Approximation Formula for Option Prices under Rough Heston Model and Short-Time Implied Volatility Behavior

: Rough Heston model possesses some stylized facts that can be used to describe the stock market, i.e., markets are highly endogenous, no statistical arbitrage mechanism, liquidity asymmetry for buy and sell order, and the presence of metaorders. This paper presents an efﬁcient alternative to compute option prices under the rough Heston model. Through the decomposition formula of the option price under the rough Heston model, we manage to obtain an approximation formula for option prices that is simpler to compute and requires less computational effort than the Fourier inversion method. In addition, we establish ﬁnite error bounds of approximation formula of option prices under the rough Heston model for 0.1 ≤ H < 0.5 under a simple assumption. Then, the second part of the work focuses on the short-time implied volatility behavior where we use a second-order approximation on the implied volatility to match the terms of Taylor expansion of call option prices. One of the key results that we manage to obtain is that the second-order approximation for implied volatility (derived by matching coefﬁcients of the Taylor expansion) possesses explosive behavior for the short-time term structure of at-the-money implied volatility skew, which is also present in the short-time option prices under rough Heston dynamics. Numerical experiments were conducted to verify the effectiveness of the approximation formula of option prices and the formulas for the short-time term structure of at-the-money implied volatility skew.


Introduction
Stochastic volatility models and jump diffusion models such as classical Heston model [1] and Merton jump diffusion model [2] have played an important role in the option pricing theory. The models have aimed at replicating the stochastic volatility effect (along with the mean-reversion) and the jump-effect as displayed in the financial market. For decades, the models have been proven to be useful in pricing options, but they are deemed as inadequate for modeling the short-time behavior of the implied volatility of the options using the real data.
It was discovered by Alòs et al. [3] that the introduction of fractional components with Hurst parameter 0 < H < 0.5 is capable of generating the term structure of at-the-money skew of order T δ for every δ > −0.5 and small time to maturity T. In addition, Fukasawa [4] managed to show that (through an example of martingale expansion's application) the stochastic volatility model with volatility term driven by fractional Brownian motion with Hurst parameter H can generate at-the-money volatility skew of order T H−0.5 for the small time to maturity T. This motivated the authors of [5] to further explore through real empirical data to verify that the use of fractional Brownian motion in a stochastic volatility model is useful and adequate in replicating the roughness behavior of model, approximation formula for option prices (transformed back to implied volatility) and short-time term structure of the at-the-money skew equation have roughly the same order of explosive behavior for the term structure of at-the-money implied volatility skew. Lastly, we give a conclusion to our work in Section 5.

Prerequisites
Consider the price model process with rough Heston dynamics: where S is the stock price, ρ is the correlation between stock return and the volatility movement, 0.5 < α = H + 0.5 < 1 , θ > 0 is the mean reversion level, λ > 0 is the rate at which the process σ 2 s reverts to the mean reversion level θ, ν > 0 is the magnitude of random movement, W * and W ⊥ are independent Brownian motion defined in a filtered probability space (Ω, F , Q) with F := F W * ∨ F W ⊥ , and σ is the volatility of the return that is positive and square-integrable function adapted to the filtration.
Let X u = log(S u ) and we describe Equation (2) in the forward variance form: where ξ t (u) := E[σ 2 u |F t ] is the forward variance curve. Following the work of Alos et al. [20], we denote the future expected variance w t (T) as w t (T) := E T t σ 2 s ds|F t (5) and the martingale as The connection between w t (T) and M t can be easily established as then it follows that dw t (T) = dM t − σ 2 t dt The Delta-Gamma-Vega relationship can be established as where B t := B(X t , w t (T)) is some function that solves the Black-Scholes equation with X being the log-spot price with the dynamics specified in Equation (3) and w depends on σ. Otherwise, B(X t , w t ) can be formulated as B(X t , w t (T)) = e X t · N(d + (w t (T))) − K · N(d − (w t (T))) (10) where N(·) is the cumulative distribution function of standard normal distribution and with the assumption of risk-free interest rate r = 0 throughout the paper.
Theorem 1 (Decomposition formula). Assume B t := B(X, w) as some function that solves the Black-Scholes equation where x is the log-spot price with the dynamics specified in Equation (3) and w depends on the σ 2 from Equation (4). Then, for all t ∈ [0, T], Proof. Applying Itô's formula to B t , we have We use Equations (3), (8) and (9) to simplify Integrating up to time T and taking conditional expectation of Equation (14) on F t , the result follows.
Theorem 1 is the decomposition formula from [20] with slight changes of the notation as corresponds to the original work in [18]. Equation (12) is exact rather than an approximation, i.e., it describes the interaction between the movement of log-stock movement and the future expected variance from time t to T. It is known that Equation (12) under conditional expectation can be difficult to compute, therefore approximation of the equation is needed. In this study, we provide an approximation formula for Equation (12) using the rough Heston model dynamics under a simple assumption. It is found that the errors are bounded and decreasing as the time to maturity gets smaller.
We consider the following assumption throughout this study: The relevance of Assumption 1 for the practical application is discussed at the end of Section 4. We prove some bounds under Assumption 1. (2) and (3) and Assumption 1; then, for all T − s ∈ [0, 1],

Approximation for Option Pricing Formula
In this section, we prove that the errors for approximation formula of rough Heston model in the range of 0.1 ≤ H < 0.5 is bounded.
Remark 1. Theorem 2 shows that the error bound does not diverge for any α ∈ [0.6, 1] (provided that λ > 0 and θ > 0). Coincidentally, for t ∈ [0, 1], we can obtain the same order convergence when α = 1 in the case of classical Heston model as shown in [18,19]. Furthermore, if Assumption 1 is satisfied, then the errors of the approximation formula can be effectively reduced when ν and T − t are small.

Second-Order Approximation and Small-Time Behavior of Implied Volatility
The outline of this section is roughly the same as that of Alòs, Elisa and De Santiago, Rafael and Vives, Josep [19], but we focus on the rough Heston model rather than classical Heston model. We expand the function f with respect to the two scales in order to obtain the second-order approximation for the implied volatility. Consider the asymptotic sequence {δ i } ∞ i=0 and the function f can be expanded as Combining the expansions f i and f , we obtain the following We thus let = ρν/α and δ = ν 2 /α with I(T, K) be the implied volatility function expanded to these two scales: as the second-order approximation to the implied volatility. Then, let v t = 1 T w t (T) as the future average volatility and V(X T , v t ) = B(X T , w t (T)) such thatV(X T , v t (T)) is the approximation formula to the option price obtained in Theorem 2. Ultimately, we havê is the Vega of the Black-Scholes formula V(X 0 , v 0 ). Similarly, we can expand the V(X 0 , v 0 ) around v 0 using the Taylor Expansion and Equation (37) as such ; then, comparing the first-order approximation of Equation (41) to Equation (40), we can obtain the followinĝ Similarly, as noted by Alòs, Elisa and De Santiago, Rafael and Vives, Josep [19], the smile effect of the implied volatility can be shown by the quadratic term d 2 The error between the option pricing formula using the approximation of the implied volatility in Equation (38) and the option price under rough Heston model is as follows: and by Theorem 2, we know that the second term of (44) has a small time (when 0 ≤ T ≤ 1) error of O(C(ν, ρ, α, λ, θ)) which is a decreasing term proportional to ρ and ν. The total errors of option prices on the approximation to the implied volatility are of the order of O Noticeably, the errors converges to zero as ρ → 0 and ν → 0. We prove some results that would assist in obtaining the short-time behavior on the approximation of implied volatility. (4); then,

Proposition 2. Assume volatility dynamics of Equation
Proof. From the definition of v 0 and Proposition 1, we can obtain the following Let lim T→0 v 0 ; we can subsequently notice that it is sufficient to prove that Then, by the above result and Fubini's theorem, we can obtain the following which is the desired result. (4); for k > −α − 1 and k = −1,

Proposition 3. Assume volatility dynamics of Equation
as T → 0.
Proof. From Proposition 1, we can obtain the following From Proposition 2 and by the Fubini's theorem again, we can obtain the following: Substitute y = s/T and notice that it can be solved using the Beta function, such that We then have Then, for k > −α − 1 and k = −1, we can deduce the following asymptotic expansion as T → 0.

Remark 2.
Proposition 2 is equivalent to the case of the classical Heston model. Furthermore, by utilizing Proposition 3, we ultimately yield the same result from [19] in the case of α = 1 for the rest of the result in this paper.

Limiting Behavior ofÎ(T, K) When T → 0
We utilize some of the derivations from [19]. The limiting behavior ofÎ(T, K) as T → 0 can be computed in term of the asymptotic formula as follows Lemma 3. Assume model (3) and (4). Then, we can obtain the asymptotic formula ofÎ 1 (T, K) aŝ as T → 0.
Proof. From Equation (A7) and Proposition 3, we can obtain the limiting behavior of U 0 when T → 0 as Then, the limiting behavior ofÎ 1 (T, K) can be obtained as following The result follows from the limiting behavior of Equation (57) and Proposition 2.
We focus on the asymptotic formula ofÎ 2 (T, K) in the next lemma. (3) and (4). Then, we can obtain the asymptotic formula ofÎ 2 (T, K) aŝ
Proof. From Equation (A7) and Proposition 3, the limiting behavior of R 0 when T → 0 can be obtained as Then, the limiting behavior ofÎ 2 (T, K) can be obtained as follows Similarly, the result follows from the limiting behavior of Equation (60) and Proposition 2.

Remark 3.
From Lemmas 3 and 4, when the option is close to maturity (T → 0), the second-order approximation to the implied volatility can be written as the asymptotic formula. Furthermore, the expression (62) becomes exact and equivalent to the result found by Alòs, Elisa and De Santiago, Rafael and Vives, Josep [19] when α = 1.

Term Structure of At-the-Money Implied Volatility Skew
This subsection is crucial in showing that the approximation formula for the implied volatility under rough Heston model is capable of replicating the explosive behavior of the implied volatility under the rough volatility model as stated in [4,5]. When the option is at-the-money, we have ln K = X 0 and this leads to d + (v 2 0 T) = v 0 2 √ T. Consequently, the term structure of the at-the-money implied volatility skew can be computed as Lemma 5. When the option is at-the-money, the term structure of implied volatility skew has the asymptotic formula of as T → 0.
Proof. From the result of Lemmas 3 and 4, we have the following as T → 0: Let ln K = X 0 and the result follows.

Remark 4.
From Lemma 5, we can also observe the following: for the uncorrelated case (ρ = 0) and correlated case (ρ = 0) in model (3) and (4), the slope of implied volatility (with respect to ln K and T → 0) is positive when lnK > X 0 and negative when ln K < X 0 . The result is slightly different than the implied volatility behavior when α = 1, where ρ = 0 yields different positive and negative region (slope is negative when ln K < X 0 − 3ρσ 2 0 /ν and positive when ln K > X 0 − 3ρσ 2 0 /ν). The empirical result in Figure 8 (page 11) of [8] shows that Lemma 5 is valid as T → 0. On a side note, Lemma 5 confirms that the term structure of at-the-money approximation of implied volatility under rough Heston model has explosive behavior of order T α−1 , i.e., it possesses an important feature which the classical Heston model does not possess. The presence of explosive behavior in the term structure of at-the-money implied volatility skew fits the empirical observation from [4,5].

Limiting Behavior of Rate of Change ofÎ(T, K) with respect to T When the Option Is At-The-Money
In this section, we discuss the behavior of the rate of change of implied volatilityÎ(T, K) with respect to T when T → 0 and the option is at-the-money.

Lemma 6.
When the option is at-the-money, the skew of the approximation of implied volatility at T → 0 has the asymptotic formula of Proof. Approximation to implied volatility when the option is at-the-money can be formulated aŝ From Lemmas 3 and 4 and Remark 3, it is easy to see that, when the option is at-the-money (ln K = X 0 ),Î(0, K) = σ 0 . To find the rate of change ofÎ(T, K) with respect to T, we use the definition of derivative as follows We solve the above expression separately. The first term of the above expression can be computed using L'Hopital's rule repetitively as follows By Proposition 3, we have the following as as as T → 0. From the previous result in Lemma 3, we know that U 0 ∼ νρσ 2 0 Γ(α+2) T α+1 as T → 0; then, along with Proposition 2, the second term of Equation (63) can be computed in terms of asymptotic formula as T → 0. Similarly, by the previous result in Lemma 3 such that R 0 ∼ ν 2 σ 2 0 2Γ(α+1) 2 (2α+1) T 2α+1 as T → 0 and Proposition 2, the third term of Equation (63) can be computed as Then, the result follows with the sum of three computed terms (74)-(76).
Remark 5. Lemma 6 satisfies the result of α = 1 from the work of Alòs, Elisa and De Santiago, Rafael and Vives, Josep [19]. Furthermore, when the option is at-the-money, we can obtain a rough approximation of implied volatility in terms of T as contrary to Remark 3 which provides constant term when the option is at-the-money. An integration of the result in Equation (66) with respect to T near T = 0 gives us the following approximation:

Numerical Experiments
Through numerical experiments, we can show that the approximation formula for the rough Heston model possesses the explosive behavior of at-the-money implied volatility skew, which is desirable in a stochastic volatility model to match the empirical implied volatility data in the market. Although Theorem 2 shows that, under Assumption 1, the approximation formula for the rough Heston model is bounded from the rough Heston model using the same set of parameters, we decided to conduct the numerical experiments using calibration approach, i.e., calibrate a new set of parameters used on approximation formula based on the artificial data given by rough Heston model. We wish to note that the components or terms of v 0 , U 0 and R 0 can be obtained using Proposition 1. The following parameters for the rough Heston model are set: Then, we calibrate the approximation Formula (40) to the rough Heston model through minimizing the squared errors of implied volatility and subsequently we obtain the following parameters: (79) Figure 1 shows a calibrated option prices from T = 0.01 to T = 0.1. After the calibration of the parameters, the approximation formula for option prices under the rough Heston model are capable of displaying a great fit to the option prices computed using Fourier inversion method with the fractional Adams scheme [21], as displayed in [11] .
The implied volatility plots are shown in Figure 2. In the plot, we can observe that the implied volatilities computed using the approximation formula (40) match the ones produced by the Fourier inversion method with the fractional Adams scheme [11,21] under the rough Heston model very well, except for at T = 0.01 where the implied volatility is understated when the option is deep out-of-money and in-the-money.  One result that sets apart from many other stochastic volatility and jump diffusion models is the short-time behavior of term-structure at-the-money implied volatility skew, i.e., derivative of implied volatility with respect to log-strike for the at-the-money call option. In Figure 3, we display the term structure of the at-the-money implied volatility skew using three different methods-Fourier inversion method with fractional Adams scheme [11,21], approximation formula for option prices (40) (transformed back to implied volatility) and Equation (63). The result is pretty much consistent with the ones demonstrated by the approximation formula for rough Heston model (40) and Equation (63). Explosive behavior on the short-time term-structure of at-the-money implied volatility skew is a remarkable feature on a stochastic volatility model and the behavior is commonly observed in the empirical data in the financial market. Figure 3. Term structure of the at-the-money implied volatility skew. The solid red line is produced using Fourier inversion method with fractional Adams scheme [11,21] under the rough Heston model with parameters stated in Equation (78), the blue circle marker is produced by the approximation formula for option prices under the rough Heston model (40) with parameters (79) and the black cross marker is produced by the Equation (63).
It is important to realize that our work is relying on Assumption 1. In terms of practical application, Assumption 1 will be fulfilled whenever the current variance process σ 2 t is equivalent or below the mean reversion level θ. In other words, the usual flat forward variance curve (E 0 (σ 2 t ) = E 0 (σ 2 0 ) for all t) frequently used by many practitioners will satisfy Assumption 1 too. Other than that, we would say that Assumption 1 will be satisfied roughly 50% of the time under the normal market condition.

Conclusions
The main contribution of this paper lies within the bounded gaps of option prices on approximation formula for the option prices of the rough Heston model under the simple Assumption 1. Due to the nature of fractional terms in the volatility component of the rough Heston model, the errors bound incurred by the decomposition formula are significantly higher than the classical version of the approximation formula for the decomposition formula of classical Heston model [18,19]. In other words, this disadvantage decreases the usability of the approximation formula in practice due to the errors incurred. Nevertheless, in the numerical experiments, we showed that it is possible to match the option prices and implied volatility quite well by re-calibrating the parameters of the approximation formula for option prices.
In addition, we propose a second-order approximation to the implied volatility of the rough Heston model. Through the approximation of implied volatility of rough Heston model, we manage to prove some important results which includes the explosive behavior of short-time term structure of at-the-money implied volatility skew using the approximation formula of option prices for the rough Heston model, which is desirable in the empirical financial market. Numerical experiments were conducted and we subsequently discovered that the approximation formula for option prices using the rough Heston model does have explosive behavior in its term-structure of at-the-money implied volatility skew and roughly at the same rate as the rough Heston model after recalibrating the parameters. Note that H(X T , w t (T))U T = 0 and, similarly, Theorem 1 enables us to write On the other hand, U t can be derived as follows and dU t as The integral T t (T − s) α (s − t) α−1 ds can be solved using the substitution of y = (s − t)/(T − t) and it corresponds to a Beta function such that and finally Meanwhile, R t can be derived as follows and dR t as Based on the previous derivations dM t , dU t , dR t , we can obtain the following Using the previous equations, we can now continue from Equation (A4) We let q s = w s (T) and notice that from Lemma 2 Consequently, we can obtain Then, we continue Suppose now we let J(α, λ) := Γ(α+1)+λ Γ(α+1) and Y := Y(α, λ, θ) = Γ(α+2)+λ(α+1) λθ . For T − t ∈ [0, 1]; using Equations (15) and (16) in Lemma 2, we can further obtain which is valid for α ∈ [0. 6,1] or H ∈ [0.1, 0.5]. Now, for T − t ∈ (1, ∞),  (17) and (18), we can deduce that A i ≤ C(ν, ρ, α, λ, θ) + C ν 2 ρ 2 Γ(2α + 1) J(α, λ)Y 1 2 (T − s) 2α+1 where the result follows from here.