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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2
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1answer
50 views

Is the Non-discounted Bachelier call option price a Martingale?

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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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
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1answer
102 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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1answer
179 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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2answers
660 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
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1answer
290 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
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1answer
224 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
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1answer
146 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
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1answer
149 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
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1answer
57 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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0answers
45 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
102 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
125 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 $\phi(t)$ ...
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2answers
235 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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2answers
93 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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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
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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
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1answer
288 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
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3answers
185 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 ...
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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
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1answer
93 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
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1answer
92 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
98 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
805 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
247 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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0answers
189 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
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 ...
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0answers
225 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
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1answer
75 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 ...
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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
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1answer
406 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
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1answer
162 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} + \...
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2answers
380 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
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1answer
773 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
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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
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1answer
223 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
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2answers
162 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 ...
1
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1answer
164 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
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1answer
145 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
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2answers
170 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
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1answer
132 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
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1answer
90 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 ...
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4answers
1k 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 ...
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1answer
100 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 ...
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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
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1answer
283 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: ...
3
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1answer
214 views

Equivalent Definitions of Self-Financing Portfolio

Consider a multi-period model with $t=0,...,T$. Suppose there is a bond with $B_0=1$ and $B_t=(1+R)^t$ and a stock with $S_0=s_0$ and $$ S_{t+1}=S_t\,\xi_{t+1}, $$ with $\xi_t$ iid random variables....
2
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1answer
61 views

Understanding the necessary and sufficient conditions for rational early exercise of a call option

I am self-studying for an actuarial exam, and I encountered the following in my text: The author states that if $PV_{t, T}\text{(Divs)} < K(1 - e^{-r(T - t)})$, early exercise is not rational. ...
4
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4answers
2k views

Difficulty understanding put-call parity for currency options

I am self-studying for an actuarial exam on models for financial economics. I am having difficulty thinking about the put-call parity for currency options, specifically how use the notation. Here is ...