Questions tagged [binomial-tree]
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161
questions
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Risk-Neutral Probability in a Binomial Tree
This question is probably very simple and I'm just missing the easy solution but I'm a bit confused so I thought I might as well try ask here.
I've been given this question:
When I tried to calculate ...
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26 views
Initial value of an investment project in a binomial real option valuation model
How do you measure the initial value of a project in a binomial tree ROV? I'm not specifically working in the valuation scene, but sort of had an interest in how the models work logically. It's not ...
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0answers
29 views
Binomial Option Pricing Model gives increasingly higher value for out-of-the-money options
I was developing the binomial option pricing model via Python, according to the explanation given on Wikipedia. After computing the errors against the pricing of real options, I find an interesting ...
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2answers
86 views
Prove the Euro call option value has positive relationship with the risk-free rate under discrete time model (Binomial tree model)
Could anyone show me how to prove that the European call option value has a positive relationship with the risk-free rate in a two-step binomial model with strike price K and different risk neutral ...
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1answer
132 views
Does CRR Model lose completeness if we add another instrument?
Consider the multiperiod binomial/CRR model with one risky asset $S^{1}$ and a numeraire $S^{0}$. By seeing that the equivalent martingale measure is uniquely determined, we obtain that the market is ...
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25 views
How do you determine the magnitude of the up/down steps in a binomial tree from the mean and variance of its increments?
I'm currently struggling with the following question;
I feel like it is a pretty simple one (i.e. just a matter of solving a pair of simultaneous equations), but I'm struggling to make it work.
Am I ...
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1answer
95 views
What's the price of a lookback call option in the arbitrage-free CRR-model?
If we consider the CRR-model in two periods, i.e. T=2. Let $S^1$ be the risky asset with $S_0^1=100$ and $S^0$ the bond with $S_0^0=1$. Furthermore, we assume the model is arbitrage-free with $y_b=-0....
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1answer
68 views
Binomial Pricing Model d and u
In the binomial pricing model, why do the magnitude of the up factor $(u)$ and down factor $(d)$ have to be multiplicative inverses? I have read from multiple sources that the reason for this is that ...
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1answer
110 views
Binomial Option pricing, paper by John C. Cox, I don't understand the calculation / choice of u.d.q
[EDIT] Question is answered, just cleaned up some clerical errors in the formulas.
[EDIT] Based on the comment I got, let me clarify, I am not stuck on the relationship between the binomial model vs ...
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0answers
76 views
Risk neutral probabilities in binomial option pricing with discrete dividends — whose argument is correct?
In trying to discover more about pricing American options with dividend payouts, I found the the post linked here. I notice two disagreeing answers when it comes to determining the replicating ...
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1answer
187 views
How to price an European put option using binomial model with dividend yield?
The initial stock price (S0) is 45, the stock volatility is 0.20 (20% per annum), and the risk-free rate is 0.02 (2% per annum). Consider a European put
option whose strike price is equal to 30, with ...
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2answers
147 views
Delta hedging for an American call option on a stock with a continuous dividend yield
Let the dividend yield be $\delta$ and $C_u, C_d$ and $S_u, S_d$ be the up and down values for the stock and the call respectively over the period $\Delta t$.
In Hull and all other resources I've ...
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0answers
78 views
Practical implementation of Vellekoop-Nieuwenhuis model/interpolation
Have read the 2006 VELLEKOOP-NIEUWENHUIS paper (Efficient Pricing of Derivatives on Assets with Discrete Dividends) (Download) many times re Discrete dividends on American Options, but remain baffled ...
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1answer
61 views
Maximal increase payoff
I am interested in the following problem.
We have a Multi-Step Binomial Model with discrete time $T=1,\dots,n$.
We also assume that the stock $S_t$ is a martingale and there is a risk-free bond with $...
2
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1answer
91 views
Risk neutral probability for stock with continuous dividend
Setting: binomial tree with one step over time $\Delta t$. I'm trying to derive the risk neutral probability for a stock which pays a continuous dividend, say $\delta$. i.e. probability $p$ such that ...
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1answer
167 views
How to price barrier options (binomial tree)
What is the easiest way to price single barrier options using binomial tree? I found This method. Is this method good or maybe should I use another one? Does this price converge to price from BS model?...
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1answer
46 views
Replication (binomial tree)
Hey what is the replication strategy on the binomial tree when I have for example 10 step model and dividend is paid at step 3? I have a well-written price tree but I do not know what the replication ...
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25 views
Reproducing a short put position using known binomial option tree
Suppose a put option follows prices according the the binomial tree I've made and posted below and consider writing a put ($S$ is the stock value, $P$ is the put value, obviously). I want to find the ...
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25 views
Backward differential equation with binomial tree
I'm trying to understand/solve the following question but I honestly don't know what it's even asking about. I've included my attempt following the picture of the question.
I would approximate the ...
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0answers
32 views
Binomial Model Strike Price Assumption
Let us have the standard single-period binomial pricing model, and denote the up and down states of the underlying by $S_u$,$S_d$ respectively. Let us say we have a call option on the underlying with ...
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76 views
How to prove that a series of random variables $Z_j = 1$ or $-1$ occurring at risk-neutral probability, converges to normal, using the CLT?
Context
When pricing options with trees, it is convenient to prove that the asset value at expiry $S_t$ be of log-normal distribution:
$$\log{S_t} = \log{S_0} + \mu T + \sigma \sqrt{\frac{T}{n}} \sum_{...
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1answer
54 views
Log-normal risk-neutral price derivation from binomial trees, not clear about step in derivation process
At page 64 of the book Concepts and practice of mathematical finance, 2nd edition by M. Joshi, paragraph 3.7.2 (Trees and option pricing - A log-normal model - The risk-neutral world behaviour) a ...
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1answer
52 views
How does $1 + R = q_u · u + q_d · d $ follow from $d ≤ (1 + R) ≤u$ in the Binomial Pricing Model?
I've been reading Tomas Bjork's 'Arbitrage theory' and it says:
To say that $d ≤ (1 + R) ≤u$ holds is equivalent to saying that $1 + R$ is a
convex combination of u and d, i.e. $1 + R = q_u · u + q_d ...
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1answer
99 views
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76 views
Expected life (Fugit) of American Option
How can I use the binomial tree pricing method for American options to determine the expected time of exercise for the option (or "Fugit")? In particular, how would I modify the algirithm ...
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1answer
88 views
Volatility input for American options
I have to price an american option on a daily basis and I have some questions regarding the CRR binomial tree model:
Is it correct to use implied volatility as an input? Or is it better to use ...
2
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1answer
129 views
Martingale Binomial Tree Process
3 step binomial tree process with $S_0=4,u=2,d=0.5,r=0.25.$ Determine the probability p and q such that the stock price process is a martingale (i.e. $E[S3]=S_0)$
I know P = 1/3 and Q = 2/3 but having ...
2
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1answer
92 views
Breakdown of Wilmott's Binomial Tree derivation of Black-Scholes equation
Hi guys,
I tried to follow the chapter of PWIQF on binomial model and got stuck when it derived the Black-Scholes (please see image). I tried to backtrack the said equations but couldn't trace back ...
2
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1answer
104 views
help with derivation of equation 8 in Derman and Kani's binomial tree for local vol
in this paper "The Volatility Smile and Its Implied Tree" - Derman and Kani 1994 i understand the derivation of all equations up to 7.
But eq 8 i cannot figure out how to derive! i have ...
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2answers
116 views
Failing to replicate Wilmott's results for binomial option pricing
I am working through Paul Wilmott introduces Quantitative Finance, 2nd ed. I am failing to reproduce one of his numerical examples and I would like to understand why.
I chapter 3, Wilmott introduces ...
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1answer
92 views
Black & Scholes formula derivation from a Binomial Tree - John C. Hull
I am reading "Option, Futures and other Derivatives" by John C. Hull, and on Appendix chapter 13, he derives BSM formula from a Binomial Tree.
When he builds U2, I just don't understood how to get ...
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1answer
96 views
Arbitrage strategy using binomial tree
Suppose that we have a one step binomial tree model for a company. Lets say that the time per step is T, and that price of the stock can go up to $p_1$ or go down to $p_2$. Suppose a T-month European ...
3
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1answer
83 views
Binomial tree with jumps
I am struggling with developing a binomial tree with jumps. although there are models such as CRR, could you suggest a book or have any idea to proceed?
Thanks,
Amir
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1answer
183 views
QuantLib convertible bond pricing generates strange delta
I am trying to generate equity delta for convertible bond using QuantLib(version 1.14) functions, but the deltas generated either using a repricing approach or by directly obtaining from the tree(code ...
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1answer
78 views
Reference of using $\mu = \frac{1}{T}(\log K - \log S_0)$ in binomial tree model
Notations: Given a binomial tree with $N$ periods and time to maturity $T,$ let $\Delta t = T / N.$
It is well-known that CRR uses the up and down multipliers as
$$u = e^{\sigma\sqrt{\Delta t}} \...
2
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0answers
78 views
Futures vs Forward pricing with different interest rates using binomial model
I'm given the aforementioned parameters for a two-step binomial model where the underlying pays no dividend, $S_0=50$ and $T=2$. With this information I was able to calculate the risk-neutral ...
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1answer
82 views
What are the relation between the risk neutral measures in binomial tree and in Black Scholes model?
I appreciate that both are the direct result of constricting a replicate portfolio using stock and bonds.
Are there deeper relationship between the two?
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1answer
538 views
Trinomial Trees for Hull-White model
I am studying trinomial trees and trying to implement them in Python to compare them to the monte carlo simulation. I searched 3-4 hours in the web; but can't find any implementation on binomial or ...
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0answers
90 views
Replication Portfolios and Binomial Option Pricing
To price a call/put option with two possible future states of the world, I understand we can price the option by essentially calculating the price of a replicating portfolio that gives the same ...
0
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1answer
354 views
How to get all the paths of a binomial tree
I'm trying to implement a pricing method for exotic options based on binomial tree's. The problem i'm having is that i'm not being able to generate all the paths of the tree. I have the following code ...
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1answer
176 views
What is the probability of a lookback option ending in the money (CRR-model)
I would like to compute the probability that a certain lookback option ends in the money, let's say that the option has the following payoff $h_N=\max\left\{0,K-\min\{S_1,...,S_N\}\right\} $ where $K$ ...
0
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1answer
376 views
Stock pricing using Binomial model
A stock is prices at $ \$100$ and follows a one-period binomial process with an up move that equals 1.05 and a down move that equals 0.97. If one million Bernoulli trials are performed and the average ...
0
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1answer
476 views
Should U and D change with the number of steps in a Binomial Tree?
In everyone's binomial trees online I see constant U and D. Even when I read Option Volatility and Pricing by Natenburg, all his diagrams use a constant U and D (where U is the upwards magnitude from ...
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0answers
62 views
One periodic binomial model
I need to look into a one-period Binomial model $(B_t, S_t)$ with interest rate $r = 0.1$ , $S_0 = 100$ and
$$
S_t= 120 \, \text{with probability}\, 0.5
$$
$$
S_t= 60\, \text{with probability}\, 0.5
$$...
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0answers
53 views
Binomial Model - completeness in presence of arbitrage
Consider a uniperiodal binomial model where I buy one bond of value $B_0$ and rate $r=0.1$, and $h$ stocks with price $S_0=5$. The value of the portfolio at time $t=0$ is
$$ V_0 = B_0 + hS_0, $$
...
2
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1answer
113 views
Definition of interest rates in binomial tree model
I'm studying financial mathematics from Shreve's text. I have two problems.
1) "for a binomial tree with three steps, where $S_0=20$, $u=1.05$, $d=.95$ and continuously compounded risk-free interest ...
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1answer
167 views
Finding distinct possible values in binomial tree
I wonder how to solve this problem.
Lets say we have a binomial tree with the following parameters:
$u=1.25,\ d = 1/u,\ T=15$.
How many distinct possible values are there for $X_{7}$?
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1answer
305 views
On pricing american put options
How come we pick the highest between the discounted weighted average (with risk neutral probabilities) and the early exercise value at each node of the binomial tree?
I dont understand why, I can ...
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3answers
264 views
Binomial model arbitrage
I've recently started studying math finance from Shreve's Stochastic calculus text. In the binomial model, there is no arbitrage $\iff d<1+r<u$. To show that no arbitrage implies $1+r<u$, ...
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1answer
110 views
Binomial model in Björk's Arbitrage Theory in Continuous Time
I am having some trouble with variable $Z$ introduced in chapter $2$ in Björk's text. In the beginning, it is the random variable that attains $u$ resp. $d$ with probabilities $p_{1}$ and $p_{2}$, i.e....
