Abstract
Recent studies have extended the Black–Scholes model to incorporate either stochastic interest rates or stochastic volatility. But, there is not yet any comprehensive empirical study demonstrating whether and by how much each generalized feature will improve option pricing and hedging performance. This paper fills this gap by first developing an implementable option model in closed-form that admits both stochastic volatility and stochastic interest rates and that is parsimonious in the number of parameters. The model includes many known ones as special cases. Based on the model, both delta-neutral and single-instrument minimum-variance hedging strategies are derived analytically. Using S&P 500 option prices, we then compare the pricing and hedging performance of this model with that of three existing ones that respectively allow for (i) constant volatility and constant interest rates (the Black–Scholes), (ii) constant volatility but stochastic interest rates, and (iii) stochastic volatility but constant interest rates. Overall, incorporating stochastic volatility and stochastic interest rates produces the best performance in pricing and hedging, with the remaining pricing and hedging errors no longer systematically related to contract features. The second performer in the horse-race is the stochastic volatility model, followed by the stochastic interest rates model and then by the Black–Scholes.
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Notes
- 1.
Amin and Ng (1993), Bailey and Stulz (1989), and Heston (1993) also incorporate both stochastic volatility and stochastic interest rates, but their option pricing formulas are not given in closed form, which makes applications difficult. Consequently, comparative statics and hedge ratios are difficult to obtain in their cases.
- 2.
There have been a few empirical studies that investigate the pricing, but not the hedging, performance of versions of the stochastic volatility model, relative to the Black–Scholes model. These include Bates (1996b, 2000), Dumas et al. (1998), Madan et al. (1998), Nandi (1996), and Rubinstein (1985). In Bates’ work, currency and equity index options data are respectively used to test a stochastic volatility model with Poisson jumps included. Nandi does investigate the pricing and hedging performance of Heston’s stochastic volatility model, but he focuses exclusively on a single-instrument minimum-variance hedge that involves only the S&P 500 futures. As will be clear shortly, we address in this paper both the pricing and the hedging effectiveness issues from different perspectives and for four distinct classes of option models.
- 3.
Here we follow a common practice to assume from the outset a structure for the underlying price and rate processes, rather than derive them from a full-blown general equilibrium. See Bates (1996a), Heston (1993), Melino and Turnbull (1990, 1995), and Scott (1987, 1997). The simple structure assumed in this section can, however, be derived from the general equilibrium model of Bakshi and Chen (1997).
- 4.
This assumption on the correlation between stock returns and interest rates is somewhat severe and likely counterfactual. To gauge the potential impact of this assumption on the resulting option model’s performance, we initially adopted the following stock price dynamics:
$$\frac{dS(t)} {S(t)} = \mu (S,t)dt +\! \sqrt{V (t)}\,d{\omega}_{S}(t) + {\sigma}_{S,R}\,\sqrt{R(t)}d{\omega}_{R}(t)\quad \!t \in [0,T],$$with the rest of the stochastic structure remaining the same as given above. Under this more realistic structure, the covariance between stock price changes and interest rate shocks is Cov t dS(t), dR(t) = σ S, R σ R R(t)S(t) dt, so bond market innovations can be transmitted to the stock market and vice versa. The obtained closed-form option pricing formula under this scenario would have one more parameter σ S, R than the one presented shortly, but when we implemented this slightly more general model, we found its pricing and hedging performance to be indistinguishable from that of the SVSI model studied in this paper. For this reason, we chose to present the more parsimonious SVSI model derived under the stock price process in (37.2). We could also make both the drift and the diffusion terms of V (t) a linear function of R(t) and ω R (t). In such cases, the stock returns, volatility and interest rates would all be correlated with each other (at least globally), and we could still derive the desired equity option valuation formula. But, that would again make the resulting formula more complex while not improving its performance.
- 5.
In making such a comparison, one should apply sufficient caution. In the BS model, the volatility delta is only a comparative static, not a hedge ratio, as volatility is assumed to be constant. In the context of the SVSI model, however, Δ V is time-varying hedge ratio as volatility is stochastic. This distinction also applies to the case of the interest-rate delta Δ R .
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Acknowledgements
We would like to thank Sanjiv Das, Ranjan D’Mello, Helyette Geman, Eric Ghysels, Frank Hatheway, Steward Hodges, Ravi Jagannathan, Andrew Karolyi, Bill Kracaw, C. F. Lee, Dilip Madan, Louis Scott, René Stulz, Stephen Taylor, Siegfried Trautmann, Alex Triantis, and Alan White for their helpful suggestions. Any remaining errors are our responsibility alone.
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Bakshi, G., Cao, C., Chen, Z. (2010). Option Pricing and Hedging Performance Under Stochastic Volatility and Stochastic Interest Rates. In: Lee, CF., Lee, A.C., Lee, J. (eds) Handbook of Quantitative Finance and Risk Management. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77117-5_37
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