Stochastic Volatility

The volatility figure used in a B-S computation is constant and derived mathematically, assuming that asset prices move according to a geometric Brownian motion. In reality, however, asset prices that are either very high or very low do not move in this way. Rather, as a price rises, its volatility increases, and as it falls, its variability decreases. As a result, the B-S model tends to undervalue out-of-the-money options and overvalue those that are deeply in the money. 

To correct this mispricing, stochastic volatility models, such as the one proposed in Hull and White (1987), have been developed. 

The market uses implied volatilities to gauge the volatility of individual assets relative to the market. The price volatility of an asset is not constant. It fluctuates with the overall volatility of the market, and for reasons specific to the asset itself When deriving implied volatility from exchange-traded options, market makers compute more than one value, because different options on the same asset will imply different volatilities depending on how close to at the money the option is. The price of an at-the-money option is more sensitive to volatility than that of a deeply in- or out-of-the-money one. 

Some of the newer models refer to parameters that are difficult to observe or measure direcdy. In practice, this limits their application much as B-S is limited. Usually the problem has to do with calibratii the model properly, which is crucial to implementing it. Galibration entails inputtii actual market data to create the parameters for calculating prices. A model for calculating the prices of options in the U.S. market, for example, would use U.S. dollar money market, futures, and swap rates to build the zero-coupon yield curve. Multifactor models in the mold of Heath-Jarrow-Morton employ the correlation coefficients between forward rates and the term structure to calculate the volatility inputs for their price calculations. 

Incorrect calibration produces errors in option valuation that may be discovered only after significant losses have been suffered. If the necessary data are not available to calibrate a sophisticated model, a simpler one may need to be used. This is not an issue for products priced in major currencies such as the dollar, sterling, or euro, but it can be a problem for other currencies. That might be why the B-S model is still widely used today, although models such as the Black-Derman-Toy and the one proposed in Brace, Gatarek, and Musiela (1994) are increasingly employed for more exotic option products. 

As noted earlier, although many practitioners use a historical volatility figure in applying the B-S model, the pertinent statistic is really the underlying asset s price volatility going forward. To estimate this future value. 

---

We overcome this inconsistency, by deriving a unified framework that directly leads to consistent cap/floor and swaption prices. Thus, in general we start from a HJM-like framework. This framework includes the traditional HIM model as well as an extended approach, where the forward rates are driven by multiple Random Fields. Furthermore, even in the case of a multifactor unspanned stochastic volatility (USV) model we are able to compute the bond option prices very accurately. First, we make an exponential affine guess for the solution of an expectation, which is comparable to the solu-... 

The implications of this new model class are in contrast to most term structure models discussed in the literature, which assume that the bond markets are complete and fixed income derivatives are redundant securities. Collin-Dufresne and Goldstein [ 18] and Heiddari and Wu [36] show in an empirical work, using data of swap rates and caps/floors that there is evidence for one additional state variable that drives the volatility of the forward rates. Following from that empirical findings, they conclude that the bond market do not span all risks driving the term structure. This framework is rather similar to the affine models of equity derivatives, where the volatility of the underlying asset price dynamics is driven by a subordinated stochastic volatility process (see e.g. Heston [38], Stein and Stein [71] and Schobel and Zhu [69]). 

Note that the unspanned stochastic volatility models are contradictory to the stochastic volatility models of Fong and Vasicek [31], Longstaff and Schwartz [56] and de Jong and Santa-Clara [24], where the bond market is complete and all fixed-income derivatives can be hedged by a portfolio solely... 

In other words, assuming a complete market stochastic volatility model implies that the short rate is modeled directly, while the traded asset (bond) has to be derived. Therefore, only the direct modeling of the bond price dynamics, together with stochastic volatility leads to an incomplete market model analog to the stochastic volatility models of equity markets". ... 

Han [34] uses a model that is a special case of our model framework, but postulating bond price dynamics that are uncorrelated with the subordinated stochastic volatility process. 

Again, we need to transform the process for the (log) bond price dynamics dXit, T) from the risk-neutral measure Q to the forward measure Tq. Thus, following section (5.1) we derive a measure transformation specially adapted to the additional innovation of the stochastic volatility. The bond price can be computed by integrating from t to 7b via... 

Finally, we analyze the impact of the stochastic volatility on the price of an option on a discount bond. Therefore, we compare the option price coming from a traditional HIM model given the average variance with... 

Note that the impact of this correlation effect is not in contradiction to the results found by Bakshi, Cao and Chen [5], Nandi [62] and Schobel and Zhu [69] for equity options. They found higher option prices given positive correlations and vice verca. On the other hand, we have a risk-neutral bond price process, where the source of uncertainty is negatively assigned (see e.g. (7.2)). Thus, assuming a USV bond model with negative correlated Brownian motions is the fixed income market analog of a stochastic volatility equity market model, with positive correlated sources of uncertainty. ... 

Like Collin-Dufresne and Goldstein [18], we started from a HJM-like term structure model, where the stochastic volatility is driven by an addi-... 

Casassus J, Collin-Dufresne P, Goldstein R (2005) Uspanned stochastic volatility and fixed income pricing. Journal of Banking and Finance 29 2723-2749. 

CoUin-Dufiesne P, Goldstein R (2002) Do Bonds Span the Fixed-Income Markets Theory and Evidence for Unspanned Stochastic Volatility. Journal of Finance 57 1685-1730. 

CoUin-Dufresne P, Goldstein R, Christopher J (2004) Can Interest Rate Volatility be Extracted from the Cross Section of Bond Yields An Investigation of Unspanned Stochastic Volatility, Working prater. 

Han B (2007) Stochastic Volatilities and Correlations of Bond Yields. Journal of Finance 62 1491-1524. 

Heston S (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies 6 327-343. 

Hull J, White A (1987) The Pricing of Options on Assets with Stochastic Volatilities. Journal of Finance 42 281-300. 

Li H, Zhao F (2006) Unspanned Stochastic Volatility Evidence from Hedging Interest Rate Derivatives. Journal of Finance 61 341-378. 

Nandi S (1998) How Important is the Correlation Between Returns and Volatility in a Stochastic Volatility Model Empirical Evidence from Pricing and Hedging S P 5(X) Index Option Market. Journal of Banking and Finance 22 589-610. 

Schobel R, Zhu JW (1999) Stochastic Volatility with an Omstein-Uhlenbeck Process an Extension. European Finance Review 3 23-46. 

Stein EM, Stein JC (1991) Stock Price Distributions with Stochastic Volatility An Analytic Approach. The Review of Financial Studies 4 727-752. 

Unspanned Stochastic Volatility and Random Field Models... 

Andersen, T., Lund, J., 1997. Estimating continuous-time stochastic volatility models of the shortterm interest rate. J. Econometrics 77, 343-377. 

All this suggests that asset-price behavior is more accurately described by nonstandard price processes, such as the jump diffusion model or a stochastic volatility, than by a model assuming constant volatility. For more-detailed discussion of the volatility smile and its implications, interested readers may consult the works listed in the References section. 

To model these phenomena, different modification of the Black Scholes model have been proposed. The most two common models are the stochastic volatility model (see Hull White process (Hull White 1987)), CIR process (see (Cox Ross 1976)) and Ornstein-Uhmenbeck process (see (Heston 1993, Stein Stein 1991)) and the jump diffusion model. 

Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with application to bond and currency options. The Review of Financial Studies 6. 

Stein, E. J. Stein (1991). Stock price distributions with stochastic volatility An analytic approach. Review of Financial Studies 4. 

Hull, ]., and A. White. 1988. An Analysis of the Bias Caused by a Stochastic Volatility in Option Pricing. Advances in Futures and Options Research 3, 29-61. Ito, K. 1951. On Stochastic DiflFerential Equations. American Mathematical Society A, 1-51. 

---