6
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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 ...
- 1,466
4
votes
Why risk neutral probabilities should be strictly greater than zero for no arbitrage condition?
There is one condition under which the risk neutral probability of an event can be zero: if the real world probability is zero. If not then any contract that pays off in that event must go down in ...
- 605
4
votes
What is Phi in Cox-Ross-Rubinstein Binomial Model?
It is explained just above the box in the original paper on page 239. The function $\Phi$ denotes the complementary binomial distribution function. Complementary means it is $\mathbb{P}(X>x)=1-F_X(...
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4
votes
Accepted
Backshifting Price Timeseries with Memory Preservation
Other than the binomial expansion, what is the relationship between $B$ and $w$?
As I understand correctly, there is no immediate relationship between the lag operator $B$ and the weights $w$. However,...
- 4,726
3
votes
What is Phi in Cox-Ross-Rubinstein Binomial Model?
Hi Mtris,
I would recommend implementing the formula above. Very easy to implement but I recommend looking up proposition 2.25 in Björk's fourth edition of Arbitrage Theory in Continuous Time. It is a ...
- 51
3
votes
Accepted
Python Numpy FFT array size limit?
Im going to hazard a guess that your problem is u**(N-i). Large exponents are notoriously poor performers, I would first look to restructure that aspect of the code ...
- 11.2k
3
votes
Accepted
Number of Time Steps in Binomial Option Pricing - Problem?
It is quite common to see non-smooth convergence in tree models and this is not specific to digital options.
The problem usually that the tree is constructed independent of the contract to be priced. ...
- 6,064
3
votes
Volatility smile risk (negative effect) on dynamically hedged portfolio?
Since all your options have the same strike, you do not have any "explicit" skew or smile exposure in your portfolio. If I had to guess, almost all of your P&L can be explained by primary ...
- 136
3
votes
Option pricing models relation between theoretical and actual price
I think you have the correct understanding. The arbitrage is only possible if the risk-neutral probability distribution of the stock is perfectly known, as it is in your simple binomial model. In ...
- 17.5k
3
votes
Accepted
Replicating an option
The pricing of options is married with the concept of a hedging strategy that replicates the effect of the option. If you can only long or short a stock that will not replicate the greeks, it only ...
- 11.2k
3
votes
Binomial model and delta hedging
Say there are just two periods: Payoff at n, and premium/price at $n-2$.
We know the current stock price, say $S_{n-2}$, and we know in the next epoch, it will either be: $S_{n-1}^u=uS_{n-2}$ (up ...
- 6,222
3
votes
Accepted
Calculate volatility under the binomial model for option pricing
From your answer to my comment, here is what I would do.
Over the horizon $[0,\Delta t]$, the BS model tells you that the expected log-return is
$$ \Bbb{E}\left[ \ln\left(\frac{S_{t+\Delta t}}{S_t}\...
- 14.7k
2
votes
Why does arbitrage free imply complete market?
You agree that the proposition is proven if the equations have a unique solution. You agree that there is a unique solution if u>d. Then we just have to show that u>d. But the definition of u and ...
- 17.5k
2
votes
call vs average of prices
The call is worth more unless the risk free rate is zero. Let $p$ be the probability of $S_0$ going up, $r$ be risk free rate, $T$ is the one time step. Then no arbitrage means
$$ S_T = S_0 \exp(r T) =...
- 241
2
votes
Failing to replicate Wilmott's results for binomial option pricing
Going through his code, seems like he has used this formula to come up with the figure 1.0604:
$u=\frac{1}{2}\left(e^{-r \delta t}+e^{(r+\sigma^2) \delta t}\right)+\sqrt{\frac{1}{4}\left(e^{-r \delta ...
- 6,222
2
votes
Failing to replicate Wilmott's results for binomial option pricing
I have checked the answer to my side, even using the alternative, the original formula $u=e^{\sigma \sqrt{\delta t}} $, I still get $1.05943$
I suspect it is not really a logic error, but more of a ...
- 341
2
votes
Accepted
Intuition behind short 1/2 stock in option value - Paul Wilmott Quant Finance Chapter 3.3
This is a classic interview question, btw.
Probabilities you mention are "subjective" hence are irrelevant for option pricing. You need to use risk neutral stock migration probabilities, ...
- 559
1
vote
Accepted
$\mathbb{Q}$ measure and $\mathbb{P}$ measure, trading strategy
As people pointed out, P and Q are different because of risk aversion. Under the P measure, the payoff in bad states are worth more; whereas Q measure is risk neutral. This is similar to why people ...
- 51
1
vote
Maximal increase payoff
This is the sum of look back call and lookback put with floating strike. You can then price it using the formulas in wikipedia:
https://en.wikipedia.org/wiki/Lookback_option
- 872
1
vote
Breakdown of Wilmott's Binomial Tree derivation of Black-Scholes equation
Earlier in the chapter, Wilmott derives the equation
$$V = \frac{V^+ - V^-}{u-v} + \frac{uV^- - vV^+}{\left(1 + r\delta t\right)\left(u - v\right)}$$
from the non-arbitrage argument $\delta\Pi = r\Pi\...
- 66
1
vote
Binomial model and delta hedging
Unlike $S$ and $f$ which are driven by the market that are out of your control, $\Delta$ is the amount of stocks $S$ that you have decided to short in the previous time step for this portfolio $\Pi$. ...
- 241
1
vote
Hedging a long position-one period from Steven Shreve Stochastic Calculus for Finance
In both of your wealth equations, there is no need to subtract 1.2 from your investment in money market because you already own the option.
Solve it you will get $\Delta=-\frac{1}{2}$ and $X_0=0$. Now,...
- 11
1
vote
How to derive Balck Scholes from the Binomial Model?
Check out Approximation of CRR as Black Scholes PDE. I show the derivation in my post there
- 1,671
1
vote
If price is a random walk, is ok to use the binomial distribution to estimate a trading strategy?
It is a good idea to make an assumption of "no informational content" on prices to have a reference level for this $H_0$ hypothesis.
The best is probably to make Monte-Carlo simulations, i.e. to ...
- 12.5k
1
vote
Yearly ytm calculation on stock using binomial model
At the terminal date, the value will be: 1.44, 0.84, 0.84, 0.49 in the four states: UU, UD, DU, and DD, respectively.
The probability of an up move is: (1.1-0.7)/(1.2-0.7)=0.8
So the probability of ...
- 6,222
1
vote
Error on Paul Wilmott Section 5.2?
I finally found my error,thanks dm63 for the explanation. I had a hard time imagining the negative position value and that it implies that I also get the interest from getting cash for the short..I ...
- 255
1
vote
Looback Put Option - finding the number of paths that reach each level
The general formula to answer this question can be found on page 105-106 of Introduction to Mathematical Finance by Pliska.
In general:
$\bar{\mathbb{P}}\left ( M_{4} \geq 4\left ( 2^{i} \right ) \...
1
vote
Real Options: Calculating the "option to switch use" using binomial lattices
hmm, Trigeorgis seems to be saying that the value of being able to switch at one of the times 1,2 and 3 is the same as the sum of being able to switch at each of the times.
This seems wrong to me ...
- 6,993
1
vote
Calculate volatility under the binomial model for option pricing
Black Scholes can be seen as the continuous limit of a binomial model when the number of steps go to infinity.
(It can be seen as a result of Donsker's theorem)
Thus it is normal that your call ...
- 2,442
1
vote
Proof of optimal exercise time theorem for American derivative security in N-period binomial asset-pricing model
I think the proof has already been provided at the end of the proof in Shreve's Theorem 4.4.5. Specifically, note that, since
\begin{align*}
\frac{1}{(1+r)^{n \wedge \tau^*}}V_{n \wedge \tau^*}.
\end{...
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