Questions tagged [binomial-tree]

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44 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
79 views

Calculating European call option, the Bjork way

We have a 3 period binomial tree with values: ...
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49 views

Risk free rate application to option pricing

We have $S_o = 50, u = 1.0606, d = 1/u, K = 54.50,$ risk free rate $r = 0.1$ per week, maturity in 9 weeks, given a binomial tree (3 steps)with the probabilities given by $q = (1+e^{r(T-t)}/u-d)$, no ...
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0answers
30 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
50 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 ...
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1answer
50 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 ...
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29 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 ...
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1answer
77 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
99 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
58 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
72 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 ...
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1answer
69 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
50 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
70 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}} \...
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32 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
63 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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63 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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31 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 ...
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55 views

Hedging a short position in the Lookback Option

SOLUTION I got the correct answer using this formula $X_2(HH)=(1+r)*[X_1(H)-\Delta_1(H)*S_1(H)]+\Delta_1(H)*S_2(HH)$ $(1+0.25)[2.24-(.06667*8)]+0.06667*16=3.20$
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1answer
76 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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35 views

Does the price of an American Put often exceed the Payoff?

According to shreve the value of a put is equivalent to or greater than the possible payoff, before a stopping time with the condition that its value equals the intrinsic value. First what does it ...
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30 views

swap discrete market model

Let be $M = (S_0,E,\Phi)$ a market where the risk-free rate is $r = 0$ and the Euribor $E$ evolves (annually) in discrete time following a three-period binomial model. Assume that $E_0 = 0.031$, the ...
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1answer
108 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 $K$ ...
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1answer
83 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 ...
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1answer
180 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
57 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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41 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 $$ V_0 = B_0 + hS_0, $$ ...
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1answer
54 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
108 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
178 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
157 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
99 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....
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1answer
52 views

A question on the binomial model

I dont understand the introduction and/or idea of the variable $X$ on page $80$ in the following handout. http://www.maths.lth.se/matstat/kurser/fmsn25masm24/ht17/Ch3.pdf Does someone know whats the ...
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2answers
88 views

Pricing of European put option with binomial model

This is an exercise from Mark Joshi's book (exercise 3.6): A stock is worth 100. Each month its value increases or decreases by precisely 10. The riskless bond is worth $e^{rt}$ at time t years with ...
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1answer
75 views

stochastic interest rate in binomial pricing model and in continuous models

Is the interest rate allowed to be truly stochastic in the binomial pricing model and in continuous models so that we are still able to switch to the risk-neutral measure? Shreve mentions multiple ...
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1answer
342 views

Binomial Option Pricing Model

This isn't homework. I'm going through sample questions for an exam. They include the answer, but no explanation. I've studied this model, but I don't know how to setup this tree to get any of the ...
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1answer
55 views

Domestic and foreign interest rate; dividends?

The spot price AUD/USD is 0.6868, strike price is 0.6915,the 6 month ATM implied volatility for AUD/USD is 7.7% p.a., for the 6 month USD deposit rate is 2.28% and the 6 month AUD deposit rate is 1.45%...
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2answers
172 views

Risk-neutral pricing and statistical arbitrages

I'm studying the martingale approach to asset pricing. Dealing with the concept of risk-neutral probability, I came up with a question about the possibility of "arbitrages in expectation". I'll be ...
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0answers
54 views

Value of portfolio with fixed discrete dividends

I know that this is a very simple question, but i want to make sure to grasp the concept of ex dividend and value of portfolio. Suppose that we have a two period binomial tree of a stock with initial ...
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1answer
118 views

American Put Option Pricing

I am trying to solve a question of American Put Option pricing as below. Build a 15-period binomial model whose parameters should be calibrated to a Black-Scholes geometric Brownian motion model with:...
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1answer
56 views

How underlying asset price variance is connected with time

I'm dealing with option pricing models and there is a statement that says the variance of underlying asset price is propotional with time $𝑉𝑎𝑟(𝑆_{𝑚+1})=𝑆_𝑚^2𝜎^2Δ𝑡$ where $\Delta t = \frac{T}{...
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0answers
57 views

One Period Binomial Option Valuation Model [closed]

My question here is how is the probability of an up move calculated by $(1+Rf-D)\over(U-D)$ derived where Rf is the risk free rate, D is the down move factor and U represents the up move factor. ...
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1answer
126 views

What's the logic behind binomial model ups and downs?

I want to understand what is the underlying logic in the calculation of u and d in a binomial model. $$ u = \exp\Bigl(\sigma \sqrt{\Delta t} \Bigr), \quad d = \exp\Bigl(-\sigma \sqrt{\Delta t} \Bigr)...
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1answer
47 views

Infinite Binomial Pricing no arbitrage

How to price a contract that pays only 1 at the first stock price drop? The stock follows an infinite binomial with no arbitrage $d<R<u$ condition. So the probability of the price going down is ...
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1answer
154 views

What happens in the binomial model if the real-world probability is $0$

Consider a binomial model. Suppose we know that the price of a stock will become a certain value at the next timestep. That is, one of the two outcomes has $0$ real-world probability. Then it should ...
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1answer
139 views

Difference between tree and lattice approach

Is there any difference between the tree and lattice approach for valuing derivatives? I was under the impression that both are the same.
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2answers
219 views

Approximation of CRR as Black Scholes PDE

I have a formula for intermediate european option price calculated at, say, m-th possible tree value. $S_n^{(m)}$ is a price at node after going up $n$ times and down $n - m$ times $V(S_n^{(m)}, t + ...
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1answer
242 views

Binomial Trees vs FDM

Binomial trees as the number of time steps is increased (or equivalently as the time step tends to 0), converge to the exact value for an option. So why do people use FDM for pricing options (for ...
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2answers
153 views

Is American option price lower than European option price?

I used to think under the same condition, the American option is always more expensive than the European option, because American option can be exercised at any time (has more rights than European ...
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1answer
308 views

Binomial Tree Option Pricing Model. Lets talk dividends and futures

I am writing an option pricing model for production use. Its not for arb or anything so it doesn't need to be 100% as accurate as possible. Just good enough for "what happens to my book if we jump 10 ...