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.

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72 views

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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1answer
53 views

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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131 views

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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87 views

Calculating European call option, the Bjork way

We have a 3 period binomial tree with values: ...
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54 views

Risk free rate application to option pricing

We have $S_o = 50, u = 1.0606, d = 1/u, K = 54.50,$ risk free rate $r = 0.1$ per week, maturity in 9 weeks, given a binomial tree (3 steps)with the probabilities given by $q = (1+e^{r(T-t)}/u-d)$, no ...
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1answer
65 views

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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39 views

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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1answer
48 views

(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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1answer
41 views

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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1answer
77 views

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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1answer
65 views

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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104 views

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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70 views

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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37 views

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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73 views

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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120 views

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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2answers
275 views

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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2answers
295 views

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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1answer
325 views

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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3answers
1k views

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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1answer
101 views

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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61 views

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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1answer
166 views

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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710 views

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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3k views

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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2answers
1k views

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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256 views

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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1answer
239 views

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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1answer
171 views

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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1answer
83 views

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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2answers
97 views

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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3answers
126 views

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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1answer
207 views

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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378 views

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-...
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107 views

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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2k views

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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1answer
5k views

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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1answer
294 views

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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220 views

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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88 views

Is it possible to approach finding the risk premium of this derivative using Ito's Lemma?

I understand the author's intended solution to the below problem, but I thought I would see if I could solve this using first principles and Ito's Lemma instead for practice. Let $V(S(t), t) = e^{rt}\...
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1answer
103 views

Is there a quick way to see why this claim $C(S, t)$ on $S$ does not satisfy the Black-Scholes PDE?

I'm self-studying for an actuarial exam on financial economics and encountered the below practice exam problem. An exam problem should typically take 5-6 minutes to complete, so I'm wondering if ...
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108 views

Why is the statement “the volatility of a $T - t$-month prepaid forward on asset X is $\sigma$” the same as “the volatility of asset X is $\sigma$”?

I'm self studying and I'm having trouble with understanding the equivalent formulations of the volatility $\sigma$ of an asset $X$, as in the below problem. In the below the problem (and the first ...
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1answer
46 views

Valuing a claim on $S^a$: This exercise/solution appears to have a mistake

The below exercise and solution was found in "Models for Financial Economics" by Abraham Weishaus. My issues are: In this problem, $S(t)$ does not satisfy the Black-Scholes framework because ...
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1answer
113 views

Simulating a stock price with Monte Carlo - Why my solution isn't equivalent to the author's

I am self-studying and I am working on the following problem: My solution is different and I'm arriving at a different answer: The parameters of the lognormal random variable $S_t/S_0$ are: $$m = \...
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1answer
1k views

Proving that the $\Delta$ of a call on a futures contract under the B-S model is $N(d_1)$

The author of my textbook says that the $\Delta$ of a call on a futures contract is $N(d_1)$ and not $e^{-rT}N(d_1)$. I wasn't convinced, so I tried to prove this. Let $F = F_{0, T}(S) = S_0e^{(r - \...
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363 views

Why doesn't the overnight profit on a delta-hedged porfolio include interest on the initial selling/buying of the option?

I am self-studying and encountered the following passage from my textbook on the market maker's overnight profit on a delta-hedged portfolio: I don't understand why their isn't a factor of $(e^{r/365}...
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371 views

How do we know that the instaneous rate of return on this option, $\gamma$ is negative?

I am self-studying models for financial economics and encountered the following problem: I don't see how the author can conclude that $\gamma = -0.62$. Let's rearrange the second to last equation: $$\...
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1answer
73 views

Why would a principal 'insist on a name' at the original price

A Dealing Certificate practice question What is a principal doing if he 'insists on a name' at the original price? Answer: He refuses the broker's compensation and demands that the transaction is ...
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333 views

How to calculate an option porfolio cost and payoff function?

There are call and put options on the same underlying asset, with the same expiry, $T$, and with strikes $K_c=(k_c^1, k_c^2, \ldots, k_c^m)$ and $K_p=(k_p^1, k_p^2, \ldots, k_p^m)$, $S_t$ is a price ...
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1answer
84 views

Is it possible to calculate the call-put parity for an option's portfolio?

Let's say I have designed an option's portfolio. The portfolio includes long as well as short positions in European-style put and call contracts based on the same underlying asset with different ...