Chapter 4 Control Variates for Stochastic Volatility Models
4.2 A Class of Stochastic Volatility Models
We consider a general class of two dimensional stochastic volatility model. Let (S)t>0 be an asset process and (V)t>0 be a volatility process. Their joint SDE is
dSt=rStdt+σf(Vt)SβtdWtS, (4.1)
dVt=α(µ−Vt)dt+ηVtγdWtV, (4.2)
dWtXdWtV =ρdt, (4.3)
wheref(v) =vξ and
γ >0 determines the process for Vt,
γ >0 whenξ 6=Z,
σ >0 is included for generality,
α, µ∈R are not required to be positive.
When ξ /∈Z,Vt is required to remain positive. When ξ ∈Z, Vt is permitted to be negative. We discuss a few characteristics of the processes.
4.2.1 Characteristics of the Processes
The parameter β determines the type of the process for St:
β= 0: St is an absolute diffusion process with stochastic volatility.
β∈(0,1): St is a CEV process with stochastic volatility.
β= 1: St is a GBM process with stochastic volatility.
The parameter ξ controls the process for Vt:
ξ= 0: Stis a GBM process with deterministic volatility.
ξ= 12: Vt is a variance process.
ξ= 1: Vt is a volatility process.
When ξ = 1, we do not require Vt to remain positive. When β < 12, one needs to impose an absorbing boundary at 0 so that SDE (4.1) has a unique solution (Andersen and Andreasen [3]). Our model nests many existing stochastic volatility models in the literature.
4.2.2 Nested Stochastic Volatility Models
Two nested and widely used stochastic volatility models are Heston [44] and SABR (Hagan et al. [41]). The Heston model (β = 1, ξ =γ = 12) has been widely used because it allows the option price to have an implied volatility smile. In addition, there is an explicit formula for European option value. The SABR model is another widely used stochastic volatility model. Its popularity is due both to the existence of a closed-form approximation of the European option implied volatility, and its consistency with the dynamics of market implied volatility smile. The latter char- acteristic renders an advantage over local volatility models in the context of trading
and hedging. To nest the SABR model, we require some manipulations of the orig- inal process.
Let Ft be the forward price on asset St and let σt be the volatility. The SABR model has the dynamics
dFt=σtFtβdWtS, (4.4)
dσt=ησtdWtV, (4.5)
E dWtSdWtV=ρdt. (4.6)
SinceFt=er(T−t)St, we can express the SABR dynamics in terms of St andσt.
dSt=rStdt+σte(β−1)r(T−t)StβdWtS. (4.7)
FixingT, we setVt=σte(β−1)r(T−t) so that, fort6T
dVt=r(1−β)Vtdt+ηVtdWtV. (4.8) We arrive at alternative dynamics for the SABR model in terms ofSt and Vt
dSt=rStdt+VtStβdWtS, (4.9)
dVt=r(1−β)Vtdt+ηVtdWtV. (4.10) This has the form of our stochastic volatility model which ξ = 1, γ = 1, µ = 0,
α=r(β−1) and V0=σ0e(β−1)rT.
Another useful model is the GARCH diffusion model (Nelson [63]). It has the parameters β = 1, ξ = 12 and γ = 1. We list other nested stochastic volatility models in the literature in Table 4.1.
β = 0 β∈(0,1) β = 1
Model ξ γ Model ξ γ Model ξ γ
1 1 SABR 1 1 H&W 1 1
Absolute 1 12 SABR- 1 12 J&S 1 12 diffusion 1 0 like 1 0 S&Z, S&S 1 0 models 12 1 LKD 1 2 1 GARCH 12 1 1 2 1 2 1 2 1 2 Heston 1 2 1 2
n/a 12 0 n/a 12 0 n/a 12 0
where (β, ξ, γ) =
(β,1, γ): Johnson and Shanno [50] (J&S).
(β,1,1): Hagan et al. [41] (SABR).
β,12, γ: Lord, Koekkoek and Dijk [57] (LKD).
(1,1,1): Wiggins [84]; Hull and White [45] (H&W) (µ= 0,ρ = 0).
(1,1,0): Stein and Stein [78] (ρ= 0) (S&S); Sch¨obel and Zhu [71] (ρ6= 0) (S&Z). 1,12,1: Nelson [63] (GARCH); Barone-Adesiet al. [9]; (and H&W, asµ= 0). 1,12,12: Heston [44]; Ball and Roma [7].
1,12, γ: Andersen and Piterbarg [6]; Lewis [56]; Ait-Sahalia and Kimmel [1]. (1,2,0): Sbai and Jourdain [70] (special case).
There are also models, however, not nested in our class. There is an exponential volatility model where (β, ξ, γ) = (1,1,0) with f(v) = exp vξby Scott [72], Ches- ney and Scott [22] and Melino and Turnbull [59], jump diffusion models such as Bates [10], general L´evy models (VG, NIG, CGMY,et cetera) and stochastic inter- est rate and higher factor models (Haastrecht et al. [82], et cetera). There are also papers which propose more general specifications, for example, Sbai and Jourdain [70] with (β, f(v), g(v)) and Bourgade and Croissant [14] with (h(S), ξ, γ).
In general, there is no explicit solution to option values in the model (4.1)-(4.3). To value derivatives, we always resort to numerical methods. PDE and lattice methods are difficult to get prices out quickly when there are more than two factors. In Monte Carlo, there are also issues with bias and convergence. The square-root process, in particular, is likely to be problematical. The CIR process, for example, admits no strong solution. If it is used to model the asset variance as in the Heston model, zero is accessible when the model is calibrated to equity options. Although exact simulation is possible (Scott [73]), it is expensive to use.
Simulation of the Heston model has been studied extensively both from the the- oretical and practical point of view (Lord et al. [57], et cetera). Despite that, effective valuation of derivatives whose payoffs depend on the whole sample path remains a challenging research question. Exact simulation methods (Broadie and Kaya [17], Glasserman and Kim [37], et cetera) are expensive. These methods are
plausible for long-step Monte Carlo but infeasible to do short stepping. Approxi- mate solutions are poor particularly when zero is accessible for Vt (Webber [83]). Therefore, fast simulation methods are necessary for the Heston model.
We propose new control variates, which we refer to as correlation CVs, in our model. The idea is to circumvent the explicit solutions for the CV candidates as they are usually unavailable other than under the simple geometric Brownian motion. To fix the idea, we first give an overview of our correlation CV method.