Scholes model $S_t = 5\exp{(\sigma W_t + \mu t)}$, $B_t = \exp{(rt)}$, where $W_t$ is Brownian motion with respect to a given measure $\mathbb{P}$. Suppose you hold a forward contract ...
Wolfy's user avatar
  • 728
0 votes
2 answers
2k views

Two-period binomial model with dividends

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.15$ and $l = 0.95$. The interest rate is $R = .05$. a.) If the asset pays 10% of its ...
Wolfy's user avatar
  • 728
27 votes
6 answers
21k views

What is a martingale?

What is a martingale and how it compares with a random walk in the context of the Efficient Market Hypothesis?
user40's user avatar
  • 2,707
14 votes
3 answers
9k views

How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?

I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
John Tyree's user avatar
14 votes
2 answers
566 views

Is a linear combination of GARCH processes also a GARCH process?

If two time series follow a GARCH process, and a third is a linear combination of them, is the third also GARCH process?
Qbik's user avatar
  • 1,018
11 votes
1 answer
7k views

What is exactly Euler's decomposition?

I have often seen the following statement in different paper: As $\sigma$ is homogeneous and of degree 1, we use Euler decomposition and write $\sigma(x)=\sum_{i=1}^n x_i \frac{\partial \sigma(x)}{\...
SRKX's user avatar
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9 votes
1 answer
4k views

application of lie groups in finance

Can some one kindly go over some of the applications and use of Lie groups in finance? The math is very rigorous and I don't fully understand it or the potential it could have. Let me share some ...
pyCthon's user avatar
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7 votes
1 answer
4k views

Abstract algebra in economics and finance

Are there any applications of abstract algebra (group theory, rings, fields etc.) in any branch of either economics or finance?
SlavicDoomer's user avatar
5 votes
4 answers
2k views

Examples of discrete math and graph theory within quantitative finance

The Wikipedia article on quants mentions discrete mathematics as a possible piece of their mathematical background. Are there good examples of problems within quantitative finance that are heavily ...
Juho's user avatar
  • 153
4 votes
1 answer
2k views

derivation of heston pde in gatheral

Following Gather (the volatility surface, chapter 2) we assume the following process: $$ dS_t = S_t(\mu_t dt+\sqrt{\nu_t}dZ^1_t)$$ $$ d\nu_t= -\lambda(\nu_t-\bar{\nu})dt+\eta\sqrt{\nu_t}dZ^2_t$$ ...
math's user avatar
  • 1,738
3 votes
1 answer
657 views

Is Black-Scholes complete?

If we have a Black-Scholes model $B_t = \exp{(rt)}$ and $S_t = S_0\exp{(\sigma W_t + \mu t)}$ then is it complete? What if $W_1$ and $W_2$ are independent Brownian motions. Then the two-stage Black-...
Wolfy's user avatar
  • 728
3 votes
1 answer
1k views

Derivation of Magrabe formula

I'm going through the following note by Davis, link. In chapter 3 he derives the Magrabe formula. I got stuck at equation $(3.16)$. We have two assets: $$dS_i(t)...
math's user avatar
  • 1,738
2 votes
1 answer
1k views

Linear combination of Payoffs using Black-Scholes

Write the payoffs in Figure 3.8 as linear combination of call options and derive a closed form formula for the Black-Scholes price, the Delta, and the Gamma of them. All the Greeks of the option are ...
Wolfy's user avatar
  • 728
2 votes
2 answers
4k views

Is there a formula for future value of a growing annuity with yearly payment growth and monthly payments?

My example is saving for college: assume a start of 0 balance deposits of 200 made monthly, every year they increase by (g) 2% to account for salary increases, first deposit made at the end of the ...
plockc's user avatar
  • 123
2 votes
1 answer
223 views

Law of One price and the Inconcistent pricing strategy

Background Information: A market satisfies the Law of One Price if every two self-financing strategies that replicate the same claim have the same initial value. An inconsistent pricing strategy is ...
Wolfy's user avatar
  • 728
2 votes
2 answers
5k views

How to derive Black-Scholes equation with dividend?

Question: The Black-Scholes equation without dividend is given by $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} -rV = ...
Idonknow's user avatar
  • 840
1 vote
0 answers
130 views

Does it make sense to apply complicated mathematics to calculate with precision when the margin of error is +/-10%? [closed]

This is more of a philosophical question than general question. Quantitative finance applies highly complicated mathematics and has attracted very smart people to this field lately given the high pay ...
curious's user avatar
  • 1,037
1 vote
1 answer
171 views

Formula for conditional expectation. Related to the Fundamental Theorems of Asset Pricing

Let $\lambda$ be a probability measure on $\Omega$ (finite), with filtration $\{\mathcal{F}_t\}$. Define $\nu(X) = \lambda\left(X\frac{d\nu}{d\lambda}\right)$, where $\frac{d\nu}{d\lambda}$ is a ...
Wolfy's user avatar
  • 728
1 vote
2 answers
2k views

Mark Joshi, Quant Interview Question problem 2.34; replicating a digital option on a 4-step symmetric binomial tree

Question: Team $A$ and team $B$, in a series of $7$ games, whoever wins $4$ games first wins. You want to bet $100$ that your team wins the series, in which case you receive $200$, or $0$ if they ...
Wolfy's user avatar
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