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Up and Down Multiplicative Factors of the Binomial Option Pricing Model

When computing these factors, according to some sources, $u=e^{rT/2+\sigma \sqrt{\Delta t}}$, where $r$ is the risk-free interest rate, $T$ is the time for maturity, and $\sigma$ is the volatility. ...
Bumblebee's user avatar
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Obtain B-S-M from a binomial tree as n goes to infinty using Lebesgue integral

My question is simple, consider a European call with payoff max(S_T-K, 0), Let's suppose that the underlying stock follows a binomial tree with up and down factors I know as we take n goes to infinity ...
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Mark Joshi, The concepts and practice of mathematical finance exercise 3.6

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 $e^{r t}$ at time $t$ ...
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Two-period binomial model probability question

I have started to work with given two period binomial model S(0)=100 u=1.25 d=0.8 r=0.05 and the market probability of stock going up each period is p=0.55. I am trying to calculate two probabilities; ...
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Stock price modelling under binomial tree model?

In binomial tree model, the stock price is modelled in the form of $S_{k\delta}=S_{(k-1)\delta}\exp(\mu\delta+\sigma\sqrt\delta Z_k)$, where $\delta$ is time invertal between two observations $S_{k\...
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Binomial option pricing model for American options on assets paying a continuous dividend yield

Let's say an asset has a continuous dividend yield of 5% (and assume interest rate is 0%). If I want to price an American call option on such an asset, I take each time step individually and construct ...
artemars's user avatar
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How to Determine Parameters in a Non-recombining Binomial Tree for Option Pricing

For a CRR recombining Binomial Tree, let the underlying stock price be $S_0$ at $t=0$ and the time interval be $\Delta t$. The nodes at $t=\Delta t$ and probabilities reaching them can be written as: $...
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Convergence in the CRR model

Under certain conditions, the option price of the CRR (Cox-Ross-Rubinstein) Binomial model converges to the Black-Scholes price as the maximal step size of the partition converges to zero (i.e. a ...
Kapes Mate's user avatar
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Why do we simply assume the risk neutral probabilities to be "0.5"?

I am aware that there was a question similar to this but my question is a little different. Firstly, in context of binomial short rate, why do we simply assume the risk neutral probabilities p=1-p=0.5?...
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Delta-hedge experiment of American Put option

I am trying to run a delta-hedge experiment for an American Put option but there's a (systematic) hedge error which I cannot seem to understand or fix. My implementation is found in the bottom of this ...
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How to fit (find $u$) in the binomial options pricing model?

In the binomial tree options pricing literature, I see frequent reference to the definition that $$ u = e^{\sigma \sqrt{n/t}} $$ I think I understand the model, but how do we derive this, i.e. how do ...
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Binomial tree convergence tree towards BS equation - Struggle with a limit

I am trying to prove that the Binomial tree pricing method converges towards the Black and Scholes model, but I am struggling on a specific step. I don't understand how the limit of p*(1-p) is ...
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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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Theory of the convergence of option prices using trees

My current understanding of the theory behind the convergence of options prices using trees is the following: Suppose $S = (S_{t})_{0\leq t\leq T}$ is the underlying process and $g(S_{t}:0\leq t\leq T)...
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Exotic options with lookback features [closed]

I am trying to value an american call option with a lookback feature. So the holder can choose to exercise either based on a fixed strike (K) or a floating strike equal to 10-day moving average (MA). ...
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Binomial tree with time dependent volatility

In the Cox approach for binomial trees, the up move $u$ and down move $d$ are given by: $u = e^{\sigma \sqrt{dt}}$ and $d = e^{-\sigma \sqrt{dt}}$. In this approach the volatility $\sigma$ is assumed ...
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Difference between closed form binomial option value and monte carlo simulation

I am trying to calculate the price of a European call option using both the the closed form expression and a monte carlo simulation. But the value's I get from both these methods are not the same: ...
56423Tree's user avatar
2 votes
1 answer
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Why does changing the step size in my Binomial Tree changes the final stock prices so much?

I am trying to price a convertible bond by using a binomial tree. For this, I wrote a binomial tree for the stock price. I noticed that changing the step size (timesteps), changes the final value of ...
Daan Commandeur's user avatar
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Noob Question - Monte Carlo vs BAPM European Option Pricing Discrepancy

It's winter break (happy new year!), and I'm trying implement a few options pricing models (bapm, tapm, monte carlo, Fast Fourier etc.) for practice. The issue: My BAPM CRR model converges to 8....
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Optimize call option purchase

If it is predicted that the price of a stock will increase from P1 to between P2 and P3 in time T (assume the distribution of the price will be evenly distributed between the range of [P2, P3] at time ...
user15482691's user avatar
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2 answers
801 views

Find the value of put option using a two-period binomial model

I've been asked to find the price of a two-month European Put Option with strike price $£40$. The price at $S_0=£30$, this can move up to $£40$ or down to $£25$ ($1/3$ chance to go up, $2/3$ chance to ...
Charlie P's user avatar
3 votes
0 answers
315 views

Cash less exercise and redemption feature in SPAC warrants

Public and private warrants of a SPAC post merger (Initial Business Combination or IBC) are often very similar. Notable differences are 1) cashless exercise of the private warrants and 2) redemption ...
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Option pricing when stock price follows binomial tree

Assume that the stock price is currently trading at $S_0$. It is known that the stock price follows a binomial tree, such that its price will be either $S_0e^{\theta_u}$ or $S_0e^{−\theta_d}$ over the ...
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Why Vasicek model on a tree is a bad choice for pricing American option on credit prepayment?

I have an American option on a credit prepayment, i.e. the holder of the option can prepay the remaining credit if the interest rate falls below the initial strike. The pricing of this option was done ...
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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 ...
Charlie P's user avatar
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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 ...
James's user avatar
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2 votes
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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 ...
Dave N.'s user avatar
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2 answers
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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 ...
James's user avatar
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1 answer
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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 ...
MinaThuma's user avatar
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1 answer
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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....
stats19's user avatar
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1 vote
1 answer
249 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 ...
Morgan Lu's user avatar
4 votes
2 answers
222 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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0 answers
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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 ...
user107224's user avatar
1 vote
1 answer
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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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2 answers
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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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0 answers
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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 ...
bizmark's user avatar
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1 answer
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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 $...
user avatar
2 votes
1 answer
593 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 ...
qp212223's user avatar
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1 answer
738 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?...
Mr.Price's user avatar
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1 answer
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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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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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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_{...
Giogre's user avatar
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1 answer
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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 ...
Giogre's user avatar
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1 answer
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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 ...
Metrician's user avatar
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1 vote
1 answer
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Calculating European call option, the Bjork way

We have a 3 period binomial tree with values: ...
simsalabim's user avatar
1 vote
0 answers
267 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 ...
Ryan J. Shrott's user avatar
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1 answer
235 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 ...
WannabeQuant's user avatar
2 votes
1 answer
411 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 ...
ya23's user avatar
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2 votes
1 answer
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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 ...
ensabahnur's user avatar
2 votes
1 answer
132 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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