# Tag Info

6

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 ...

5

The one-factor Hull-White model is given by $$dr(t) = (\theta(t) - \alpha\; r(t))\,dt + \sigma\, dW(t)\,\!.$$ The zero curves are only sufficient for the calibration of the parameter $\theta(t)$, which is given in terms of them by $$\theta\mathrm{(t)=}\frac{\partial f(0,t)}{\partial T}+\alpha f(0,t)+\frac{\sigma^2}{2a}(1-e^{-2\alpha t}),$$ where ...

3

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. ...

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

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), ...

3

honestly your question is hard to understand. Are these two questions the same? "Does fitting sub-optimal option exercise strategies to market data yield better option values?" "which modeling approach leads to better predictions and better relative value measures?" I think you want to ask 1 and I think it is similar to Setting the r in put-call parity? ...

2

Time is expressed in fractions of year in the GBM formula. Therefore, $T=1$ year and $\Delta t = 1/m$. Considered that you have $253$ observations, I would use $m = 253$, so the second option as Drew suggested. In general, using 253 or 365 days in a year depends on how you consider reality: do you think that when markets are closed (i.e. weekends) the price ...

2

The second one will be the best estimate. Also, a smaller timestep usually corresponds to a smaller bias. But I agree, the answer is not obvious. You should be careful about increasing $T$ though, because for negative drifts there is a threshold value ($2\mu + \sigma^2 < 0$) beyond which the variance of the price process stops increasing. It's an ...

2

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.

2

Here is how I would approach such a calibration. Assuming we have the necessary market data one can easily construct the emprical distribution of the arrival rate. Let $\lambda_{emp}(\delta)$ be the empirical distribution. Then one can define a metric by $$m(k,A,N)=\sum_{i=1}^N |\lambda_{emp}(i)-\lambda^a(i)|$$ After you have decided upon a suitable ...

2

At long maturities, the real problem tends more to be model error than volatility estimation: over that kind of time period most companies undergo significant capital structure changes, for which there are very few models.

2

GMM method is a powerful method to calibrate historically, only. Also, the historical Calibration is used in the banking industry for forecast an asset’s performances and not for replicating them. Mathematically, it's known that historical vs options calibration is equivalent to observing an asset through two different probabilities (historical vs the ...

2

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 ...

1

This is pretty straight forward: The market prices vanilla options via implied volatility. You can like it or not like it but that is the way it is. So, the fair price of the option is the equivalent of the implied vol via BS. Now, if you believe the true price of an option should be different from the traded market price and you figure out that you have ...

1

It depends what you want volatility for. Theory will tell you that: "Implied variance of short maturity ATM options is approximately equal to the expectation of the realised integrated variance of the underlying over the life of the option and under the risk neutral measure" In math: $\sigma^2_{ATM}\approx E^Q\left(\frac{1}{T}\int_0^T\sigma^2_t dt\right)$ ...

1

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 ...

1

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 ...

1

Given that you have swap rates and Cap prices (ATM, I assume), you can back out the IVs for the time periods using by bootstrapping. Strictly speaking, you would need Caplet prices for the given strikes. In such a case, You would look at the shortest dated cap and (assume) it is made up of only one caplet. You can then use black's formula and back out ...

1

Thanks to my research leader, I found what I missed. $V_{0,1}$ is vol of swaption that matures at $T_0$ which is not 0 (as I thought), rather it is maturity of the first libor. So $V_{0,1}$ is the closest available point on market. And now this is all clear with table on page 323 in section 7.4. $V_{0,2}$ is realy vol of swaption that matures at $T_0$=1y ...

1

Here's a decent study of calibration performance using fast fourier transforms versus other techniques. It concludes Gaussian quadrature works better than other techniques. http://www.frankfurt-school.de/dms/publications-cqf/CPQF_Arbeits6.pdf Edit: AZhu points out the link above is dead and that a working link is ...

1

Dealing with model error under stochastic volatility (in a more formal way) you could use the UVM (Uncertain Volatility Framework). Here are what i think are the most seminal references: Avellenada et al (1995) Pricing And Hedging Derivative Securities In Markets With Uncertain Volatilities http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.3736 ...

1

In reality, you needn't bring exotics into consideration to think about this issue. Consider the case of a shop that has fundamental analysts but also trades options on those equities. The fact that the fundamental analysts trade stocks means they think those prices are somehow "wrong". So of course it seems from their point of view that the options ...

1

The starting point for your analysis should be to convert all your options bid/ask prices to implied volatilities, which are the invariant (time-homogeneous, i.i.d.) process in your case. Even though you do not have the dividend yield and other factors necessary to back out the implied volatility, so long as you use the same assumptions (and they are ...

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