23

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 ...


8

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,...


6

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 and Journals" the author formulate his version of the question as follows: It is worth mentioning one last aspect ... of the ‘institutional ecology’ in ...


5

Because vanilla derivatives with European exercise depend only on total variance , not on it's dynamics in time. If you have a simpler model (like interpolation of these total variances from your volatility surface) you don't have as much of unobservable parameters stochastic volatility models have. Having more parameters (which many times would need to be ...


4

Assume that: $$ S_0^1(1+r)\leq a,b $$ Arbitrage for a portfolio $V_t$ is defined as: $$V_0\leq0, \quad P(V_1\geq0)=1, \quad P(V_1>0)>0$$ Consider borrowing at rate $r$ to buy the risky asset such that $V_0=0$. Then, assuming $a\not= b$: $$\begin{align} \min_{\omega}V_1(\omega)=a-S_0^1(1+r)\geq 0 \\ \max_{\omega}V_1(\omega)=b-S_0^1(1+r)> 0 \end{...


4

It is indeed no rounding error, but follows from the way Yahoo computes the adjusted price: it does not reflect the actual returns of the investor. Just look at August 17 and 20. The actual close prices were 10.75 and 9.95. On August 20 the company went ex-dividend for an amount 0.4508. The return on that day is $\frac{P_t+D_t}{P_{t-1}} -1 = \frac{9.95+0....


4

You may show it as follows: \begin{align*} f_{t,T}&= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{T-t}}-1\\ &=e^{\frac{1}{T-t} \left[\ln (1+r_T)^T - \ln (1+r_t)^t \right]} -1\\ &\approx e^{\frac{1}{T-t} \left[(1+r_T)^T-1 - \big((1+r_t)^t -1\big)\right]} -1\\ &=e^{\frac{1}{T-t} \left[(1+r_T)^T - (1+r_t)^t\right]} -1\\ &\approx 1+ \...


4

I'm assuming you're talking about a European option. I did a similar problem for my homework recently, I used the in-out parity for pricing the up and in barrier option. Basically European Option = Knock up and in Option + Knock up and out option You can price the up and out easily using Binomial and use BS formula for pricing the European Option, then ...


4

1 USD today will be worth 1.04 USD in 1 year. Similarly, 1.7 CHF today will be worth 1.7459 CHF in 1 year. As a result, we can expect the USD/CHF rate in one year to simply be 1.7459/1.04 = 1.67875.


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 ...


3

Under Black-Scholes assumption for the 2 assets $S_1$ and $S_2$ with volatilities $\sigma_{1,2}$ and correlation $\rho$ the value of this option has an explicit expression which is the Margrabe formula To quote the result explicitly Introducing $\sigma = \sqrt{\sigma_1^2 + \sigma_2^2 - 2 \sigma_1\sigma_2\rho}$, Margrabe's formula states that the fair price ...


3

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. Then to answer your question is suffices to notice that $$\frac{Y_u}{\sqrt{1+Y_u^2}}=\tanh(U_t)$$ which is bounded therefore your expression is finite ...


3

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 $h=h(u,x)=f(g(u),x)$ is defined with variables $u$ and $x$. Then Ito's formula states that $$ {\rm d}h(u,X_u)=\frac{\partial h}{\partial u}(u,X_u)\,{\rm d}u+\frac{\partial h}{\...


3

$t$ is fixed to simply apply Ito Lemma to $h(s,X_s)$ with the function $h: (s,x)\rightarrow f(t-s,x)$ and you get your answer. There's nothing special about it, I think you are a bit confused by the change of variable $s\rightarrow(t-s)$. @hypernova has laid out the complete steps below for you.


3

Note that we can write $S_1(\omega)$ as a convex combination of $\alpha$ and $\beta$ with \begin{equation} S_1(\omega) = \frac{\beta-S_1(\omega)}{\beta-\alpha} \alpha + \frac{S_1(\omega) - \alpha}{\beta-\alpha} \beta \end{equation} Since $h$ was a convex function then by definition \begin{equation} h(S_1(\omega)) \leq \frac{\beta-S_1(\omega)}{\beta-\alpha} ...


3

For the house, there are reputational advantages of publishing (“ we have smart quants”). This may not outweigh the loss of competitive information, depending on the material published. As you imply, the main benefit seems to accrue to the quants themselves, in terms of enhancing their own brand. Perhaps this paradox arises from the question of what is ...


3

Moving averages of prices are closely related to moving averages of price differences. In particular, if the price is a cumulative sum of historical price differences, $$ p_t = \sum_{j=0} \delta p_{t-j} $$ then a moving average of prices with weights $w_k$ can be written as a moving average of price differences with weights $v_k$ $$ \sum_{k=0}w_k p_{t-k} =...


3

Broadly you're asking about directional versus relative value strategies. There are lots of directional approaches, but I've yet to see many discussed publicly in non-generic ways (I mean, if they work, why would anyone talk about them?). As others have noted, trend following is a notable example. I'd consider a lot of equity factor approaches as a ...


3

The optimal investment strategy depends on the investment goals, or equivalently your utility function (which the investment strategy is supposed to maximize). The forward will trade at $\mathbf{E}^*_0(F_T)$ in the market when you invest at $t=0$. If you buy your maximum volume $M$, then gain/loss at $T$ is given by $M(F_T-\mathbf{E}^*_0(F_T))$ (which is ...


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

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 ...


2

The parameter $\xi$ represents your strategy, namely the quantity you hold in your portfolio of each security $S^0$, $S^1$ and $S^2$. Consider the following strategy: $${\xi}=(\xi^1,\xi^2,\xi^3)=(1.5,1,-0.5)$$ Then: $$\begin{align} & t=0: && \xi\bar{S}_0=\xi^0S_0^0+\xi^1S_0^1+\xi^2S_0^2 = 1.5+2-3.5=0 \\ & t=1: && \xi\bar{S}_1(\omega_1)...


2

Both @alexprice and @FunnyBuzer have some good points, and I have upvoted them. I think I have enough to add here that I'll make another answer entry. First off, @AFK was fairly correct that you do not need stochastic volatility for vanilla (European exercise) option pricing, since (as he says and alexprice elaborates) you just interpolate the surface of ...


2

I think that the main advantage of using a stochastic volatility model is to produce a consistent volatility smile. Let's consider the pricing formulas for the normal and lognormal volatilities: $$dS_t=\sigma dW_t\Rightarrow \mathbb{E}[(S_T-K)^+]=(S_t-K)\Phi\left(\frac{s-S_t}{\sigma\sqrt{\Delta t}}\right)+\sigma\sqrt{\Delta t}\phi\left(\frac{s-S_t}{\sigma\...


2

Your reasoning for the first property does not look correct or at least I do not understand it. Your arguments for the second property seem sound. But your wording of the second property is a bit fuzzy. You should state this more clearly, for example: $C(1,\ldots,1,u_j,1,\ldots,1) = u_j$ for all $u_j\in [0,1]$ and $j\in 1,\ldots, d.$ You don't mention it ...


2

There are different types of publication by banks paid publication: this is what is called “research” where bank analysts or quants offer the bank client various sorts of market insights or reviews. This is paid research and usually under condition that client does not broacast it to others. Of course some very famous ones get so popular that they ...


2

At $t_1$, this payoff can be priced using the Margrabe formula as used for pricing an exchange option. See Margrabe Formula here Using the notations in the question and those used the hyperlinked document above - $Price_{t_1} = P_{t_1}e^{(\mu_P-r)\tau}\Phi(d_+) -HR \times G_{t_1}e^{(\mu_G-r)\tau}\Phi(d_-) \tag{1}$ $Price_0$ is the discounted value of $...


2

Find the conditions under which: $E_{0}^{*}[\max (P_{T} - HR\times G_T, 0)] = \max (P_{0} - HR\times G_0, 0)$ We have a no-brainer solution - the condition that the drift and volatility of both $P$ and $G$ is zero, which means $P$ and $G$ are constants in time. Second valid condition - the option is deep in the money or deep out of the money, such that ...


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

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 ...


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