26 votes
Accepted

What mathematical theory is required for high frequency trading?

Hah! There is no such thing as the “rigorous mathematical underpinning” of high frequency trading - because HFT, like all trading, is not primarily a mathematical endeavour. It’s true that many ...
  • 5,638
17 votes

What mathematical theory is required for high frequency trading?

I would argue, taking a note from John von Neumman, that quantitative finance lacks rigorous underpinnings. Von Neumann warned in 1953 that many things that look like proofs in economics and finance ...
  • 4,123
16 votes
Accepted

Long Gamma vs Vega

Long gamma is being long realized volatility. Long vega is being long implied volatility. Long gamma positions benefit when realized volatility goes up or the actual underlying has volatility. Long ...
  • 5,452
15 votes

What is the difference between pull to par and roll down in both mathematics and conceptual?

Pull-to-par just says that a bond's (clean) price will converge towards its face value as the bonds approaches maturity. There is nothing really interesting about pull-to-par - a bond's (clean) price ...
  • 5,638
13 votes

What is the difference between pull to par and roll down in both mathematics and conceptual?

Pull-to-par says that the bond's price will gradually converge toward par (100% of face value) when yield is unchanged. This process is also known as accretion for a bond trading at a discount (since ...
  • 10.9k
13 votes

Stop-loss start-gain paradox: Why is it a 'paradox'?

I am one of the two authors of the paper. The continuity in time of the path of the underlying suggests that at every trading time, the strategy is self-financing. In fact, if the underlying random ...
11 votes
Accepted

Mathematical equation relating $\frac{dV}{dS}$ to $\frac{dV}{dK}$

If your working modelling assumptions are such that the dynamics of the log price process $\ln(S_t)$ is space homogeneous, you have that the price of a European vanilla option is itself a space-...
  • 14.1k
11 votes
Accepted

What is an adapted process

Let $\{X_t\}$ be a stochastic process and $\mathcal{F}$ be a filtration. The intuitive idea is that for $\{X_t\}$ to be adapted, it can't reveal what's unknowable (according to the filtration). By ...
  • 6,384
11 votes
Accepted

Relationship between Vega and Gamma in Black-Scholes model

Consider any option, vanilla or exotic. In between fixing dates it satisfies the Black & Scholes PDE (for simplicity zero interest rate and dividends) $$ \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 ...
10 votes
Accepted

Does financial math benefit society?

It is not financial mathematics in general, but a scientific approach that is beneficial: quantitative views and open objective tools make transactions more transparent. It decreases information ...
  • 10.7k
10 votes

Implied Volatility of stock on Think or Swim

What they gave you is Newton's formula. If you have a function $f(x)$ then you can find the value $x_0$ such that $f(x_0) = 0$ by this method. It uses the derivative $f'$ which in your case is the ...
  • 13.3k
10 votes
Accepted

Periodic functions when determining No Arbitrage price

It is, of course, possible to price such a contract in a no-arbitrage market. Indeed, if $f$ is a sufficiently smooth function, then you can price all contracts paying $f(S_T)$. Note that your ...
  • 14k
9 votes

Long Gamma vs Vega

Vega (denoted by $\nu$ in what follows) is the first order sensitivity of the option price with respect to volatility $\sigma$. Gamma (denoted by $\Gamma$ in what follows), is the second order ...
  • 14.1k
9 votes

Throwing a dice and risk neutral probability

the information you provided is not sufficient to deduce risk neutral probabilities. You have to provide something like a price process from which risk neutral probabilities can be computed. Here are ...
  • 1,386
8 votes
Accepted

Finding arbitrage opportunity

Generally speaking, let us consider a problem where you have a series of simple payoffs $f_{K_i}(S_T)$ of strike $K_i$, $i \in I$, that depend on the value of $S_T$ at time $T$, as well as a more ...
8 votes

What the expectation of S^2 is from GBM?

As Sanjay said, you can apply Itô's Lemma to $f(t,x)=x^2$ and obtain \begin{align*} \mathrm{d} S^2_t=\left(2\mu S_t^2+\sigma^2S_t^2\right)\mathrm{d}t+\left(2\sigma S_t^2\right)\mathrm{d}W_t. \end{...
  • 14k
8 votes
Accepted

Forward skew generated by Local Vol model

We can demonstrate this via a pricing experiment using QuantLib-Python. I've defined several utility functions in the code block at the bottom of the answer that you will need to replicate the work. ...
  • 2,856
8 votes
Accepted

Inverse Covariance Matrix Transformation from CAPM

This is the result of the Sherman-Morrison inversion for the sum of an invertible matrix and an outer product. You will find this (and many other helpful methods) in the Matrix Cookbook. Specifically, ...
  • 5,903
7 votes
Accepted

Understanding the solution of this integral

Let $\tau = T-t$. Then \begin{align*} S_T = S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, Z}, \end{align*} where $Z$ is a standard normal random variable, independent of $\mathcal{F}...
  • 20.5k
7 votes

What are the benefits of publishing papers in mathematical finance/trading?

There is an interesting article "How Derivatives and Risk Models Really Work: Sociological Pricing and the Role of Co-Ordination" by R. Rebonato answering your question. In section "3.8 Conferences ...
  • 1,644
6 votes
Accepted

Replicating a portfolio with a certain payoff function

A general hedging strategy Let assume that $S_1(t)$ and $S_2(t)$ are the price processes of your 2 stocks and that they follow a Geometric Brownian Motion (GBM): $$\forall \, i \in \{1,2\}, dS_i(t) =...
6 votes
Accepted

Cadlag Property of Jump Proccesses

Intuitively, cadlag expresses the fact that we know a jump has occurred after the fact, but we never have advance knowledge that the jump is about to occur (i.e no knowledge of the starting point for ...
  • 9,647
6 votes
Accepted

Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 4

Under the risk-neutral measure the discounted (under some numéraire) price process is a martingale. If we have a bank account with dynamics $dB_t = r B_t dt$ then the discounted asset $X_t = \frac{S_t}...
  • 1,048
6 votes
Accepted

Application of Ito's lemma

Consider OP's general formula $f(g(t),X_t)$. In case of ambiguity, let us claim that $f=f(t,x)$ is defined with variables $t$ and $x$, $g=g(s)$ is defined with the variable $s$, and $h=h(u,x)=f(g(u),...
  • 386
6 votes
Accepted

How to check if $ E [\exp \{ \int_0^t \frac{Y_u^2}{1+Y_u^2}du \}]< \infty $

If you make the change of variable $Y_t = \sinh U_t$ and apply Ito then you immediately get $$dU_t = 2dW_t$$ so the solution of your SDE is $$Y_t = \sinh\left(2W_t + C\right)$$ with $C$ a constant. ...
  • 2,137
6 votes
Accepted

Zero coupon bond calculations

The notations in the snapshot are pretty messy. I prefer to proceed as follows. Let $X_t = -\int_t^T f(t, u)du$. Note that \begin{align*} f(t, u) - f(0, u) = \frac{\partial }{\partial u}\left(\int_0^...
  • 20.5k
6 votes

Quantitative finance for physicists

Physicists typically know PDEs but not stochastic calculus I have a masters in physics, so have a reasonable idea of the usual skillsets a physicist will know (at least at undergraduate level), and ...
  • 1,359
6 votes

Relationship between Vega and Gamma in Black-Scholes model

I'll give a heuristic "proof" for general European claims which will cause mathematicians to feel sick, but which physicists / practitioners would probably be quite happy work with: Write ...
6 votes

Interpolation of $\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$

That is a tricky question because interpolation seems to be ok if you need one point $\tau$ between $t_k$ and $t_{k+1}$ but it is not. The difficulty arise a direct way if you want two points inside $[...
  • 10.7k

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