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1answer
49 views

European Markovian option

Background information: Consider a European contingent claim with payoff $V(S_T)$, where $V: \mathbb{R}_+\rightarrow \mathbb{R}$ is a function which assigns a value to the payoff based on the price of ...
0
votes
1answer
72 views

Arrow-Debreu Model and Risk-Neutral Probabilities

Consider one period Arrow-Debreu model with $N = 2$ and $M = 4$ shown in Figure 3.5 and take $R = 0$. a.) Show that any risk neutral probability $\hat{\pi} = (\hat{\pi}_1, \hat{\pi}_2, \hat{\pi}_3, ...
0
votes
1answer
52 views

Put-Call Parity Application

In the binomial model, how that the Delta of a call option $\Delta^{call}$ and the Delta of a put option $\Delta^{put}$ with the same maturity and strike satisfy $$\Delta^{call}_t - \Delta^{put}_t = ...
3
votes
2answers
132 views

Risk-Neutral Probabilities, Trinomial Model

My professor has many grammatical mistakes and errors in his questions, so apologies ahead of time. I am just trying to understand what he wants for this question, In trinomial model, let $S_0 = 1$, ...
0
votes
1answer
29 views

Binomial Model, Number of nodes from $t = 0$ to $t = n$

How many paths are there in a binomial model from time $t = 0$ to time $t = n$? How many nodes (states) are there? Intutively it seems that there are $2^n$ paths and $2n - 1$ nodes. But I am not sure ...
3
votes
1answer
89 views

Modeling Financial Assets

Let $\tilde{W}_t := (1+R)^{-t}W_t$ and $\tilde{S}_t := (1+R)^{-t}S_t$ be respectively discounted wealth process and discounted asset price. Then, show that $$\tilde{W}_t = w_0 + ...
7
votes
7answers
691 views

Proof that no trading system always wins

I am pondering on the existence/impossibility of a trading system (or algorithm) that ALWAYS ends up winning money, no matter how the price of a futures moves. In a context where one can go long or ...
0
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0answers
29 views

Return, STD and CAPM based on Continuously compound return on daily prices

Mission: For some ETF, Get 1, 3, 5 years: Return STD CAPM parameters (alpha, beta) Reference if I calculated correctly: Yahoo finance performance & risk data Raw data: Daily adj. close ...
0
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0answers
15 views

Question in the proof of “Optimization of conditional value-at-risk”

I'm reading the paper "Optimization of conditional value-at-risk" by Rockafellar and Uryasev. The state two theorems within the paper which are proven in the appendix. Let me introduce some notation ...
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1answer
22 views

How to calculate 5 years return & STD for ETF?

I want to calculate by-myself 5 year return & STD for SPY ETF. What I did: Downloaded to Excel from yahoo finance historical data for the ETF (daily Adj. Close) from ...
4
votes
1answer
97 views

Clarify a derivation in Pat Hagan's Convexity Conundrums

I am looking for help in understanding the algebraic derivation to go in between some of the lines in Pat Hagan's famous Convexity Conundrums paper e.g. how he goes from 3.4a to 3.5a.
3
votes
1answer
80 views

Derivation of Magrabe formula

I'm going through the following note by Davis, link. In chapter 3 he derives the Magrabe formula. I got stuck at equation $(3.16)$. We have two assets: ...
3
votes
1answer
163 views

Value of European Call equals Value of American Call, Question on Explanation/Proof

I am reading S. Shreve, Stochastic Calculus for Finance, Vol. I. There he proves that American Call Options have the same value as European Call Options. In the proof he uses that for a Call option ...
1
vote
1answer
91 views

Stochastic calculus: what am I doing wrong?

it is just the computation of a second moment but however is creating debate !!... Can someone spot the error?
2
votes
0answers
39 views

Show that in an arbitrage-free and non-redundant market a certain set is compact

Some notation: We consider a financial market with $d+1$ assets, the $0$-th asset is considered the risk-free asset, the others are the risky ones. The vector $\overline \pi \in \mathbb R^{d+1}$ ...
2
votes
1answer
82 views

Pricing digital options in discrete time

I am stuck in this exercise from my textbook: Consider a one-period market model with $N+1$ assets: a bond, a stock and $N-1$ call options. The prices of the bond are $B_0=1$ and $B_1 = 1+r$, ...
3
votes
2answers
311 views

Book recommendation: math toolkit for quantitative finance and statistics

I am looking for a book which teaches mathematical topics which are relevant to master quantitative finance and statistics. Please note, I do not mean a book which would explain how math is applied ...
1
vote
1answer
49 views

Desperate for help with simple derivative

Can someone help explain how differentiating the following with respect to $x$: $$ \frac{1}{2} \alpha \mathbf{x}^T \Sigma \mathbf{x} + (\mathbf{\mu} - R\mathbf{1})\mathbf{x} $$ Yields the following: ...
3
votes
2answers
67 views

Conditional expectation of a non stochastic process

In an example I was working through it was shown that $W_{t}^{2} - t$ was a martingale with respect to the Brownian motion filtration $\mathcal{F}_{s}^{W}$ with $t>s$. Everything was fine except a ...
1
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1answer
54 views

Is it possible that some types of financial systems can resonate?

Financial systems can certainly be modeled using the same tools physicists use to model dynamic physical systems. The validity of such is evidenced by models such as that developed by Black and ...
0
votes
1answer
102 views

How can we write swap as a chain of FRA's

For the rest of my question I use the notation from Brigo. The discounted payoff of a receiver interest rate swap (RFS) at $t<T_{\alpha}$, where $T_{\alpha}$ is the first resetting date, is given ...
2
votes
1answer
69 views

$\frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+$ at time $T_i$ is equivalent to a payment of $(A-p(T_{i-1},T_i))^+$ at time $T_{i-1}$

How can I show that payment of $\frac{1}{p(T_{i-1},T_i)}(A-p(T_{i-1},T_i))^+$ at time $T_i$ is equivalent to a payment of $(A-p(T_{i-1},T_i))^+$ at time $T_{i-1}$ ? Where A is a deterministic ...
1
vote
1answer
133 views

derivation of heston pde in gatheral

Following Gather (the volatility surface, chapter 2) we assume the following process: $$ dS_t = S_t(\mu_t dt+\sqrt{\nu_t}dZ^1_t)$$ $$ d\nu_t= -\lambda(\nu_t-\bar{\nu})dt+\eta\sqrt{\nu_t}dZ^2_t$$ ...
1
vote
1answer
160 views

Arrow-Debreu Price in “The Volatility Smile and its implied Tree”

I was reading the old, but still interesting paper "The volatility smile and its implied tree" by Derman and Kani. I have a two questions about the derivation of the $2n+1$ equations, both of them ...
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votes
3answers
332 views

How is stock data objectively different to this random walk?

I have a random walk that is generated as so using python, numpy, and matplotlib ...
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vote
2answers
88 views

Weighting with restrictions, but no clear objective function?

I have 40 shares in an index and I want to weight them based on their market value, define the known value as $x_i$ In the traditional way, the weight of each share is calculated as: $w_i = x_i / ...
3
votes
2answers
234 views

Is linear programming important for quant?

I'm thinking about taking a course on Linear and Convex Programming, but I don't know how useful it is in the real world finance. Which areas in finance is mathematical programming used?
1
vote
0answers
71 views

Understanding Price Elasticities in Discrete Choice Models (Derivative)

I'am in the midst of a paper on mutual fund product differentiation by Li and Qiu. Here, the authors model the utility an investor derives from investing in a mutual fund using a Discrete Choice Model ...
9
votes
2answers
1k views

Why Ito calculus?

Coming from physics, I am used to the fact that the Ito interpretation of most natural stochastic equations is wrong, and one should be using Stratonovich calculus instead (of course they are ...
-1
votes
1answer
89 views

Math basics of Equally-weighted Risk contributions

i'm writing my BA Thesis about "Equally-weighted Risk contributions". Can anyone recommend math books for further understanding of Risk contributions?
2
votes
1answer
103 views

Distribution of minimum of hazard functions

Suppose I have two random variables, $X_1$ and $X_2$, that are independent (but not identically distributed) and assume both have hazard functions $\lambda_1(s)$ and $\lambda_2(s)$, for $s > 0$. ...
3
votes
3answers
236 views

Budget Constraint in Sharpe Ratio Optimization

I am a math student and I am trying to understand the budget constraint in Sharpe Ratio optimization for portfolio design. Recall the budget constraint requires that the sum of the portfolio weights ...
3
votes
1answer
259 views

PDE pricing of barrier options in BS

Path-dependent options in BS framework is intuitive to price with monte-carlo under risk-neutral measure, however it appears that several kinds can be priced with PDEs. I understand how does the story ...
4
votes
1answer
105 views

backward Kolmogorov equations - Markov properties

I'm a physicist who's research has lead him into the theory of stochastic differential equations. If this question is not appropriate for this forum, please feel free to delete it. So I've been ...
3
votes
2answers
372 views

Why is that a risk averse consumer buys the optimum insurance when there is actuarially fair insurance?

I think I understand the fact that when marginal utilities of the same function are equal (a consequence of the actuarially fair insurance), the independent variables in it must be equal -- right? But ...
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0answers
75 views

Sampling and/or asymptotic distribution of a function

Assume we have the following function: $$f(p) = \frac{1}{(1-p)d}\ln\left(\frac{1}{T}\sum_{t=1}^{T}\left[\frac{1+X_t}{1+Y_t} \right]^{1-p} \right)$$ where $d$ is a constant $T$ is a constant $X_t$ ...
0
votes
1answer
37 views

How to rightfully balance the share of the organization between departments after variable changes?

This is an abstracted version of the problem I'm facing and I have to tell you first, my question might not be precise and or even correct, so I hope you understand and in that case can improve the ...
9
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2answers
586 views

Is it possible to understand financial theory without mathematics?

I am trying to develop a short course on financial theory, covering the fundamentals of forward and options pricing, and 'efficient market' theory. I want to reduce the amount of mathematics to a ...
3
votes
0answers
58 views

FTAP in the model independent case, paper by Schachermayer

I have a question about the following paper by Beatrice Acciaio, Mathias Beiglböck, Friedrich Penkner, Walter Schachermayer. At the very beginning of the paper, on page 3, there are two definitions ...
1
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0answers
123 views

What is the right group of durations?

It seems that the group of durations commonly used in quantitative analyse is $\mathbf{R}$ but it seems to me that $\mathbf{R_+^*}$ could also be an interesting choice. While I am not aware of ...
1
vote
1answer
81 views

Get discount factors with limited knowledge?

I am facing the problem of just having this information: 6% coupon bond with 2.5 years to maturity, traded at a 100% clean price 4% coupon bond with 1.5 years to maturity, traded at a 98% clean price ...
1
vote
1answer
57 views

Why is the discount function non increasing if pure cash holdings are feasible?

I am struggeling with the question, for example lets take a swap with rate of 3.2 for one year and 3.6 for 2 years and Discount Factor 0.96899 for the first year and 0.93158 for the second year. ...
7
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0answers
194 views

Proving the asymptotic distribution of Manipulation-Proof Performance Measure (MPPM) (Paper by Goetzmann et al.)

In Goetzmann et al.'s (2007) paper, the authors derive a "Manipulation-Proof Performance Measure" (MPPM), which is a performance measure that is impervious to performance manipulation by fund ...
0
votes
1answer
743 views

Determine state price vectors?

I have 3 states with two assets, stocks and bonds. The bond has a payoff of 1 in every state of the world. And the stock has a current price of $S_0 = 100$ and ...
1
vote
0answers
112 views

Does it make sense to apply complicated mathematics to calculate with precision when the margin of error is +/-10%? [closed]

This is more of a philosophical question than general question. Quantitative finance applies highly complicated mathematics and has attracted very smart people to this field lately given the high pay ...
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0answers
94 views

How to show that the risk contribution function is or is not injective?

Assume a portoflio $w \in \mathbb{R}^n$, you can get the total risk contribution $\psi_i$ of asset $i$ by doing: $$\psi_i = w_i \frac{\partial \sigma(w)}{\partial w_i}= \frac{1}{\sigma(w)} \left[ ...
7
votes
1answer
1k views

What is exactly Euler's decomposition?

I have often seen the following statement in different paper: As $\sigma$ is homogeneous and of degree 1, we use Euler decomposition and write $\sigma(x)=\sum_{i=1}^n x_i \frac{\partial ...
4
votes
1answer
147 views

What is the analytic value of an asset's risk contribution, if $n=2$?

The marginal risk contribution of asset $i$ is defined by Roncalli in his paper on ERC as follows: $$\frac{\partial \sigma(x)}{\partial x_i} = \frac{1}{\sigma(x)} \left( w_i \sigma_i^2 + ...
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votes
2answers
209 views

What's the first time-integral of price called?

In general I'm wondering about the names of time-derivatives of price. E.g. in physics the first few time-derivatives of position are: f(x) = displacement f'(x) = velocity f''(x) = acceleration ...
2
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0answers
733 views

How to correctly construct a value- and equally weighted portfolio consisting of property-types?

A problem of which I couldn’t find the answer on the forum is about the construction of equally-weighted and value-weighted portfolio. I want to compute the equally-weighted property-type portfolio ...