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.
89
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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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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 ...
2
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0
answers
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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vote
0
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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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votes
1
answer
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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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1
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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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1
answer
89
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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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1
answer
790
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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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0
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86
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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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votes
0
answers
60
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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 ...
- 141
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vote
2
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1k
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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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vote
0
answers
97
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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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1
answer
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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
answer
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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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1
answer
115
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Calculating European call option, the Bjork way
We have a 3 period binomial tree with values:
...
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1
answer
311
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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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1
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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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1
answer
102
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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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1
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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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1
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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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1
answer
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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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1
answer
83
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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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0
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602
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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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2
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383
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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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2
answers
777
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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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1
answer
647
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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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1
answer
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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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1
answer
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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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2
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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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1
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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 ...
1
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1
answer
456
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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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1
answer
203
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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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votes
1
answer
149
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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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3
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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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542
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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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How to derive the Greek theta from Black-Scholes solution formula?
Which are the steps to compute the theta greek from the BS solution:
$$c(t, x) = xN(d_+(T-t,x)) - K e ^{-r(T-t)}N(d_-(T-t,x))$$
with:
$$ d_\pm (T-t, x) = \dfrac{1}{\sigma \sqrt{T-t}} \left[ \ln \...
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1
answer
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Z-Spread vs Discount Margin
I'm comparing two types of discounting: Z-Spread and Discount Margin.
Reading the article by O'Kane Credit Spread Explained I found Z-Spread is used for fixed rate notes meanwhile Discount Margin, ...
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314
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What is a notation '1' in risk neutral probabilities paper?
I'm reading the paper by Zhao et al (2008) and have a problem with used definitions in the text on the page 1535.
First, we generate a sample, $R$, of a given size from the distribution (21). Let $\...
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A more mathematically rigorous explanation for why in the B-S model, the expected return on a call goes down as the stock price goes up
A problem asks whether the following statement is true assuming the Black-Scholes Framework:
The expected return on a call option goes up as the stock price goes up.
The solution is:
The statement ...
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