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Questions tagged [homework]

Homework questions for students studying Quantitative Finance or a similar subject.

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1answer
36 views

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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1answer
35 views

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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0answers
24 views

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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1answer
49 views

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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1answer
97 views

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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0answers
113 views

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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0answers
65 views

Black-Schole PDE for european put option under Cox-Ingersoll-Ross model

For a european put option, starting from the classical Black-Scholes PDE (assuming constant rate), how do we come up with the Black-Scholes PDE under the Cox-Ingersoll-Ross model (CIR) such as the ...
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45 views

Solve Poisson-SDE

for an assignment I have to solve stochastic differential equations involving Poisson processes. However, in class we only went from $F_t \rightarrow dF_t$ using Ito formula for Poisson processes, ...
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0answers
304 views

Pricing options using a binomial tree

The past few months, I have been taking the financial engineering course offered by Columbia. It is a great course but there is a huge disconnect between the theory they teach and the questions then ...
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2answers
102 views

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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98 views

Ito's & correlated Brownian proof

$W_{1}(t)$ , $W_{2}(t)$ are correlated Brownian motion with quadratic-var $\langle W_{1}, W_{2}\rangle_{t}=\rho.$ $Z_{t}=e^{\int_{0}^{t}W_{1}(s)ds}+e^{W_{1}(t)}$ and $X_{t}=\cos(W_{2}(t))$ Find: $$...
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1answer
101 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 $\phi(t)$ ...
2
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1answer
307 views

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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1answer
37 views

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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427 views

Changing the numeraire, equivalent martingale measures

Why do we have $\overline Y_t\cdot X_t^1$ instead of $\overline Y_t\cdot X_T^1$ in the middle equation below ? (The page is from the book ''Stochastic Finance'' by Hans Föllmer and Alexander ...
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2answers
3k views

Strike / delta relationship for FX options

I am tryinto find out how to go from delta to strike. If wee look at the bloomberg I am looking at 1M ATM volatility. I have included the Bloomberg data as a picture where we have following ...
2
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1answer
200 views

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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1answer
2k views

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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1answer
141 views

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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1answer
132 views

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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2answers
61 views

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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1answer
200 views

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 $...
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46 views

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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1answer
49 views

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 ...
3
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1answer
141 views

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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0answers
146 views

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 ...
5
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1answer
161 views

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 ... ...
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0answers
90 views

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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1answer
173 views

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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0answers
190 views

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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1answer
114 views

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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0answers
240 views

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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1answer
227 views

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 ...