7.4 Calibration

Calibration of stochastic volatility models can be done in two conceptually different ways. One way is to look at a time series of historical data. Estimation methods such as Generalized, Simulated, and Efficient Methods of Moments (respectively GMM, SMM, and EMM), as well as Bayesian MCMC have been extensively applied, for a review see Chernov and Ghysels (2000). In the Heston model we could also try to fit empirical distributions of returns to the marginal distributions specified in (7.4) via a minimization scheme. Unfortunately, all historical approaches have one common flaw - they do not allow for estimation of the market price of volatility risk $ \lambda(t,v,S)$. However, multiple studies find evidence of a nonzero volatility risk premium, see e.g. Bates (1996). This implies in turn that one needs some extra input to make the transition from the physical to the risk neutral world. Observing only the underlying spot price and estimating stochastic volatility models with this information will not deliver correct derivative security prices. This leads us to the second estimation approach. Instead of using the spot data we calibrate the model to derivative prices.

We follow the latter approach and take the smile of the current vanilla options market as a given starting point. As a preliminary step, we have to retrieve the strikes since the smile in foreign exchange markets is specified as a function of the deltas. Comparing the Black-Scholes type formulas (in the foreign exchange market setting we have to use the Garman and Kohlhagen (1983) specification) for delta and the option premium yields the relation for the strikes $ K_i$. From a computational point of view this stage requires only an inversion of the cumulative normal distribution.

Next, we fit the five parameters: initial variance $ v_0$, volatility of variance $ \sigma$, long-run variance $ \theta$, mean reversion $ \kappa$, and correlation $ \rho$ for a fixed time to maturity and a given vector of market Black-Scholes implied volatilities $ \{\hat{\sigma}_i\}_{i=1}^n$ for a given set of delta pillars $ \{\Delta_i\}_{i=1}^n$. Since we are calibrating the model to derivative prices we do not need to worry about estimating the market price of volatility risk as it is already embedded in the market smile. Furthermore, it can easily be verified that the value function (7.12) satisfies:

$\displaystyle \textrm{HestonVanilla}(\kappa,\theta,\sigma,\rho,\lambda,r_d,r_f,v_0,S_0,K,\tau,\phi)= \phantom{(U,U,U)(U,U,U)}$      
$\displaystyle =\textrm{HestonVanilla}\left(\kappa+\lambda,\frac{\kappa}{\kappa+\lambda}\theta,\sigma,\rho,0,r_d,r_f,v_0,S_0,K,\tau,\phi\right),$     (7.28)

which means that we can set $ \lambda=0$ by default and just determine the remaining five parameters.

After fitting the parameters we compute the option prices in Heston's model using (7.12) and retrieve the corresponding Black-Scholes model implied volatilities $ \{\sigma _i\}_{i=1}^n$ via a standard bisection method (a Newton-Raphson method could be used as well). The next step is to define an objective function, which we choose to be the Sum of Squared Errors (SSE):

$\displaystyle \textrm{SSE}(\kappa,\theta, \sigma, \rho, v_0) = \sum_{i=1}^n \{\hat{\sigma}_i-\sigma _i(\kappa,\theta, \sigma, \rho, v_0) \}^2.$     (7.29)

We compare volatilities (rather than prices), because they are all of comparable magnitude. In addition, one could introduce weights for all the summands to favor at-the-money (ATM) or out-of-the-money (OTM) fits. Finally we minimize over this objective function using a simplex search routine to find the optimal set of parameters.

Figure 7.3: Left panel: Effect of changing the volatility of variance (vol of vol) on the shape of the smile. For the red dashed ``smile'' with triangles $ \sigma =0.01$, and for the blue dotted smile with squares $ \sigma =0.6$. Right panel: Effect of changing the initial variance on the shape of the smile. For the red dashed smile with triangles $ v_0=0.008$ and for the blue dotted smile with squares $ v_0=0.012$.
\includegraphics[width=0.7\defpicwidth]{STFhes03a.ps} \includegraphics[width=0.7\defpicwidth]{STFhes03b.ps}


7.4.1 Qualitative Effects of Changing Parameters

Before calibrating the model to market data we will show how changing the input parameters affects the shape of the fitted smile curve. This analysis will help in reducing the dimensionality of the problem. In all plots of this subsection the solid black curve with circles is the smile obtained for $ v_0=0.01$, $ \sigma = 0.25$, $ \kappa = 1.5$, $ \theta = 0.015$, and $ \rho = 0.05$.

Figure 7.4: Left panel: Effect of changing the long-run variance on the shape of the smile. For the red dashed smile with triangles $ \theta =0.01$, and for the blue dotted smile with squares $ \theta =0.02$. Right panel: Effect of changing the mean reversion on the shape of the smile. For the red dashed smile with triangles $ \kappa =0.01$, and for the blue dotted smile with squares $ \kappa =3$.
\includegraphics[width=0.7\defpicwidth]{STFhes04a.ps} \includegraphics[width=0.7\defpicwidth]{STFhes04b.ps}

First, to take a look at the volatility of variance (vol of vol), see the left panel of Figure 7.3. Clearly, setting $ \sigma$ equal to zero produces a deterministic process for the variance and hence volatility which does not admit any smile. The resulting fit is a constant curve. On the other hand, increasing the volatility of variance increases the convexity of the fit. The initial variance has a different impact on the smile. Changing $ v_0$ allows adjustments in the height of the smile curve rather than the shape. This is illustrated in the right panel of Figure 7.3.

Effects of changing the long-run variance $ \theta$ are similar to those observed by changing the initial variance, see the left panel of Figure 7.4. This requires some attention in the calibration process. It seems promising to choose the initial variance a priori and only let the long-run variance vary. In particular, a different initial variance for different maturities would be inconsistent.

Changing the mean reversion $ \kappa$ affects the ATM part more than the extreme wings of the smile curve. The low deltas remain almost unchanged whereas increasing the mean reversion lifts the center. This is illustrated in the right panel of Figure 7.4. Moreover, the influence of mean reversion is often compensated by a stronger volatility of variance. This suggests fixing the mean reversion parameter and only calibrating the remaining parameters.

Figure 7.5: Left panel: Effect of changing the correlation on the shape of the smile. For the red dashed smile with triangles $ \rho =0$, for the blue dashed smile with squares $ \rho =-0.15$, and for the green dotted smile with rhombs $ \rho =0.15$. Right panel: In order for the model to yield a volatility skew, a typically observed volatility structure in equity markets, the correlation must be set to an unrealistically high value (with respect to the absolute value; here $ \rho =-0.5$).
\includegraphics[width=0.7\defpicwidth]{STFhes05a.ps} \includegraphics[width=0.7\defpicwidth]{STFhes05b.ps}

Finally, let us look at the influence of correlation. The uncorrelated case produces a fit that looks like a symmetric smile curve centered at-the-money. However, it is not exactly symmetric. Changing $ \rho$ changes the degree of symmetry. In particular, positive correlation makes calls more expensive, negative correlation makes puts more expensive. This is illustrated in Figure 7.5. Note that for the model to yield a volatility skew, a typically observed volatility structure in equity markets, the correlation must be set to an unrealistically high value.


7.4.2 Calibration Results

Figure 7.6: The market smile (solid black line with circles) on July 1, 2004 and the fit obtained with Heston's model (dotted red line with squares) for $ \tau $ = 1 week (top left), 1 month (top right), 2 months (bottom left), and 3 months (bottom right).
\includegraphics[width=0.7\defpicwidth]{STFhes06a.ps} \includegraphics[width=0.7\defpicwidth]{STFhes06b.ps} \includegraphics[width=0.7\defpicwidth]{STFhes06c.ps} \includegraphics[width=0.7\defpicwidth]{STFhes06d.ps}

We are now ready to calibrate Heston's model to market data. We take the EUR/USD volatility surface on July 1, 2004 and fit the parameters in Heston's model according to the calibration scheme discussed earlier. The results are shown in Figures 7.6-7.8. Note that the fit is very good for maturities between three and eighteen months. Unfortunately, Heston's model does not perform satisfactorily for short maturities and extremely long maturities. For the former we recommend using a jump-diffusion model (Cont and Tankov; 2003; Martinez and Senge; 2002), for the latter a suitable long term FX model (Andreasen; 1997).

Figure 7.7: The market smile (solid black line with circles) on July 1, 2004 and the fit obtained with Heston's model (dotted red line with squares) for $ \tau $ = 6 months (top left), 1 year (top right), 18 months (bottom left), and 2 years (bottom right).
\includegraphics[width=0.7\defpicwidth]{STFhes06e.ps} \includegraphics[width=0.7\defpicwidth]{STFhes06f.ps} \includegraphics[width=0.7\defpicwidth]{STFhes06g.ps} \includegraphics[width=0.7\defpicwidth]{STFhes06h.ps}

Figure 7.8: Term structure of the vol of vol (left panel) and correlation (right panel) in the Heston model calibrated to the EUR/USD surface as observed on July 1, 2004.
\includegraphics[width=0.7\defpicwidth]{STFhes06i.ps} \includegraphics[width=0.7\defpicwidth]{STFhes06j.ps}

Performing calibrations for different time slices of the volatility matrix produces different values of the parameters. This suggests a term structure of some parameters in Heston's model. Therefore, we need to generalize the Cox-Ingersoll-Ross process to the case of time-dependent parameters, i.e. we consider the process:

$\displaystyle dv_t=\kappa(t)\{\theta(t)-v_t\}\,dt+\sigma(t)\sqrt{v_t}\,dW_t$ (7.30)

for some nonnegative deterministic parameter functions $ \sigma(t)$, $ \kappa(t)$, and $ \theta(t)$. The formula for the mean turns out to be:

$\displaystyle \textrm{E}(v_t)= g(t)=v_0e^{-K(t)}+\int_0^t\kappa(s)\theta(s)e^{K(s)-K(t)}\,ds,$ (7.31)

with $ K(t)= \int_0^t\kappa(s)\,ds$. The result for the second moment is:
$\displaystyle \textrm{E}(v_t^2)=v_0^2e^{-2K(t)}+\int_0^t \{2\kappa(s)\theta(s)+\sigma^2(s)\} g(s)e^{2K(s)-2K(t)}\,ds,$     (7.32)

and hence for the variance (after some algebra):
$\displaystyle \textrm{Var}(v_t)=\int_0^t\sigma^2(s)g(s)e^{2K(s)-2K(t)}\,ds.$     (7.33)

The formula for the variance allows us to compute forward volatilities of variance explicitly. Assuming known values $ \sigma_{T_1}$ and $ \sigma_{T_2}$ for some times $ 0<T_1<T_2$, we want to determine the forward volatility of variance $ \sigma_{T_1,T_2}$ which matches the corresponding variances, i.e.

    $\displaystyle \int_0^{T_2}\sigma_{T_2}^2g(s)e^{2\kappa(s-T_2)}\,ds =$ (7.34)
    $\displaystyle =\int_0^{T_1}\sigma_{T_1}^2g(s)e^{2\kappa(s-T_2)}\,ds
+\int_{T_1}^{T_2}\sigma_{T_1,T_2}^2g(s)e^{2\kappa(s-T_2)}\,ds.$  

The resulting forward volatility of variance is thus:

$\displaystyle \sigma_{T_1,T_2}^2=\frac{\sigma_{T_2}^2H(T_2)-\sigma_{T_1}^2H(T_1)}{H(T_2)-H(T_1)},$ (7.35)

where

$\displaystyle H(t)=\int_0^tg(s)e^{2\kappa s}\,ds =\frac{\theta}{2\kappa}e^{2\ka...
...pa}(v_0-\theta)e^{\kappa t} +\frac{1}{\kappa}\left(\frac{\theta}{2}-v_0\right).$ (7.36)

Assuming known values $ \rho_{T_1}$ and $ \rho_{T_2}$ for some times $ 0<T_1<T_2$, we want to determine the forward correlation coefficient $ \rho_{T_1,T_2}$ to be active between times $ T_1$ and $ T_2$ such that the covariance between the Brownian motions of the variance process and the exchange rate process agrees with the given values $ \rho_{T_1}$ and $ \rho_{T_2}$. This problem has a simple answer, namely:

$\displaystyle \rho_{T_1,T_2}=\rho_{T_2}, \ \ \ T_1\leq t\leq T_2.$ (7.37)

This can be seen by writing the Heston model in the form:
$\displaystyle dS_t$ $\displaystyle =$ $\displaystyle S_t \left( \mu\,dt+\sqrt{v_t}\,dW_t^{(1)} \right)$ (7.38)
$\displaystyle dv_t$ $\displaystyle =$ $\displaystyle \kappa(\theta-v_t)\,dt +\rho\sigma\sqrt{v_t}\,dW_t^{(1)}+\sqrt{1-\rho^2}\sigma\sqrt{v_t}\,dW_t^{(2)}$ (7.39)

for a pair of independent Brownian motions $ W^{(1)}$ and $ W^{(2)}$. Observe that choosing the forward correlation coefficient as stated does not conflict with the computed forward volatility.

As we have seen, Heston's model can be successfully applied to modeling the volatility smile of vanilla currency options. There are essentially three parameters to fit, namely the long-run variance, which corresponds to the at-the-money level of the market smile, the vol of vol, which corresponds to the convexity of the smile (in the market often quoted as butterflies), and the correlation, which corresponds to the skew of the smile (in the market often quoted as risk reversals). It is this direct link of the model parameters to the market that makes the Heston model so attractive to front office users.

The key application of the model is to calibrate it to vanilla options and afterward employ it for pricing exotics, like one-touch options, in either a finite difference grid or a Monte Carlo simulation (Wystup; 2003; Hakala and Wystup; 2002). Surprisingly, the results often coincide with the traders' rule of thumb pricing method. This might also simply mean that a lot of traders are using the same model. After all, it is a matter of belief which model reflects the reality most suitably.