6 votes

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

one of the most fundamental results states that the binomial model converges towards the Black Scholes model if the step size $\Delta t$ converges to zero. The Black Scholes model is an option ...
Cettt's user avatar
  • 1,446
5 votes

Black Derman Toy model: from tree to differential equation

From the gentleman and scholar Emanuel Derman. Emanuel states "the last two pages answer the question asked". https://www.dropbox.com/s/cg299qsbquuqdru/TwitterNotesOnBDT.2017.pdf?dl=0&m= Please ...
Paul Portesi's user avatar
5 votes
Accepted

Approximation of CRR as Black Scholes PDE

Assuming continuously compounded returns for a multi-period model with $N$ being the number of periods: \begin{cases} &\log u \quad \text{with probability q}\\ &\log d \quad \text{with ...
FunnyBuzer's user avatar
  • 1,012
5 votes

Binomial Option Pricing Model

Note that the tree is recombining. You have $u=1.2$ and $d=0.8$ with $ud=0.96$. Your tree for the asset price reads as At time zero: 100 At time one: 80 or 120 At time two: 64 or 96 or 144 The ...
Kevin's user avatar
  • 15.7k
5 votes

What are the relation between the risk neutral measures in binomial tree and in Black Scholes model?

There is a deeper relationship between the two risk-neutral measures. Take any event in the binomial model with a finite number of steps and calculate the risk-neutral probability of it. Take the ...
Peter Carr's user avatar
5 votes
Accepted

Does CRR Model lose completeness if we add another instrument?

I answer from a general discrete time/discrete state model point of view. This includes the binomial tree model as a special case. In finite dimensions, you can interpret asset payoffs and returns as ...
Kevin's user avatar
  • 15.7k
4 votes

Why don't real-world probabilities affect the price of a call in a 1-step binomial model?

"But just for fun, let's say Pr(S1=Su)=1% and Pr(S1=Sd)=99%, in which case, on average, the call at time 1 would be worth 0.01*10 = 0.1$. How would anyone be willing to pay 9.28$ for that ? I'm ...
Kiwiakos's user avatar
  • 4,327
4 votes
Accepted

Why don't real-world probabilities affect the price of a call in a 1-step binomial model?

It's a pitty that you don't show in your question how you get to your value for $c_0$ but the idea is that you build a portfolio $X_0 = \Delta S_0 - \lambda$ and you infer the values for $\Delta$ and $...
SRKX's user avatar
  • 11.1k
4 votes
Accepted

Difference in formulas for u & d in Binomial trees

there are many different trees. The first one, the CRR tree, used $$ u = e^{\sigma\sqrt{h}} $$ and $d = 1/u.$ However, you can take any real-world drift and still get the same prices in the limit so ...
Mark Joshi's user avatar
  • 6,913
4 votes
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Put-Call Parity on Currency and Binomial Trees

You have forgotten the combinatorial factors for binomial probabilities on your terms. You need $$ {n\choose k} p^n(1-p)^{n-k},$$ not just $$ p^n(1-p)^{n-k}.$$ The second term should have a factor of $...
spaceisdarkgreen's user avatar
4 votes
Accepted

Why my implementation of CRR model does not converge?

from the look of it your discounting is incorrect because as you increase M you should discount with 1/(1+r0*t) (assuming r0=0.0214 is the annual interest rate where as you seem to discount by 1/(1+r0*...
Ezy's user avatar
  • 2,187
4 votes

Binomial Trees vs FDM

Actually recombining binomial trees are only a particular case of an explicit FDM scheme. But they have obvious limitations, the foremost being that they cannot accomodate local volatilities. Also 1/2 ...
Antoine Conze's user avatar
4 votes

Risk-neutral pricing and statistical arbitrages

What you say is perfectly true and there is no contradiction. Arbitrage means risk free profit , so your ‘statistical arbitrage’ is not arbitrage at all. It just says that if you take risk, your ...
dm63's user avatar
  • 16.9k
4 votes
Accepted

Reference of using $\mu = \frac{1}{T}(\log K - \log S_0)$ in binomial tree model

It appears that the motivation for $\mu = (\log K - \log S_0)/T$ may be that K is in the middle of the tree at $T$. I could see how this may improve accuracy since K is where the ‘action’ is. @...
dm63's user avatar
  • 16.9k
4 votes
Accepted

How to price barrier options (binomial tree)

This may not be answering your question - but it is worth noting that valuing barrier options on a binomial / trinomial tree is at best problematic. It is difficult to enforce the boundary conditions ...
Marco's user avatar
  • 139
4 votes

Optimize call option purchase

Assuming the options available to you are priced using the Black-Scholes model and because your predicted prices of the stock at time $T$ are evenly distributed between $P_2$ and $P_3$ where $P_3 \ge ...
Alper's user avatar
  • 1,036
4 votes
Accepted

Why does changing the step size in my Binomial Tree changes the final stock prices so much?

You only got one minor bug, but let me explain why the range increases. Let us denote $n:=timesteps$, then You are looping one iteration too little when filling your $S$ matrix array, causing you to ...
Pontus Hultkrantz's user avatar
3 votes
Accepted

Explanation on the application of CLT in bionomial tree model

Thanks to P.Windridge's comment, I can now answer my own question. Indeed the convergence to standard normal in question can follow from a triangular array version of CLT called the Lindeberg-Feller ...
Vim's user avatar
  • 903
3 votes
Accepted

Trinomial tree VBA code

Joe, I wrote this a while ago and it could be cleaned up a little. It is for European Calls and Puts. I have a couple of lines commented out. I was probably going to add American pricing in but ...
amdopt's user avatar
  • 4,738
3 votes
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Deriving $u$ and $d$ coefficients using binomial tree approach

We assume that $u=e^x$ and $d = e^{-x}$. Note that \begin{align*} u &\approx 1+ x +\frac{x^2}{2}, \textrm{ and}\\ d &\approx 1- x +\frac{x^2}{2}. \end{align*} Substituting these into your ...
Gordon's user avatar
  • 21.1k
3 votes
Accepted

Replication strategy of European call option

You are at the beginning of a period and the stock price, worth $S$, can evolve in either of the 2 states: $S_u = u S$ or $S_d = d S$. The part you don't understand is related to forming so-called ...
Quantuple's user avatar
  • 14.6k
3 votes
Accepted

Demonstration of Ito's correction term/lemma in binomial tree

Actually it is quite simple to demonstrate Ito's correction term in a binomial tree. Details can be found in my new paper (p. 8-10): von Jouanne-Diedrich, Holger: Ito, Stratonovich and Friends (April ...
vonjd's user avatar
  • 27.4k
3 votes

How would I exploit arbitrage if risk-neutral pricing doesn't hold? (Option Pricing)

To rule out arbitrage in the one-period model, we must assume $$ 0 < d < 1+r < u, $$ where $u$ is the up-factor, $d$ is the down-factor and $r$ is the risk-free interest rate. This chain of ...
bcf's user avatar
  • 2,798
3 votes
Accepted

How to price and find a replicating portfolio for a call spreads using a two-period binomial model?

Quick answer The payoff you mention is that of a call spread, i.e. long a call $C_1$ struck at $K_1$ and short a call $C_2$ struck at $K_2$, with $K_2>K_1$. The price of the instrument is ...
Quantuple's user avatar
  • 14.6k
3 votes

Is there an error in this problem on pricing an asset using the true probability of an up move?

Your formula for $p$ is $$p = \frac{e^{{(\alpha - \delta})h} - d}{u - d},$$ where $\alpha$ is not expected return on stock but continuous risk free rate, i.e. 1%. If you use $\alpha$ as 1%, you will ...
Neeraj's user avatar
  • 2,228
3 votes

What discount rate to use when valuing binomial option with real probabilities

The Pricing equations are derived from duplicating portfolios consisting of underlying and a risk free asset. This means that the price of your option is relative only to the price of the underlying. ...
Lennart_R's user avatar
  • 181
3 votes
Accepted

Is American option price lower than European option price?

You compare the result of an analytical solution (european call) with the numerical solution for the american option. It seems as if you use to few steps to calculate your American option price. Just ...
JohnDoe's user avatar
  • 268
3 votes
Accepted

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

If I understand your question correctly, another way to word it is: if an event that has probability 0 under the physical measure $\mathbb{P}$, how can it have a positive probability under the risk-...
AdB's user avatar
  • 704
3 votes
Accepted

Binomial model arbitrage

In theory, we do not suppose there are transaction costs (or costs for short selling or even buying a security). In practice, effectively, you will have to pay the people that lend you the security ...
JeanGuillaume's user avatar
3 votes

Should U and D change with the number of steps in a Binomial Tree?

You can choose $U=e^{(r-\sigma^2/2)\delta t + \sigma \sqrt{\delta t}}$ and $D=e^{(r-\sigma^2/2)\delta t - \sigma \sqrt{\delta t}}$ to have the binomial model converge to the BS model when $\delta t=T/...
Antoine Conze's user avatar

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