Deterministic Volatility Models
Summary of Deterministic Volatility Literature
There are two avenues in the research on deterministic volatility literature. The deterministic volatility functions include those of Ncube (1996), Dumas, Fleming and Whaley (1998), Pena, Rubio, and Serna (1999), and Derman (1999). The other is a lattice based approach to modeling implied volatility. Lattice based approaches are implied binomial/volatility trees of Rubinstein (1994), Derman and Kani (1994), Derman, Kani, and Zou (1996), Derman and Kani (1997); trinomial trees of Derman, Kani, and Chriss (1996); and finite difference methods of Clewlow and Grimwood (1997).
Deterministic Volatility Functions
Deterministic volatility functions are derived from the option prices partial differential equation studied by DFW (1998) and PRS (1999) or a deterministic function of known variables by Ncube (1996). DFW investigate models of implied volatility deterministically in an attempt to capture the effects of time varying volatility in the S&P 500 index. Their findings conclude that the more flexible a deterministic model the more reliable the estimated structure, and that the longer the forecast period the less accurate the predictive capabilities the model becomes relative to the BLACK-SCHOLES implied volatility. They also find that the hedging performance of the BLACK-SCHOLES is superior relative to their deterministic volatility model. PRS investigate the determinants of the smile based on the methodology of DFW using the Spanish IBEX futures contracts on 35 European stocks. Their research chooses models based on the Adjusted R2 of each model investigated and test for linear sneers in the IVEX option market. They find that the Spanish IBEX is representative of a quadratic relationship or smile. Ncube (1996) proposes an alternative model that provides insights into modeling volatility and utilizes different econometric estimation techniques. Ncube estimates volatility using panel data or pooled cross sectional time series methodology of Fixed Effects Model (FEM) and Random Components Model (RCM). Ncube compares these models with ordinary least squares. He finds the FEM out performs the other models, in forecasting the volatility of the FTSE 100 Index options and has smaller pricing errors than the BSOPM.
The DVF proposed by Ncube, DFW, and PRS allow volatility to vary through time dependent upon the known factors of time to expiration and strike price.
Current Research on Deterministic Volatility Functions
The implied volatility as expressed by Ncube is ln(si,t)=f(ti, Ki) for i=1,…, n where n is the number of options at different strike prices at time t, ti is the time to expiration in years for option i, and Ki is the strike price for option i. The natural logarithm is used to ensure the volatility is always positive. Ncube’s formal model is
si,t=s0 exp{b1t i+b2t i2+b3K i+b4Di+Ui}, (1)
where D is a dummy variable for a call or put (D=1 or D=0 respectively), s0 is the intercept, bi are parameters to be estimated, and Ui is a random error with zero mean and constant variance. The dummy variable for a call or a put is included to allow for puts and calls to have different volatilities (Ncube, 1996, Abken and Nandi, 1996 and Derman, Kani, and Zou, 1996). This DVF is a parabolic relationship in time to expiration but not commodity level.
Ncube compares estimation methods of ordinary least squares with pooled cross sectional time series estimation techniques. These techniques are fixed effects model and random components model and both are utilized to increase efficiency of parameter estimates. The Ncube (1996) model delineates cross sections by contract quarter (Mar., …, Dec.) and strike price (S = KLow, …, KHigh) and is defined as
. (2)
These variables in (2) are as previously defined. This specification defines a cross section for each expiration month each year at each strike price. The number of cross sections increases as new options at different strikes and expirations are introduced.
Dumas, Fleming, and Whaley (1998) study deterministic volatility functions of the S&P 500 European index options. The data consists of options traded between June 1988 and December 1993 that are traded in the last 30 minutes with expirations between 6 and 100 days and an absolute moneyness of 10% on CME S&P 500 futures options. Dividend adjustments are made based on data from the S&P 500 bulletin. The risk free rate is interpolated for between day settlements based on Treasury Bills from the Wall Street Journal.
DFW posit that when the local volatility is a deterministic function of the underlying asset price and time and the Black-Scholes backward equation allows for deriving local implied volatilities. The Black-Scholes partial differential equation for the option C(F,t) is
, (3)
where F is the forward price of the S&P 500 index and s2 (F,t) is the local volatility for the forward price at time t. The local implied volatility solutions from equation (3) are found using the Crank-Nicholson finite difference method. These solutions are used to estimate the DFM in several forms as follows:
Model 0: s = max (0.01,a0),
Model 1: s = max (0.01,a0 + a1K + a2K2),
Model 2: s = max (0.01,a0 + a1K + a2K2 + a3t +a 5Kt), and
Model 3: s = max (0.01,a0 + a1K + a2K2 + a3t + a4t2 + a5Kt).
The variables K and t are strike price and time to expiration respectively, and ai are estimated parameters. A minimum value of 0.01 is imposed to prevent negative volatilities. Model 0 is the standard Black-Scholes constant volatility model, model 1 incorporates effects of the strike price, and models 2 and 3 incorporate time to maturity effects along with strike price. Models 1-3 are quadratic in the strike price and models 2-3 are a parabola in the strike price and time to expiration. The parabolic relationship attempts to capture the curvature that exists in the volatility smile. DFW analyze calls and puts separately using ordinary least squares estimation methods.
DFW use a Crank-Nicholson finite difference method to estimate local implied volatilities. As Chriss (1997) notes, this is different from estimating the implied volatility using Black-Scholes or Black-76. As defined in the introduction, the local volatility is the expected instantaneous volatility over the next period. The local volatility and the implied volatility may be equivalent under appropriate assumptions as shown by (Derman and Kani, 1997) and Feinstein (1989). DVF rely on these assumptions and empirical findings by DFW do not support assuming local volatility and implied volatility as being equal. This is where IVT are useful. They do not assume that the volatilities are equivalent, rather an output of IVT is a tree of local volatilities. Nandi (1996) states that DFW's findings do not support DVF. This is not the case because DFW are investigating the local volatility not the global measure of expected volatility or implied volatility.
Model 3 was found to provide the best fit with actual option prices as measured by the root mean squared error. The major finding of DFW is that Black-Scholes hedge ratios are more reliable than the DVF hedge ratios. They find the DVF performs in sample relatively well in weekly evaluations while out of sample performance is weak in the weekly and biweekly evaluations as measured by "hedging error" or the difference between difference in actual and model prices. Time to expiration variables are an important source of over-fitting of DVF. Analysis of different hedging horizon show that short time to expiration has the worse performance with improvements in intermediate and long term options. Chriss (1997) points out that these results are surprising given the better "fit" their model 3 provides.
Pena, Rubio, and Serna (1999) extend the work of DFW and study the determinants of the volatility smile. Their work uses a new data source, the Spanish IBEX Futures option that is a European option on the underlying futures. The implied volatilities for each option are calculated with Black-76 including adjustments for volatility trading days and calendar days for interest rates pursuant to the findings by French (1984). The period of investigation is January 1994 - April 1996. Options included in the data are only prices traded in the last 45 minutes with at least 5 days to expiration and options that were nearest to expiration in the March, June, September, and December option cycle and an absolute moneyness of 14%. The Treasury Bill repurchase rate is used a proxy for the risk free rate of return. PRS extend the models of DFW to capture quadratic smiles and estimate the following models
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Levels Models |
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with the variables U and D introduced to capture asymmetries in the smile. These variables and the variable X are defined as
X=K/F, 
, and 
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Models 1 and 1a are the constant volatility model; 2 and 2a are a linear sneer and reduce to the constant case if the commodity equals the strike. Model 3 and 3a is the quadratic symmetric smile, and models 4 - 6 and 4a - 6a are asymmetric linear models that are skewed left (4 and 4a) or right (5 and 5a) while models 6 and 6a are asymmetric quadratic models. Based on Adjusted R2, Models 3 and 3a are chosen for additional analysis that is to determine factors that effect the shape of the smile. This is performed using the DVF estimated parameters and regressing variables that have theoretical basis for influencing the volatility of an option. The variables are day of the week effects, interest rate effects, market momentum, historical volatility, transaction costs, time effects, and volume effects. Greater detail is provided on these variables in following sections describing the methodology for this research. The major findings of PRS are that transaction costs influence the smile and historical volatility and time to expiration have significant impacts on the volatility smile. The degree of market momentum and historical volatility are interrelated. As the market tends to increase and the volatility of the commodity increases, the smile becomes flatter implying that OTM calls and puts are valued more symmetrically.
Derman (1999) provide several heuristic rules for adjusting volatility. These rules are compared across past historical volatility regimes and specify guidelines for implementing the appropriate rule. These rules are consistent with Wiggin (1987) discussion of relative volatility. Derman (1999) linearly parameterizes the volatility skew or smile for implied volatility of S&P 500 options that is index dependent and time to expiration independent that is similar to the derivation presented earlier with commodity level and time dependence. Index level dependence is also termed S-dependence of implied volatility for a given index level, strike and time to expiration. The linear model for liquid options is
.
The variables are SK.T-t for implied volatility at strike, K, and time to expiration T-t, Satm.T-t is the implied volatility for the at-the-money option for the same time to maturity, K-St is the S-dependence relationship, and b is the slope of the skew. The slope, b, is an annual percentage of implied volatility per strike term. The S-dependence of implied volatility is important for obtaining appropriate hedge ratios and current option values. This parameterization allows for three invariance rules of implied volatility. An invariance rule is implemented based on a perceived volatility regime similar to the assumption of a terminal distribution in the IBT to be discussed in the next section.
The first rule is the Sticky-Strike rule based on (??) above and is defined as
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This rule implies that the implied volatility has no dependence on the index level with the term on b being only a moneyness reference. Implications of this rule are that each strike, K, has its own implied local volatility that is constant for the term of the option, and that changes in the option's value are only attributable to changes in moneyness.
The second rule is the Sticky-Delta Rule where delta is defined as the first derivative of the option pricing formula with respect to the commodity value[1]. It is derived from a sticky-moneyness relationship. The Sticky-Moneyness rule is
.
Moneyness is defined as K/S -1 and measures the percentage a call (put) option is in (out) or out (in) the money. If the index level is close to the strike, the Sticky-Delta rule results in the form
.
This relationship is that the implied volatility is constant at each moneyness level. An example of this rule is a 15% out of the money option's volatility on day t at a given index level will be the same as a 15% out of the money option's volatility on day t+1 or any other trading day.
The final rule, and the only rule that is a formal valuation rule, is the Sticky Implied Tree. This rule allows for non constant volatility that is seen in a skewed binomial tree. The Sticky Implied Tree rule is
.
This rule implies that at-the-money option has implied volatility that will decrease twice as rapidly with a decrease in the index level. The delta exposure of hedges under this rule is smaller because an increase in strike or index level have a decreasing volatility. Table ?? contains a summary of these regime rules.
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