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.
64
questions
0
votes
0answers
26 views
Positive carry with negative yielding bonds when repo is negative
Could someone please explain to me how positive carry is achieved when the repo rate is negative?
For example I can see the German repo rate is -0.57% and the 2 year German bund is -0.78%. So to ...
1
vote
0answers
41 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 ...
2
votes
2answers
169 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-...
2
votes
2answers
101 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 ...
1
vote
1answer
65 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.
1
vote
3answers
307 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
66 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
$$C_t = \...
1
vote
0answers
48 views
What should I learn/know before reading Investments by Bodie Kane Marcus?
I hope this is the appropriate place to post this. If not, I would really appreciate if someone could redirect me to the right site.
I've been seeing a lot of recommendations for the book, ...
3
votes
1answer
118 views
Introduction of a stochastic discount factor in martingale pricing
The example below is taken from Björk (2009). Let Radon-Nikodym derivative be
$$L=\frac{dP}{dQ} \;\; \text{on} \; \mathcal F$$
or written analogously
$$P(A) = \int_AL(\omega)dQ(\omega) \;\; \text{for ...
5
votes
1answer
348 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 ...
5
votes
2answers
1k 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
597 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
232 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
169 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
155 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
64 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
50 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
109 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
146 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
264 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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vote
2answers
98 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
4k 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
191 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
79 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
98 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
100 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
102 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
913 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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1answer
272 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
225 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: $$\...
-1
votes
1answer
73 views
Why would a principal 'insist on a name' at the original price
A Dealing Certificate practice question
What is a principal doing if he 'insists on a name' at the original price?
Answer:
He refuses the broker's compensation and demands that the transaction is ...
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0answers
242 views
How to calculate an option porfolio cost and payoff function?
There are call and put options on the same underlying asset, with the same expiry, $T$, and with strikes $K_c=(k_c^1, k_c^2, \ldots, k_c^m)$ and $K_p=(k_p^1, k_p^2, \ldots, k_p^m)$, $S_t$ is a price ...
0
votes
1answer
78 views
Is it possible to calculate the call-put parity for an option's portfolio?
Let's say I have designed an option's portfolio. The portfolio includes long as well as short positions in European-style put and call contracts based on the same underlying asset with different ...
1
vote
2answers
1k views
Trading liquidity risk
I am trying to understand trading liquidity risk $\cdots$ "Trading liquidity risk occurs when an entity is unable to buy or sell a security at the market price due to a temporary inability to find a ...
0
votes
1answer
470 views
How to derive the formula for risk-neutral probability for a Standard Binomial Tree (Forward Tree)
Consider a standard binomial tree. Let $u = e^{(r - \delta)h + \sigma\sqrt{h}}$ and $d = e^{(r - \delta)h - \sigma\sqrt{h}},$ where $\delta$ is the continuously compounded dividend yield, $h$ is the ...
2
votes
1answer
164 views
Calculating the annual return on an option using a replicating porfolio
I am self-studying and encountered the following problem:
My idea was to calculate the price of the put using a replicating portfolio, then use the formula:
$$Pe^{\gamma h} = S\Delta e^{\alpha h} + \...
0
votes
2answers
434 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
883 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
235 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
180 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
194 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
154 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
190 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
111 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 ...
