11
votes
Do basket options have a closed form valuation formula?
I'm not completely certain from your question, but I'm going to assume you have a basket of $n$ stocks with prices $S_0(t)$ to $S_n(t)$, and you want to price an option with payoff at $C(\tau)$ at ...
- 2,886
10
votes
Accepted
How to simulate Levy processes
You have many different options. Firstly, you know the characteristic function for the log stock price and, using inversion, you can recover the (inverse) distribution and density function and ...
- 14.9k
8
votes
Accepted
How are Brownian Bridges used in derivatives pricing in practice?
Yes, the term Brownian Bridge seems to be used loosely. I assume you are talking about continuously monitored barriers by the way, since you mention the probability of the barrier being crossed in ...
- 511
6
votes
Pricing a log-contract using Monte Carlo
By definition, the payoff of a log-contract of maturity $T$ writes
$$ \phi(S_T) = \ln\left(\frac{S_T}{S_0}\right) $$
Let $\Pi_t$ denote the $t$-value of such a contingent claim. We are interested in ...
- 14.3k
6
votes
Simulating trading strategies
Is there a place online where you can simulate strategies programmatically?
Your best choice is most likely a service such as Quantopian or QuantConnect. Quantopian provides equity and futures data ...
- 1,162
6
votes
Accepted
negative values in geometric brownian motion
I agree with wrong formula in simulation, but think i understand the question.
Here's my take on it:
The reason SDE may seem to allow a negative value of x is because dW can be a large negative ...
- 111
6
votes
Risk Neutral and Real World Valuations using Monte Carlo
You probably wonder whether $\mathbb{E}^\mathbb{P}[P_T\mid\mathcal{F}_t]= \mathbb{E}^\mathbb{Q}[P_T\mid\mathcal{F}_t]$. Note the $T$ as index, i.e. the future unknown payoff and not the current price $...
- 14.9k
6
votes
Accepted
Simulating covariance matrices with nonzero correlation
What does 'simulate a covariance matrix' mean?
If the question means, generate an arbitrary correlation matrix for 1000 stocks, then we can choose any symmetric matrix with all 1s down the diagonal, ...
- 2,886
6
votes
Interpolation of $\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$
That is a tricky question because interpolation seems to be ok if you need one point $\tau$ between $t_k$ and $t_{k+1}$ but it is not.
The difficulty arise a direct way if you want two points inside $[...
- 11k
5
votes
Exploding Libor Rates in Libor Market Model
this is a well-known problem. One solution is to make volatility zero when rates exceed a certain high level.
It's less problematic than it looks because any cash-flows generated will be divided by ...
- 6,823
5
votes
Accepted
Model reference price of Limit order book
This reference price is also sometimes called intrinsic price. One of the simplest ways to improve it in regards to the mid-price (assuming you have the depth data) is the following:
define a ...
- 962
5
votes
How to generate simulated stock price from historical data using R?
This approach is rather crude. It only takes the mean and volatility of the historical returns and assumes a very simple model. I'm not sure if you have much experience with Time Series, but your ...
- 51
5
votes
Terminal Variance in the Heston Model
From the equations of the model it is clear that $v_t$ is the instantaneous variance of the log-returns, not the terminal annualised variance of the log-asset price.
Put differently, you are you ...
- 14.3k
5
votes
Python libraries for Monte Carlo simulations?
Try Quantlib https://www.quantlib.org, it comes with everything you need.
- 103
5
votes
Accepted
Simulating from a multivariate clayton copula
Since I think this is of interest for other people, I will post the approach I found:
First, let $C_n(u_1,\ldots,u_n)$ be a $n$ - dimensional Clayton copula with generator function $F$ and inverse $F^...
- 373
5
votes
Accepted
What is actually going on in Monte-Carlo simulation for Mortgage backed securities?
In my understanding, the mortgage prepayment option, at any point in time, is a function of the value of the mortgage from that point in time forward. This value, in turn, is a function of the future ...
- 6,175
5
votes
Reliable random number generation for Monte Carlo
You have misunderstood the statement in Matsumotos original paper. The original Mersenne twister guarantees, over its period of $2^{19937}-1$ (a number which I am sure you will agree is larger than ...
- 707
5
votes
Correct Monte Carlo simulation of local volatility models
[I think] the problem is with the SDE, rather than the numerical scheme
At a glance, and as I commented, I think the issue you are coming up against stems more from the underlying SDE rather than the ...
- 1,369
5
votes
Accepted
backtesting guide for research
This was too long for a comment, so I'm writing it as an answer. I have provided some interesting literature that will give you insight into the common pitfalls of backtesting algorithmic trading ...
- 4,108
5
votes
Accepted
Why we introduce correlations between Wiener processes?
Suppose we model two stocks by
\begin{align*}
\text{d}S_1 &= \mu_1S_1\text{d}t+\sigma_1S_1\text{d}W_1 \\
\text{d}S_2 &=\mu_2S_2\text{d}t+\sigma_2 S_2\text{d}W_2
\end{align*}
where $\text{d}W_1\...
- 14.9k
5
votes
Accepted
Optimize interest rate swap calculations in Monte Carlo Simulation
Yes, it's possible. You can create the 100 swaps and their engines beforehand and only change the curves. If you're using the same discount curve for all swaps, you can even create just one engine ...
- 5,988
4
votes
how to derive critical values for augmented Dickey–Fuller test (ADF) using Monte Carlo method?
The ADF test assumes the DGP
$$
\Delta y_t = \alpha +\beta t +\gamma y_t +\delta_1 \Delta y_{t-1}+\cdots +\delta_k \Delta y_{t-k}+\epsilon_t
$$
The parameters are estimated using OLS on a sample of ...
- 4,267
4
votes
Simulating a path of bond yields by Monte Carlo (Python)
I do not know Python but this is what I would do in Excel (I am assuming you are familiar with Excel and can then translate the steps into Python:
Pick a time series of Bond Yields which has $n$ ...
- 5,517
4
votes
Accepted
Monte Carlo - Multivariate Simulation of Returns
You should apply it to the covariance matrix and from that compute the correlation matrix. Here's an example correlating 3 random normal variables.
Let:
$$
\bf Y \sim \mathcal N(0, \Sigma)
$$
where ...
- 7,293
4
votes
How to model High/Low prices for Stocks with Monte Carlo
Some time ago I wrote an Octave C++ function to do just what you want and blogged about it on my blog. The link to the relevant post is https://dekalogblog.blogspot.com/2011/08/creation-of-synthetic-...
- 2,084
4
votes
Shifted Log-Normal model
Let us assume we are interested in some (forward) rate $F_t=F(t,T)$ which we assume is log-normally distributed:
$$\text{d}F_t=\sigma F_t\text{d}W_t$$
However, we observe market rates can in practice ...
- 7,468
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