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1answer
50 views

Preparation for interview: influx of power of the moon

I am preparing myself for an interview for a quantitative analyst position and one of the sample questions asked in previous examinations was: "Suppose the moon were to disintegrate, and fall to ...
3
votes
1answer
76 views

The Relation Between the Ricci flow and the Black-Scholes-Merton Equation

Grisha Perelman once wrote that The Ricci-flow equation, a type of heat equation, is a distant relative of the Black-Scholes equation that bond traders around the world use to price stock and ...
0
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1answer
50 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 ...
1
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0answers
31 views

Arrow-Debreu Equilibrium Pricing

I have this problem in asset pricing that I don't know how to solve. Here it is: Consider an economy with a complete set of Securities and $N$ states of the world Tomorrow. Assume that there are two ...
3
votes
2answers
225 views

Understanding the solution of this integral

The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price): $$\int_{z^*} ...
0
votes
1answer
44 views

Payoff of option

Consider the payoff $g(S_T)$ shown the figure: I believe the payoff represented as a linear combination of the payoffs of some options with different strike and same maturity $T$ is $$g(S_T) = ...
1
vote
0answers
48 views

Replicating American call option

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$,$u = 1.2$, and $l=0.8$. The interest rate for both periods is $R = .05$ a.) If the ...
2
votes
1answer
105 views

Linear combination of Payoffs using Black-Scholes

Write the payoffs in Figure 3.8 as linear combination of call options and derive a closed form formula for the Black-Scholes price, the Delta, and the Gamma of them. All the Greeks of the option are ...
1
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2answers
43 views

Perpetual American options

Formulate and solve the free boundary problem for the perpetual American options with the following payoffs. a.) $(S - K)_{+} + a$ where $a > 0$ b.) $(K - S)_{+} + a$ where $a > 0$ ...
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0answers
67 views

Two-period binomial model for American option

Consider a two-period binomial model for a risk asset with each period equal to a year and take $S_0 = 1$, $u = 1.5$, and $l = 0.6$. The interest rate for both periods is $R = .1$. a.) Price ...
2
votes
1answer
50 views

Trinomial model converges to Black-Scholes weakly

Consider risk-neutral trinomial model with $N$ periods presented by $$S_{(k+1)\delta}H_{k+1}, \ \ \text{for} \ \ k=0,\ldots,N-1$$ where $\delta:=\frac{T}{N}$ and $\{H_k\}_{1}^{N}$ is a sequence ...
2
votes
1answer
139 views

Black-Scholes Equation with dividend

Consider a European option with payoff $$g(S_T) = S_T^{-5}e^{10S_T}$$ Assume that the interest rate is $r = .1$ and the underlying asset satisfies $S_0 = 2, \sigma = .2$, an pays dividend at ...
1
vote
1answer
103 views

Pricing of Black-Scholes with dividend

Consider the payoff $g(S_T)$ shown in the figure below. Consider Black-Scholes model for the price of a risky asset with $T = 1$, $r = .04$, and $\sigma = .02$ and dividends are paid quarterly with ...
1
vote
1answer
48 views

Symmetric probability and subjective return

Let $\{Z_k\}_{k=1}^{N}$ be a sequence of i.i.d. random variables with the following distribution $$Z_k = \begin{cases} \alpha &\text{with probability} \ \hat{\pi}\\ -\beta &\text{with ...
1
vote
1answer
65 views

Find the solutions of the ODE from SDE

Consider the SDE $$dS_t = rS_t dt + \sigma S_t dB_t \ \ \ \text{where} \ r \ \text{and} \ \sigma \ \text{are constants}$$ a.) Find the ODE for the function $V(x)$ such that $e^{-rt}V(S_t)$ is ...
1
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0answers
35 views

Find the PDE for a function that makes it a martingale

Given the SDE, find the PDE for the function $V(t,x)$ such that $V(t,S_t)$ is a martingale. $dS_t = \kappa(m - S_t)dt + \sigma\sqrt{S_t}dB_t$ where $\kappa$,$m$, and $\sigma$ are constants. ...
0
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2answers
66 views

Two-period binomial model with dividends

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.15$ and $l = 0.95$. The interest rate is $R = .05$. a.) If the asset pays 10% of its ...
0
votes
3answers
76 views

Linear combination of payoffs of bull and bear spreads

Write the following payoffs as linear combination of call options with different strikes and possibly some cash and give the closed form formula for them. Attempted solution: The payoff for the bear ...
0
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1answer
44 views

Replicating option strategies

I was curious if there was any references to replicating option strategies i.e. bull spread, bear spread, butterfly, strangle, straddle, etc...? Also what is the insight into replicating of these ...
0
votes
1answer
32 views

two-period binomial model, with price that is path-dependent

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.03$ and $l = 0.98$. How do you price a look-back option with ...
4
votes
1answer
101 views

How to price and find a replicating portfolio for a call spreads using a two-period binomial model?

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.03$ and $l = 0.98$. a.) If the interest rate for both periods is $R = .01$, find the ...
3
votes
1answer
44 views

Use no dominance to show that the price of the call option satisfies the inequality

Assumption 2.1 - If the payoff $P$ of a financial instrument is non negative, then the price $p$ of the financial instrument is non negative. Assume $C$ is just the price of the call option, and ...
0
votes
1answer
51 views

Convexity in Markovian contingent claim

Background information: I believe we can use Jensen's Inequality here Show that if the payoff function $V(S_T)$ is a convex function on $S_T$, then the Markovian European contingent claim with ...
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2answers
65 views

Black-Scholes and Markovian contingent claim

Background information: Proposition 4.1 - For a European Markovian contingent claim, the Black-Scholes price satisfies $$\Theta(\tau,S) = -\frac{\sigma^2 S^2}{2}\Gamma(\tau,S) - rS\Delta(\tau,S) + ...
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votes
3answers
104 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 = ...
0
votes
1answer
54 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
81 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, ...
3
votes
2answers
177 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
96 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 + ...
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votes
7answers
756 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 ...
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0answers
35 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
19 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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votes
1answer
25 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
98 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
85 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: ...
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5answers
7k views

What is a martingale?

What is a martingale and how it compares with a random walk in the context of the Efficient Market Hypothesis?
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3answers
356 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 ...
3
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1answer
235 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 ...
76
votes
10answers
83k views

How can I go about applying machine learning algorithms to stock markets?

I am not very sure, if this question fits in here. I have recently begun, reading and learning about machine learning. Can someone throw some light onto how to go about it or rather can anyone share ...
1
vote
1answer
95 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?
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2answers
386 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 ...
2
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0answers
49 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
95 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 $B_1 = 1+r$, ...
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2answers
95 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
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2answers
336 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
53 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
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2answers
75 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 ...
0
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1answer
133 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 ...
1
vote
1answer
56 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 ...