ADOL - Markovian approximation of rough lognormal model
AADOL - Markovian approximationof rough lognormal model
Peter Carr and Andrey Itkin
Tandon School of Engineering, New York University, 12 Metro Tech Center, 26th floor, Brooklyn NY 11201, USA
April 22, 2019 I n this paper we apply Markovian approximation of the fractional Brownian motion(BM), known as the Dobric-Ojeda (DO) process, to the fractional stochastic volatilitymodel where the instantaneous variance is modelled by a lognormal process with driftand fractional diffusion. Since the DO process is a semi-martingale, it can be representedas an Itô’s diffusion. It turns out that in this framework the process for the spot price S t is a geometric BM with stochastic instantaneous volatility σ t , the process for σ t is also ageometric BM with stochastic speed of mean reversion and time-dependent colatility ofvolatility, and the supplementary process V t is the Ornstein-Uhlenbeck process with time-dependent coefficients, and is also a function of the Hurst exponent. We also introducean adjusted DO process which provides a uniformly good approximation of the fractionalBM for all Hurst exponents H ∈ [0 , but requires a complex measure. Finally, the cha-racteristic function (CF) of log S t in our model can be found in closed form by usingasymptotic expansion. Therefore, pricing options and variance swaps (by using a forwardCF) can be done via FFT, which is much easier than in rough volatility models. Introduction
It was discovered in [Gatheral et al., 2014] that for a wide range of assets, historical volatility time-seriesexhibit a behavior which is much rougher than that of a Brownian motion. It was also shown thatdynamics of log-volatility is well modeled by a fractional Brownian motion with Hurst parameter oforder 0.1. Note that in the literature there exist various opinions whether the Hurst index should beless than 1/2 (short memory) or above 1/2 (long memory) depending on the particular asset class. Asmentioned in [Funahashi and Kijima, 2017], it is well known that i) the decrease in the market volatilitysmile amplitude is much slower than that predicted by the standard stochastic volatility models, and(ii) the term structure of the at-the-money volatility skew is well approximated by a power-law function1 a r X i v : . [ q -f i n . M F ] A p r DOL - Markovian approximation of rough lognormal model with the exponent close to zero. These stylized facts cannot be captured by standard models, and while(i) has been explained by using a fractional volatility model with Hurst index
H > /
2, (ii) is provento be satisfied by a rough volatility model with
H < / As mentioned, this paper aims to construct a rough lognormal model by replacing the fractional Brownianmotion, driving the instantaneous volatility, with a similar process first introduced in [Dobrić and Ojeda, 2009].To provide a short description of this process, which further for the sake of brevity we call the DO process,below we follow [Dobrić and Ojeda, 2009, Conus and Wildman, 2016, Wildman, 2016].The DO process is a Gaussian Markov process with similar properties to those of a fractional Brownianmotion, namely its increments are dependent in time. The DO process is defined by first considering thefractional Gaussian field Z = Z H ( t ) , ( t, H ) ∈ [0 , ∞ ) × (0 ,
1) on a probability space (Ω , F , P ) defined bycovariance (compare this with a standard fractional BM where α H,H = 1, and H = H ) E [ Z H ( t ) Z H ( t ) ] = α H,H h | t | H + H + | s | H + H − | t − s | H + H i , (1) α H,H = − ηπ ξ ( H ) ξ ( H ) cos h π H − H ) i cos h π H + H ) i , H = H = 1 ,ξ sin ( πH ) ≡ α h ≡ α H , H + H = 1 ,ξ ( H ) = [Γ(2 H + 1) sin( πH )] / , η = Γ( − ( H + H )) , ξ = [Γ(2 H + 1)Γ(3 − H )] / . Here Γ( x ) is the Gamma function, [Abramowitz and Stegun, 1964]. Obviously, if H = H , Z H is afractional Brownian motion, and so if H = H = 1 / Z H exists.Further [Dobrić and Ojeda, 2009] are seeking for a process of the form ψ H ( t ) M H ( t ) that in somesense approximates fractional Brownian motion, assuming that ψ H ( t ) is a deterministic function of time,and M H ( t ) is a stochastic process. They construct M H ( t ) as follows. On the Gaussian field Z define M H ( t ) , t ∈ [0 , ∞ ) as M H ( t ) = E [ Z H ( t ) |F Ht ] , (2)where F Ht is a filtration generated by a sigma-algebra Z H ( r )) , ≤ r ≤ t . It is proved in [Conus and Wildman, 2016,Dobrić and Ojeda, 2009], that M H ( t ) is a martingale with respect to ( F Ht ) t ≥ . It is also shown that M H ( t ) Page 2 of 19
DOL - Markovian approximation of rough lognormal model is a Gaussian centered process with independent increments and covariance E [ M H ( t ) M H ( s )] = c H α H ¯ B (3 / − H )( s ∧ t ) − H , (3) c H = α H H Γ(3 / − H )Γ( H + 1 / , where ¯ B ( x ) = B ( x, x ) , B ( x, y ) is the Beta function.The coefficient ψ H ( t ) could be determined by minimizing the difference E [( Z H ( t ) − ψ H ( t ) M H ( t )) ] toprovide ψ H ( t ) = E [ Z H ( t ) M H ( t )] E [ M H ( t )] , (4)and, as shown in [Dobrić and Ojeda, 2009], in the closed form ψ H ( t ) = Γ(3 − H ) c H Γ (3 / − H ) t H − . (5)To summarize this construction, it introduces the DO process V H ( t ) , t ∈ [0 , ∞ ] defined as V H ( t ) = ψ H ( t ) M H ( t ) where ψ H ( t ) is given in Eq. (4), and M H ( t ) - in Eq. (2) with H + H = 1.The most useful property of the DO process is that it is a semi-martingale, and can be represented asan Itô’s diffusion. This means, see again [Dobrić and Ojeda, 2009, Wildman, 2016], that there exists aBrownian motion process W t , t ∈ [0 , ∞ ) adapted to the filtration F Ht , such that dV H ( t ) = 2 H − t V H ( t ) dt + B H t H − / dW t , (6) B H = 2 − H csc ( πH )Γ(2 − H )Γ (3 / − H ) Γ( H ) . In contrast to [Conus and Wildman, 2016] where the DO process was used as noise in the Black-Scholesframework, here we apply it for modeling dynamics of the instantaneous variance. The main advantage ofsuch a model as compared with the rough volatility models is that the semi-martingale property of theDO process allows utilization of the Itô’s calculus.Also in a recent paper [Harms, 2019] it is shown that fractional Brownian motion can be representedas an integral over a family of the Ornstein-Uhlenbeck (OU) processes. The author proposes numericaldiscretizations which have strong convergence rates of arbitrarily high polynomial order. He uses thisrepresentation as the basis of Monte Carlo schemes for fractional volatility models, e.g. the rough Bergomimodel. Thus, the DO process can be considered as a particular case of the construction in [Harms, 2019].However, as we show below, using the DO approximation of the fractional Brownian motion providessome additional tractability, while is less accurate. As ψ ( t ) in Eq. (4) is determined by minimization of variance of the process Y H ( t ) = Z H ( t ) − ψ H ( t ) M H ( t ), letus derive an explicit representation of this minimal value E [ Y H ( t ) |F Ht ]. As shown in [Dobrić and Ojeda, 2009], E [ Z H ( t ) M H ( t )] = E [ Z H ( t ) Z H ( t )] = α H t. (7)Now, using Eqs. (3), (4) and (7) one can derive E [ Y H ( t )] = t H − n E [ Z H ( t ) M H ( t )] o E [ M H ( t )] = d H t H = d H E [ Z H ( t )] , (8) d H = 1 − H Γ(3 − H )Γ( H + 1 / / − H ) . Page 3 of 19
DOL - Markovian approximation of rough lognormal model
The last expression indicates that for H ∈ [0 . ,
1] the process V H approximates Z H with a relative L error at most at 12%, see Fig.1 in [Dobrić and Ojeda, 2009]. At lower H the discrepancy is biggerand can reach 80-100% at small H . However, based on the survey presented in Introduction, the Hurstexponent could vary for various markets, and the region H < . V H ( t ) = ψ H ( t ) M H ( t ) + i d H t H = V H ( t ) + i d H t H , (9)with i be an imaginary unit. The ADO process inherits a semi-martingale property from V H ( t ). Also ψ H ( t ) as it is defined in Eq. (4), still minimizes the difference E [ Y H ( t )] = E [( Z H ( t ) − V H ( t )) ]. Finally,the minimum value of this difference is E [ Y H ( t )] = E [ { Z H ( t ) − ( ψ H ( t ) M H ( t ) + i d H t H ) } ] = 0 . (10)However, this requires an extension of the traditional measure theory into the complex domain, see, e.g.,[Carr and Wu, 2004].As from the definition, V H ( t ) = V H ( t ) − i d H t H , Eq. (6) can be transformed to d V H ( t ) = (cid:20) i Hd H t H − + 2 H − t V H ( t ) (cid:21) dt + B H t H − / dW t , (11)with the same Brownian motion as in Eq. (6). In other words, the ADO process can also be representedas an Itô’s diffusion. If H < / F Ht due to theadjustment made. However, as we use this process for modeling the instantaneous variance, it should notbe a martingale. Hence, the only property we need is that the ADO process is a semi-martingale, and itcan be represented as an Itô’s diffusion in Eq. (11).As mentioned in [Conus and Wildman, 2016], the term 1 /t in the drift of V H ( t ) causes explosion ofthe DO process at t = 0. To remedy this issue, they define a modified process, in which the drift is 0 untiltime t = (cid:15) >
0. Here we exploit this idea for the ADO process as well.
One of the most popular stochastic volatility (SV) models of [Heston, 1993] introduces an instantaneousvariance v t as a mean-reverting square-root process correlated to the underlying stock price process S t .The model is defined by the following stochastic differential equations (SDEs): dS t = S t ( r − q ) dt + S t √ vdW (1) t (12) dv t = κ ( θ − v t ) dt + ξ √ v t dW (2) t ,S t (cid:12)(cid:12) t =0 = S , v t (cid:12)(cid:12) t =0 = v . where W (1) and W (2) are two correlated Brownian motions with the constant correlation coefficient ρ , κ isthe rate of mean-reversion, ξ is the volatility of variance v (vol-of-vol), θ is the mean-reversion level (thelong-term run), r is the interest rate and q is the continuous dividend. All parameters in the Heston modelare assumed to be time-independent, despite this assumption could be relaxed, [Benhamou et al., 2010].As mentioned in Introduction, analysis of the market data reveals a rough nature of the impliedvolatility. Therefore, to take this into account in [El Euch and Rosenbaum, 2016] a fractional versionof the Heston model was introduced. The authors consider the case H ∈ [0 , /
2] where their roughHeston model is neither Markovian, nor a semi-martingale. An alternative rough Heston models is
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DOL - Markovian approximation of rough lognormal model proposed in [Guennoun et al., 2014]. The main result obtained in [El Euch and Rosenbaum, 2016] is thatthe characteristic function of the log-price in rough Heston models exhibits the same structure as thatone in the classical Heston model. However, the corresponding Riccati equation, see eg, [Rouah, 2013], isreplaced by a fractional Riccati equation. This equation doesn’t have an explicit solution anymore, butcan be solved numerically by transforming it to some Volterra equation.In addition to the Heston model, a similar model but written in terms of the volatility, rather thanvariance, was also given some attention in the literature. The model is defined by the following SDE: dS t = S t ( r − q ) dt + S t σ t dW (1) t (13) dσ t = κ ( θ − σ t ) dt + ξσ t dW (2) t ,S t (cid:12)(cid:12) t =0 = S , σ t (cid:12)(cid:12) t =0 = σ . Thus, it is a mean-reverting lognormal model for the instantaneous volatility σ t . This model is a flavor of afamous SABR model of [Hagan et al., 2002], and is also advocated, e.g., in [Sepp, 2016]. The latter paperclaims that working with the model dynamics for σ t is more intuitive, and provides a clearer interpretationof the parameters in terms of the log-normal SABR model, which is well understood by practitioners. Asfar as the market data is concerned, [Sepp, 2016] makes a reference to [Christoffersen et al., 2010] whoexamined the empirical performance of the Heston, lognormal and 3/2 SV models using market data onVIX, the implied volatility of S&P500 options, and the realized volatility of S&P500 returns. It was foundthat the lognormal model outperforms the others. For more discussion, again see [Sepp, 2016].As far as pricing options under the lognormal SV model is concerned, in contrast to the Heston model,the former is not affine. Therefore, yet a closed-form solution for the characteristic function for thelog S T is not known, while some approximations were reported in the literature. In [Lewis, 2000] thischaracteristic function is constructed assuming θ = 0 and by using a Hypergeometric series expansion. In[Sepp, 2016] an approximate solution is constructed by using a series expansion in the centered volatilityprocess Y t = σ t − θ .The main idea of this paper, however, is to propose a tractable version of the rough lognormal model.For doing that we achieve the following steps:1. For the instantaneous volatility process instead of the fractional Brownian motion we use the ADOprocess.2. Similar to [Lewis, 2000] we assume the mean-reversion level θ = 0. This, however, can be relaxed,see discussion at the end of this paper.Also, further for simplicity of notation we will use symbols V t instead of V H ( t ), and ν ( t ) = B H t H − / .Then, assuming real-world dynamics (i.e., under measure P ), our model could be represented as dS t = S t µdt + S t σ t dW (1) t (14) dσ t = σ t [ − κ + ξD v ] dt + σ t ξν ( t ) dW (2) t ,d V t = D v dt + ν ( t ) dW (2) t ,D v = (cid:20) i Hd H t H − + 2 H − t V t (cid:21) t>(cid:15) ,S t (cid:12)(cid:12) t =0 = S , σ t (cid:12)(cid:12) t =0 = σ , V t (cid:12)(cid:12) t =0 = V , where µ is the drift. This model is a two-factor model (actually, we introduced three stochastic variables S t , σ t , V t , but two of them: σ t and V t are fully correlated).The model in Eq. (14) is a stochastic volatility model where the speed of mean-reversion of theinstantaneous volatility σ t is stochastic, but fully correlated with σ t . In the literature there have been Page 5 of 19
DOL - Markovian approximation of rough lognormal model already some attempts to consider an extension of the Heston model by assuming the mean-reversion level θ to be stochastic, see [Gatheral, 2008, Bi et al., 2016]. In particular, in [Gatheral, 2008] it is shown thatsuch a model is able to replicate a term structure of VIX options. However, to the best of our knowledge,stochastic mean-reversion speed has not been considered yet. In what follows, for the sake of brevity wecall it ADOL - the adjusted DO lognormal model. Also in our model the vol-of-vol is time-dependent.To use this model for option pricing, the stock price S t should be a martingale under the risk-neutral measure Q . Then, for instance, for the Heston model an additional restriction was proposedin [Heston, 1993] that the market price of volatility risk is λ √ v t , where λ = const . This is dictatedby tractability (while a financial argument is also available, see [Wong and Heyde, 2006] and referencestherein), because then the SDE for v t has the same functional form under P and Q assuming both measuresexist.It can be seen from Eq. (14) that under the ADOL model the process for S t is a geometric Brownianmotion with stochastic instantaneous volatility σ t , the process for σ t is also a geometric Brownian motionwith stochastic speed of mean reversion and time-dependent vol-of-vol, and the process for V t is the (OU)process with time-dependent coefficients. As by definition in Eq. (14) the drift D v vanishes at t = 0, themean-reversion speed of σ t at the origin becomes − k , i.e. is well-defined ∀ H ∈ [0 , To price options written on the underlying stock price S t which follows the ADOL model, a standardapproach can be utilized, [Gatheral, 2006, Rouah, 2013]. Consider a portfolio consisting of one option V = V ( S, σ, V , t ), ∆ units of the stock S , and φ units of another option U = U ( S, σ, V , t ) that is used tohedge the volatility. The dollar value of this portfolio isΠ = V + ∆ S + φU. (15)The change in the portfolio value d Π could be found by applying Itô’s lemma to dV and dU , and assumingthat the continuous dividends are re-invested back to the portfolio d Π = dV + ∆ dS + φdU + ∆ qSdt, (16)= ( ∂V∂t + 12 σ S ∂ V∂S + 12 ξ σ ν ( t ) ∂ V∂σ + 12 ν ( t ) ∂ V∂ V + ρSξσ ν ( t ) ∂ V∂S∂σ + ρSσν ( t ) ∂ V∂S∂ V + ξσν ( t ) ∂ V∂ V ∂σ ) dt + φ ( ∂U∂t + 12 σ S ∂ U∂S + 12 ξ σ ν ( t ) ∂ U∂σ + 12 ν ( t ) ∂ U∂ V + ρSξσ ν ( t ) ∂ U∂S∂σ + ρSσν ( t ) ∂ U∂S∂ V + ξσν ( t ) ∂ U∂ V ∂σ ) dt + ( ∂V∂S + φ ∂U∂S + ∆ ) dS + ( ∂V∂σ + φ ∂U∂σ ) dσ + ( ∂V∂ V + φ ∂U∂ V ) d V + ∆ qSdt. Based on Eq. (14), the last three terms in Eq. (16) in the explicit form could be re-written as ( ∂V∂S + φ ∂U∂S + ∆ ) dS + ( ∂V∂σ + φ ∂U∂σ ) dσ + ( ∂V∂ V + φ ∂U∂ V ) d V (17) Page 6 of 19
DOL - Markovian approximation of rough lognormal model = ( ∂V∂S + φ ∂U∂S + ∆ ) h Sµdt + SσdW Q ,t i + ( ∂V∂σ + φ ∂U∂σ ) σ h − κ + ξ ¯ D v i dt + ( ∂V∂ V + φ ∂U∂ V ) ¯ D v dt + ν ( t ) dW Q ,t ( (cid:20) ∂V∂ V + φ ∂U∂ V (cid:21) + ξσ (cid:20) ∂V∂σ + φ ∂U∂σ (cid:21) ) . To make this portfolio riskless, the risky terms proportional to increments of the Brownian Motionsmust vanish. This implies that the hedge parameters are∆ = − ∂V∂S − φ ∂U∂S , (18) φ = − (cid:20) ξσ ∂V∂σ + ∂V∂ V (cid:21) (cid:20) ξσ ∂U∂σ + ∂U∂ V (cid:21) − . Also a relative change of the risk free portfolio is the interest earned with the risk free interest rate, i.e. d Π = r Π dt. (19)With allowance for Eq. (18), Eq. (16) could be represented in the form d Π = ( A + φB ) dt . Therefore,Eq. (19) can be transformed to A + φB = r ( V + ∆ S + φU ) . (20)Using the definition of φ in Eq. (18), this could be re-written as A − rV + ( r − q ) S ∂V∂S ξσ ∂V∂σ + ∂V∂ V = B − rU + ( r − q ) S ∂U∂S ξσ ∂U∂σ + ∂U∂ V . (21)The left-hand side of this equation is a function of V only, and the right-hand side is a function of U only. This could be only if both sides are just some function f ( S, v, V , t ) of the independent variables.Accordingly, using the explicit expression for A from Eq. (21) we obtain the ADOL PDE0 = ∂V∂t + 12 σ S ∂ V∂S + 12 ξ σ ν ( t ) ∂ V∂σ + 12 ν ( t ) ∂ V∂ V (22)+ ρSξσ ν ( t ) ∂ V∂S∂σ + ρSσν ( t ) ∂ V∂S∂ V + ξσν ( t ) ∂ V∂ V ∂σ + ( r − q ) S ∂V∂S + ( ¯ D v − f ) ∂V∂ V + σ h − κ + ξ ( ¯ D v − f ) i ∂V∂σ − rV. To proceed, we need to choose an explicit form of f ( S, v, V , t ). We consider two options. The first onerelies on a tractability argument and suggests to choose f = ¯ D v + λ , where, similar to [Heston, 1993], λ is the market price of volatility risk and is constant. However, with this choice the risk-neutral drift of σ t becomes − ( κ + ξλ ) σ t dt , i. e., the stochastic volatility σ t doesn’t depend on V t . In such a model onlythe vol-of-vol term is a function of t and the Hurst exponent H , so this is a stochastic volatility modelwith the time-dependent vol-of-vol. This makes this model not rich enough for our purposes, despite it istractable. Therefore, in what follows we ignore this choice. For the reference, pricing options using thetime-dependent Heston model is considered in [Benhamou et al., 2010] by using an asymptotic expansionof the PDE in a small vol-of-vol parameter, and a similar method could be applied in this case as well.The other construction we introduce in this paper is the choice f = ¯ D v + λ + m ( t ) V with m ( t ) besome function of time t . Since in Eq. (14) the drift of σ t is already a linear function of V t under a physical Page 7 of 19
DOL - Markovian approximation of rough lognormal model measure, the proposed construction either keeps it linear under the risk-neutral measure. With thisdefinition Eq. (22) takes the form0 = ∂V∂t + 12 σ S ∂ V∂S + 12 ξ σ ν ( t ) ∂ V∂σ + 12 ν ( t ) ∂ V∂ V + ρSξσ ν ( t ) ∂ V∂S∂σ (23)+ ρSσν ( t ) ∂ V∂S∂ V + ξσν ( t ) ∂ V∂ V ∂σ + ( r − q ) S ∂V∂S − [ λ + m ( t ) V ] ∂V∂ V − [ κ + ξ ( λ + m ( t ) V )] σ ∂V∂σ − rV. As by Girsanov’s theorem, [Karatzas and Shreve, 1991] dW (1) t = dW Q ,t − γ ( t ) dt, (24) dW (2) t = dW Q ,t − γ ( t ) dt, with W Q , W Q be the corresponding Brownian motions under measure Q , a necessary condition for thismeasure to exist is µ − ( r − q ) = σ t (cid:18) ργ ( t ) + q (1 − ρ γ ( t ) (cid:19) , which ensures that the discounted stock price is a local martingale under measure Q , see eg, [Wong and Heyde, 2006].Accordingly, by using the same argument, one can see that the PDE in Eq. (23) corresponds to thefollowing model under the risk-neutral measure Q dS t = S t ( r − q ) dt + S t σ t dW Q ,t (25) dσ t = − [ κ + ξ ( λ + m ( t ) V t )] σ t dt + σ t ξν ( t ) dW Q ,t ,d V t = − [ λ + m ( t ) V ] dt + ν ( t ) dW Q ,t ,S t (cid:12)(cid:12) t =0 = S , σ t (cid:12)(cid:12) t =0 = σ , V t (cid:12)(cid:12) t =0 = V . When this model is used for option pricing, and with parameters obtained by calibration of the model tomarket options prices, one is already in the risk-neutral setting. Then, as explained in [Gatheral, 2006],that allows setting the market price of volatility risk λ equal to zero. So in what follows we set λ = 0.The model for σ t in Eq. (25) in a certain sense is similar to that introduced in [Benth and Khedher, 2016]who considered a generalized OU process by letting a mean-reversion speed to be stochastic, and, in partic-ular, a Brownian stationary process. As our process V t is also a time-dependent OU process, it may attainnegative values, so the mean-reversion rate could become negative. However, in [Benth and Khedher, 2016],the authors are able to show the stationarity of the mean, the variance, and the covariance of the process(the process σ t in our notation) when the average speed of mean-reversion is sufficiently larger than itsvariance. Explicit conditions for these results to hold are also derived in that paper. log S T under the ADOL model One of the main reasons that the Heston model is so popular is that the characteristic function of log S T in this model is know in closed form. Then any FFT based method, [Carr and Madan, 1999, Lewis, 2000,Fang and Oosterlee, 2008], can be used to price European, and even American, [Lord et al., 2007], optionswritten on the underlying stock S t .Let us denote T to be the option maturity, and use the representation of the characteristic function E [ e iu log S T | S, v, V ] = e iu log S ψ ( u ; x, σ, V , t ), where ψ ( u ; x, σ, V , τ ) = E [ e iu log x ] and x = log S T /S . It isknown that as per Feynman-Kac theorem, [Shreve, 1992], ψ ( u ; x, σ, V , τ ) solves a PDE similar to Eq. (23) Page 8 of 19
DOL - Markovian approximation of rough lognormal model but with no discounting term rV ∂ψ∂t + 12 σ ∂ ψ∂x + 12 ξ σ ν ( t ) ∂ ψ∂σ + 12 ν ( t ) ∂ ψ∂ V + ρξσ ν ( t ) ∂ ψ∂x∂σ (26)+ ρσν ( t ) ∂ ψ∂x∂ V + ξσν ( t ) ∂ ψ∂ V ∂σ + (cid:18) r − q − σ (cid:19) ∂ψ∂x − m ( t ) V ∂ψ∂ V − ( κ + ξm ( t ) V ) σ ∂ψ∂σ , subject to the initial condition ψ ( u ; x, σ, V , T ) = 1.We will search the solution of this PDE in the form ψ ( u ; x, σ, V , t ) = e i ux z ( u ; t, σ, V ) , (27)where z ( u ; t, σ, V ) is a new dependent variable. Substituting Eq. (27) into Eq. (26) yields0 = ∂z∂t + 12 ξ ν ( t ) σ ∂ z∂σ + 12 ν ( t ) ∂ z∂ V + ξν ( t ) σ ∂ z∂ V ∂σ (28)+ [ − ( κ + ξm ( t ) V ) + i uρξν ( t ) σ ] σ ∂z∂σ + [i uρν ( t ) σ − m ( t ) V ] ∂z∂ V + (cid:20) − u (i + u ) σ + i u ( r − q ) (cid:21) z, which should be solved subject to the initial condition z ( u ; T, σ, V ) = 1.To the best of our knowledge this PDE doesn’t have a closed form solution. However, an approximatesolution can be constructed. In particular, in what follows we assume the vol-of-vol parameter ξ to be small.More rigorously, observe that in the second line of Eq. (25) the term Ψ = ξν ( t ) dW Q ,t is dimensionless.As dW Q ,t ∝ / (2 √ t ), and ν ( t ) = B H t H − / , we have Ψ ∝ ξB H t H / ξB H T H / t/T ) H . Suppose weconsider only time intervals 0 ≤ t ≤ T , hence 0 ≤ ( t/T ) H ≤
1. Then our assumption on ξ being smallmeans that ξB H T H / (cid:28)
1, or ξ (cid:28) B H T H . (29)Obviously, this condition is too strong when we consider time intervals t (cid:28) T , because then ( t/T ) H is alsosmall. However, for relatively small maturities and small H the latter could be violated even for small t . Therefore, we prefer not to rely on the smallness of ( t/T ) H even if it does take place, and considerEq. (29) as the definition of the small parameter.With allowance for this assumption we construct the solution of Eq. (27) as follows. Let us representthe solution of Eq. (27) as a series z ( u ; t, σ, V ) = ∞ X i =0 ξ i z i ( u ; t, σ, V ) , (30)where ξ is a small parameter in a sense of Eq. (29). Substituting this representation into Eq. (27) yields0 = ∞ X i =0 ξ i ∂z i ∂t + ∞ X i =0 ξ i L z i (31)+ 12 ∞ X i =0 ξ i +2 ν ( t ) σ ∂ z i ∂σ + ∞ X i =0 ξ i +1 " ν ( t ) σ ∂ z i ∂ V ∂σ + ( m ( t ) V + i uρν ( t ) σ ) σ ∂z i ∂σ , L = 12 ν ( t ) ∂ ∂ V − κσ ∂∂σ + [i uρν ( t ) σ − m ( t ) V ] ∂∂ V + (cid:20) − u (i + u ) σ + i u ( r − q ) (cid:21) . It is clear that terms in the second line of Eq. (31) have a higher order in ξ , and as such don’tcontribute, e.g., into the zero order solution. But for higher order approximations they appear as sourceterms. In other words, the terms in the second line have no influence on Green’s function of Eq. (31).This fact makes finding the solution of Eq. (31) much easier. Page 9 of 19
DOL - Markovian approximation of rough lognormal model
In the zero order approximation on ξ Eq. (31) transforms to0 = ∂z ∂t + L z . (32)This equation could be solved in a few steps. First, we make a change of the dependent variable z ( u ; t, σ, V ) y ( u ; t, σ, V ) exp h a ( t ) + γ ( t ) σ + β ( t ) σ V i , (33) α ( t ) = − i u ( r − q )( t − T ) , β ( t ) = − i ρu ν ( t ) , γ ( t ) = − u [1 + u (1 − ρ ) ]4 κ (cid:16) − e κ ( t − T ) (cid:17) . With the new variable y ( u ; t, σ, V ) Eq. (32) transforms to0 = ∂y ∂t + 12 ν ( t ) ∂ y ∂ V − m ( t ) V ∂y ∂ V − κσ ∂y ∂σ + i ρσ V ν ( t ) + ν ( t )[ κ + m ( t )] ν ( t ) y , (34)and should be solved subject to the initial (terminal) condition y ( u ; T, σ, V ) = e − β ( T ) σ V .Second, we introduce a new independent variable σ g = σ V , and also will search the solution forthe dependent variables y ( u ; t, g, V ) in the form y ( u ; t, g, V ) = Y ( u ; t, g ) Y ( u ; t, V ) . (35)It turns out that after some algebra Eq. (34) in the new variables could be represented in the form1 Y ∂Y ∂t − κgY ∂Y ∂g + i ρg ν ( t ) + ν ( t )[ κ + m ( t )] ν ( t ) = − Y ∂Y ∂t + m ( t ) V Y ∂Y ∂ V − n ( t ) Y ∂ Y ∂ V . (36)This equation has to be solve subject to the terminal condition Y ( t, g ) Y ( t, V ) = e − β ( T ) g . Hence, we mayimpose the independent terminal conditions for Y and Y as Y ( u ; T, g ) = e − β ( T ) g , Y ( u ; T, V ) = 1 . (37)A standard approach tells that since the LHS of Eq. (36) is a function of ( t, g ) only, and the RHS ofEq. (36) is a function of ( t, V ) only, both parts must be a function of t only. In our case we can choosethis function to be zero. This splits Eq. (36) into two independent equations ∂Y ∂t = κg ∂Y ∂g − i ρg ν ( t ) + ν ( t )[ κ + m ( t )] ν ( t ) Y , Y ( u ; T, g ) = e − β ( T ) g , (38) ∂Y ∂t = m ( t ) V ∂Y ∂ V − ν ( t ) ∂ Y ∂ V , Y ( u ; T, V ) = 1 . The first equation in Eq. (38) is a first order PDE (of the hyperbolic type), and it can be easily solved inclosed form to get Y ( u ; t, g ) = exp (cid:20) − β ( T ) ge k ( t − T ) − i gρu Z tT ν ( t )[ k + m ( t )] + ν ( t ) ν ( t ) dt (cid:21) . (39)The second equation is a convection-diffusion PDE of the type 3.8.7.4 in [Polyanin, 2002] which can bereduced to the Heat equation. For instance, this can be done by doing a change of independent variables Y ( u ; t, V ) = e α ( t ) V + τ ( t ) w ( τ, ς ) , (40) α ( t ) = e R tT m ( t ) dt , τ ( t ) = − Z tT ν ( s ) α ( s ) ds, ς = α ( t ) V + 2 τ ( t ) , . Page 10 of 19
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In particualr, by this transformation the terminal point t = T is mapped to τ = 0. However, due to theterminal condition Y ( u ; T, V ) = 1 the solution is just a constant Y ( u ; t, V ) = 1 , ∀ t ∈ [0 , T ].Thus, combining all the above expressions into Eq. (33), we obtain z ( u ; t, σ, V ) = exp h α ( t ) + γ ( t ) σ + ¯ β ( t ) σ V i , (41)¯ β ( t ) = β ( t ) − β ( T ) e k ( t − T ) − i ρu Z tT ν ( t )[ k + m ( t )] + ν ( t ) ν ( t ) dt = i ρu " e k ( t − T ) ν ( T ) − ν ( t ) − Z tT k + m ( t ) ν ( t ) dt . With this expression, the final representation of the CF in Eq. (27) reads ψ ( u ; x, σ, V , t ) (cid:12)(cid:12)(cid:12) t =0 = exp h i ux + a (0) + γ (0) σ + ¯ β (0) σ V i , (42) a (0) = i u ( r − q ) T, γ (0) = − u [1 + u (1 − ρ ) ]4 κ (cid:16) − e − κT (cid:17) , ¯ β (0) = i ρu " e − kT ν ( T ) − ν (0) + Z T k + m ( t ) ν ( t ) dt . As by definition ν ( t ) = B H t H − / , the expression for ¯ β (0) is well-defined only for H < /
2. Then1 /ν (0) = 0. To construct higher order approximations in ξ we need to determine Green’s function of Eq. (32). In oursetting the Green function G ( σ, σ , V , V , t ) should vanish at the boundary of the domain Dom( σ )Dom( V ) =[0 , ∞ ] × [ −∞ , ∞ ] , and at t = T it should be G ( σ, σ , V , V , T ) = ∆( σ − σ )∆( V − V ).The key point in determining the Green function of Eq. (32) is the representation Eq. (35). Here wefurther modify it by making a change of variables y ( u ; t, g, V ) = e α ( t ) V + τ ( t ) w ( u ; τ, ω ) w ( u ; τ, ς ) , ω = e κt g, (43)and τ ( t ) , α ( t ) , ς are defined in Eq. (40).Accordingly, in new variables Eq. (36) takes the form − a ( t ) ω + ∂ τ w ( u ; τ, ω ) w ( u ; τ, ω ) = − ∂ τ w ( u ; τ, ς ) + ∂ ω w ( u ; τ, ς ) w ( u ; τ, ς ) , (44) a ( t ) = 2i ρu ν ( t )( κ + m ( t )) + ν ( t ) ν ( t ) e − κt − R tT m ( t ) dt , and t = t ( τ ). This dependence can be obtained as an inverse of τ ( t ) defined in Eq. (40).Again, the LHS of Eq. (44) is a function of ( τ, ω ) only, while the RHS is a function of ( τ, ς ) only.Therefore, both sides could be only some function of τ, i.e., f ( τ ). In our case we can put f ( τ ) = 0.As based on Eq. (43), the solution of the PDE for e − α ( t ) V− τ ( t ) y ( u ; t, g, V ) can be represented as aproduct w ( u ; τ, ω ) w ( u ; τ, ς ), the Green function G ( ω, ω , ς, ς , τ ) can also be factorized, so G ( ω, ω , ς, ς , τ ) = G ( ω, ω , τ ) G ( ς, ς , τ ) . (45)The function G ( ς, ς , τ ) is the Green function of the Heat equation ∂w ( u ; τ, ς ) ∂τ = ∂ w ( u ; τ, ς ) ∂ω , ς ∈ ( −∞ , ∞ ) , (46) Page 11 of 19
DOL - Markovian approximation of rough lognormal model with G ( ς, ς ,
0) = δ ( ς − ς ). It is well-known and reads, [Polyanin, 2002] G ( ς, ς , τ ) = 12 √ πτ e − ( ς − ς )24 τ . (47)For the second equation ∂w ( u ; τ, ω ) ∂τ = a ( t ( τ )) ωw ( u ; τ, ω ) , ω ∈ ( −∞ , ∞ ) , (48)the Green function can be found directly to obtain G ( ω, ω , τ ) = e ω R τ a ( t ( k )) dk δ ( ω − ω ) [1 − Θ( − τ ) + Θ(0)] , (49)where Θ( τ ) is the Heaviside theta-function, [Abramowitz and Stegun, 1964]. To construct the solution in the first order, we keep first two terms in Eq. (30), and ignore all terms O ( ξ ).Thus, in this approximation z ( u ; t, σ, V ) = z ( u ; t, σ, V ) + ξz ( u ; t, σ, V ). Then from Eq. (31) we obtain0 = ∂z ∂t + L z + Φ z , (50) Φ = ν ( t ) σ ∂ ∂ V ∂σ + [ m ( t ) V + i uρν ( t ) σ ] σ ∂∂σ . Thus, this equation acquires almost the same form as Eq. (32), but with two important changes. First, ithas an additional source term Φ z . Second, as the terminal condition z ( u ; T, σ, V ) = 1 is already satisfiedby the zero-order approximation z ( u ; t, s, V ), this equation has to be solved subject to the vanishinginitial condition z ( u ; T, σ, V ) = 0.It is well known from the theory of PDEs, e.g., see [Polyanin, 2002], that the general solution ofEq. (50) can be represented as z ( u ; t, σ, V ) = Z ∞−∞ Z ∞ z ( u ; T, σ , V ) G ( σ, σ , V , V , T − t ) dσ d V (51) − Z tT Z ∞−∞ Z ∞ Φ ( u ; k, σ , V ) G ( σ, σ , V , V , k − t ) dσ d V dk,Φ ( u ; k, σ , V ) = Φ z ( u ; k, σ , V ) , where z ( u ; T, σ , V ) is the terminal condition, and G ( σ, σ , V , V , t ) is the Green function of the homoge-neous counterpart of Eq. (50). Since in our case z ( u ; T, σ , V ) = 0, the first integral in Eq. (50) disappears.Also, as the homogeneous counterpart of Eq. (50) has exactly same structure as Eq. (32), the Greenfunction G ( σ, σ , V , V , t ) can be transformed to those found in Section 5.2.In more detail, computation of the second integral in Eq. (51) can be done as follows. First, we re-writeit as I = Φ I , I = Z Tt Z ∞−∞ Z ∞ z ( u ; k, σ , V ) G ( σ, σ , V , V , k − t ) dσ d V dk. (52)Second, we represent z ( u ; t, σ, V ) in variables w ( u ; τ ( t ) , ω ) , w ( u ; τ, ς ) using a series of transformations Page 12 of 19
DOL - Markovian approximation of rough lognormal model presented in Section 5.1 z ( u ; t, σ, V ) = exp h a ( t ) + γ ( t ) σ + β ( t ) σ V i y ( u ; t, σ, V ) (53)= exp " a ( t ) + γ ( t ) (cid:18) g V (cid:19) + β ( t ) g y ( u ; t, g, V )= exp " a ( t ) + ω e − κt γ ( t ) α ( t )[ ς − τ ( t )] + β ( t ) e − κt ω + ς − τ ( t ) w ( u ; τ ( t ) , ω ) w ( u ; τ ( t ) , ς )= exp " a ( t ) − τ ( t ) + ω e − κt γ ( t ) α ( t )[ ς − τ ( t )] + ¯ β ( t ) e − κt ω + ς . Then using the Green functions found in Eq. (49), Eq. (47), we obtain I = Z τ Z ∞−∞ exp " a ( t ) − χ + ω e − κt γ ( t ) α ( t )( ς − χ ) + f ( t ) ω + ς − ( ς − ς ) χ − τ ) G ( χ ) (54) · − Θ( τ − χ ) + Θ(0)2 p π ( χ − τ ) dς dχ,f ( t ) = ¯ β ( t ) e − κt + Z χ − τ a ( t ( k )) G ( t ( k )) dk, G ( t ) = 1 dτ ( t ) /dt , where t = t ( χ ) and t = t ( k ) are the inverse of the function τ ( t ). Also, according to Eq. (40), ∂τ ( t ) ∂t = − ν ( t ) α ( t ) . Switching back from τ to t , we obtain I = Z Tt − Θ( t − χ ) + Θ(0)2 p π ( χ − t ) e a ( χ ) − χ + f ( χ ) ω J ( t, ς, ω ; χ ) dχ, (55) J ( t, ς, ω ; χ ) = Z ∞−∞ exp " ω e − κχ γ ( χ ) α ( χ )( ς − χ ) + ς − ( ς − ς ) χ − t ) dς ,f ( χ ) = ¯ β ( χ ) e − κχ + Z χ − t a ( k ) dk. As follows from the definition of γ ( χ ) in Eq. (33), γ ( χ ) ≤
0. Therefore, the second integral in Eq. (55) iswell-defined.Finally, to obtain z ( u ; t, σ, V ), in Eq. (55) we set t = 0, substitute ω = e κχ σ V , ς = α ( χ ) V + 2 τ ( χ ),and apply operator Φ to the result. In the second order approximation on ξ Eq. (31) transforms to0 = ∂z ∂t + L z + Φ z + Φ z , (56) Φ = 12 ν ( t ) σ ∂ ∂σ . Again, this equation acquires the same form as Eq. (50), but with a slightly different source term.Also, similar to Eq. (50), as the terminal condition z ( u ; T, σ, V ) = 1 is already satisfied by the zero-order approximation z ( u ; t, s, V ), Eq. (56) should be solved subject to the vanishing terminal condition z ( u ; T, σ, V ) = 0. Page 13 of 19
DOL - Markovian approximation of rough lognormal model
Thus, the solution at this step is given by Eq. (51) with Φ ( u ; k, σ , V ) = Φ z ( u ; k, σ , V ) + Φ z ( u ; k, σ , V ) , (57)where the first integral in Eq. (51) again vanishes due to the terminal condition. The second integralcan be computed in the same way as this was done for the first-order approximation, as the the Greenfunction of the homogeneous PDe is already known.In principle, the higher order approximations could be constructed in a similar way, as the Greenfunction doesn’t change, but only the source term. The higher order PDEs take the form0 = ∂z i ∂t + L z i + Φ z i − + Φ z i − , i > , (58)and should be solved subject to the vanishing terminal condition z i ( u ; T, σ, V ) = 1. Again, the solution isgiven by Eq. (51) with Φ ( u ; k, σ , V ) = Φ z i − ( u ; k, σ , V ) + Φ z i − ( u ; k, σ , V ) . (59) Looking closely at the integrand of J (0 , ς, ω ; χ ) in Eq. (55), one can observe that it behaves as follows:1. Suppose we consider options with maturities T < ≤ χ ≤ T , at ς far away from ς theterm ( ς − ς ) χ − t ) is large. Therefore, for these regions of ς the integrand almost vanishes.2. Also, at large | ς | where ( ς − x ) is also large, the term ω e − κχ γ ( χ ) α ( χ )( ς − χ ) = − σ V u [1 + u (1 − ρ ) ]4 κ (cid:16) − e κ ( χ − T ) (cid:17) . α ( χ )( ς − χ )2is small for σ, V , u fixed.Thus, the integrand of J (0 , ς, ω ; χ ) has a bell shape with a maximum close to the point ς = ς ∗ whichsolves the equation ∂ ς " ω e − κχ γ ( χ ) α ( χ )( ς − χ ) + ς − ( ς − ς ) χ − t ) = 0 . (60)Indeed, consider an example with the explicit form of the function m ( t ) = %t π , %, π ∈ R , π ≥
0, sothe SDE for V t in Eq. (25) is mean-reverting. With this m ( t ) one can find that γ ( t ) = B H π ) e − % π T π " t H E − H π , − %t π π ! − T H E − H π , − %T π π ! , (61)where E ( k, z ) is the exponential integral function, [Abramowitz and Stegun, 1964]. Let’s also use thevalues of our model parameters given in Table 1. κ H T σ V ρ u % π Table 1:
Parameters of the test.
Page 14 of 19
DOL - Markovian approximation of rough lognormal model
Figure 1:
Integrand of J (0 , ς, ω ; χ ) as func-tion of ( ς , χ ) at ≤ χ ≤ T . Figure 2:
Integrand of J (0 , ς, ω ; χ ) as func-tion of ( ς , χ ) at . ≤ χ ≤ T . Now the integrand of J (0 , ς, ω ; χ ) can be computed explicitly, and the result is presented in Fig. 1.The bell shape of this function could be clearly seen at small T . To make sure a similar shape could beseen at large T , we zoom-in this plot in χ , and the result is presented in Fig. 2 which justifies the previousobservation.The bell shape of the integrand implies that the integral J (0 , ς, ω ; χ ) an be computed approximatelyin closed form. Indeed, the maximum of the integrand approximately corresponds to the point ς = ς where ς = α ( χ ) V + 2 τ ( χ ). Then, using the representation of the integrand in the form f ( χ, ς, ς ) = k ( χ )( ς − χ ) + ς − ( ς − ς ) χ , (62) k ( χ ) = γ ( χ ) α ( χ ) σ V , we expand f ( χ, ς, ς ) into series on ς around ς to obtain f ( χ, ς, ς ) = a + a ( ς − ς ) + a ( ς − ς ) + O (( ς − ς ) ) , (63) a = ς + k ( χ )( ς − χ ) , a = 1 − k ( χ )( ς − χ ) , a = − χ + 3 k ( χ )( ς − χ ) . Then J (0 , ς, ω ; χ ) = r π − a e a − a a , (64)which exists if a < a ( χ ) computed in this experiment which turns out to be negative forall values of 0 ≤ χ ≤ T . Then Fig. 3 presents a percentage difference between the value of J (0 , ς, ω ; χ )computed numerically and using Eq. (64). The difference is about 5 bps, so in out test this aproximationworks pretty well.Alternatively, Eq. (60) is a quartic algebraic equation which can be solved in closed form. Denotingthis solution by ς ∗ and using it instead of ς in Eq. (63), we obtain another approximation. As the CF of the log S T is known in closed form (in our case this is an approximation of the exact solutionconstructed by using power series in ξ ), pricing options can be done in a standard way by using FFT, Page 15 of 19
DOL - Markovian approximation of rough lognormal model
Figure 3:
Difference in % between J (0 , ς, ω ; χ ) obtained by us-ing numerical integration andEq. (64). Figure 4:
Function a ( χ ) computed for ourexperimant. [Carr and Madan, 1999, Lewis, 2000, Fang and Oosterlee, 2008]. In turn, pricing variance swaps can bedone by using a forward CF, similar to how this is done in [Itkin and Carr, 2010]. Using the forward time t the forward characteristic function is defined as φ t,T = E Q [exp(i uχ t,T ) | S , σ ] , (65)where χ t,T = χ T − χ t , and χ t = log S t . Then under a discrete set of observations of the stock price attimes t i , i ∈ [1 , N ], the quadratic variation Q N ( x ) of S t is given by, [Itkin and Carr, 2010] Q N ( s ) = 1 T N X i =1 E Q h ( χ t i − χ t i − ) i = 1 T N X i =1 E Q h χ t i ,t i − i = − T N X i =1 ∂ φ t i ,t i − ( u ) ∂u (cid:12)(cid:12)(cid:12) u =0 . (66)As χ t i ,t i − = log S t i − log S t i − = x T − t i − − x T − t i , Q N ( s ) in Eq. (66) can be computed in a way similar tohow this was done in Section 5.Therefore, the proposed model could be useful, e.g., for pricing options and swaps as, on the one hand,it catches some properties of rough volatility, but, on the other hand, is more tractable. We underline, thatour approach allows the CF to be found as the solution of the PDE in Eq. (26). This PDE, in general, canbe solved numerically. But in this paper we provide a closed-form series solution obtained by assumingthe vol-of-vol ξ to be small. This condition is defined in Eq. (29), so, as can be seen, it is a function of theHUrst exponent H and time to maturity T . The function f ( H, T ) = 2 / ( B H T H ) for various values of H and T is represented in Fig. 5. As the range H ∈ [0 , .
3] is reported in the literature to be important, were-plot this graph in Fig. 6 by zooming into this area.Overall, the values of f ( H, T ) look reasonable as compared with those reported in the literature,for instance, for the Heston model. In other words, the values of the vol-of-vol parameter ξ , found bycalibration of the Heston model to market prices of European vanilla options, could be of the order ofmagnitude to obey Eq. (29) for H > . T > .
1. However, this definitely should be justified byindependent calibration of the ADOL model to those market data. This calibration would require solvingthe PDE in Eq. (26) numerically, to not rely on the assumption Eq. (29). These results will be reportedelsewhere.Another assumption we made when deriving Eq. (14) is that the mean-reversion level θ = 0. With alittle algebra it can be checked, that relaxing this assumption adds an extra term to Eq. (28) which is κθ ∂z ∂σ . Accordingly, in the definition of operator L in Eq. (31) this also adds an extra term κθ ∂∂σ . The Page 16 of 19
DOL - Markovian approximation of rough lognormal model “ Figure 5:
Function f ( H, T ) for H ∈ [0 , and T ∈ [0 , , years Figure 6:
Function f ( H, T ) for H ∈ [0 , . and T ∈ [0 , , years next step is to make a change of variable σ σ − θ . As θ is assumed to be constant, Eq. (34) could againbe replicated if in Eq. (33) we add an extra term ¯ γ ( t ) s , i.e. z ( u ; t, σ, V ) y ( u ; t, σ, V ) exp h a ( t ) + ¯ γ ( t ) s + γ ( t ) σ + β ( t ) σ V i , (67)¯ γ ( t ) = θu ( u + 1) κ (cid:16) e k ( t − T ) − (cid:17) . Then construction of the solution remains the same.
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