22
A general model (with continuous paths) can be written
$$
\frac{dS_t}{S_t} = r_t dt + \sigma_t dW_t^S
$$
where the short rate $r_t$ and spot volatility $\sigma_t$ are stochastic processes.
In the Black-Scholes model both $r$ and $\sigma$ are deterministic functions of time (even constant in the original model). This produces a flat smile for any expiry $...
5
Here "dynamics" means the assumed future behaviour of the spot process, namely that it follows the SDE
$$ dS/S = r dt + \sigma_{loc}(S,t) dW_t .$$
There are various ways to see that these dynamics are unrealistic.
One is to look for time homogeneity. In normal cases, you expect the market to follow the same rules in one week and in one year from today. ...
3
We assume that, under the risk-neutral measure $Q$,
\begin{align*}
dP(t, T) = P(t, T)(r_t + \sigma(t, T)dW_t),
\end{align*}
where $\{W_t, \, t \ge 0\}$ is a standard Brownian motion. Then
\begin{align*}
dL(t) &= \frac{1}{T-S}\bigg(\frac{dP(t, S)}{P(t, T)} -\frac{dP(t, S)}{P^2(t, T)}dP(t, T) \\
&\qquad + \frac{dP(t, S)}{P^3(t, T)} \langle dP(t, T), \,...
3
KeSchn and I pointed out in the comments that this it is not possible to represent all stock dynamics using the Generalized Black Scholes model. For example, there can be jumps at random moments and not just at random moments but also jumps of random size. These jumps can affect either $\mu_t$ or $\sigma_t$. Models with too many sources of randomness are not ...
3
Assuming we are talking about Pearson correlation, then we may apply the triangle inequality. Let $\rho(X,Y)$ denote the correlation between $X$ and $Y$. Then,
$(1-\rho(X,Z))^{1/2}\le (1-\rho(X,Y))^{1/2} + (1-\rho(Y,Z))^{1/2}$
3
I totally missed the coining of the term "Approximate Dynamic Programming" as did some others. Also, in my thesis I focused on specific issues (return predictability and mean variance optimality) so this might be far from complete. That's enough disclaiming.
Let's start with an old overview: Ralf Korn - Optimal Portfolios. Kenneth Judd - Numerical Methods ...
3
This book might be what you are looking for:
Theory of Financial Risk and Derivative Pricing. From Statistical Physics to Risk Management by J.-P. Bouchaud and M. Potters
As one reviewer from amazon wrote:
Econophysics (the application of techniques developed in the
physical sciences to economic, business and financial problems) has
emerged as a ...
2
To me, that smelled like dynamic programming too. After implementing a dynamic programming solution according to http://www.cs.rpi.edu/~magdon/courses/cf/notes/optimal.pdf and other sources from the same author, it dawned on me that dynamic programming might not really be necessary at all.
In the end, what you want is to put all your value into the single ...
2
Hurst exponents are most often used in identifying trends in time series.
It's been quite a while, but I read this book years ago and this sort of thing is addressed therein (albeit, in a somewhat superficial manner as typical for any trading-centric modeling). Might be worth checking this out.
https://www.amazon.com/Chaos-Order-Capital-Markets-...
1
The problem was a missing $W_t$ in the equation for correlation. I've updated the above code and did a rerun. We have now the following allocation which is much closer to the Infanger paper.
> solmat
US Stock Int Stocks Corp Bonds Gvnt Bond Cash
50000 5.043872e-01 0.089871441 0.40574133 2.745030e-08 1.788550e-09
55000 4.050341e-...
1
The "rmgarch" package in R requires specifying univariate GARCH models before a DCC (or asymmetric DCC, aDCC) can be fitted. The workaround is to specify models that essentially "do nothing", e.g. a GARCH model with $\alpha=0.00001$ and $\beta=0.99999$ and variance targetting at the unconditional variance. These models will produce roughly constant ...
1
I think in this case no fancy normalization techniques are implied. At least from what I understand from the cited part, they just scale the variables so that they are equal to 100 in the base period (end of preceding year) - something like computing a deflator, commonplace in macro analysis.
1
You can use a for-loop on your correlation series.
for i=1:2000
simulation=copularnd('t',rho(i),NU,N));
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