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.
61
questions
2
votes
2answers
58 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 ...
0
votes
1answer
49 views
How does buying a CDX and then taking a short CDS position generates alpha?
Can someone please explain to me how buying a CDX and then taking a short CDS position generates alpha? I am so confused.
1
vote
3answers
127 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 ...
2
votes
1answer
53 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
$$C_t = \...
1
vote
0answers
47 views
What should I learn/know before reading Investments by Bodie Kane Marcus?
I hope this is the appropriate place to post this. If not, I would really appreciate if someone could redirect me to the right site.
I've been seeing a lot of recommendations for the book, ...
3
votes
1answer
104 views
Introduction of a stochastic discount factor in martingale pricing
The example below is taken from Björk (2009). Let Radon-Nikodym derivative be
$$L=\frac{dP}{dQ} \;\; \text{on} \; \mathcal F$$
or written analogously
$$P(A) = \int_AL(\omega)dQ(\omega) \;\; \text{for ...
5
votes
1answer
222 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 ...
4
votes
2answers
806 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{|\...
1
vote
1answer
336 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 ...
-1
votes
1answer
226 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 ...
1
vote
1answer
150 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 ...
2
votes
1answer
150 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 ...
2
votes
1answer
59 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 ...
1
vote
0answers
47 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 ...
1
vote
3answers
103 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 ...
0
votes
1answer
128 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 $\phi(t)$ ...
0
votes
2answers
240 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-...
1
vote
2answers
96 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 ...
1
vote
0answers
1k 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 \...
2
votes
1answer
3k 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, ...
2
votes
1answer
289 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 $\...
2
votes
3answers
189 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 ...
1
vote
1answer
77 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}\...
2
votes
1answer
96 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 ...
0
votes
1answer
93 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 ...
0
votes
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 ...
0
votes
1answer
99 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 = \...
1
vote
1answer
832 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 - \...
0
votes
1answer
249 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}...
2
votes
0answers
192 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: $$\...
-1
votes
1answer
72 views
Why would a principal 'insist on a name' at the original price
A Dealing Certificate practice question
What is a principal doing if he 'insists on a name' at the original price?
Answer:
He refuses the broker's compensation and demands that the transaction is ...
0
votes
0answers
228 views
How to calculate an option porfolio cost and payoff function?
There are call and put options on the same underlying asset, with the same expiry, $T$, and with strikes $K_c=(k_c^1, k_c^2, \ldots, k_c^m)$ and $K_p=(k_p^1, k_p^2, \ldots, k_p^m)$, $S_t$ is a price ...
0
votes
1answer
76 views
Is it possible to calculate the call-put parity for an option's portfolio?
Let's say I have designed an option's portfolio. The portfolio includes long as well as short positions in European-style put and call contracts based on the same underlying asset with different ...
1
vote
2answers
1k views
Trading liquidity risk
I am trying to understand trading liquidity risk $\cdots$ "Trading liquidity risk occurs when an entity is unable to buy or sell a security at the market price due to a temporary inability to find a ...
0
votes
1answer
413 views
How to derive the formula for risk-neutral probability for a Standard Binomial Tree (Forward Tree)
Consider a standard binomial tree. Let $u = e^{(r - \delta)h + \sigma\sqrt{h}}$ and $d = e^{(r - \delta)h - \sigma\sqrt{h}},$ where $\delta$ is the continuously compounded dividend yield, $h$ is the ...
2
votes
1answer
164 views
Calculating the annual return on an option using a replicating porfolio
I am self-studying and encountered the following problem:
My idea was to calculate the price of the put using a replicating portfolio, then use the formula:
$$Pe^{\gamma h} = S\Delta e^{\alpha h} + \...
0
votes
2answers
389 views
Understanding the payoff of currency options
I am self-studying for an actuarial exam and I am having a hard time understanding what happens when a currency option pays off.
Consider the below problem. The payoff at $C_u$ would be $\max(x_u - ...
3
votes
1answer
801 views
Understanding the relationship between the Black-Scholes formula and a replicating portfolio
I'm self-studying and I'm considering the below example. The specific example is not especially relevant, but I included it for reference.
I'm trying to understand the relationship between a ...
0
votes
1answer
63 views
Clarification on this author's solution for this problem on lognormal stock distribution
I am self-studying from a manual on financial economics, and I am trying to completely wrap my head around this solution:
I'm trying to fill in the in-between steps of this solution based on first ...
2
votes
1answer
226 views
Is this a poorly written example, or could volatility in fact be negative?
I'm self-studying and I encountered the following example. It seems to suggest that volatility is negative in this example. I was under the impression that volatility can never be negative, both from ...
1
vote
2answers
163 views
Monte Carlo Accuracy - Antithetic Variate Method
I'm self studying for an actuarial exam and I am curious about a property of the antithetic variate method for increasing the Monte Carlo price accuracy (i.e. For every random draw of $z$, also ...
2
votes
1answer
169 views
Why are the greeks for the underlying stock 0 with the exception of delta?
In my textbook that I am self-studying from it is given that (assuming the Black-Scholes framework):
$\Delta_{stock} = \partial S / \partial S = 1$
All other Greeks for the underlying stock = 0
I ...
1
vote
1answer
146 views
How does this statement about the price of a prepaid forward on a stock follow?
I am self-studying for an actuarial exam on financial economics. This statement in the following problem/solution seems to imply that the prepaid forward price on a stock is the same as the prepaid ...
4
votes
2answers
173 views
What's the explanation for the formula for the volatility of a stock / volatility of the continuously compounded return of a stock?
I am self-studying for an actuarial exam, Models for Financial Economics.
It's stated as a given in my manual that $\sigma$ is the volatility of the stock, $\sqrt{\text{Var}(\ln(S_t/S_0))}$ and that ...
3
votes
1answer
134 views
Is there an error in this problem on pricing an asset using the true probability of an up move?
I'm self-studying for an actuarial exam and I encountered the following problem:
The true probability of an up move, $p$, must satisfy: $$p = \frac{e^{{(\alpha - \delta})h} - d}{u - d},$$
where $\...
1
vote
1answer
93 views
Calculating the price of a call and put using multinomial trees and risk-neutral probabilities
I am self-studying for an actuarial exam and I encountered this example. The books shows one method of solving using a replicating portfolio, and then shows this solution involving risk-neutral ...
8
votes
4answers
2k views
Self study references for a Mathematician
I just finished my undergraduate (BSc) degree in Pure Mathematics & Applied Mathematics. I am starting my postgraduate degree in Pure Mathematics in a month's time.
I am considering pursuing a ...
1
vote
1answer
101 views
Option analysis
Assume zero dividend and that the strike price for a European call option on a stock at a fixed maturity T and strike price K is given by C(K).Suppose that $C(K)=e^{-k}$ for all $K\geq 0$ ,then, I ...
2
votes
0answers
74 views
Modelling the Cost of Risk
I would like to read something about the cost of risk. Could anyone recommend some reference about how it is calculated or modelled?
3
votes
1answer
286 views
Risk Manager must-know list
What are the products, concepts, and models a risk manager must know? I'm not looking for an exhaustive list, but rather a general list as the one in Paul & Dominic's Guide To Quant Careers:
...
