Questions tagged [homework]
Homework questions for students studying Quantitative Finance or a similar subject.
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One Period Risk Neutral Probability for Caplet
I am studying some financial modeling put together by the Society of Actuaries in the USA. In it, the following practice problem was given:
Find the Risk Neutral price of an at-the-money interest ...
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Price of financial assets at $t=0$ in Black-Scholes framework
Given the share price equation
$$
dS_t=rS_tdt+\sigma S_tdW_t
$$
working in the framework of Black-Scholes model, find the price at $t=0$ of the following two financial assets:
(a) The asset pays at $t=...
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Monte Carlo Pricing of Barrier Options - can't figure out where I'm wrong
I'm trying to price a simple Up-and-out Barrier option using Monte Carlo; haven't even implemented the variance reduction but it's already glitching.
The code seems right, but I'm not sure where it's ...
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Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?
Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete.
(a) $\beta_{t}=e^{t}, S_{t}...
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Replicating call option in market which only trades stock and forward contracts
I am having a bit of trouble with a problem I've been given.
Consider a market which only trades a stock and forward contracts. There's only time 0 and 1.
Initial stock price S_0 is 10, the forward ...
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How to use Girsanov theorem for complicated RN derivatives?
Let $W_t$ be a Brownian motion under probability measure $\mathbb{P}$. Let $X_t$ be defined as follows.
$$\mathrm{d}X_t = a \mathrm{d}t + 2\sqrt{ X_t} \mathrm{d}W_t.$$
Also define: $$L_t = \exp\left(-\...
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Portfolio returns, volatility and weights of capital
I would just like to check if I've done these questions right, I feel like I might have used the complete wrong methods to get my answers.
I've been given information on 3 stocks:
I've filled in the ...
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Options On Earthquakes
As a financial innovation, the options market is introducing Options contracts based on
California Earthquakes. In your own words, discuss the following:
True or False?
“The sellers of Options on ...
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vote
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Which strategy did the fund most likely follow?
The following time series represents the return stream of a real hedge fund. Which
strategy did the fund most likely follow?
I did this data superimposition on Matlab. So, there should be some data ...
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ARCH; Expectation and Variance
I have got the following question that I am struggling to answer.
The stock return $S_t$ follows the following DL model, with $Z_t$ being a dependent variable explaining the stock return:
$S_t = \...
user38671
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vote
0
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Pricing forward start Cliquet option with implied volatility with Dupire
I have the following implied volatility matrix of a stock index downloaded the 15th February 2019, the value of the stock was 3188.44 at the time:
...
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1
answer
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Calculating value of bond
The bond has a facevalue of 40 and maturity of 20 years. It produces 0 coupon payments during the first 6 years but pays coupons of 2 annually during the last 14 years. The discount rate is 7%.
The ...
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1
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Calculating beta when holding market portfolio
Suppose that CAPM holds and that you hold a portfolio of the market portfolio and the risk-free asset with weights equal to 0.74 and 0.26 respectively. What is the beta of your portfolio?
My ...
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Use of second similar European Option as control variate to simulate a European option
I understand the idea and math behind the concept of control variate for the sake of variance reduction, but I struggle to apply it to option pricing.
I need to simulate an European option of a stock ...
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1
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For a market with a bank and risky assets $S_1, S_2$ with different volatility, what should be the short interest rate in this market?
Let there be two assets $S_1$ and $S_2$ s.t.for $\sigma_1 \neq \sigma_2$
$$dS_{1t}=\mu_1 S_{1t}dt+ \sigma_1S_{1t}dB_t \\dS_{2t}=\mu_2 S_{2t}dt+ \sigma_2 S_{2t}dB_t$$ .
If there exists a bank, what ...
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Basic question on Ito integrals
$Let \space X(t) =\begin{cases}
2, \qquad\text{if} \space 0\le t \le 1 \\
3, \qquad\text{if} \space 1 < t \le 3 \\
-5, \qquad\text{if}\space 3 < t \le 4
\end{cases}
$
or in one forumala $...
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Mean Variance Optimization of 2000 pairs of securities (Python)
I would like to take the opportunity to ask for your help on an assignment I'm trying to complete.
For this 'Modern Robo Advisory' course we are asked to solve a (target) goal-based investment ...
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vote
2
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Cause of difference in theoretical vs observed value of a (call) option under the Black-Scholes model?
I am currently considering the price $C_0$ of a call option on a stock $S$ with
$$
S_0 = 1 \\
K = 1.1 \\
r = 1\% \\
T = 1
$$
Based on the Black-Scholes formula, I have deduced that $C_0 = 0.356$.
...
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Self finance conditions - proof check
Find expressions for the process $\psi=(\psi(t),\ 0\leq t\leq T)$ , so the portfolio $(\phi,\ \psi)$ is self-financing when:
(1) $\phi(t)= \int_{0}^{t}S_{s}ds $
(2) $\phi(t)=S_{t}$
where $\phi(t)$ is ...
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1
answer
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How to take the differential of a stochastic integral?
Denote
$$X_t = \int^t_0\sigma e^{-k(t-s)}dW_s$$
here $W_s$ is the Brownian motion, $k,\sigma$ are constants.
I want to calculate $d X_t$ and the variance $Var[X_t].$ I know how to take the ...
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Linear programming cash match portfolio - how to formulate?
How would you formulate this linear program in standard form? (ie objective function and constraints).
any help would be appreciated. I don't understand how to formulate this without having an ...
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2
answers
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Strike / delta relationship for FX options
I am trying to find out how to go from delta to strike. If we look at the Bloomberg I am looking at 1M ATM volatility. I have included the Bloomberg data as a picture where we have following ...
user25295
2
votes
1
answer
383
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Using crude Monte Carlo
Background Information:
The crude Monte Carlo algorithm for the arithmetic Asian call option is $$Y = e^{-rT}(\overline{S}_A - K)^{+}$$ and the control is $$C e^{-rT}(\overline{S}_G - K)^{+}$$
The ...
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Monte Carlo European Option Pricing
I've written code below that simulates GBM paths for determining the price of a given European call option and put option. The stock is priced at 150 USD, strike price at 155 USD, risk-free rate was ...
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1
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323
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A forward Monte Carlo method for American Options Pricing
I am trying to implement the forward Monte Carlo algorithm from the paper "A Forward Monte Carlo Method for American Options Pricing" by Daniel Wei-Chung Miao and Yung-Hsin Lee. I am a little bit ...
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Use random-shift Halton sequence to obtain 40 independent estimates for the price of a European call
Background Information:
Random-shift Halton sequence: Consider the first six Halton vectors in dimension $2$, using base $2$ and $3$:
$$\begin{bmatrix}
1/2\\
1/3
\end{bmatrix}, \begin{bmatrix}
1/4\\
...
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Bjork exercise 7.6: Claim that depends on $T_1$ and $T_0$
See the solution to Exercise 7.6 here.
The solution calculates $E^Q (S(T_1)/S(T_0))$ and then just plugs that into the risk neutral valuation formula. But why? The risk neutral valuation formula ...
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709
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How to define the $f$ function to apply Ito's lemma?
\begin{equation}
Z(t) = \exp (a W(t))
\end{equation}
I am asked to find $dZ$. I am pretty sure it can be done using Ito's lemma. But in all my textbook (Bjork) examples Ito's lemma is giving from a $...
user25295
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Arbitrage and completeness in multiperiod model?
Given a 2-period market with above stock price process along with a riskfree stock with a return of 5%, how do I determine whether the market is arbitrage-free and complete when I only have knowledge ...
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1
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construct portfolio offering risk free profit
Have trouble understanding this question, seems quite open ended.
Assume that $S(0)$ is the current rate of exchange for foreign currency. Assume that and $K_n$ and $K_f$ are rates of return on home ...
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Coupon bond pricing problem with reinvestment
The three year bond has face value USD 100, and pays USD 5 coupons annually, the last one at maturity. Assume that the continuously compounding rate is 7%.
(a) Find the price of this bond.
(b) ...
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Is there an efficient method or technique to find an arbitrage between two FX dealers?
Crossposted on Mathematics SE
I was able to solve the following problem and find the arbitrage but only after spending a long time on it and trying out different possibilites. Is there a method or ...
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School project about Black Scholes with stochastic volatility
In a university project I am looking at Black Scholes model with a stochastic volatility. I’m still not quite sure about my focus (I am in the beginning 'Idea phase'). I want to explain the theory ...
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12-month rate calculation for Problem 4.23 in Hull's Options, Futures, and Other Derivatives
From Hull's Options, Futures, and Other Derivatives, 8th ed., problem 4.23:
Excerpt from Problem 4.23
The cash prices of six-month and one-year Treasury bills are 94.0 and 89.0 ...
Calculate the six-...
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Need help understanding basics of cash flow engineering
I'm studying Financial Engineering, a subject I'm completely new to.
I'm using Principles of Financial Engineering 3rd Edition and trying to solve the exercises ...
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1
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Asset Liability Management Test Topic Interpretation
I will write a test based on Excel and one of the topics is "The Asset Liability related analysis: including the input assumptions generation, constraints, portfolio optimization analysis and results ...
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What is the arbitrage opportunity in Arrow-Debreu One Period market Model
The one period market model is made of 4 securities(A, B, C, D) and has 4 future states.
Assume the market model is complete. and the state prices are (-2, 2, 4, 8).
Given that I dont know the payoff ...
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Pricing options with two assets
I'm studying for a test and am stuck on this practice question:
With interest rates equal to 0, two different stocks $S_1$ and $S_2$, both valued at \$1 today, can be worth \$2 or \$0.50 at some ...
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Finding the dynamics of a dividend paying asset under arbitrary numeraire
Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
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Floor and Cap problem
So I have a problem from Marcel Finan's "A Basic Course in the Theory of Interest and Derivative Markets." We are going over floors and caps, covered puts and covered calls.
Consider the following ...
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