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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.

2
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1answer
50 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
38 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
98 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
106 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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0answers
41 views

Relationship between two Brownian motions generated by equivalent martingale measures

If $Q^1$ and $Q^2$ is equivalent martingale measures ($Q^1\sim Q^2$) and the following condition holds: $$\frac{dQ^2}{dQ^1}=(\frac{X_T}{X_0})^q$$ for some positive constant $q$ and the $Q^1$ dynamics ...
0
votes
2answers
156 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
86 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
901 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, ...
3
votes
1answer
283 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
165 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
69 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}\...
3
votes
1answer
79 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
60 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
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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 ...
0
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1answer
94 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
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1answer
541 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
195 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
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0answers
153 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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0answers
93 views

How cyclical should a rating system be and why?

I am reading about different approaches for building a rating system and came across the terms Through-the-Cycle (TTC) and Point-in-Time (PIT). They are both extreme cases of whether a rating system ...
-1
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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
193 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
67 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
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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 ...
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0answers
229 views

How to get the optimal risky portfolio with two risky assets?

If I don't have the risk free rate, how can I get the optimal portfolio? Given: E(rB)=0.1 E(rC)=0.16 Risk Aversion coefficient=4 var(rB)=0.15 var(rC)=0.2 Corr(rB,rC)=0.05
0
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1answer
315 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
161 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
308 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
559 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
62 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 ...
3
votes
1answer
192 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
138 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
vote
1answer
135 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
125 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 ...
3
votes
2answers
150 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 ...
4
votes
1answer
121 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
68 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 ...
1
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1answer
97 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 ...
3
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0answers
70 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
262 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
174 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
59 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 ...
0
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1answer
83 views

Problem solving using the put-call parity

I am self-studying for an actuarial exam on financial economics. I encountered this problem, and I am having difficulty seeing why the statement underlined is true: How do we know that $P(60) - C(60) ...
4
votes
2answers
113 views

Computing $\gamma$ and $\mu$ at the efficient frontier

Consider the condition which the weights of any portfolio belonging to the efficient frontier satisfy: \begin{equation} \gamma\boldsymbol{wC} = \boldsymbol{m} - \mu\boldsymbol{u}\end{equation} ...
0
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1answer
111 views

Where am I making a mistake in my calculation of profit on a short-sale?

I am studying financial math and here is a problem and the solution from the author: Here are my calculations: The short sale is $200\cdot24.82 = 4964$. Now half of this amount will be taken for a ...
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0answers
200 views

stochastic calculus and multidimentional itos lemma

I am considering a number of assets (N) in a portfolio. each asset follows a geometric Brownian motion process therefore the stochastic differential equation is dS(i) = S(i)μdt + S(i)σdX(i). The ...
3
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1answer
748 views

Derive OIS rate from IRS rate and Fed Funds/Libor basis spread

For example I have 7Y interest rate swap rate and 7Y Fed funds/Libor basis spread. What is the step-by-step procedure to derive OIS rate from these two?
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2answers
164 views

Expected Utility

We know that under certainty, any increasing monotone transformation of a utility function is also a utility function representing the same preferences. Under uncertainty, we must restrict this ...