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76
votes
10answers
80k 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 ...
30
votes
9answers
7k 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?
26
votes
5answers
3k 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 ...
18
votes
6answers
2k views

How random are financial data series?

Pseudorandom number generators are often tested using e.g. a test suite like Diehard tests or Dieharder. If one would run these tests e.g. on stock market time series or other financial data, would ...
12
votes
1answer
922 views

Any recommendations for textbooks for an undergraduate course in mathematical finance? [closed]

I'll teach an introductory course on mathematical finance in the near future. The course is intended to entertain and broaden some well-prepared advanced undergrad mathematics majors, some physics ...
11
votes
5answers
7k 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?
10
votes
2answers
1k views

Why Ito calculus?

Coming from physics, I am used to the fact that the Ito interpretation of most natural stochastic equations is wrong, and one should be using Stratonovich calculus instead (of course they are ...
9
votes
1answer
271 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?
9
votes
2answers
599 views

Is it possible to understand financial theory without mathematics?

I am trying to develop a short course on financial theory, covering the fundamentals of forward and options pricing, and 'efficient market' theory. I want to reduce the amount of mathematics to a ...
8
votes
1answer
330 views

When pricing options, what precision should I work with?

I'm wondering if there's any point at all in double-precision calculations, or whether it's ok to just do everything in single-precision, seeing how the difference on non-Tesla GPUs for single and ...
8
votes
1answer
562 views

Modified Durations of Different Noncallable Bonds and function of Maturity

I'm hoping someone could help me understand this subject better. Basically I am reading a book and it shows a table ...
8
votes
0answers
198 views

Proving the asymptotic distribution of Manipulation-Proof Performance Measure (MPPM) (Paper by Goetzmann et al.)

In Goetzmann et al.'s (2007) paper, the authors derive a "Manipulation-Proof Performance Measure" (MPPM), which is a performance measure that is impervious to performance manipulation by fund ...
7
votes
4answers
714 views

What are the options for a mathematician to break into QF without working for a fund?

I have a degree in mathematics, and I've worked as a statistician and done some programming work. I'm very strong in my math/stats/programming background and have browsed some QF books, and I'm very ...
7
votes
1answer
2k 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 ...
7
votes
7answers
730 views

Proof that no trading system always wins

I am pondering on the existence/impossibility of a trading system (or algorithm) that ALWAYS ends up winning money, no matter how the price of a futures moves. In a context where one can go long or ...
7
votes
1answer
1k 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 ...
6
votes
2answers
3k 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 ...
5
votes
3answers
349 views

How is stock data objectively different to this random walk?

I have a random walk that is generated as so using python, numpy, and matplotlib ...
4
votes
2answers
712 views

Implications of the Riemann hypothesis in finance?

I was recently at a seminar by a top hedge fund manager at a top university for finance students, when one of the finance professors asked him what do you think are important areas of research for my ...
4
votes
2answers
304 views

What is Heston's equation?

This paper mentions the elliptic Heston operator: $Av:= -\frac y2(v_{xx}+2\rho\sigma v_{xy} + \sigma^2v_{yy}) - (c_0 - q - \frac y2)v_x + \kappa(\theta -y)v_y + c_0v$. Then boundary value problem ...
4
votes
1answer
98 views

Clarify a derivation in Pat Hagan's Convexity Conundrums

I am looking for help in understanding the algebraic derivation to go in between some of the lines in Pat Hagan's famous Convexity Conundrums paper e.g. how he goes from 3.4a to 3.5a.
4
votes
1answer
98 views

How to price and find a replicating portfolio for a call spreads using a two-period binomial model?

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.03$ and $l = 0.98$. a.) If the interest rate for both periods is $R = .01$, find the ...
4
votes
1answer
148 views

What is the analytic value of an asset's risk contribution, if $n=2$?

The marginal risk contribution of asset $i$ is defined by Roncalli in his paper on ERC as follows: $$\frac{\partial \sigma(x)}{\partial x_i} = \frac{1}{\sigma(x)} \left( w_i \sigma_i^2 + ...
4
votes
1answer
109 views

backward Kolmogorov equations - Markov properties

I'm a physicist who's research has lead him into the theory of stochastic differential equations. If this question is not appropriate for this forum, please feel free to delete it. So I've been ...
3
votes
2answers
202 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^*} ...
3
votes
2answers
336 views

Book recommendation: math toolkit for quantitative finance and statistics

I am looking for a book which teaches mathematical topics which are relevant to master quantitative finance and statistics. Please note, I do not mean a book which would explain how math is applied ...
3
votes
2answers
167 views

Risk-Neutral Probabilities, Trinomial Model

My professor has many grammatical mistakes and errors in his questions, so apologies ahead of time. I am just trying to understand what he wants for this question, In trinomial model, let $S_0 = 1$, ...
3
votes
2answers
242 views

Is linear programming important for quant?

I'm thinking about taking a course on Linear and Convex Programming, but I don't know how useful it is in the real world finance. Which areas in finance is mathematical programming used?
3
votes
3answers
268 views

Budget Constraint in Sharpe Ratio Optimization

I am a math student and I am trying to understand the budget constraint in Sharpe Ratio optimization for portfolio design. Recall the budget constraint requires that the sum of the portfolio weights ...
3
votes
2answers
71 views

Conditional expectation of a non stochastic process

In an example I was working through it was shown that $W_{t}^{2} - t$ was a martingale with respect to the Brownian motion filtration $\mathcal{F}_{s}^{W}$ with $t>s$. Everything was fine except a ...
3
votes
2answers
381 views

Why is that a risk averse consumer buys the optimum insurance when there is actuarially fair insurance?

I think I understand the fact that when marginal utilities of the same function are equal (a consequence of the actuarially fair insurance), the independent variables in it must be equal -- right? But ...
3
votes
1answer
42 views

Use no dominance to show that the price of the call option satisfies the inequality

Assumption 2.1 - If the payoff $P$ of a financial instrument is non negative, then the price $p$ of the financial instrument is non negative. Assume $C$ is just the price of the call option, and ...
3
votes
1answer
284 views

PDE pricing of barrier options in BS

Path-dependent options in BS framework is intuitive to price with monte-carlo under risk-neutral measure, however it appears that several kinds can be priced with PDEs. I understand how does the story ...
3
votes
1answer
4k views

Hurst Exponent Calculation

I am trying to calculate the Hurst Exponent using Excel. I am facing a problem where the exponent value sometime goes beyond 1. Can someone share a link / material so that it will help me to calculate ...
3
votes
1answer
96 views

Modeling Financial Assets

Let $\tilde{W}_t := (1+R)^{-t}W_t$ and $\tilde{S}_t := (1+R)^{-t}S_t$ be respectively discounted wealth process and discounted asset price. Then, show that $$\tilde{W}_t = w_0 + ...
3
votes
1answer
85 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: ...
3
votes
1answer
221 views

Value of European Call equals Value of American Call, Question on Explanation/Proof

I am reading S. Shreve, Stochastic Calculus for Finance, Vol. I. There he proves that American Call Options have the same value as European Call Options. In the proof he uses that for a Call option ...
3
votes
1answer
65 views

The Relation Between the Ricci flow and the Black-Scholes-Merton Equation

Grisha Perelman once wrote that The Ricci-flow equation, a type of heat equation, is a distant relative of the Black-Scholes equation that bond traders around the world use to price stock and ...
3
votes
0answers
59 views

FTAP in the model independent case, paper by Schachermayer

I have a question about the following paper by Beatrice Acciaio, Mathias Beiglböck, Friedrich Penkner, Walter Schachermayer. At the very beginning of the paper, on page 3, there are two definitions ...
3
votes
1answer
192 views

Foward-start option pricing

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and is generated by $1 d $- ...
3
votes
0answers
389 views

Monty Hall Model

Given a fixed time period,say 3 days, the stock/market can go up,down or stay sideways. A hedge fund can long, short or use rangebound(options strategy) to bet for that 3 days closing level. Hedge ...
2
votes
1answer
103 views

Distribution of minimum of hazard functions

Suppose I have two random variables, $X_1$ and $X_2$, that are independent (but not identically distributed) and assume both have hazard functions $\lambda_1(s)$ and $\lambda_2(s)$, for $s > 0$. ...
2
votes
1answer
103 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 ...
2
votes
1answer
94 views

Pricing digital options in discrete time

I am stuck in this exercise from my textbook: Consider a one-period market model with $N+1$ assets: a bond, a stock and $N-1$ call options. The prices of the bond are $B_0=1$ and $B_1 = 1+r$, ...
2
votes
1answer
49 views

Trinomial model converges to Black-Scholes weakly

Consider risk-neutral trinomial model with $N$ periods presented by $$S_{(k+1)\delta}H_{k+1}, \ \ \text{for} \ \ k=0,\ldots,N-1$$ where $\delta:=\frac{T}{N}$ and $\{H_k\}_{1}^{N}$ is a sequence ...
2
votes
1answer
137 views

Black-Scholes Equation with dividend

Consider a European option with payoff $$g(S_T) = S_T^{-5}e^{10S_T}$$ Assume that the interest rate is $r = .1$ and the underlying asset satisfies $S_0 = 2, \sigma = .2$, an pays dividend at ...
2
votes
1answer
72 views

$\frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+$ at time $T_i$ is equivalent to a payment of $(A-p(T_{i-1},T_i))^+$ at time $T_{i-1}$

How can I show that payment of $\frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+$ at time $T_i$ is equivalent to a payment of $(A-p(T_{i-1},T_i))^+$ at time $T_{i-1}$ ? Where A is a deterministic ...
2
votes
0answers
46 views

Show that in an arbitrage-free and non-redundant market a certain set is compact

Some notation: We consider a financial market with $d+1$ assets, the $0$-th asset is considered the risk-free asset, the others are the risky ones. The vector $\overline \pi \in \mathbb R^{d+1}$ ...
2
votes
0answers
771 views

How to correctly construct a value- and equally weighted portfolio consisting of property-types?

A problem of which I couldn’t find the answer on the forum is about the construction of equally-weighted and value-weighted portfolio. I want to compute the equally-weighted property-type portfolio ...
2
votes
0answers
185 views

Measure change in a bond option problem

This is not a homework or assignment exercise. I'm trying to evaluate $\displaystyle \ \ I := E_\beta \big[\frac{1}{\beta(T_0)} K \mathbf{1}_{\{B(T_0,T_1) > K\}}\big]$, where $\beta$ is the ...