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Pricing and hedging long-term options. (English) Zbl 0989.91041
The authors study the difference between the short-term and long-term options. The following model that incorporates stochastic volatility, stochastic interest rates $R(t)$ and random jumps is considered. Let the spot interest rate follow a square-root diffusion $$dR(t)=[\theta_{R}-k_{R}R(t)] dt+\sigma_{R}\sqrt{R(t)} dw_{R}(t)$$ where $k_{R}, \theta_{R}/k_{R}$ and $\sigma_{R}$ are respectively the speed of adjustment, the long-run mean, the volatility coefficient of process $R(t)$; $w_{R}(t)$ is a standard Brownian motion. The underlying stock is assumed to pay a constant dividend yield, denoted by $\delta$, and its price $S(t)$ changes, under risk-neutral measure, according to the jump-diffusion stochastic differential equation $$dS(t)/S(t)=[R(t)-\delta-\lambda \mu_{J}]dt+ \sqrt{V(t)} dw_{S}(t)+J(t) dq(t),$$ where $V(t)$ also follow a square-root equation $$dV(t)=[\theta_{V}-k_{V}V(t)] dt+ \sigma_{V}\sqrt{V(t)} dw_{V}(t).$$ The intensity of the jump component is measured by $\lambda$, the size of percentage price jumps at time $t$ is represented by $J(t)$, which is lognormal, identically and independently distributed over time with unconditional mean $\mu_{J}$; $q(t)$ is a Poisson counter with $$P\{dq(t)=1\}=\lambda dt, \quad P\{dq(t)=0\}=1-\lambda dt.$$ Finally, let $\text{Cov}_{t}[dw_{S}(t),dw_{V}(t)]\equiv \rho dt$, $q(t)$ and $J(t)$ be uncorrelated with each other or with $w_{S}(t)$ and $w_{V}(t)$. The option pricing formula for a European put option is derived for the considered model. The authors study the option deltas and state-price densities under alternative models for short-term and long-term options. A description of the regular and LEAPS S&P 500 option data is provided and the estimation of structural parameters by using the method of simulated moments is presented. The difference between information in short-term and long-term options is studied. The hedging of the underlying stock portfolio and evaluation of the relative effectiveness of the underlying asset, short-term and medium-term options in hedging LEAPS are presented.

##### MSC:
 91B28 Finance etc. (MSC2000)
Full Text:
##### References:
 [1] Ait-Sahalia, Y., Lo, A., 1998. Nonparametric estimation of state-price densities implicit in financial prices. Journal of Finance 53, 499--548. [2] Amin, K.; Jarrow, R.: Pricing options on risky assets in a stochastic interest rate economy. Mathematical finance 2, 217-237 (1992) · Zbl 0900.90097 [3] Amin, K.; Ng, V.: Option valuation with systematic stochastic volatility. Journal of finance 48, 881-910 (1993) [4] Bailey, W.; Stulz, R.: The pricing of stock index options in a general equilibrium model. Journal of financial and quantitative analysis 24, 1-12 (1989) [5] Bakshi, G.; Cao, C.; Chen, Z.: Empirical performance of alternative option pricing models. Journal of finance 52, 2003-2049 (1997) [6] Bakshi, G.; Chen, Z.: An alternative valuation model for contingent claims. Journal of financial economics 44, No. 1, 123-165 (1997) [7] Bakshi, G.; Chen, Z.: Equilibrium valuation of foreign exchange claims. Journal of finance 52, No. 2, 799-826 (1997) [8] Bates, D.: The crash of 87: was it expected? the evidence from options markets. Journal of finance 46, 1009-1044 (1991) [9] Bates, D.: Jumps and stochastic volatility: exchange rate processes implicit in deutschemark options. Review of financial studies 9, No. 1, 69-108 (1996) [10] Bates, D., 1996b. Testing option pricing models. In: Maddala, G.S., Rao, C.R. (Eds.), Handbook of Statistics, Vol. 14: Statistical Methods in Finance, North-Holland, Amsterdam, pp. 567--611. [11] Bates, D., 1999. Post-87 crash fears in S & P 500 futures options. Journal of Econometrics, this issue. · Zbl 0942.62118 [12] Black, F.; Scholes, M.: The pricing of options and corporate liabilities. Journal of political economy 81, 637-659 (1973) · Zbl 1092.91524 [13] Bollerslev, T.; Mikkelsen, H.: Modeling and pricing long memory in stock market volatility. Journal of econometrics 73, 151-184 (1996) · Zbl 0960.62560 [14] Bollerslev, T., Mikkelsen, H., 1999. Long-term equity anticipation securities and stock market volatility dynamics. Journal of Econometrics, this issue. · Zbl 0933.62125 [15] Breeden, D.; Litzenberger, R.: Prices of state contingent claims implicit in option prices. Journal of business 51, 621-652 (1978) [16] Broadie, M., Detemple, J., Ghysels, E., Torres, O., 1999. American options with stochastic volatility and stochastic dividends: A nonparametric investigation. Journal of Econometrics, this issue. · Zbl 1122.91323 [17] Comte, F.; Renault, E.: Long-memory continuous time models. Journal of econometrics 73, 101-149 (1996) · Zbl 0856.62104 [18] Cox, J.; Ingersoll, J.; Ross, S.: A theory of the term structure of interest rates. Econometrica 53, 385-408 (1985) · Zbl 1274.91447 [19] Cox, J.; Ross, S.: The valuation of options for alternative stochastic processes. Journal of financial economics 3, 145-166 (1976) [20] Ding, Z.; Granger, C.; Engle, R.: A long memory property of stock market returns and a new model. Journal of empirical finance 1, 83-106 (1993) [21] Ding, Z.; Granger, C.: Modeling volatility persistence of speculative returns: a new approach. Journal of econometrics 73, 185-215 (1996) · Zbl 1075.91626 [22] Duffie, D.; Singleton, K.: Simulated moments estimation of Markov models of asset prices. Econometrica 61, 929-952 (1993) · Zbl 0783.62099 [23] Gouriéroux, C., Monfort, A., 1996. Simulation based econometric methods. Core Lecture Series. [24] Ghysels, E., Harvey, A., Renault, E., 1996. Stochastic volatility. In: Maddala G.S., Rao, C.R. (Eds.), Handbook of Statistics, Vol. 14: Statistical Methods in Finance, North-Holland, Amsterdam. [25] Heston, S.: A closed-form solution for options with stochastic volatility with applications to Bond and currency options. Review of financial studies 6, 327-343 (1993) [26] Hull, J.; White, A.: The pricing of options with stochastic volatilities. Journal of finance 42, 281-300 (1987) [27] Jackwerth, J., Rubinstein, M., 1997. Recovering stochastic processes from option prices. Working paper, University of California, Berkeley. [28] Madan, D., Chang, E., 1996. The variance gamma option pricing model. Working Paper, University of Maryland. [29] Melino, A.; Turnbull, S.: Pricing foreign currency options with stochastic volatility. Journal of econometrics 45, 239-265 (1990) · Zbl 1126.91374 [30] Melino, A.; Turnbull, S.: Misspecification and the pricing and hedging of long-term foreign currency options. Journal of international money and finance 14, 373-393 (1995) [31] Merton, R.: Theory of rational option pricing. Bell journal of economics 4, 141-183 (1973) · Zbl 1257.91043 [32] Merton, R.: Option pricing when the underlying stock returns are discontinuous. Journal of financial economics 4, 125-144 (1976) · Zbl 1131.91344 [33] Nandi, S., 1996. Pricing and hedging index options under stochastic volatility. Working Paper, Federal Reserve Bank of Atlanta. [34] Ross, S., 1996. Hedging long-run commitments: exercises in incomplete market pricing. Working Paper, Yale School of Management. [35] Rubinstein, M., 1985. Nonparametric tests of alternative option pricing models using all reported trades and quotes on the 30 most active CBOE options classes from August 23, 1976 through August 31, 1978, Journal of Finance 455--480. [36] Rubinstein, M.: Implied binomial trees. Journal of finance 49, 771-818 (1994) [37] Scott, L.: Option pricing when the variance changes randomly: theory, estimators, and applications. Journal of financial and quantitative analysis 22, 419-438 (1987) [38] Scott, L., 1997. Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: application of fourier inversion methods. Mathematical Finance 7, 413--426. · Zbl 1020.91030 [39] Stein, E.; Stein, J.: Stock price distributions with stochastic volatility. Review of financial studies 4, 727-752 (1991) [40] Wiggins, J.: Option values under stochastic volatilities. Journal of financial economics 19, 351-372 (1987)