10

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 ...


5

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 ...


5

"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 ...


5

I suspect the link is describing autocallable notes of some sort. Typically investors (the buyers of these notes) get high coupon rates in return for selling downside protection, i.e. they sell put options (usually with barrier) to the bank to finance these coupons. That means that as the index moves down approaching the put option barrier the banks become ...


5

The specific quote you reference from the article is from a section explaining why/how the current environment came to be. Note that 'current environment' in the context of this article is almost 16 years ago as the article is dated October 2004. The specific paragraph from that section is referring to the long end of the volatility curve where banks were ...


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

Interest rate options should be priced with risk neutral methods regardless of your opinion of interest rate trends. If you have a view on interest rates, you can express it by taking a delta position. If you were to bias your option prices , you would just end up paying the wrong price for the option.


4

CDS quotes are observable. But none of: probabilities of default, hazard rates, loss given default/recovery, etc are observable. To get some kind of (risk-neutral) probabilities of default, many people make a lot of assumptions, in particular, that the hazard rate is constant (or if you're lucky enough to have CDS quotes at more than one tenor, then ...


3

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 ...


3

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} \...


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

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

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

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


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

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

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

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 ...


2

To calculate the forward price $F$ of a zero coupon bond at t=4, note that arbitrage considerations imply that $$Z(0,10)= Z(0,4) F$$. This essentially means that investing in a 4 year zero coupon bond together with a forward contract to invest from year 4 to year 10 must be the same as investing in a 10 year bond. So you need to first calculate Z(0,4) ...


2

Preamble It is in general true that structured products can be decomposed in simpler products (linear and options for example). Regardless of the decomposition of the specific product, it is typical for the bank to "buy" some rights from the customer. This is the way to grant a higher (expected) yield in structured products: there is no free lunch, if ...


2

As people in the comments noted, signal broadly refers to a trigger variable that denotes an investment decision. This is normally a boolean variable (i.e. 0 or 1) but could be continuous (0 to 1) or any other range (e.g. -1/0/1 sell/hold/buy), depending on what your execution algo might dictate. Just wanted to add that this terminology comes from the ...


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