9
The Hull-White model can represents the risk free rate as a stochastic process, that is, in terms of expected return and volatility. The zero curve only gives you expected returns and you have to find a source to calibrate volatility, as FQuant told you.
Common volatility sources used for this calibration are historical series of the zero curve or ...
9
Concerning your first question, this depends on what curve, currency, etc. you are interested in. The general method for constructing yield curves is called bootstrapping which allows you to derive spot, zero-coupon rates from the known price of coupon-bearing instruments $-$ such as bonds or swaps. In general:
You start picking short-term (typically less ...
8
We assume that the process $\{r(t), \, t \ge 0\}$ satisfies an SDE of the form
\begin{align*}
dr(t) = \big( \theta(t) - a(t) r(t) \big)dt + \sigma(t) dW_t, \quad t > 0,
\end{align*}
where $\{W_t, \, t \ge 0 \}$ is a standard Brwonian motion. Note that
\begin{align*}
d\left(e^{\int_0^t a(u) du}r(t) \right) &=a(t) e^{\int_0^t a(u) du}r(t) dt + e^{\int_0^...
7
The one-factor Hull-White model is given by
$$dr(t) = (\theta(t) - \alpha\; r(t))\,dt + \sigma(t)\, dW(t)\,\!.$$
The zero curves are only sufficient for the calibration of the parameter $\theta(t)$, which is given in terms of them by
$$\theta\mathrm{(t)=}\frac{\partial f(0,t)}{\partial T}+\alpha f(0,t)+\frac{\sigma^2}{2\alpha}(1-e^{-2\alpha t}),$$
...
7
I assume that for approximating the second derivative of the call price $C (K,T) $ at the bounds of the strike domain (see first 2 "if" cases of the last for loop of your code) you tried to set up boundary conditions.
On the right bound, your approximation $C(K+\Delta K, T) \approx 0$ could make sense for $K \to \infty$ or at least big enough.
On the left ...
6
The most used equity volatility models in the industry are the Black-Scholes model (including its time dependent version) and the local volatility model. It always come along with stochastic rates, discrete dividends and quanto effects (a must-have when pricing even simple payoffs) so the calibration/pricing process is much more involved than what you might ...
6
The typical approach is: you only use option data from the last day. Furthermore, you only include those points that are liquid enough. One approach to this is to weigh the modelling error of an option by its bid-ask spread and vega.
Using data from multiple days is not a good approach, because you might have options with the same strike but different ...
6
I find your approach to calibration (training an ANN to learn the inverse function f-1 from a training set of 'market_prices = f(model_parameters)' interesting, novel (at least this is the first time I am hearing about it) and definitely worth investigating further. If you make it work, you have almost instantaneous calibration and a methodology applicable ...
6
It is not cheating. It allows you to make your results (e.g. prices, calibrated parameters) 'reproducible' which is good. However, fixing the seed can hide convergence issues. When the variance of your Monte Carlo estimator is large, picking different seeds could yield drastically different results. So be careful. In practice you can obviously solve this by ...
5
Gatheral and Jacquier discuss this issue in section 4 of the paper. Instead of using the raw parameterization of the SVI, they use the natural parameterization of the total implied variance:
$$ w(k) = \Delta + \frac{\omega}{2} \left\{ 1 + \zeta \rho (k - \mu) + \sqrt{(\zeta (k-\mu) + \rho)^2 + (1-\rho^2)} \right\} (\text{p. 61 of the published paper}) $$
In ...
5
You can find the derivation of the Heston characteristic function (its Fourier Transform) in Gatheral (2006).
Using the characteristic function, you can optimize the model on the prices. There are multiple approaches to optimize, among others pattern search (which is very slow) and stochastic optimization (randomly jump around and stop after n iterations), ...
5
I highly recommend you to stick with the error function (RMSE) value minimization approach. I love MC techniques for this and related problem solving and thus do not recommend you to use anything else because of its simplicity and transparency. It comes down to using the right discretization function and to possibly implement variance reduction approaches.
...
5
I know two papers explaining how to calibrate this kind of models, and one of them explain the impact of the quality of the fit on a pricing model:
Aït-Sahalia, Y. (2002, January). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 (1), 223-262.
Azencott, R., Y. Gadhyan, and R. Glowinski (...
5
Maybe it would help you to think of it the following way.
The strike $\sigma^2(T)$ of a fresh-start variance swap of maturity $T$ in the Heston model only depends on parameters $(v_0,\theta,\kappa)$, see related question here. More specifically
\begin{align}
\sigma^2(T) &= \Bbb{E}_0^\Bbb{Q}\left[ \frac{1}{T} \langle \ln S\rangle_T \right] \\
&= \...
5
You are an investment bank. You trade a multitude of vanilla and exotic options. You want to make sure the option prices you quote as a client are arbitrage-free with respect to liquid option prices quoted in the market $-$ and also consistent between the different trading desks within your bank.
Basically you want to avoid other market participants taking ...
4
Given the Ho-Lee interest rate model of the form
\begin{align*}
dr_t = \theta_t dt + \sigma dW_t,
\end{align*}
the price at time $t>0$ of a zero-coupon bond, with maturity $T$ and unit face, has the form
\begin{align*}
B(t, T) &=E\Big(e^{-\int_t^T r_s ds} \mid r_t \Big)\\
&=e^{-(T-t)r_t - \int_t^T (T-u)\theta_u du + \frac{\sigma^2}{6}(T-t)^3}.
\...
4
I will just answer your first question as I do not know the details of SSVI.
Total variance is more intrinsic than volatility.
The BS formula can be rewritten in terms of 3 parameters: the log-strike (log-moneyness would be more accurate) $k$, the total variance $w$ and the discount factor. Volatility never appears without a $\sqrt{T}$. It is just there ...
4
Market practitioners do the following: Correlation is calibrated most often by looking at historical correlations between liquid par swap rate pairs. One could look at implied correlations within options on the yield curve (eg 10 yr minus 2yr) also.
Swaption calibration should be done by comparing straddle prices in the market to prices produced by the ...
4
I am intrigued by this question because it gets at the heart of so many grey areas of the financial system in which it becomes almost impossible to know how many assets derive their values from some unseen or ill-prescribed, but presumed extant, underlying process.
Calibration can be interpreted as means of deriving an expectation which is the ...
3
For short maturity SPX option chain, the analytic form of the V-shape volatility smile has been fully worked out in my latest paper on SSRN. You can take a look.
3
I would say
Take log of first equation to get rid of dependence on $x_t$
Apply Kalman filter equations to estimate parameters
I believe Conrad and Kaul (1988) J of Business do exactly what you describe.
3
The main difference is that one approach assumes that a certain dynamical structure properly describes the underlying instrument, while the other approach is really only a re-writing of the price in terms of an implied volatility.
Implied volatility
Implied volatility really only needs two things: the underlying stock price and the call option price (apart ...
3
Doesn't the Heston model have some Fourier transform formulae for pricing vanillas? I think one could use those to calibrate to the vanillas. Can't provide references at this moment, on the road.
Edit: check out http://www.visixion.com/dok/Visixion_Calibrating_Heston.pdf -- I haven't read this closely but it sounds familiar
3
Jim Gatherals Book deals with the models you mention and gives an intuitive understanding about calibration and issues that arise. Mostly basic stuff, but very useful if you're just starting out. Also very understandable without an extensive math background.
3
I would suggest you to add spreads to the implied hazard rates, spreads that you regress on the macroeconomic factors. Then you stress by calculating the spreads corresponding to the stressed factors.
3
It seems that implicitly you have a multi-objective optimization in mind, hence of course it may happen that you are not able to achieve all the objectives simultaneously. Let's say that output of a more general model is $f(x,y)$ so that the output of the first model is $f(x,0) = f_0(x)$. Denoting market prices by $m_k$ which in your case means $m_1 = A$ and ...
3
Hi am having to write as an 'answer' as am new to forum.
We used stochastic intensity models on desk from a while back. Generally Black-Karasinski to avoid negative hazard rates (and for useful features such as mean reversion). Now in your choice of structural approach with lognormal jumps as some respondents have pointed out you will have to simulate to ...
3
EDIT: My reasoning below seems to be wrong. The process as you write it tends to infinity if $a$ is big enough and positive and if $\lambda_0$ is positive.
I would not call this process non-meanreverting OU. It is just an Ito process of a simple form. If we remove the stochastic part then we get
$$
d\lambda_t = a \lambda_t dt
$$
with the solution (if $\...
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