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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 ...
James Bender's user avatar
0 votes
0 answers
98 views

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=...
Tyrell's user avatar
  • 101
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0 answers
391 views

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 ...
NotYoAvgJoe's user avatar
10 votes
2 answers
1k views

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}...
randorando's user avatar
1 vote
0 answers
162 views

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 ...
woiddiow's user avatar
1 vote
0 answers
160 views

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(-\...
Bravo's user avatar
  • 671
0 votes
1 answer
252 views

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 ...
Charlie P's user avatar
6 votes
1 answer
500 views

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 ...
AS07's user avatar
  • 61
1 vote
0 answers
49 views

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 ...
Dexter's user avatar
  • 111
1 vote
0 answers
44 views

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 = \...
user avatar
1 vote
0 answers
417 views

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: ...
RandowMalk's user avatar
0 votes
1 answer
86 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 ...
armara's user avatar
  • 123
1 vote
1 answer
139 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 ...
armara's user avatar
  • 123
2 votes
0 answers
86 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 ...
Scholesterie25's user avatar
1 vote
1 answer
77 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 ...
Focus's user avatar
  • 274
9 votes
1 answer
209 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 $...
FFSU's user avatar
  • 91
1 vote
0 answers
336 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 ...
Mehdi's user avatar
  • 31
1 vote
2 answers
343 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 $C_0 = 0.356$. ...
M Smith's user avatar
  • 439
0 votes
1 answer
317 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 $\phi(t)$ is ...
user531618's user avatar
2 votes
1 answer
1k 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 ...
user6703592's user avatar
-1 votes
1 answer
73 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 ...
wu54656213's user avatar
3 votes
2 answers
9k views

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 ...
user avatar
2 votes
1 answer
385 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 ...
Wolfy's user avatar
  • 728
0 votes
1 answer
5k 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 ...
Marcus L's user avatar
1 vote
1 answer
325 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 ...
Wolfy's user avatar
  • 728
-1 votes
1 answer
328 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\\ ...
Wolfy's user avatar
  • 728
0 votes
2 answers
116 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 ...
Marinab's user avatar
  • 11
0 votes
1 answer
714 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 $...
user avatar
1 vote
0 answers
67 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 ...
Malik's user avatar
  • 11
1 vote
1 answer
74 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 ...
foshizzle's user avatar
  • 432
2 votes
1 answer
305 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) ...
idknuttin's user avatar
  • 203
3 votes
2 answers
372 views

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 ...
idknuttin's user avatar
  • 203
1 vote
0 answers
232 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 ...
Sanjay's user avatar
  • 1,667
6 votes
1 answer
239 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 ... Calculate the six-...
buruzaemon's user avatar
1 vote
0 answers
121 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 ...
Newtt's user avatar
  • 121
1 vote
1 answer
221 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 ...
nouveau's user avatar
  • 63
1 vote
0 answers
275 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 ...
agent251's user avatar
2 votes
1 answer
165 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 ...
user108's user avatar
  • 33
4 votes
0 answers
355 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 ...
Good Guy Mike's user avatar
0 votes
1 answer
326 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 ...
Eleven-Eleven's user avatar