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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 people who work in HFT have a mathematical background, but that’s because the tools of applied math and statistics are useful when analysing the large amounts of ...

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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 depended on problems that were yet to be solved in mathematics, and where economists were assuming solutions into existence. As the problems were solved in math,...

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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 specific payoff has no optionality and that the payoff may be negative. Bakshi and Madan (2000) discuss the economic meaning of a derivative paying $\cos(S_T)$ in the ...

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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{align*} Thus, $(S_t^2)$ is again a geometric Brownian motion and hence, for each time point $t$ log-normally distributed with drift $2\mu+\sigma^2$ and volatility $... 8$\max(B_T,S_T)=\max(0,S_T-B_T)+B_T,$so this is just a call option (with strike$B_T$) plus$B_T.$8 the information you provided is not sufficient to deduce risk neutral probabilities. You have to provide something like a price process from which risk neutral probabilities can be computed. Here are some examples: Example1: Consider a game where you pay 1 and you win 6 in case a six is thrown and 0 otherwise. So in financial mathematics terms we have a ... 6 The notations in the snapshot are pretty messy. I prefer to proceed as follows. Let$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^t \frac{\sigma^2(s, u)}{2} ds - \int_0^t \sigma(s, u) d W_s \right). \end{align*} Then \begin{align*} r_t = f(t, t) = f(0, t) + \frac{\partial }{\partial u}\... 4 We assume that the stock price process\{S_t,\,t>0\}$satisfies, under the real-world probability measure$P, an SDE of the form \begin{align*} dS_t=S_t\big((\mu-q)dt+\sigma dW_t\big), \end{align*} where\{W_t, \, t >0\}$is a standard Brownian motion. Here, we need to consider the total return asset$e^{qt}S_t, that is, the asset with the dividend ... 4 In physics (statistical physics), this angle bracket is used to represent average, for example, here is the notation from Van Kampen’s book: And in stochastic calculus, the quadratic variation is usually represented by the same angle brackets. But like he noted the context should make clear which one is meant. In the equation you have referenced, an ... 4 If you are happy to try the brute force approach, then here are the relevant formulae: In ordinary calculus, you have the product rule for the differential of two variables: $$d \left( x_1 x_2\right)=x_1 dx_2+x_2 dx_1$$ The general version of this for differential of products of n variables is: $$d \prod_{i=1}^{n}{x_i}=\sum_{i=1}^{n}{ \prod_{j=1 j \ne i}^... 4 American options (on any underlying) is the first that comes to mind. They are often priced using a tree-based algorithm to determine if there is a benefit to early exercise anywhere along the life of the option. Any options that has non-linearity or trigger conditions (binary, knock-in, etc.) are also candidates for numerical models. 4 The US Dollar Index is the ratio of the US dollar (USD) to a geometric basket of six major foreign currencies – the Euro (EUR), the Japanese yen (JPY), the pound sterling (GBP), the Canadian dollar (CAD), the Swedish kroner (SEK) and the Swiss franc (CHF). The countries that use these currencies constitute the bulk of international trade with the United ... 4 Let's stick to a discrete market for simplicity. So, you have a finite number of states in this type of model. The first fundamental theorem of asset pricing says that the absence of arbitrage in such markets imply the existence of (not necessarily unique) risk-neutral measure and vice-versa. The reason it works in the second direction (the existence of a ... 3 Classic asset price model in the continuous-time limit using a Wiener process notation can be written as$$ dS_t=\mu S_tdt+\sigma S_t dX where S_t is the stock price (not the stock return) and dX is an independent random variable with normal distribution. If we eliminate the drift (\mu = 0) and only focus on randomness as asked in your question we ... 3 In a nutshell, this is the "variance drag" problem. The mechanics of how you short something matter, and it's relevant to the discussion of levered/inverse ETFs that behave differently from classic/vanilla positions. Consider an XYZ future at 100. A day later it's 1% up, at 101. Two days later, it's up 1% again, at 102.1. If I go long, I make 2.1 profit. ... 3 Here is one recipe, in case you can live with Spearman rank correlation. (Which you should: linear correlation is often not appropriate in the non-normal case. And in the normal case, there is almost no difference between the two correlation types.) Generate samples of your k features with all the desired attributes. These samples may be random or ... 3 You should consider the stages of the default process instead of a binary "default", where there are various points the borrower is able to cure the loan. In a traditional credit model, the general process is to predict the state of the loan and then predict transitions between stages over the life of the loan. This is done by simulating macro variables (... 3 Let \mathrm{d}r_t=\mu(t,r_t)\mathrm{d}t+\sigma(t,r_t)\mathrm{d}W_t be a model for the short rate under the risk-neutral measure \mathbb{Q}. Starting from the bond PDE \begin{align*} P_t + \mu(t,r) P_r + \frac{1}{2}\sigma(t,r)^2P_{rr} -rP=0, \end{align*} subject to P(T,T)=1 whose general solution is P(t,T)=\mathbb{E}^\mathbb{Q}\left[e^{-\int_t^T r_u\... 3 Actually this is just the Black-Scholes SDE with zero drift and -\frac{1}{2}\sigma^2 volatility. If you plug that into the well known solution, you get S_t=S_0e^{\frac{1}{8}\sigma^4t-\frac{1}{2}\sigma^2 W_t} but let's calculate it with Ito's formula. Choose f(x)=\log(x), then we have f'(x)=\frac{1}{x} and f''(x)=-\frac{1}{x^2}. Inserting in Ito's ... 3 Suppose that you are riskless asset with return r_{ft} and a risky asset with return r_t and conditional volatility \sigma_t(r_t) := \sqrt{V_t(r_t)}. We build a portfolio using weights (w_1, w_2) \in \mathbb{R}, or as you wrote it w_t := w_{1t}, w_{2t} := 1 - w_t. This portfolio will have a time t return of r_{pt}. Its volatility is given by ... 3 I'm inclined to downvote this question for not being relevant in particular to quantitative finance nor programming but I will give you the benefit of the doubt. I too am new to this community yet I can tell you that we are all deeply interested and passionate about the topic. Some of us are undergrad, some postgrads, some make their career out of this and ... 2 Optimal stochastic control. Hamilton jacobi bellman 2 This is pretty standard fare for a Stats 101 course, so as to rationale, etc. you might benefit from picking up a textbook or otherwise do some reading on this. In brief though, hypothesis testing allows us to assess the likelihood sample estimates are different than theorized values in the absence of actual population values. In the cases above, with a ... 2 Let c_t be the price of an European call with maturity T and D_{t,T} the discount factor from T to t. We assume deterministic rates. Then note that for s<t\leq T:\begin{align} E^Q_s\left(c_t\right)&=E^Q_s\left(E^Q_t\left(D_{t,T}(S_T-K)^+\right)\right) \\[3pt] &=E^Q_s\left(D_{t,T}(S_T-K)^+\right) \\[3pt] &=E^Q_s\left(\frac{D_{s,t}... 2 The intuition is that the price process is diffusive in nature. Over time the distribution of possible prices for the underlying spreads out (i.e. the variance in the possible price 1 year from now is much larger than 1 day from now). So you can think of a humped bell curve distribution flattening out over time. This is exactly how heat behaves. Areas where ... 2 The only difference in the derivation when you have a dividend-yield paying stock lies in the value of the Riskless Portfolio\Pi_t$. The financial meaning here is the key: to delta-hedge your option you buy a quantity$\Delta$of the stock$S$, and only the stock is paying you the dividend, so you have to add this contribution in time to your hedge. The ... 2 There isn't a single answer to this question. It strongly depends on your goals and why it is missing. If you have a long enough time-series, you will find large numbers of missing data points. The NYSE used to maintain a post for companies that did not trade weekly not so long ago. However, unless you have cause to believe there was a reason for it to ... 2 This one's not too difficult. Because the p() of the boom and bust regimes are a 50:50, the vol remains 10% Where you vary the regime probabilities, life gets only a little more complicated. You have four scenarios, as per above. The mean is the sum of the scenario probability * payoff. The variance is sum of the scenario probability * (scenario payoff - ... 2 Seems like the total law of variance problem:$V\left[Y\right]=E\left[ V\left[Y \mid X \right] \right]+V\left[ E\left[Y \mid X \right] \right]$Mean on the other hand will be just the iterated expectation problem:$E\left[Y\right]=E\left[ E\left[Y \mid X \right]\right]$2 Note that for the replicating portfolio to be self-financing it suffices that (1): $$\lambda_t=\frac{V_t-h_tS_t}{B_t}$$ where I have changed the notation by designating by$B_t$the money market account: $$B_t=B_0e^{rt}$$ Hence, because the portfolio is self-financing, its dynamics are:$\$\begin{align} dV_t&=\left(\frac{V_t-h_tS_t}{B_t}\right)dB_t+...

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