6.7.3 Implied Volatility Modeling Under No-Arbitrage
6.7.3.3 IV Smoothing Using Local Polynomials
As an alternative to smoothing in the call price domainBenko et al.(2007) suggest to directly smooth IV by means of constrained local quadratic polynomials. This implies minimization of the following (local) least squares criterion
˛0min;˛1;˛2
Xn iD1
˚ei˛0˛1.xi x/˛2.xi x/22
Kh.xxi/; (6.40)
wheree is observed IV. We denote byKh.xxi/ D h1Kxx
hi
and by K a kernel function – typically a symmetric density function with compact support, e.g.
K.u/D 34.1u2/1.juj 1/, the Epanechnikov kernel, where1.A/is the indicator function of some set A. Finally,h is the bandwidth which governs the trade-off between bias and variance, see H¨ardle (1990) for the details on nonparametric regression. SinceKhis nonnegative within the (localization) windowŒxh; xCh, points outside of this interval do not have any influence on the estimatorb.x/.
No-arbitrage conditions in terms of IV are obtained by computing (6.9) for an IV adjusted BSM formula, seeBrunner and Hafner(2003) among others. Expressed in forward moneynessxDX=F this yields for the convexity condition
@2CBSM
@x2 DerTp T '.d1/
( 1
x2bT C 2d1
xbp T
@b
@xC d1d2
b @b
@x 2
C@2b
@x2 )
(6.41) whered1andd2are defined as in (6.4) and (6.5).
The key property of local polynomial regression is that it yields simultaneously to the regression function its derivatives. More precisely, comparing (6.40) with the Taylor expansion ofb shows that
b.xi/D˛0; b0.xi/D˛1;b00.xi/D2˛2: (6.42)
Based on this factBenko et al.(2007) suggest to miminize (6.40) subject to erTp
T '.d1/ 1
x2˛0T C 2d1˛1
x˛0
pT C d1d2
˛0 .˛1/2C2˛2
0; (6.43)
with
d1D ˛20T =2log.x/
p
T ; d2Dd1˛0
pT :
This leads to a nonlinear optimization problem in˛0; ˛1; ˛2.
The case of the entire IV surface is more involved. Suppose the purpose is to estimateb.x; T /for a set of maturitiesfT1; : : : ; TLg. By (6.11), for a given value x, we need to ensureb2.x; Tl; / b2.x; Tl0/;for allTl < Tl0. Denote byKhx;hT
.xxi; TlTi/a bivariate kernel function given by the product of the two univariate kernel functionsKhx.xxi/andKhT.T Ti/. Extending (6.40) linearly into the time-to-maturiy dimension then leads to the following optimization problem:
min˛.l/
XL lD1
Xn iD1
Khx;hT.xxi; Tl Ti/
nei˛0.l/
˛1.l/.xix/˛2.l/.TiT /˛1;1.l/.xi x/2
˛1;2.l/.xix/.Ti T / o2
(6.44) subject to
pTl'.d1.l//
1 x2˛0.l/Tl
C2d1.l/˛1.l/
x˛0.l/p Tl
Cd1.l/d2.l/
a0.l/ ˛12.l/C2˛1;1.l/
0;
d1.l/D ˛20.l/Tl=2log.x/
˛0.l/p Tl
; d2.l/Dd1.l/a0.l/p
Tl; l D1; : : : ; L 2Tl˛0.l/˛2.l/C˛20.l/ > 0 lD1; : : : ; L
˛20.l/Tl < ˛02.l0/Tl0; Tl < Tl0:
The last two conditions ensure that total implied variance is (locally) nondecreasing, since @@T2 > 0can be rewritten as2T ˛0˛2C˛20 > 0for a givenT, while the last conditions guarantee that total variance is increasing across the surface. From a computational view, problem (6.44) calculates for a givenx the estimates for all givenTlin one step in order to warrant thatbis increasing inT.
The approach by Benko et al. (2007) yields an IV surface that respects the convexity conditions, but neglects the conditions on call spreads and the general price bounds. Therefore the surface may not be fully arbitrage-free. However, since
convexity violations and calendar arbitrage are by far the most virulent instances of arbitrage in observed IV data occurring the surfaces will be acceptable in most cases.
References
A¨ıt-Sahalia, Y., Bickel, P. J., & Stoker, T. M. (2001). Goodness-of-fit tests for regression using kernel methods. Journal of Econometrics, 105, 363–412.
A¨ıt-Sahalia, Y., & Lo, A. (1998). Nonparametric estimation of state-price densities implicit in financial asset prices. Journal of Finance, 53, 499–548.
An´e, T., & Geman, H. (1999). Stochastic volatility and transaction time: An activity-based volatility estimator. Journal of Risk, 2(1), 57–69.
Audrino, F., & Colangelo, D. (2009). Semi-parametric forecasts of the implied volatility surface using regression trees. Statistics and Computing, 20(4), 421–434.
Benko, M., Fengler, M. R., H¨ardle, W. & Kopa, M. (2007). On extracting information implied in options. Computational Statistics, 22(4), 543–553.
Bharadia, M. A., Christofides, N., & Salkin, G. R. (1996). Computing the Black-Scholes implied volatility – generalization of a simple formula. In P. P. Boyle, F. A. Longstaff, P. Ritchken, D. M.
Chance & R. R. Trippi (Eds.), Advances in futures and options research, (Vol. 8, pp. 15–29.).
London: JAI Press
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.
Brenner, M., & Subrahmanyam, M. (1988). A simple formula to compute the implied standard deviation. Financial Analysts Journal, 44(5), 80–83.
Britten-Jones, M., & Neuberger, A. J. (2000). Option prices, implied price processes, and stochastic volatility. Journal of Finance, 55(2), 839–866.
Brockhaus, O., Farkas, M., Ferraris, A., Long, D., & Overhaus, M. (2000). Equity derivatives and market risk models. London: Risk Books.
Brunner, B., & Hafner, R. (2003). Arbitrage-free estimation of the risk-neutral density from the implied volatility smile. Journal of Computational Finance, 7(1), 75–106.
Carr, P., & Wu, L. (2003). Finite moment log stable process and option pricing. Journal of Finance, 58(2), 753–777.
Castagna, A., & Mercurio, F. (2007). Building implied volatility surfaces from the available market quotes: A unified approach. In I. Nelken (Ed.), Volatility as an asset class (pp. 3–59). London:
Risk Books.
Cont, R., & da Fonseca, J. (2002). The dynamics of implied volatility surfaces. Quantitative Finance, 2(1), 45–60.
Corrado, C. J., & Miller, T. W. (1996). A note on a simple, accurate formula to compute implied standard deviations. Journal of Banking and Finance, 20, 595–603.
Dumas, B., Fleming, J., & Whaley, R. E. (1998). Implied volatility functions: Empirical tests, Journal of Finance, 53(6), 2059–2106.
Feinstein, S. (1988). A source of unbiased implied volatility. Technical Report 88–89, Federal Reserve Bank of Atlanta.
Fengler, M. R. (2005). Semiparametric modeling of implied volatility, Lecture Notes in Finance.
Berlin: Springer.
Fengler, M. R. (2009). Arbitrage-free smoothing of the implied volatility surface. Quantitative Finance, 9(4), 417–428.
Fengler, M. R., & Wang, Q. (2009). Least squares kernel smoothing of the implied volatility smile.
In W. H¨ardle, N. Hautsch & L. Overbeck (Eds.), Applied Quantitative Finance (2nd ed.). Berlin:
Springer.
Fengler, M. R., H¨ardle, W., & Villa, C. (2003). The dynamics of implied volatilities: A common principle components approach. Review of Derivatives Research, 6, 179–202.
Fengler, M. R., H¨ardle, W., & Mammen, E. (2007). A semiparametric factor model for implied volatility surface dynamics. Journal of Financial Econometrics, 5(2), 189–218.
Gatheral, J. (2004). A parsimonious arbitrage-free implied volatility parameterization with appli- cation to the valuation of volatility derivatives, Presentation at the ICBI Global Derivatives and Risk Management, Madrid, Espa˜na.
Gatheral, J. (2006). The volatility surface: A practitioner’s guide, New Jersey: Wiley.
Gouri´eroux, C., Monfort, A., & Tenreiro, C. (1994). Nonparametric diagnostics for structural models. Document de travail 9405. CREST, Paris.
Green, P. J., & Silverman, B. W. (1994). Nonparametric regression and generalized linear models.
In: Monographs on statistics and applied probability (Vol. 58). London: Chapman and Hall.
Hafner, R., & Wallmeier, M. (2001). The dynamics of DAX implied volatilities. International Quarterly Journal of Finance, 1(1), 1–27.
Hagan, P., Kumar, D., Lesniewski, A., & Woodward, D. (2002). Managing smile risk. Wilmott Magazine, 1, 84–108.
H¨ardle, W. (1990). Applied nonparametric regression. Cambridge, UK: Cambridge University Press.
H¨ardle, W., Okhrin, O., & Wang, W. (2010). Uniform confidence bands for pricing kernels. SFB 649 Discussion Paper 2010–03. Berlin: Humboldt-Universit¨at zu.
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.
Hodges, H. M. (1996). Arbitrage bounds on the implied volatility strike and term structures of European-style options. Journal of Derivatives, 3, 23–35.
Isengildina-Massa, O., Curtis, C., Bridges, W., & Nian, M. (2007). Accuracy of implied volatility approximations using “nearest-to-the-money” option premiums. Technical report, Southern Agricultural Economics Association.
Kahal´e, N. (2004). An arbitrage-free interpolation of volatilities. RISK, 17(5), 102–106.
Latan´e, H. A., & Rendelman, J. (1976). Standard deviations of stock price ratios implied in option prices. Journal of Finance, 31, 369–381.
Lee, R. W. (2004). The moment formula for implied volatility at extreme strikes. Mathematical Finance, 14(3), 469–480.
Li, S. (2005). A new formula for computing implied volatility. Applied Mathematics and Computation, 170(1), 611–625.
Malz, A. M. (1997). Estimating the probability distribution of the future exchange rate from option prices. Journal of Derivatives, 5(2), 18–36.
Manaster, S., & Koehler, G. (1982). The calculation of implied variances from the black-and- scholes model: A note. Journal of Finance, 37, 227–230.
Mercurio, F., & Pallavicini, A. (2006). Smiling at convexity: Bridging swaption skews and CMS adjustments. RISK, 19(8), 64–69.
Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Manage- ment Science, 4, 141–183.
Obł´oj, J. (2008). Fine-tune your smile: Correction to Hagan et al., Wilmott Magazine, 35, 102–104.
Randall, C., & Tavella, D. (2000). Pricing financial instruments: The finite difference method. New York: Wiley.
Rebonato, R. (2004). Volatility and Correlation, 2nd edn, Wiley.
Reiner, E. (2000). Calendar spreads, characteristic functions, and variance interpolation. Mimeo.
Reiner, E. (2004). The characteristic curve approach to arbitrage-free time interpolation of volatility. Presentation at the ICBI Global Derivatives and Risk Management, Madrid, Espa˜na.
Rogers, L. C. G., & Tehranchi, M. (2009). Can the implied volatility surface move by parallel shifts?. Finance and Stochastics, 14(2), 235–248.
Roper, M., & Rutkowski, M. (2009). On the relationship between the call price surface and the implied volatility surface close to expiry. International Journal of Theoretical and Applied Finance, 12(4), 427–441.
Rosenberg, J. (2000). Implied volatility functions: A reprise. Journal of Derivatives, 7, 51–64.
Rubinstein, M. (1994). Implied binomial trees. Journal of Finance, 49, 771–818.
Shimko, D. (1993). Bounds on probability. RISK, 6(4), 33–37.
Steele, J. M. (2000). Stochastic calculus and financial applications. Berlin: Springer.
Stineman, R. W. (1980). A consistently well-behaved method of interpolation. Creative Comput- ing, 6(7), 54–57.
Tehranchi, M. (2009). Asymptotics of implied volatility far from maturity. Journal of Applied Probability, 46(3), 629–650.
Tehranchi, M. (2010). Implied volatility: Long maturity behavior. In R. Cont (Ed.), Encyclopedia of quantitative finance. New York: Wiley.
Wang, Y., Yin., H., & Qi, L. (2004). No-arbitrage interpolation of the option price function and its reformulation. Journal of Optimization Theory and Applications, 120(3), 627–649.
West, G. (2005). Calibration of the SABR model in illiquid markets. Applied Mathematical Finance, 12(4), 371–385.
Wolberg, G., & Alfy, I. (2002). An energy-minimization framework for monotonic cubic spline interpolation. Journal of Computational and Applied Mathematics, 143, 145–188.
Interest Rate Derivatives Pricing with Volatility Smile
Haitao Li
Abstract The volatility “smile” or “skew” observed in the S&P 500 index options has been one of the main drivers for the development of new option pricing models since the seminal works of Black and Scholes (J Polit Econ 81:637–654, 1973) and Merton (Bell J Econ Manag Sci 4:141–183, 1973). The literature on interest rate derivatives, however, has mainly focused on at-the-money interest rate options. This paper advances the literature on interest rate derivatives in several aspects. First, we present systematic evidence on volatility smiles in interest rate caps over a wide range of moneyness and maturities. Second, we discuss the pricing and hedging of interest rate caps under dynamic term structure models (DTSMs). We show that even some of the most sophisticated DTSMs have serious difficulties in pricing and hedging caps and cap straddles, even though they capture bond yields well.
Furthermore, at-the-money straddle hedging errors are highly correlated with cap- implied volatilities and can explain a large fraction of hedging errors of all caps and straddles across moneyness and maturities. These findings strongly suggest the existence of systematic unspanned factors related to stochastic volatility in interest rate derivatives markets. Third, we develop multifactor Heath–Jarrow–Morton (HJM) models with stochastic volatility and jumps to capture the smile in interest rate caps. We show that although a three-factor stochastic volatility model can price at-the-money caps well, significant negative jumps in interest rates are needed to capture the smile. Finally, we present nonparametric evidence on the economic determinants of the volatility smile. We show that the forward densities depend significantly on the slope and volatility of LIBOR rates and that mortgage refinance activities have strong impacts on the shape of the volatility smile. These results provide nonparametric evidence of unspanned stochastic volatility and suggest that the unspanned factors could be partly driven by activities in the mortgage markets.
H. Li ()
Professor of Finance, Stephen M. Ross School of Business, University of Michigan, Ann Arbor, MI 48109
e-mail:htli@umich.edu
J.-C. Duan et al. (eds.), Handbook of Computational Finance, Springer Handbooks of Computational Statistics, DOI 10.1007/978-3-642-17254-0 7,
© Springer-Verlag Berlin Heidelberg 2012
143