11
votes
Tick Imbalance Bars - Advances in Financial Machine Learning
Question 1. Actually, the assumption of trade data format is that you have timestamp, size and price (not bid/ask) of trade. Sometimes, trades(ticks) are included to Level 1 data (also called BBO) ...
6
votes
Measure of a Brownian motion = normal distribution?
It is correct that
$$
\mathbf{P}(t^{-1/2}W(t) \in[a,b])=Φ(b)−Φ(a), \forall t\in(0,\infty)
$$
due to the stationary increments property of the Wiener process and the fact that you normalized the ...
- 71
4
votes
Accepted
Is the Non-discounted Bachelier call option price a Martingale?
Let $P(t,T)$ denote the time $t$ price of a zero-coupon bond maturing at time $T$ and $\mathbb{Q}_T$ be the associated equivalent martingale measure which uses $P(t,T)$ as numeraire. Then, for any $\...
- 16.3k
4
votes
Show a process is Martingale
Assuming standard BS dynamics for $S_t$, you have
$\frac{Z_t}{Z_0}\equiv(\frac{S_t}{S_0})^p = \exp{((rt-\frac{\sigma^2}{2}t+\sigma W_t)(1-\frac{2r}{\sigma^2}))}$
Now, this is a lognormally ...
- 1,416
4
votes
issue with benchmarks in "standard securities calculation methods"
90.422798 is not the correct value for price for Benchmark 18A. If you were the original purchaser of the book in 1993 you would have received an errata correcting that benchmark. If you have a ...
- 41
3
votes
Show a process is Martingale
As a starting point: for price dynamics $dS(t) = rS(t)dt + \sigma S(t) dW^\mathbb{Q}(t)$, to show that $Z(t)/Z(0)$ is a positive mean 1 Q-martingale, use Itô's formula to get:
$dZ(t)=p \sigma Z(t)dW^\...
- 187
3
votes
Accepted
Is a wiener proces measurable? (exercise from Bjork)
I think you mixed several things up. I will try to help you out.
Everything started with your claim that $\Bbb E \bigl[W(T) \mid \mathcal F_t \bigr] = 0$ which is wrong!
if $W$ is a Brownian notion,...
- 1,476
3
votes
Accepted
Risk neutral modelling of a stock
The risk neutral measure is used to price assets (e.g. derivatives) and not to base your investment decisions on.
In the first part of you question your simulation gives you the Risk-Neutral ...
- 1,667
3
votes
Accepted
Why is the statement "the volatility of a $T - t$-month prepaid forward on asset X is $\sigma$" the same as "the volatility of asset X is $\sigma$"?
Almost, indeed
The volatility of an asset over a horizon $[t,T]$ indeed refers to the standard deviation of the log-return observed over that period:
$$ \sqrt{\text{Var}\left(\ln\left(\frac{X_T}{X_t}\...
- 14.8k
3
votes
Accepted
What is a notation '1' in risk neutral probabilities paper?
The 1 you are referring to is a vector of ones
The expression $(1 + R)^T \backslash 1$ appears to be shorthand for a MATLAB equation such as:
...
- 7,024
3
votes
Accepted
Probability and statistics in Quantitative Finance
Imho that's more of a probability question than finance really.
If you take $N$ attempts, then the probability of at least one (or more) failures is the complementary probability of never failing on ...
- 1,671
3
votes
Tick Imbalance Bars - clarification on T index
There is an open source hedge fund project which is implementing the ideas contained in the book and which has a github where you can see their code implementation of tick bars. Personally I always ...
- 2,214
3
votes
How does buying a CDX and then taking a short CDS position generates alpha?
One can argue that the theoretical fair value (intrinsic value) of a credit index is just the sum of values of its component CDSs on the same names and with the same terms and conditions (maturity, ...
- 13.2k
3
votes
Black-Scholes-Merton formula and option pricing
Suppose the distribution of the stock price $S_t$ is positively skewed and thus, assigns more weight (higher probability) to outcomes with high stock prices. The exercise price of an OTM call lies to ...
- 16.3k
2
votes
Interpolating the swap curve
Which swap curve are you trying to interpolate? And what swap are you trying to price? The 60d to 1y60d swap (1y long, starting at 60d), or now to 1y60d (not a usual length)?
Really by interpolating ...
- 3,719
2
votes
Interpolating the swap curve
You can calculate the forward starting rates by hand in Excel using the relationship of spot and forward rates. Ignoring any daycount intricacies, we can say that:
$$(1 + r_{2y})^2 = (1 + r_{1y}) * (...
- 1,986
2
votes
Introduction of a stochastic discount factor in martingale pricing
Let ${\mathcal{F}} = 2^{\Omega}$ and let $\mathcal{G} =\left\{ \Omega, \emptyset, \{1\}, \{2,3\} \right\}$. Both are sigma-algebras of subsets of $\Omega$.
The book/question is confusing since the ...
- 666
2
votes
Tick Imbalance Bars - Advances in Financial Machine Learning
Question 1: Only tick data works for this method of bar generation, which means you actually need to provide the successful transactions. Bid and ask won't work.
Question 2: T value is generated at ...
2
votes
Accepted
Relationship between CML and SML
In equilibrium, all securities and portfolios (i.e. convex combinations of securities) lie on the SML, which plots expected return as a function of beta. Note that outside of equilibrium, if a ...
- 714
2
votes
Accepted
A more mathematically rigorous explanation for why in the B-S model, the expected return on a call goes down as the stock price goes up
I think you nearly got there but made a few mistakes in the application of l'Hopital's rule.
First Limit
In the first case, you got
\begin{eqnarray}
\lim_{S_0 \rightarrow \infty} \Omega & = &...
- 6,094
2
votes
Accepted
Is it possible to approach finding the risk premium of this derivative using Ito's Lemma?
Assume that under the real world measure
$$ dS_t/S_t = (\alpha-\delta) dt + \sigma dZ_t^\Bbb{P} \tag{1} $$
Under the EMM $\Bbb{Q}$ one then needs to have (fundamental theorem of asset pricing: in the ...
- 14.8k
2
votes
Accepted
Is there a quick way to see why this claim $C(S, t)$ on $S$ does not satisfy the Black-Scholes PDE?
You already associated the valuation function ins A, B, C and E with the corresponding products. In order to explicitly exclude D, you don't have to compute all the derivatives but just note that
\...
- 6,094
2
votes
Accepted
Valuing a claim on $S^a$: This exercise/solution appears to have a mistake
Under the Black-Scholes' framework, the underlying stock price price process $\{S_t, \, t\ge 0\}$ satisfies an SDE of the from
$$ dS_t = S_t (\alpha dt + \sigma dW_t),$$
where $\{W_t, \, t\ge 0\}$ is ...
- 21.3k
2
votes
Accepted
Simulating a stock price with Monte Carlo - Why my solution isn't equivalent to the author's
you got a typo. It should be 40.886 in your last equation. Then $\sigma$ should match.
Also, If $\alpha$ means annualized log return, it should be
$\mu\,t = \...
- 712
2
votes
Accepted
Characteristic function and distribution of a random variable
The characteristic function (chf) defines the distribution function in a unique correspondence.
For $X$ Gaussian with mean $0$ and variance $\sigma^2$ the chf $E[e^{i u X}]$ which is given by
$$
e^{-\...
- 13.8k
2
votes
Accepted
Black-Scholes-Merton formula and option pricing
I am giving answer to my question.
If the stock price log returns distribution is skewed to the right, then $mode<median<mean$ in most of the cases.
The strike price of an OTM calls lies to the ...
2
votes
Martingale Binomial Tree Process
Under Martingale framework you can admit ,without loss of generality, to be under an arbitrage free market. By the way the martingale process is the discounted spot, you then need to use $$\exp^{-3*0....
- 21
2
votes
Characterizing distribution of a stochastic intergal
It was shown in another question that
\begin{align}
\int^T_0 f(t) Z_t dt &= Z_T \int^T_0 f(u) du - \int^T_0 \Bigl( \int^s_0 f(u) du \Bigr) dZ_s\\
&= Z_T \cdot F(T) - \int^T_0 F(s) \cdot dZ_s
\...
- 3,056
2
votes
How to derive the weights of tangency portfolio?
The derivation is simple but quite tedious. The tangency portfolio is found by maximizing the slope of the capital allocation line (CAL). The slope $S_p$ of the CAL is given by:
$\begin{align}S_p=\...
- 321
1
vote
Bootstrap zero curve source of information
I would recommend you start with the basics and only then go to detailed examples when understanding bootstrapping.
Important things to remember:
The source of information when building a curve are ...
- 5,895
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