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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) ...
Alexandr  Proskurin's user avatar
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 ...
phantagarow's user avatar
5 votes
Accepted

Is this a poorly written example, or could volatility in fact be negative?

You seem to use the term "volatility" to describe two very different quantities: (1) the diffusion coefficient of your SDE and (2) the standard deviation of the log-returns under your modelling ...
Quantuple's user avatar
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4 votes
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Why are the greeks for the underlying stock 0 with the exception of delta?

In the black scholes model, today's stock price, risk free rate and stock volatility are considered independent variables. They are inputs to the model. Hence the cross partial derivatives are zero. ...
dm63's user avatar
  • 17.1k
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 ...
Ivan's user avatar
  • 1,386
4 votes
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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 $\...
Kevin's user avatar
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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 ...
user50494's user avatar
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^\...
AlexAbrahams's user avatar
3 votes
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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,...
Cettt's user avatar
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3 votes
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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}\...
Quantuple's user avatar
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3 votes
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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: ...
Matthew Gunn's user avatar
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3 votes
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How does this statement about the price of a prepaid forward on a stock follow?

It is a very badly worded question in my humble opinion. There are three "prices" to contend with. (1) If you want to buy a stock and pay for it now, you pay the current stock price S. (2) If you ...
nbbo2's user avatar
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3 votes
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What's the explanation for the formula for the volatility of a stock / volatility of the continuously compounded return of a stock?

The standard starting point with modelling a stock price process is to use the Black-Scholes model for the stock price. This simply asserts that the changes in the stock price are described by the ...
oliversm's user avatar
  • 1,389
3 votes

Is there an error in this problem on pricing an asset using the true probability of an up move?

Your formula for $p$ is $$p = \frac{e^{{(\alpha - \delta})h} - d}{u - d},$$ where $\alpha$ is not expected return on stock but continuous risk free rate, i.e. 1%. If you use $\alpha$ as 1%, you will ...
Neeraj's user avatar
  • 2,238
3 votes

Difficulty understanding put-call parity for currency options

It costs 0.03 dollars for the option to (sell 1 pound/buy 1.5 dollars. Now divide everything by 1.5: It costs 0.02 dollars for the option to (sell 2/3 pound / buy 1 dollar). Now convert to pounds ...
dm63's user avatar
  • 17.1k
3 votes
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Understanding the relationship between the Black-Scholes formula and a replicating portfolio

It is my understanding that a replicating portfolio for a put involves short selling stock and lending money. You cannot statically replicate an option. So this is not true in general, you'll need to ...
Quantuple's user avatar
  • 14.6k
3 votes
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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 ...
Sanjay's user avatar
  • 1,657
3 votes
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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 ...
ZRH's user avatar
  • 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 ...
babelproofreader's user avatar
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, ...
Dimitri Vulis's user avatar
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 ...
Kevin's user avatar
  • 15.9k
2 votes
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Clarification on this author's solution for this problem on lognormal stock distribution

Yes, your steps are valid This is a wrong use of the term "quantile". Here you need to compute a probability (through the normal cdf) and not a quantile (i.e. the value of a random variable ...
Quantuple's user avatar
  • 14.6k
2 votes
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Monte Carlo Accuracy - Antithetic Variate Method

No, you can have $$ \frac{1}{2n}\sum_{i=1}^{2n} C(S^i_T,K,T) = 0 $$ First off, there's the obvious case where $n=1$ and $u_1 = 0.5$ More generally, for options way out of the money it is common to ...
Brian B's user avatar
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2 votes
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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 & = &...
LocalVolatility's user avatar
2 votes
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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 ...
Quantuple's user avatar
  • 14.6k
2 votes
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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 \...
LocalVolatility's user avatar
2 votes
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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 ...
Gordon's user avatar
  • 21.1k
2 votes
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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 = \...
Will Gu's user avatar
  • 712
2 votes
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How to derive the formula for risk-neutral probability for a Standard Binomial Tree (Forward Tree)

Here's one algebraic way to derive it: $$ \frac{(1 - e^{-\sigma\sqrt{h}})(1 + e^{\sigma\sqrt{h}})}{(e^{\sigma\sqrt{h}} - e^{-\sigma\sqrt{h}})(1 + e^{\sigma\sqrt{h}})} = \frac{1 - e^{-\sigma\sqrt{h}} + ...
KarolisR's user avatar
  • 693
2 votes
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Calculating the annual return on an option using a replicating porfolio

Your computation of $\Delta$ is correct. However, your computation of the cash amount is wrong. You choose the cash amount $\beta$ that you need to initially lend or borrow such that in the up state, ...
LocalVolatility's user avatar

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