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 ...
- 1,436
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 ...
- 101
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 ...
- 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 ...
- 15.1k
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 ...
- 126
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 ...
- 15.1k
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 ...
- 4,307
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 $...
- 11k
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 ...
- 6,853
4
votes
Accepted
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 $...
- 605
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*...
- 2,177
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 ...
- 5,622
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 ...
- 16.2k
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.
@...
- 16.2k
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 ...
- 129
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 ...
- 1,028
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 ...
- 1,256
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 ...
- 883
3
votes
Accepted
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 ...
- 21k
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 ...
- 14.5k
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 ...
- 27.3k
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 ...
- 14.5k
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 ...
- 2,218
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 ...
- 2,778
3
votes
Accepted
pricing american calls on non dividend paying stocks
The argument that the American and European call are worth the same is model independent. So it holds for the binomial model. So there is no need to check to
see if the early exercise occurs because ...
- 6,853
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 ...
- 4,728
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.
...
- 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 ...
- 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-...
- 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 ...
- 516
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