Questions tagged [self-study]
A routine question from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for self-study questions.
93
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Expectation of the realized volatility
I was reading Zhang and Wang 2023 and I have some doubts regarding it. The realized Stochastic Volatility Model is expressed as follows:
$$\begin{matrix}
y_t = \exp \big( \frac{h_t}{2} \big) \...
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covariance between squared returns and past returns
Let $y_t = \sqrt{h_t} \epsilon_t$ where $\epsilon_t\overset{ iid}{\sim} N(0,1)$
$h_t = \alpha_0 +\alpha_1 y_{t-1}^2+\beta_1 h_{t-1}$ with $\alpha_0>0, \alpha_1>0, \beta_1<1,\alpha_1+\beta_1&...
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Notation in Advances in Finance ML for time interval
I was reading Prof. Lopez book Advances in Finance ML and he uses a notation I could not understand. In 4.2 (Overlapping Outcomes) he says:
In Chapter 3 we assigned a label $y_i$ to an observed ...
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Proof: Deterministic Ito Integral (Thomas Mikosh Chapter 2)
I'm referencing Elementary Stochastic Calculus with Finance in View by Thomas Mikosch between chapters of Shreve's Volume II text. In one section Mikoshch text makes the following claim without proof:
...
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how to calculate pdf and cdf for an Ornstein-Uhlenbeck process
I have the
Task. For Ornstein-Uhlenbeck process generate a path and plot a)
cumulative distribution (cdf), b) density function (pdf), c) calculate the 95%-quantile.
My solution.
From the literature we ...
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0
answers
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Applications of a certain type of stochastic processes in quantitative finance [duplicate]
A compound Poisson random vector $Y$ is well defined in this site in wikipidia.
Nothing prevents me from compound strictly stationary stochastic processes instead of compound random vectors. The ...
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126
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Optimal consumption process [Munk (2011)]
I'm trying to solve problem 4.4 in Munk (2011). The problem is as follows:
Assume the market is complete and $\xi = (\xi_{t})$ is the unique state-price deflator.
Present value of any consumption ...
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141
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How to test a risk model?
I'm reading the Barra risk model handbook (2004) available online and trying to understand the methodology. I've read a few materials on portfolio theory, so I can get at least the theoretical ideas ...
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1
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142
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Implication of unique risk neutral measure
I'm reading Shreve Stochastic Calculus II, theorem 5.4.9 (Second fundamental theorem of asset pricing),
This is the part that confuses me :
suppose there is only one risk-neutral measure. This ...
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Literature recommendations regarding sentiment in the stock market
I plan to write a paper about the influence of investors sentiment on the stock market.
I would like to look specifically at the question of what has an impact on what: does sentiment influence ...
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Calculate options prices based on given options and spread prices
Suppose you know the following information:
Futures price on a stock is 66
70 strike straddle is trading at 27
50-60 put spread is trading at 2.5
50-60-70 put butterfly is trading at 0.2
Assume ...
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2
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How to derive the weights of tangency portfolio?
I am well aware of this formula but I could not find how to derive this. Of course, I failed to derive (or prove) it by myself. I will appreciate if you guys provide me a good, detailed derivation.
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Vega of derivative when volatility is stochastic?
What is Vega for a derivative when the volatility of the underlying asset stochastic process itself?
When the value of the derivative is $V_d$ vegais $\partial V_d/\partial\sigma$. Consider for ...
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How to compute this current value using no arbitrage condition?
Suppose $X_t$ is a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. $X_0$ is known. You have a machine that produces something worth $X_t$ at random times $t$ generated by a Poisson ...
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Trouble Calibrating a Vasicek Model
I have simulated some data according to a Vasicek process and I am then trying to apply ordinary least squares (OLS) regression analysis to see how accurate the estimated model parameters are from the ...
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Real domestic return
I would like to calculate the real domestic return of a foreign asset
What I know
Real price is $$P_{Real, t} = \frac{P_{Nominal, t}}{CPI_t}$$
where CPI is consumer price index.
And I know that the ...
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Show that portfolio's percentage contribution to loss (PCL) equals PCR (risk)
I came across this question during self study on a quantitative book (Question 3.6 on Page 75 of Quantitative Equity Portfolio Management: Modern Techniques and Applications By Edward E. Qian, Ronald ...
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Characterizing distribution of a stochastic intergal
characterize the distribution of $\int_0^T f(t)Z_tdt$. In
particular, verify that it is a Gaussian distribution and compute its moments.
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1
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issue with benchmarks in "standard securities calculation methods"
I wonder if anyone is using the benchmark cases in "Standard securities calculation methods" issued by Securities Industry Association (Vol 1, 3rd ed.) to calibrate their implementations for ...
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132
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Calculating European call option, the Bjork way
We have a 3 period binomial tree with values:
...
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374
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Martingale Binomial Tree Process
3 step binomial tree process with $S_0=4,u=2,d=0.5,r=0.25.$ Determine the probability p and q such that the stock price process is a martingale (i.e. $E[S3]=S_0)$
I know P = 1/3 and Q = 2/3 but having ...
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Optimizing Portfolio Return by Targeting Variance
I understand Markowitz and targeting returns to minimize our variance. I know this optimization problem well and its constraints. However when the reverse scenario is to be considered I get very ...
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(Self-study) Futures, bonds, and arbitrage
I'm currently self studying futures, so I'm sorry if this questions comes off a bit stupid. I'm currently reading a book by Walsh, J.B. Knowing the Odds: An Introduction to Probability.
I quote this ...
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142
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the relationship between VaR(0.05) and mean?
What is the meaning of the difference between the quantile of prob=0.05 and mean for a sample form a specific distribution?
In other words, I would like to understand the relationship between ...
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Most liquid index options?
I need to work with option prices in my master's thesis. Specifically, I investigate index options (S&P 500). Which kind of options could you recommend to use? I have seen that there are options ...
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Should he choose long position or short position? [closed]
On July 2, 1997, a a company is worry about the value of its Yen income over the next few weeks and makes a decision to hedge its risk by taking a position in the futures market. Right now, a futures ...
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calculation of theoretical value of futures contract [closed]
we form a stock index by using only two stocks in the index.
One of the stocks is the Stock-A. The current selling price of the stock-A is 103 dollars and the second stock is the stock-B. The current ...
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How can I calculate returns for three investment strategy?
Assume that the price of DF stock went from a price of $104 on March 2 to 146 on April 1.
With a current stock price of 146, there is a call option available on the DF stock with an exercise price ...
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112
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Calculate 6 month- return for an investment [closed]
Assume that the price of DF stock went from a price of $104 on March 2 to 146 on April 1.
With a current stock price of 146,
Invest all of your amount 14,600 in the DF stock (buy 100 shares)
...
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Question about the writing a call option on an existing portfolio of stocks [closed]
My question is Please discuss about the following statement
“ the advantages and disadvantages of writing a call option on an existing portfolio of stocks”
Note that
I read an article nearly ...
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735
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Difference between spread duration & IR duration for a fixed rate bond
I am struggling to comprehend the difference in impact between spread duration & IR for a fixed rate bond when yields move.
I know that both measures would be the same for a fixed rate bond but ...
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Black-Scholes-Merton formula and option pricing
If the distribution is skewed to the right,Black-Scholes overprices out-of-the-money puts and in-the-money calls. It underprices in-the-money puts and out-of-the-money calls.
How?
Stock price log-...
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Carry & roll - question regarding the repo transaction
Could someone please explain the carry and roll trade that a lot of traders are doing with negative euro debt?
I read an example that they borrow in the repo market then buy a longer dated bond to ...
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How does buying a CDX and then taking a short CDS position generates alpha? [closed]
Can someone please explain to me how buying a CDX and then taking a short CDS position generates alpha? I am so confused.
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Interpolating the swap curve
Does anyone know how I can calculate the swap rate in between main tenors for specific dates? For example: what is the implied swap rate in 1 year, 60 days time.
Is there an easy way to do this in ...
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Is the Non-discounted Bachelier call option price a Martingale? [duplicate]
My math finance professor once said someting that I can't make sense of. Hope you can answer:
For a foward process the non-discounted price for a European call option under Bachelier is
$$C_t = \...
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What should I learn/know before reading Investments by Bodie Kane Marcus?
I hope this is the appropriate place to post this. If not, I would really appreciate if someone could redirect me to the right site.
I've been seeing a lot of recommendations for the book, ...
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Introduction of a stochastic discount factor in martingale pricing
The example below is taken from Björk (2009). Let Radon-Nikodym derivative be
$$L=\frac{dP}{dQ} \;\; \text{on} \; \mathcal F$$
or written analogously
$$P(A) = \int_AL(\omega)dQ(\omega) \;\; \text{for ...
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Tick Imbalance Bars - clarification on T index
I have been trying to learn quant related things on my own. I recently picked up a book called "Advances in Financial Machine Learning" by Marcos Lopez De Prado. I am having difficulty understanding ...
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Tick Imbalance Bars - Advances in Financial Machine Learning
I would really appreciate if any of you can clarify the following questions. I have been struggling to understand it on my own.
$b_t=\begin{cases}b_{t-1}, & \text{if}\ \Delta p_t = 0 \\ \frac{|\...
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Relationship between CML and SML
I am referring to the book Sharpe et al. (1998), Investments, 6th Edition. I am trying to wrap my head around some lines from the book, pertaining to Security Market Line.
It reads:
Earlier it was ...
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Probability and statistics in Quantitative Finance
Certain types of traders attempt to repeatedly buy and sell the same asset for a profit over a short time period, such as high-frequency “market makers”. For example, if you can repeatedly sell a ...
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Measure of a Brownian motion = normal distribution?
Consider some model where the process increments are normally distributed, e.g. Vasicek:
$$dr(t) = \left(\theta - ar(t)\right)dt + \sigma dW(t).$$
We usually say that $W(t)$ is a Brownian motion ...
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Risk neutral modelling of a stock
Suppose a stock $S$ follows
$$dS(t) = \alpha(t)S(t)dt + \sigma(t)S(t)dW(t),$$
where $W(t)$ is a Brownian motion under $P$. Also suppose there is a short rate process $r(t)$. My question would be is ...
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Characteristic function and distribution of a random variable
This is exercise 4.3 in Bjork, Arbitrage Theory in Continous Time.
$$
X_t = \int^t_0 \sigma(s)dW_s
$$
$\sigma$ is a deterministic function and $W_t$ is brownian motion.
I am asked to find the ...
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Bootstrap zero curve source of information
I'm trying to understand the bootstrap methodology to construct a zero curve from a par curve in detail. I'm looking for a good source of information, preferably with a detailed example, that ...
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Need help to interpret the definition of a diffusion process
https://studentportalen.uu.se/uusp-filearea-tool/download.action?nodeId=1134155&toolAttachmentId=218130
In these lecture notes at page 15 and 16 I am looking at the definition of diffusion ...
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Self finance conditions - proof check
Find expressions for the process $\psi=(\psi(t),\ 0\leq t\leq T)$ , so the portfolio $(\phi,\ \psi)$ is self-financing when:
(1) $\phi(t)= \int_{0}^{t}S_{s}ds $
(2) $\phi(t)=S_{t}$
where $\phi(t)$ is ...
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2
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554
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Show a process is Martingale
$$Z(t)=(\frac{S(t)}{H})^p$$where $S$ has a standard Black-scholes Dynamics for a stock, $H$ is a postive constant and $p =1 - \frac{2r}{\sigma^2}$. How can I show that $Z(t)/Z(0)$ is a postive Q-...
user32098
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Is a wiener proces measurable? (exercise from Bjork)
I will claim $$E[W(T) \vert F_t] = 0$$ for $t<T$. Anyway, in an exercise in Bjork the results requires that $$E[W(t) \vert F_t] = 0$$ But why? Isn't $W(t)$ measurable at time $t$ and hence not ...
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