7

There is a lot of ways to understand why stationarity allows to apply usual time series analysis. Here is one more. Very often, the theoretical justification of what you do in time series need to be able to identify the mean formula and the expectation: $$\frac{1}{N}\sum_{n=1}^N X_n \underset{N\rightarrow +\infty}{\longrightarrow} \mathbb{E} X, $$ where the ...


6

Bond Price Dynamics I do not know the source of the bond dynamics you show above but seeing how we are dealing with an affine model there is a very elegant way to derive those. Due to the model being affine the bond price is given by $$P(t,T)=A(t,T)e^{-r(t)B(t,T)}$$ you can find the exact formulas for $A(t,T)$ and $B(t,T)$ in this document (or just read ...


5

Let define $\mathbb{Q}$ and $\mathbb{P}$ two equivalent probabilities on a filtered space $(\Omega,(\mathcal{F}_t)_{t\geq 0})$ Let define $Z_T=\frac{d\mathbb{Q}}{d\mathbb{P}}$ restricted to $\mathcal{F}_T$ measurable events. It means that for $X_T$ being $\mathcal{F}_T$ measurable we have: $$\mathbb{E}^{\mathbb{Q}}[X_T] = \mathbb{E}^{\mathbb{P}}\left[...


5

Aside from the independence requirement for the increments, that is, the independence of $X_{s+t}-X_s$ and $\mathcal{F}_s$, you can check whether the increment $X_{s+t}-X_s$ has the distribution of $N(0, t)$. In fact, note that \begin{align*} X_{s+t}-X_s &= (\sqrt{s+t}-\sqrt{s}) Z\\ &\sim N\left(0,\, (\sqrt{s+t}-\sqrt{s})^2\right), \end{align*} which ...


4

This thread will inevitably close because it doesn't meet community guidelines, but I respect your passion in this field and my best suggestion for you is that if you're trying to emulate a MFE education, go look up the course listings of any reputable MFE program, and then look into the sites for those (past) classes and see the recommended readings and ...


4

1) In an academic sense could it be enough to use ML to create a new factor portfolio? The original FF papers (92,93) said something deep because they contradicted the dominant theory of the day. When you say in an academic sense, you may not get much respect from serious academics if you data mine a factor these days. However, as a statistical exercise, ...


4

"I need to get an algo or a formula to determine to right quantity to trade each time I place the pair (limit_buy_order, limit_sell_order)." Actually, you need a formula for determination of the optimal prices, not quantities. For example, if the market goes down and you have long positions in inventory, you should reduce ask price to attract more buy ...


3

There are lots of different sources out there where you can find various quantitative strategies. Usually, different blog aggregators like https://www.r-bloggers.com post on their websites interesting new approaches and techniques. You can also find great deal of information on Quantopian's dedicated page here: https://www.quantopian.com/posts/trading-...


3

If you allow $X_t$ to be two dimensional then a model with a stock price $X_t^1$ and its variance process $X_t^2$ (stochastic volatility) would fit your definition. In such cases to my knowledge we often don't have a closed form of the density of $X_T^1$ but in some cases we have a closed form of the Laplace transform. An example is the Heston model.


3

It depends on your knowledge and skills. Any book that attempts to cover a wide range of financial product is most likely not very technical. You should choose a book that suits your purpose. For example, if you're interested in interest rates modelling, you should consider something like Interest Rate Models - Theory and Practice: With Smile, Inflation and ...


3

Saying that you can't analyze something as is does not make it garbage. You can't eat flour "as-is", but that doesn't mean you throw it out. In order to use "standard" analysis tools, you must first transform the series into something compatible. Some examples of such a transformation include k-th order differences or a log transformation. These ...


3

In my estimation, you are best-served by creating these sheets from scratch. There are a number of reasons for this: You will thoroughly understand the underlying machinations of each calculation You can customize to your specific needs, and so on... If you are looking for some decent introductory texts, I have benefited from Moyer Excel Templates. More ...


3

In the simple case, you have as per first equation on your last slide: $\frac{P(t,T_0)}{P(t,T)}=1+\delta F(t,T_0, T)$ The continuous time equivalent, assuming constant piecewise rate, as per your question, is: $\frac{P(t,T_0)}{P(t,T)}=e^{y (T_0,T) \delta}$ Taking log of both sides, and rearranging: $\frac{1}{\delta} \ln {\frac{P(t,T_0)}{P(t,T)}}=y (T_0,...


2

I would create separate estimates for volume and choice of debt instrument. There are tools to estimate these simultaneously but I do not see a compelling advantage here. I assume the volume is conditional on the choice of debt issuance so you might start by predicting choice of debt issuance and use this as an input to the volume model. The volume model ...


2

@amber - As a general hint: try to solve a small-scale case first. Pick a two- or better three-asset $(\mu,\Sigma)$ and construct the objectives. Construct the "skewness tensor" (it's not a matrix, you can think of it like a "cube" or something - consult this book on how you can actually represent it as a matrix, or Fabozzi et al's textbook for an ...


2

I would recommend "Active Portfolio Management" from Richard Grinold and Ronald Kahn. The book builds up most theories used in portfolio composition with much detail.


2

For Question 1, let $\phi$ be a replicating strategy, that is, $V_T(\phi) = X$. Then for any two martingale measures $u$ and $v$, from the First Fundamental Theorem of Asset Pricing, \begin{align*} E_u\left(X\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right) = V_t(\phi), \end{align*} and \begin{align*} E_v\left(X\frac{S_t^0}{S_T^0}\mid \mathcal{F}_t\right) = V_t(\...


2

We assume that $V_0(\phi)<0$; otherwise, we can consider the strategy $-\phi$. Then, we buy extra $-V_0(\phi)/S_0^0$ share of the risk-free asset $S^0$, from the $k+1$ assets $S^0, S^1,\ldots, S^k$, which is the deposit or money-market account, and hold until maturity $T$, that is, we consider the trading strategy $\psi$, where \begin{align*} \psi_i = \...


2

You basically need to show ii') and iii'), as they automatically imply ii) and iii). Note that, since \begin{align*} \frac{dQ}{dP}\big|_T = \exp\Big(-\gamma W_T - \frac{1}{2} \gamma^2 T\Big), \end{align*} we obtain that \begin{align*} \zeta_t &= E_P\left(\frac{dQ}{dP}\big|_T \mid \mathcal{F}_t \right)\\ &=E_P\left(\exp\Big(-\gamma W_T - \frac{1}{2} \...


2

Assume the law of one price. We show that there does not exist an inconsistent pricing strategy. Suppose that $\phi$ is an inconsistent self-financing trading strategy, that is, $V_T(\phi)\equiv 0$ and $V_0(\phi) < 0$. Consider another self-financing trading strategy $\psi$ that does not nothing, that is, without holding any of the underlying assets. Then ...


2

There is a procedure for finding $T^*$ starting from the portfolio $T$ on the efficient frontier, such that $cov(T^*,T)=0$: From the point $T$ draw a line thorough the point $C$ (which represents the global minimum variance portfolio or GMVP) until it intersects the Y axis at a point $R_z$. Draw a horizontal line from this point until it intersect the ...


2

It seems part of the instruction is there to trouble you. If you have a contract forcing you to buy a stock $S$ at $t=5$ for 2\$, then the value of your contract at maturity is by definition $S_5 -2$. My guess is the question has a follow-up where they as you what the value is at time $t=0$. In this case you can simply create a replicating portfolio, buy ...


2

all (STIR) short term interest rate futures are cash settled [see comment, STIR in this context is -IBOR futures which are the most common in the largest markets] If a party sells 5 contracts at a price of 98.50, and at settlement the EDSP (exchange delivery settlement price) (which is derived from 3M US LIBOR) is, say, 98.40 then the bank has made a profit ...


2

First of all it is martingale not martangale. Secondly it is numeraire not numerator. It sounds like you need to study the basics of risk- neutral pricing. A hint would be that the ratio of two asset prices has no expected drift in the appropriate measure. You are supposed to find the implied probability of the up- move and the down move assuming in (b) ...


2

There is no credit risk because the client pledges the underlying shares as collateral to the funded collar. This is not explained in the article. The structure is built in such a way that the value of the loan + derivative package is always less than the value of the shares. This can be done by for example lending an amount equal to the discounted put ...


2

See also utility indifference pricing (Henderson, V., & Hobson, D. (2004) Utility Indifference Pricing - An Overview http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.321.994&rep=rep1&type=pdf is a good reference) for examples where utility functions can be used to price derivatives on non tradable assets, such as stock options on non ...


1

If you are not interested in correlations, etc. I'd just make a column for each rebalance date, let's say you want to rebalance at the end of each year for 50 years, you'd have 50 columns, and a row for each number of simulations. Let's say 10,000 (each row is going to be the same, but rand() will make the result different for each. The first cell would be ...


1

In your setting, $\beta = B_0 B_1^{-1}$. Then \begin{align*} E_Q(S_1/B_1) &= Q(u)S_1(u)/B_1 + Q(d)S_1(d)/B_1\\ &=\frac{\beta^{-1}S_0 - S_1(d)}{S_1(u) - S_1(d)}S_1(u)/B_1 + \frac{S_1(u) - \beta^{-1}S_0}{S_1(u) - S_1(d)}S_1(d)/B_1\\ &=\frac{\beta^{-1}S_0S_1(u)/B_1 - S_1(d)S_1(u)/B_1 + S_1(u)S_1(d)/B_1-\beta^{-1}S_0S_1(d)/B_1}{S_1(u) - S_1(d)}\\ &...


1

You are almost there. Note that, to have an expectation of the form $(1.15)$, you need to treat $X\beta$ as a random variable together. That is, the respective probabilities $Q(u)$ and $Q(d)$ should apply to the corresponding realizations $X(u)\beta(u)$ and $X(d)\beta(d)$. Specifically, continuing from your last step, \begin{align*} V_0(X) &=\left(\frac{...


1

Thanks for everybody answering my question! Here is my understanding. If a process $\Delta(t)$ is nonrandom, then one could tell what the values will be for all time $t$ when one is standing at $t=0$. On the other hand, if such process is random, then one stands at $t=0$ he cannot see anything in the future. Moreover, the randomness of a simple process ...


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