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 ...
- 5,971
22
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 ...
- 6,672
17
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 ...
- 5,971
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 ...
- 4,329
15
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 ...
- 251
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 ...
- 11.9k
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-...
- 14.8k
11
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 ...
- 13.8k
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 ...
- 7,024
11
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 ...
- 14.8k
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 ...
- 16.3k
10
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.
...
- 3,056
10
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 ...
- 5,702
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 ...
- 1,476
9
votes
Accepted
Preferred Option pricing model
General Comment: In industry, you're effectively an engineer/mechanic. You choose the best tool for the job, and there is no 1 tool that works with everything because they all have different benefits ...
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 ...
- 8,219
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{...
- 16.3k
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, ...
- 7,140
7
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 ...
- 11.8k
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 ...
- 1,714
7
votes
Accepted
Explanation for Different Piecewise Yield Term Structures from QuantLib Python
When you bootstrap a curve, you get discount factors/zero rates for the maturities of the instruments you supplied. So in practice, you get points, and not a "curve".
After you have built ...
- 5,895
7
votes
Accepted
Parameters in Nelson-Siegel model and Nelson-Siegel-Svensson model
The Nelson-Siegel model has four parameters: $\beta_0$, $\beta_1$, $\beta_2$, and $\lambda$. These parameters have the following restrictions:
$\beta_0$, $\beta_1$ and $\beta_2$ can be any real ...
- 1,779
6
votes
Accepted
Why are some utility functions widely used?
These are a natural and easiest (most tractable mathematically) choice.
A utility function is defined up to a positive affine transformation: economically there is no difference between the utility ...
- 5,120
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) =...
- 8,219
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}...
- 1,096
6
votes
Accepted
Determine the right order size with market making strategy
"I need to get an algo or a formula to determine to right quantity to trade each time I place the pair (limit_buy_order, limit_sell_order)."
Actually, you need a formula for determination of the ...
- 451
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),...
- 406
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.
...
- 2,207
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^...
- 21.3k
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