# Tag Info

### 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) ...

### 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 ...
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### 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 ...
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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. ...
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### 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 ...
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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,...
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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: $$\sqrt{\text{Var}\left(\ln\left(\frac{X_T}{X_t}\... • 14.6k 3 votes Accepted ### What is a notation '1' in risk neutral probabilities paper? The 1 you are referring to is a vector of ones The expression (1 + R)^T \backslash 1 appears to be shorthand for a MATLAB equation such as: ... • 6,944 3 votes Accepted ### How does this statement about the price of a prepaid forward on a stock follow? It is a very badly worded question in my humble opinion. There are three "prices" to contend with. (1) If you want to buy a stock and pay for it now, you pay the current stock price S. (2) If you ... • 11.3k 3 votes Accepted ### What's the explanation for the formula for the volatility of a stock / volatility of the continuously compounded return of a stock? The standard starting point with modelling a stock price process is to use the Black-Scholes model for the stock price. This simply asserts that the changes in the stock price are described by the ... • 1,389 3 votes ### Is there an error in this problem on pricing an asset using the true probability of an up move? Your formula for p is$$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 ... • 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 ... • 17.1k 3 votes Accepted ### 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 ... • 14.6k 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 ... • 1,657 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 ... • 2,159 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, ... • 12.3k 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 ... • 15.9k 2 votes Accepted ### 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 ... • 14.6k 2 votes Accepted ### Monte Carlo Accuracy - Antithetic Variate Method No, you can have$$ \frac{1}{2n}\sum_{i=1}^{2n} C(S^i_T,K,T) = 0 $$First off, there's the obvious case where n=1 and u_1 = 0.5 More generally, for options way out of the money it is common to ... • 14.9k 2 votes Accepted ### A more mathematically rigorous explanation for why in the B-S model, the expected return on a call goes down as the stock price goes up I think you nearly got there but made a few mistakes in the application of l'Hopital's rule. First Limit In the first case, you got \begin{eqnarray} \lim_{S_0 \rightarrow \infty} \Omega & = &... • 6,044 2 votes Accepted ### Is it possible to approach finding the risk premium of this derivative using Ito's Lemma? Assume that under the real world measure$$ dS_t/S_t = (\alpha-\delta) dt + \sigma dZ_t^\Bbb{P} \tag{1} $$Under the EMM \Bbb{Q} one then needs to have (fundamental theorem of asset pricing: in the ... • 14.6k 2 votes Accepted ### Is there a quick way to see why this claim C(S, t) on S does not satisfy the Black-Scholes PDE? You already associated the valuation function ins A, B, C and E with the corresponding products. In order to explicitly exclude D, you don't have to compute all the derivatives but just note that \... • 6,044 2 votes Accepted ### Valuing a claim on S^a: This exercise/solution appears to have a mistake Under the Black-Scholes' framework, the underlying stock price price process \{S_t, \, t\ge 0\} satisfies an SDE of the from$$ dS_t = S_t (\alpha dt + \sigma dW_t),$$where \{W_t, \, t\ge 0\} is ... • 21.1k 2 votes Accepted ### Simulating a stock price with Monte Carlo - Why my solution isn't equivalent to the author's you got a typo. It should be 40.886 in your last equation. Then \sigma should match. Also, If \alpha means annualized log return, it should be \mu\,t = \... • 712 2 votes Accepted ### How to derive the formula for risk-neutral probability for a Standard Binomial Tree (Forward Tree) Here's one algebraic way to derive it:$$ \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}} + ...
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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, ...