A Market Driver Volatility Model via Policy Improvement Algorithm
AA MARKET DRIVER VOLATILITY MODEL VIA POLICY IMPROVEMENTALGORITHM
JUN MAEDA AND SAUL D. JACKA
Abstract.
In the over-the-counter market in derivatives, we sometimes see large numbers oftraders taking the same position and risk. When there is this kind of concentration in the mar-ket, the position impacts the pricings of all other derivatives and changes the behaviour of theunderlying volatility in a nonlinear way.We model this effect using Heston’s stochastic volatility model modified to take into account theimpact. The impact can be incorporated into the model using a special product called a marketdriver, potentially with a large face value, affecting the underlying volatility itself. We derivea revised version of Heston’s partial differential equation which is to be satisfied by arbitraryderivatives products in the market. This enables us to obtain valuations that reflect the actualmarket and helps traders identify the risks and hold appropriate assets to correctly hedge againstthe impact of the market driver. Introduction
Japan has one of the largest equity derivatives markets in the world. According to the Bankfor International Settlements, Japan had $378 billion in face value of equity-linked contracts outof the worldwide total of $5,445 billion as of September 13, 2015 [1]. A common underlying forequity-linked derivatives products in the country is the price-weighted Nikkei Stock Average Index(Nikkei 225) published by Nikkei Inc. Since the country is the world’s third largest economy byGDP, people generally assume that the market is liquid enough to trade freely any desired position.However, in the first author’s experience, this is not quite true. Long-dated volatility (especiallybetween 2 and 5 years) is generally priced low due to the fact that the most of the traders inthe market already own vega (sensitivity to volatility) from selling (usually in significant sizes) astructured product called an ’auto-callable’ to their clients. Therefore it is generally difficult tosell vega in the market. We will discuss this exotic case in more detail in a future paper, and fornow, we will focus on explaining our model in a simpler context in this paper. The importantfact to note is that there is a position that affects the pricings and risks of all the existing andpotential derivatives products in the market.In order to understand the background of our model, we first formulate a toy example.Assume that there are only 2 traders, A and B, in the over-the-counter (OTC) market. If Awants to buy volatility, A needs to buy it from B and vice versa. If A buys $10 million of vega
Date : December 5, 2016.
Key words and phrases. stochastic volatility model, Heston model, semilinear parabolic partial differential equa-tions, policy improvement algorithm, Hamilton-Jacobi-Bellman equation. a r X i v : . [ q -f i n . P R ] D ec JUN MAEDA AND SAUL D. JACKA from B, the vega that B holds is decreased by the same amount. Generally, traders don’t want toown so much risk on one side, so they might want to hedge the risk a little. Since the only marketparticipants are A and B, they need to reverse what they previously traded in order to hedgethemselves. This does not make much sense in this case, as there are only 2 market participants,but even if we assumed more participants in the market, this is still essentially what is happening:overall, the market vega is maintained and does not change whatever A and B do. Whatever Agains, B loses and vice versa.Now introduce a new market participant C. Let us assume that C is not a participant in theOTC market but only buys vega from A and B as their client to hedge against market risk, anddoesn’t otherwise hedge the position (we could think of C as an insurance company, for example).If C buys $10 million of vega from A, then A is now short the risk, so may want to buy some backin the market to hedge himself. A needs to buy it from B, of course, as C doesn’t sell any vega.The important point is that the OTC market whose only participants are A and B is now short$10 million of vega overall. The market now would like to buy some vega back. This generallydrives the volatility of the underlying security or index higher.We elaborate this point in more detail. The demand and supply of vega could be, in general,directly converted to the supply and demand of volatility. It is easier to think of this in theBlack-Scholes framework. If there is more demand for vega than supply, more people want to buyvega. The way they accomplish this is to buy plain vanilla calls and puts, which are positive vegaproducts. If more people buy these products, the prices of the products move higher. Given otherparameters are fixed, this price increase could only be explained by the increase in the underlyingvolatility. This is why the actual market participants refer to ’buying (selling) volatility’ when theyare actually buying (selling) vega. These phrases will be used with the same meanings hereafter.
Remark . The corresponding volatility level is implied volatility as opposed to realized (or his-torical) volatility.Up to this point, volatility movement is just a matter of demand and supply. Now supposethat the derivative product that C bought has big second order risks, like vanna (the derivative ofvega with respect to stock price) and volga (the derivative of vega with respect to volatility). Forexample, if the product is long vanna, vega increases when the underlying stock moves higher. Inthis case, A gets shorter vega just from the market movement and he needs to buy it in the marketto rehedge himself. However, if B has the same position, B gets shorter vega as well, so neitherof them are interested in selling any more vega. This will make the volatility even higher. Notethat in this situation, what is moving the volatility is just the change in the risk of the productthat was already traded, not a new trade. We call the special product (of which the risks affectthe dynamics of supply and demand of the volatility) the market driver .In order to model the example above, we posit a simple and easy-to-use model which is just anextension of the Heston model, one of the most popular stochastic volatility models. The core ofour model is a semilinear parabolic partial differential equation (PDE) that we retrieve to price the
MARKET DRIVER VOLATILITY MODEL VIA PIA 3 market driver. Once we obtain the valuation of the market driver, we use a linear parabolic PDE,which is very similar to those of Black-Scholes and Heston, to price other derivatives products.As mentioned earlier, we are more interested in the case where the market driver is of a specificexotic type because we think its risk feedback effect is more prominent in practice. We will handlethis problem in a future paper and concentrate now on the case when it is of plain vanilla type.The problem statement so far may remind some readers of the ’feedback effect’ of optionswhich is now a somewhat mature field. The Black-Scholes model with a feedback effect modelsthe prices of derivative products affected by delta hedging executed by program traders [5, 19, 21].It was a field which attracted a lot of interests in the 1990s. However, as far as we know, notmuch work has been done since then. Although the research in this paper was done separatelyfrom the studies done in the field, our ideas are very similar in the sense that some trade affectsother option pricing. We are (in a way) incorporating the feedback effect in a stochastic volatilityframework. The key difference, however, is that we are not applying the feedback effect of theunderlying asset (stock), but instead, that of the underlying volatility . In the earlier models, theeffect impacts the volatility passively via program traders trading the underlying asset. On theother hand, our model incorporates the effect directly in the dynamics of the volatility.It may not look natural to incorporate a feedback effect in the volatility as it is not a tradableasset. However, from the first author’s experience, supply and demand effects of the volatilitydo exist in the actual market and we think our model reflects, at least qualitatively, the actualmarket dynamics with the market driver. We believe that our model is more in line with marketpractitioners’ perspectives than the classical feedback model.One of the difficulties in the earlier feedback models is that they model the realized (historical)volatility rather than the implied volatility. Hedging delta of derivative products by dynamicallytrading the underlying asset does affect the implied volatility, but only that of short maturity.Behaviour of the current stock price has little impact over the long-dated implied volatility. De-pending on the sign of the vanna of the market driver, it is possible, for example, that evenwhen the realized volatility increases with a large drop in the stock price, the long-dated impliedvolatility go lower. This cannot be modelled in the classic feedback model.One of the benefits of our model is that the nonlinear PDEs that we derive can be approximatedby a series of linear ones. The PDEs derived in the classic feedback model are generally ofquasilinear type, where the nonlinearity occurs in the highest order of the equations. On the otherhand, although we need a pair of PDEs, one for the market driver and the other for a generalderivative product, our PDEs are at most of semilinear parabolic type, where the nonlinearityoccurs in lower order of the equations. This enables us to apply a linear approximation algorithmcalled the Policy Improvement Algorithm (PIA) in which the approximated solution convergesquickly to the actual solution of the semilinear PDE.The reason why we introduce the PIA is that it enables us to reuse the setup for the Hestonmodel. The Heston model has already been implemented in practice and is widely used. It is
JUN MAEDA AND SAUL D. JACKA convenient to use the existing setup, whenever possible, to calculate the solutions of the newmodel. We also note that in the course of our research, we encountered some cases where wehad a convergence in numerical solution using the PIA, but not using the finite difference method(FDM): the PIA seems to have better convergence in numerical solution than the FDM.The rest of the paper is organized as follows: Section 2 explains the new model in detail. Wewill establish the existence and uniqueness of the solution to our PDEs in Section 3. In Section 4,we transform the nonlinear PDE to an HJB equation. The PIA is then described in Section 5. InSection 6, we give a numerical example to see how valuations and risks, which are very importantfor day-to-day hedging for traders, change in our model from those in Heston’s model. We alsosee in this section how PIA-approximated solutions converge to that of the nonlinear PDE. Wegive our conclusions in Section 7.2.
The Market Driver Model
We start by briefly reviewing Heston’s stochastic volatility model [7]. The stochastic differentialequations for the stock price and the variance are:(2.1) dS = µSdt + √ vSdW dv = κ (¯ v − v ) dt + η √ vdW (cid:104) dW , dW (cid:105) = ρdt .Here, S denotes the underlying stock price and v the variance of the underlying. W and W are Wiener processes with correlation ρ , µ is the drift of the stock, κ > v is the mean variance, and η is thevolatility of the variance.Since v only takes positive values, we usually require the model to satisfy Feller’s condition foravoiding the origin [13]:(2.2) 2 κ ¯ v > η .With this setup, the value V of a derivatives product with its satisfies Heston’s PDE: ∂V∂t + rS ∂V∂S + κ (cid:16) ¯ v − ωv (cid:17) ∂V∂v + 12 vS ∂ V∂S + 12 vη ∂ V∂v + vSηρ ∂ V∂S∂v − rV = 0(2.3)with appropriate initial (or terminal, if we are calculating backwards in time) and boundaryconditions. Equation (2.3) is a second order linear parabolic PDE. Here, ω is some constant forvolatility risk premium and r is the interest rate.Let us now assume that there is some distinguished product (called the market driver) withvalue denoted by F . MARKET DRIVER VOLATILITY MODEL VIA PIA 5
Using this F , our revised model is written as(2.4) dS = µSdt + √ vSdW dv = κ (¯ v − v + Q ∂F∂v ) dt + η √ vdW d (cid:104) W , W (cid:105) t = ρdt with some coefficient Q .Note that the only change made to the Heston SDE (2.1) is the term κQ ∂F∂v in the secondequation. A simple justification for this is that the vega (in this paper, we use the term ’vega’ forthe derivative of the valuation with respect to variance, whereas it usually means the derivativeof the valuation with respect to volatility) of the market driver impacts supply and demand ofthe variance and causes the shift in its mean. We are only adding this adjustment to the varianceSDE. If we want to, we could, of course, similarly add ’delta’ (derivatives of valuation with respectto the underlying stock price) adjustment in the SDE for the stock price S in (2.4). However, wedo not do this since i) deltas of derivatives products are generally low, so in order to have a largeimpact on the stock price, the face value traded on the position needs to be massive, which is notrealistic and ii) the stock market is more liquid than the OTC derivatives market, in the sensethat there are more people with different incentives in trading and many more people have accessto the market (for example, personal investors can easily trade stocks, whereas they might needto satisfy additional requirements in order to trade derivatives. It is even harder for them to beable to trade in the OTC market due to size requirements, credit issues, and other restrictions).A sufficient condition for the variance not to go negative is derived by comparing the 2 processes v and v (cid:48) starting at the same value:(2.5) (cid:40) dv = κ (¯ v − v + Q ∂F∂v ) dt + η √ vdW dv (cid:48) = κ { ¯ v − v (cid:48) + min( Q ∂F∂v ) } dt + η √ v (cid:48) dW .Since we will be working in a bounded domain, Proposition 5.2.18 in [12] shows that v (cid:48) ≤ v almost surely. Applying Feller’s condition (2.2) on v (cid:48) , if(2.6) 2 κ (cid:110) ¯ v + min (cid:16) Q ∂F∂v (cid:17)(cid:111) > η ,then v (cid:48) > v > positive variancecondition .If we follow the usual argument, we obtain the following PDE for the value V of a derivative: ∂V∂t + rS ∂V∂S + κ (cid:16) ¯ v − ωv + Q ∂F∂v (cid:17) ∂V∂v + 12 vS ∂ V∂S + 12 vη ∂ V∂v + vSηρ ∂ V∂S∂v − rV = 0.(2.7) JUN MAEDA AND SAUL D. JACKA
Since F is also the value of a specific derivative, we can substitute V = F in (2.7) and obtaina nonlinear PDE for F : ∂F∂t + rS ∂F∂S + κ (cid:16) ¯ v − ωv + Q ∂F∂v (cid:17) ∂F∂v + 12 vS ∂ F∂S + 12 vη ∂ F∂v + vSηρ ∂ F∂S∂v − rF = 0.(2.8)Note that given F , the differential equation (2.7) is a second order parabolic PDE that is linearin V as in the Heston model. On the other hand, the differential equation (2.8), is semilinear.3. Partial Differential Equations
We recall some theorems from the theory of PDEs. For more detail, refer to [14].We take a bounded, open, and connected domain E in R which is bounded away from theaxes. We further assume that ∂ E is C α (cid:48) for some α (cid:48) >
0. Let Q T = E × (0 , T ), D = ∂ E , D T = { ( x, y, t ) | ( x, y ) ∈ D , t ∈ [0 , T ] } , and Γ T = D T ∪ { ( x, y, t ) | ( x, y ) ∈ E , t = 0 } . We impose ψ as our initial and boundary conditions and assume it satisfies the compatibility condition, i.e. ψ ( x, y, t ) ∈ C ( Q T ).We define the differential operator L by − Lu : = rxu x + κ ( v − αy ) u y + 12 x yu xx + 12 η yu yy + ηρxyu xy = a ij u ij + b i u i (3.1)under the Einstein summation convention.We reparameterize time-to-go t backwards by replacing t → T − t and rewrite (2.8) in generalform:(3.2) u t + Lu + ru − κQu y = 0.The PDE (3.2) is uniformly parabolic as it satisfies(3.3) ν | ξ | ≤ a ij ξ i ξ j ≤ ν | ξ | ∀ ( x, y ) ∈ E , ∀ ξ ∈ R for some ν , ν > H β,β/ ( Q T ) for some 0 < β <
1, it also has bounded first spatial derivatives in Q T , andits second order spatial derivatives and first order time derivative belong to H γ,γ/ ( Q T ) for somenonnegative and nonintegral number γ . MARKET DRIVER VOLATILITY MODEL VIA PIA 7
By substituting this solution in the coefficients of the PDE (2.7), Corollary 1 in Section 3.5 onpage 74 of [6] affirms the existence and uniqueness of the solution to the linear PDE for suitableinitial and boundary conditions.
Remark . We require the positive variance condition (2.6) to be satisfied in order to ensure that v is nonnegative. Theorem 6.2 of Chapter V from [14] affirms that F y is bounded, but as far asthe statement of the theorem goes, we don’t have an explicit expression of it. For that reason,it is not easy to show that (2.6) is satisfied in general. In the case where the market driver withvalue F is of plain vanilla type with Q >
0, enforcing Feller’s condition (2.2) is sufficient for thepositive variance condition (2.6) to be satisfied since ∂F/∂y ≥
0, and therefore Q ( ∂F/∂y ) ≥ Control Problem
From now on, we focus on solving (3.2). We can apply various numerical methods, for example,the FDM, to calculate the solution numerically. If we were to do this, we would need additionalresources to implement it in actual trading and in some cases, it may not be easy to do so.One of the difficulties may originate from the fact that even though it’s semilinear, it’s still anonlinear PDE that we are dealing with. Applying the model to actual trading becomes morestraightforward with a help of the Policy Improvement Algorithm (PIA).It is easy to see that (3.2) can be rewritten as(4.1) inf π ∈ R (cid:18) u t + Lu + ru − πu y + π κQ (cid:19) = 0.Note that this is the HJB equation to minimize(4.2) V π ( x, y, t ) = E (cid:20) (cid:90) τ ∧ t e − rs f π ( Z z,πs , t − s ) ds + e − r ( τ ∧ t ) g ( Z z,πτ ∧ t , t ∧ τ ) (cid:21) under the controlled process Z z,πt := ( X, Y π ) T with SDEs(4.3) dX = µXdt + √ Y XdW dY π = κ (¯ v − Y π + π/κ ) dt + ηρ √ Y π dW + η √ Y π (cid:112) − ρ dW d (cid:104) W , W (cid:105) t = 0with Z z,π = z = ( x, y ) T . Here, f π = π / κQ , g = ψ is the initial and boundary conditionsintroduced in Section 3, and τ is the first hitting time of the boundary of the domain.Our problem is now converted into the HJB equation for the following controlled initial/boundaryproblem: JUN MAEDA AND SAUL D. JACKA (4.4) inf π ∈ R (cid:18) u t + Lu + ru − πu y + π κQ (cid:19) = 0 ( x, y, t ) ∈ E × (0 , T ) u ( x, y, t ) = inf π V π ( x, y, t ).From the positive variance condition (2.6),(4.5) π > η − κ ¯ v is sufficient for Y not to go below zero.5. Policy Improvement Algorithm
We now give a detailed formulation of the PIA and the proof of convergence. For more detail,refer to [10] and [11].Let (Ω , F , ( F t ) t ≥ , P ) be a filtered probability space satisfying the usual conditions that supportsa 2-dimensional ( F t ) t ≥ - Wiener process W = ( W t ) t ≥ .For any process Y = ( Y t ) t ≥ , define(5.1) τ E ( Y ) := inf { t ≥ Y t ∈ ∂ E} .Let A ( z, T ) := { Π = (Π t ) t
For any Markov policy π that is Lipschitz on compacts in R , the followingholds: V g, E ,π ∈ C , ( Q T ) and it satisfies (5.8) L π V g, E ,π − rV g, E ,π + f π = 0 .Proof. It suffices to prove that V g, E ,π satisfies (5.8) in every open ball U with U ⊂ Q T . Let U besuch an open ball with centre ζ and radius (cid:96) . Let z ∈ U and define τ as the first time the process Z z,π hits the boundary of U . For every n ∈ N , define U n as the closed ball with centre ζ andradius (cid:96) − n , and let τ n be the first time the process Z z,π hits the boundary of U n .Let v ∈ C , ( U ) ∩ C ( U ) be the unique solution of the initial boundary value problem(5.9) (cid:40) L π v − rv + f π = 0 v | D T = V g, E ,π | D T .The existence and uniqueness is guaranteed by Corollary 1 on page 71 in [6] and Lemma A.5in the Appendix. The partial derivatives of v are H¨older continuous by the same corollary. Let n be large enough such that z ∈ U n , and for every n ≥ n , define the process ( J n ) n ≥ n by(5.10) J nt := (cid:90) t ∧ τ n e − rs f π ( Z z,πs , t − s ) ds + e − r ( t ∧ τ n ) v ( Z z,πt ∧ τ n , t − t ∧ τ n )and(5.11) J t := (cid:90) t ∧ τ e − rs f π ( Z z,πs , t − s ) ds + e − r ( t ∧ τ ) v ( Z z,πt ∧ τ , t − t ∧ τ ).Ito’s formula on [0 , τ n ] and the differential equation for v yield J nt = v ( z, t ) + (cid:90) t ∧ τ n e − rs ( f π − rv + L π v )( Z z,πs , t − s ) ds + (cid:90) t ∧ τ n e − rs ( ∇ v ) T σ π dW s = v ( z, t ) + (cid:90) t ∧ τ n e − rs ( ∇ v ) T σ π dW s .(5.12)Hence J n is a local martingale, and since it is clearly a bounded process, it is a uniformlyintegrable martingale. Thus the Dominated Convergence Theorem yields(5.13) v ( z, t ) = lim n →∞ E ( J n ) = lim n →∞ E ( J nt ) = E ( J t ).From the initial and boundary conditions for v , we obtain J t = (cid:90) t ∧ τ e − rs f π ( Z z,πs , t − s ) ds + e − r ( t ∧ τ ) v ( Z z,πt ∧ τ , t − t ∧ τ )= (cid:90) t ∧ τ e − rs f π ( Z z,πs , t − s ) ds + e − r ( t ∧ τ ) V g, E ,π ( Z z,πt ∧ τ , t − t ∧ τ )= E (cid:18) (cid:90) t ∧ τ e − rs f π ( Z z,πs , t − s ) ds + e − r ( t ∧ τ ) g ( Z z,πt ∧ τ , t ∧ τ ) (cid:12)(cid:12)(cid:12)(cid:12) F S (cid:19) .(5.14)The last equality in (5.14) follows from Lemma A.1. We conclude: v ( z, t ) = E (cid:18) (cid:90) t ∧ τ e − rs f π ( Z z,πs , t − s ) ds + e − r ( t ∧ τ ) g ( Z z,πt ∧ τ , t ∧ τ ) (cid:19) = V g, E ,π ( z, t )(5.15) ♦ We now describe the algorithm. Let π be a Markov policy that is Lipschitz on compacts in R . The algorithm is defined as follows:(5.16) L u i − π i ( u i ) y + f π i = 0 π i +1 ( z, t ) = arg min p ∈ A ( L p u i ( z, t ) − ru i ( z, t ) + f p ( z, t )),where the differential operator L is defined as(5.17) L := − ∂∂t − L using L in (5.7). It is important to note that L is independent of π i . In our problem, (5.16)can be further calculated as MARKET DRIVER VOLATILITY MODEL VIA PIA 11 (5.18) (cid:40) L u i − π i ( u i ) y + π i / κQ = 0 π i +1 ( z, t ) = 2 κQ ( D y u i ),where D y denotes the partial differential operator with respect to y .We already know from Section 3 that the solution of the semilinear PDE (3.2) exists uniquelywith bounded spatial derivatives, so instead of A in (5.2), we can take a subset of A of which thecontrols are uniformly bounded. Also, note that D y u i expresses the vega of the (approximated)market driver which we assumed to be a plain vanilla. As mentioned in Section 1, since plainvanilla options are positive vega products and Q >
0, we know that π i is nonnegative from thedefinition in (5.18). If we assume that (2.2) is satisfied, then we see that the condition (4.5) isalso satisfied. This precludes Y from becoming negative.In order to apply the PIA, we need to check if the algorithm (5.18) satisfies the criteria ofthe PIA. The only criterion needed to be verified is the uniform Lipschitz condition on π i . Thefollowing lemma proves this. Lemma 5.2. { π i } i defined in (5.18) is uniformly Lipschitz continuous.Proof. From the Schauder estimate, we have(5.19) (cid:107) u i +1 (cid:107) α ≤ C ( (cid:107) g (cid:107) α + (cid:107) f π n (cid:107) α ),where C only depends on the H¨older norms of the coefficients of L π , the domain Q T , and ν in (3.3). If g is continuous, we can approximate it uniformly in 2 + α norm by the Weierstrassapproximation theorem as mentioned on page 71 in [6]. In our specific problem, f π n = π n / κQ ,so (cid:107) f π n (cid:107) α is uniformly bounded thanks to the uniform boundedness of π i ∈ A . As the righthand side of (5.19) is uniformly bounded, ( u i ) i is uniformly bounded in 2 + α norm, hence π i isuniformly Lipschitz continuous from the second equation in (5.18). ♦ The PIA tells us that the u i in (5.18) converges and the limit function is V g, E which is C , and satisfies the HJB equation (4.4) in Q T .We will see later in the actual numerical example that the convergence to the solution happensfast. In the case of a plain call option as the market driver, we get a numerical solution very closeto that of the semilinear PDE with only 1 iteration. Proposition 5.3. (cid:107) u i +2 − u i +1 (cid:107) α ≤ CκQ (cid:107) u i +1 − u i (cid:107) α Proof.
By definition and Proposition 5.1(5.20) (cid:40) L u i +2 − π i +2 ( u i +2 ) y + π i +2 / κQ = 0 L u i +1 − π i +1 ( u i +1 ) y + π i +1 / κQ = 0. Subtracting these 2 equations and setting v i +2 := u i +2 − u i +1 ,(5.21) L v i +2 − π i +2 ( v i +2 ) y − ( π i +2 − π i +1 ) / κQ = 0.Since v i is 0 on the parabolic boundary, from the Schauder estimate: (cid:107) v i +2 (cid:107) α ≤ C (cid:107) π i +2 − π i +1 κQ (cid:107) α = CκQ (cid:107) ( v i +1 ) y (cid:107) α ≤ CκQ (cid:107) v i +1 (cid:107) α (5.22) ♦ Proposition 5.3 shows that if the approximation of the solution is close enough to the classicalsolution of the semilinear PDE, { u i } i converges quadratically to the solution. In other words,Proposition 5.3 shows the quadratic local convergence of the solutions of the PIA to the classicalsolution. Corollary 5.4. (5.23) (cid:107) u i +1 − u i (cid:107) α ≤ ( CκQ (cid:107) u − u (cid:107) α ) i − (cid:107) u − u (cid:107) α Proof.
Use Proposition 5.3 and induction. ♦ Numerical Simulation
We now numerically investigate how the pricing and risks change with our model. We assumethat a large amount of 2 year, 120 strike call is owned by investors outside the OTC market. Wefirst price this structure using (2.8). Then, substituting this solution in (2.7), we price a differentderivatives product, a 2 year, 100 strike call. We compare the results with the ones obtained fromthe Heston model. We used the explicit FDM method. We note that with sufficiently fine meshin the discretization, the numerical solutions converge to the analytic ones. Hereafter, we refer tothe 2 year, 120 strike call as 120 call and 2 year, 100 strike call as 100 call or at-the-money (ATM)call.We use the parameters in Table 1.Note that Feller’s condition (2.2) is met and
Q >
0. From Remark 2, the positive variancecondition (2.6) is therefore satisfied.We take our domain E to be a round rectangle and denote by S min , S max , v min , and v max the minimum and maximum values of the variables in the domain. In this example, we took S min = 0 . S max = 200, v min = 0 . v max = 1 . , F H the value F calculated in the Heston model and by F N the value calculated in the new model. Similarly, wedenote by V H and V N the corresponding values for an arbitrary V . MARKET DRIVER VOLATILITY MODEL VIA PIA 13
Parameter Value Q r ρ -0.7571 η ω v κ Table 1.
Parameters for numerical simulation.As in [7], for the calculation in the Heston model, we use the initial and boundary conditions:(6.1) F H ( S, v,
0) = max(0 , S − K ) + ( S, v ) ∈ Ω F H ( S min , v, t ) = 0 ( S = S min ) ∂F H ∂S ( S max , v, t ) = 1 ( S = S max ) ∂F H ∂t − rS ∂F H ∂S + rF H − κ ¯ v ∂F H ∂v = 0 ( v = v min ) F H ( S, v max , t ) = S ( v = v max ).The solution to the initial-boundary problem of Heston’s PDE with conditions (6.1) is contin-uous up to the boundary, so we can use the value of F H as the boundary condition for F N . Thisway, the values of F H and F N match on the parabolic boundary.We use corresponding boundary conditions for V .With the parameters in Table 1, the drift in the second SDE of (2.4) is shifted by κQ ∂F N ∂v ,which in this case is calculated as 0 . × . × .
188 = 0 . κv .The result for the 120 call (which in our case is the market driver) is shown in Table 2.Risks Value Delta Vega Vanna VolgaHeston 2.6058 35.378% 70.940 3.8766 119.001New Model 3.5121 42.457% 77.188 2.4132 -535.557 Table 2.
Summary for 120 call at S = 98 .
255 and v = 0 . Table 3.
Summary for at-the-money (ATM) call at S = 98 .
255 and v = 0 . The results are for S = 98 .
255 and v = 0 . t = T = 2. In volatility convention(i.e. standard deviation, as traders usually prefer this over variance), this value of v is equivalentto σ = √ v = 17 . Table 4.
Implied volatility calculated based on the risk calculated in the Heston modelFrom Table 4, we see that the volatility is higher, and the increments against the Heston volatilitiesare different for different structures. The result of Table 4 shows a skewness of the impact themarket driver has on the volatility.To understand how large this difference in the implied volatility is, we can assume that thevega traders maintain ranges between ± $10 million. With 3% difference in volatility as shown inTable 4, if they are short $10 million of vega, their mark-to-market loss would be -$30 million. Iftheir goal is to raise $100 million of profit in a year, then this loss already corresponds to 30% ofthe annual target.Figure 1 and Figure 2 show a simulation of the processes of the stock price and the volatility.As mentioned in Section 1, this model prices-in not only the initial impact when some bigposition is traded with clients, but also the adjusted impact due to the change in the risk of themarket driver. The risks change as the market moves, therefore the way traders hedge optionschanges under the new model. This is reflected in the graphs of the delta, vanna, and volga riskscalculated in the new model compared to the ones calculated in the Heston model in Figure 3.The difference in each risk is plotted in Figure 4.For example, when we check the delta on Table 2 and Table 3, the values are higher in the newmodel. This is because traders lose money when the stock price goes higher. To explain this inmore detail, when the stock price goes higher, the vega of the 120 call gets larger since the stockprice gets closer to the strike 120. This makes the traders in the OTC market get shorter in vega,hence they will even be more eager to buy the volatility in the market. This shifts the volatilityhigher. The consequence of this is that the traders will lose in mark-to-market because the valueof the call they are short is greater now due to the spike in volatility. The new model anticipatesthis and asks the traders to buy more stocks beforehand so that they are hedged from this event. MARKET DRIVER VOLATILITY MODEL VIA PIA 15
Stock Price and Vega time p r i c e v ega Figure 1.
Simulation of the SDEs (2.4) for the first 6 months starting from S =100 and v = 0 .
04 with F being the value of the 2Y 120 call. We plotted boththe stock price processes of the Heston (dotted line) and of the new model (solidline) on the upper half of the graph. The largest difference in absolute value of therealizations of the two price processes is 0.7336, which corresponds to 73.36 basispoints to the initial stock price. We used the drift µ = 0 .
05. The lower graphshows how the vega of the call in the new model changes over time.We now see what happens when we apply the PIA to the semilinear case in calculating thevalue of 120 call. We take π ≡ × − Table 5.
Largest differences in absolute value between the numerical solutions ofthe approximated linear PDE and the original semilinear PDE. The figures couldbe regarded as the differences in percentage against the initial price of the stockas it is set to 100.In Figure 6, we show a magnification of Figure 5 centered around the stock price where we sawthe largest difference, which happened to be at-the-money.
Volatility time v o l ( % ) Figure 2.
The volatility processes on the same simulation as in Figure 1, wherethe dotted line corresponds to that of the Heston model and the solid line to thatof the new model.We see in Figure 6 that the numerical solution of the semilinear PDE is different from that ofthe Heston model (0th iteration), but the 1st iteration in the PIA already brings the solution veryclose to that of the semilinear PDE. This is also implied by the result from Table 5. This meansthat the numerical solution of the semilinear PDE is well approximated by a series of linear PDEs.This is good news as we don’t have to create a separate program to calculate the solution to thenew model, but can just reuse the same program for the Heston model with modified coefficients.The PIA also appears to have better convergence compared to the explicit FDM on a Dirichletboundary value problem of a second order semilinear elliptic PDE.7.
Conclusions
We introduced a new model which reflects the impact of a large position that is skewing thevolatility market. We also introduced the Policy Improvement Algorithm. The algorithm lets ushandle a semilinear PDE as a series of linear PDEs and at the same time keep the calculationload similar to that when we run the FDM on the original semilinear problem, thanks to thefast convergence of the iterations. This enables us to easily implement the new model in practiceby reusing the resources used for the Heston model which has already been widely used in theindustry.
MARKET DRIVER VOLATILITY MODEL VIA PIA 17
Delta (%)
Stock Price D e l t a ( % ) Vanna
Stock Price V ega Volga
Variance V ega Delta (%)
Stock Price D e l t a ( % ) Vanna
Stock Price V ega Volga
Variance V ega Figure 3.
Risks of the calls; Top 3 charts are for 120 call and the bottom 3 arefor ATM call. The solid lines indicate the risks calculated in the new model andthe dotted line the corresponding risks calculated in the Heston model.We only used a single product as a market driver, but we might try to extend this to the casewhen it is of a portfolio of several products. We only used a plain vanilla option as the marketdriver, but we should also be able to extend the model to be used for more exotic options. Thedifficulty then is to show the existence and uniqueness of the solution to the semilinear PDE(2.8) and to check if the solution satisfies the positive variance condition (2.6). If so, then bysubstituting this solution in the coefficient of the linear PDE (2.7), we can solve for the values ofother derivatives products as in the case of the Heston model. It only takes relatively small effortto allow for the market asymmetry and to get the correct risks driven by the market driver.The other difficulty in applying the model to actual trading appears in the calibration process.We assumed that we knew all the parameters including the detail of the market driver, but it maybe challenging to recover these in the actual market, especially with more freedom in the modelthan in the Heston model and with limited market information. - Delta (%)
Stock Price D e l t a ( % ) - - Vanna
Stock Price V ega - - - - Volga
Variance V ega - - Delta (%)
Stock Price D e l t a ( % ) - - - Vanna
Stock Price V ega - - - - Volga
Variance V ega Figure 4.
The difference plotted between the values in the new model and theHeston model from Figure 3.
Acknowledgements
We would like to thank Aleksandar Mijatovi´c for reading the draft and providing helpful sug-gestions and insights.
References
Finite Difference Methods in Financial Engineering: A Partial Differential Equation Ap-proach . John Wiley & Sons, Inc..[3] Evans, L.C. 2010.
Partial Differential Equations , American Mathematical Society, second edition.[4] Fleming, W.H. and Soner, H.M. 1993.
Controlled Markov Processes and Viscosity Solutions , Springer-Verlag.[5] Frey, R. and Stremme, A. 1997.
Market Volatility and Feedback Effects from Dynamic Hedging , MathematicalFinance, vol. 7(4), pp. 351-374.[6] Friedman, A. 1964.
Partial Differential Equations of Parabolic Type , Prentice-Hall.[7] Heston, S.L. 1993.
A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bondand Currency Options , Review of Financial Studies, vol. 6(2), pp. 327-343.[8] Hull, J.C. 2012.
Options, Futures, and Other Derivatives . Pearson, eighth edition.
MARKET DRIVER VOLATILITY MODEL VIA PIA 19
Value of 120 Call
Stock Price V a l ue Heston1st Iteration2nd Iteration3rd Iteration4th Iteration
Figure 5.
PIA results for 120 call. Dotted line is the solution using Finite Dif-ference Method (FDM) directly on the semilinear PDE. v is taken as v = 0 . [9] Ikeda, N. and Watanabe, S. 1981. Stochastic Differential Equations and Diffusion Processes , North Holland,North-Holland Mathematical Library.[10] Jacka, S.D., Mijatovi´c, A., and ˇSiraj, D.
Policy Improvement Algorithm for Continuous Finite Horizon Problem ,forthcoming.[11] Jacka, S.D., Mijatovi´c, A., and ˇSiraj, D.
Policy Improvement Algorithm for Controlled Multidimensional Dif-fusion Processes , forthcoming.[12] Karatzas, I. and Shreve, S.E. 1998.
Brownian Motion and Stochastic Calculus , Springer Science+BusinessMedia, Inc., second edition.[13] Karlin, S. and Taylor, H.M. 1981.
A Second Course in Stochastic Processes , Academic Press Inc.[14] Ladyˇ z enskaja, O.A., Solonnikov, V.A., and Ural’ceva, N.N. 1968. Linear and Quasi-linear Equations of Para-bolic Type , American Mathematical Society.[15] Lieberman, G.M. 1996.
Second Order Parabolic Differential Equations . World Scientific Publishing Co. Pte.Ltd.[16] Lin, S. 2008.
Finite Difference Schemes for Heston Model . Master’s thesis.University of Oxford.[17] Lindvall, T. and Rogers, L.C.G. 1986.
Coupling of Multidimensional Diffusions by Reflection , The Annals ofProbability, vol. 14(3), pp. 860-872.[18] Lord, R., Koekkoek, R., and Dijk, D. 2010.
A Comparison of Biased Simulation Schemes for Stochastic VolatilityModels , Quantitative Finance, vol. 10(2), pp. 177-194.[19] Platen, E. and Schweizer, M. 1998.
On Feedback Effects from Hedging Derivatives , Mathematical Finance, vol.8(1), pp. 67-84.[20] Rockafellar, R.T. 1970.
Convex Analysis , Princeton University Press.[21] Sircar, K.R. and Papanicolaou, G. 1998.
General Black-Scholes Models Accounting for Increased Market Volatil-ity from Hedging Strategies , Applied Mathematical Finance, vol. 5(1), pp. 45-82.
80 90 100 110 120
Value of 120 Call
Stock Price V a l ue Heston1st Iteration2nd Iteration3rd Iteration4th Iteration
Figure 6.
Magnification around at-the-money of Figure 5. We see that the firstiteration already approximates well the numerical solution to the semilinear PDE. [22] Smith, G.D. 1985.
Numerical Solution of Partial Differential Equations: Finite Difference Methods . OxfordApplied Mathematics and Computing Science Series. Oxford, third edition.[23] Soner, H. 2007.
Stochastic Representations for Nonlinear Parabolic PDEs in Handbook of Differential Equations ,Evolutionary Equations, vol. 3, Chapter 6, Elsevier B.V.[24] Tavella, D., and Randall, C. 2000.
Pricing Financial Instruments: The Finite Difference Method . John Wiley& Sons, Inc.
Appendix. A. Lemmas.
The lemmas stated here are more or less those in [10] and [11]. We only modifythem to fit our problem. We show them here, however, so that this paper is self-contained.A property that forms the basis of the following lemmas is that processes controlled by Markovpolicies are strong Markov processes (Theorem 4.20 in [12]).
Lemma A.1.
For every Markov policy π , z ∈ E , < t < T , and any stopping time S that isalmost surely less than t ∧ τ Ω , E (cid:18) (cid:90) t ∧ τ e − rs f π ( Z z,πs , t − s ) ds + e − r ( t ∧ τ ) g ( Z z,πt ∧ τ , t ∧ τ ) (cid:12)(cid:12)(cid:12)(cid:12) F S (cid:19) = (cid:90) S e − rs f π ( Z z,πs , t − s ) ds + e − r S V g, E ,π ( Z z,π S , t − S ) . (A.1) MARKET DRIVER VOLATILITY MODEL VIA PIA 21
In particular, the process ( (cid:82) T (cid:48) e − rs f π ( Z z,πs , t − s ) ds + e − rT (cid:48) V g, E ,π ( Z z,πT (cid:48) , T (cid:48) )) T (cid:48) ≤ T is a uniformly integrable martingale.Proof. Let τ = τ E ( Z z,π ) and τ S := τ ◦ θ S = τ E ( Z z,π · + S ), where θ is the shift operator. Then τ S = τ −S holds almost surely, and we obtain E (cid:18) (cid:90) t ∧ τ e − rs f π ( Z z,πs , t − s ) ds + e − r ( t ∧ τ ) g ( Z z,πt ∧ τ , t ∧ τ ) (cid:12)(cid:12)(cid:12)(cid:12) F S (cid:19) = (cid:90) S e − rs f π ( Z z,πs , t − s ) ds + E (cid:18) (cid:90) t ∧ τ S e − rs f π ( Z z,πs , t − s ) ds + e − r ( t ∧ τ ) g ( Z z,πt ∧ τ , t ∧ τ ) (cid:12)(cid:12)(cid:12)(cid:12) F S (cid:19) = (cid:90) S e − rs f π ( Z z,πs , t − s ) ds + E (cid:18) (cid:90) t ∧ τ −S e − r ( s + S ) f π ( Z z,πs + S , t − ( s + S )) ds + e − r (( t −S ) ∧ ( τ −S )+ S ) g (cid:0) Z z,π ( t −S ) ∧ ( τ −S )+ S , ( t − S ) ∧ ( τ − S ) + S , (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) F S (cid:19) = (cid:90) S e − rs f π ( Z z,πs , t − s ) ds + e − r S E (cid:18) (cid:90) ( t −S ) ∧ τ S e − rs f π ( Z z,πs + S , t − ( s + S )) ds + e − r (( t −S ) ∧ τ S ) g (cid:0) Z z,π ( t −S ) ∧ τ S + S , ( t − S ) ∧ τ S + S , (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) F S (cid:19) = (cid:90) S e − rs f π ( Z z,πs , t − s ) ds + e − r S E x (cid:18)(cid:26) (cid:90) ( t −S ) ∧ τ e − rs f π ( Z z,πs , t − ( s + S )) ds + e − r ( t −S ) ∧ τ g (cid:0) Z z,π ( t −S ) ∧ τ , ( t − S ) ∧ τ (cid:1)(cid:27) ◦ θ S (cid:12)(cid:12)(cid:12)(cid:12) F S (cid:19) = (cid:90) S e − rs f π ( Z z,πs , t − s ) ds + e − r S E Z z,π S (cid:18) (cid:90) ( t −S ) ∧ τ e − rs f π ( Z z,πs , t − ( s + S )) ds + e − r ( t −S ) ∧ τ g (cid:0) Z z,π ( t −S ) ∧ τ , ( t − S ) ∧ τ (cid:1)(cid:19) = (cid:90) S e − rs f π ( Z z,πs , t − s ) ds + e − r S V g, E ,π ( Z z,π S , t − S ) ♦ By taking expectation on both sides of (A.1), we retrieve a corollary which is so-called Bellman’sprinciple.
Corollary A.2.
For every Markov policy π , z ∈ E , < t < T , and stopping time S which isalmost surely less than or equal to t ∧ τ Ω , V g, E ,π ( z, t ) = E (cid:18) (cid:90) S e − rs f π ( Z z,πs , t − s ) ds (cid:19) + e − r S E ( V g, E ,π ( Z z,π S , t − S )) . (A.2)We now use the method of mirror coupling [17]. Lemma A.3.
For every Lipschitz Markov control and small enough (cid:15) > , there exists δ > suchthat the following holds for every z , z ∈ E : if (cid:107) z − z (cid:107) < δ then there exist processes ˜ Z z ,π and ˜ Z z ,π that have the same laws as Z z ,π and Z z ,π respectively such that (cid:107) ˜ Z z ,πt − ˜ Z z ,πt (cid:107) ≤ G τ t on t < ρ δ and ˜ Z z ,πt = ˜ Z z ,πt on t ≥ ρ for every t ≥ , where ρ c := inf (cid:8) t ≥ ; (cid:107) ˜ Z z ,π − ˜ Z z ,π (cid:107) = c (cid:9) , (inf φ = ∞ ) for any c ≥ , G is the squared Bessel process of dimension (cid:15) started at (cid:107) z − z (cid:107) , and ( τ t ) t ≥ is a stochastic time change with the property τ t ≤ tν , t ≥ . For the proof of Lemma A.3, we refer to [11].
Lemma A.4.
For every Lipschitz Markov policy π , the function V g, E ,π ( · , t ) is continuous withbounded initial condition.Proof. Let (cid:15) > δ ≤ δ . For (cid:107) z − z (cid:107) ≤ ˆ δ , we calculate | V g, E ,π ( z , t ) − V g, E ,π ( z , t ) | . | V g, E ,π ( z , t ) − V g, E ,π ( z , t ) | = (cid:12)(cid:12)(cid:12)(cid:12) E (cid:18) (cid:90) t ∧ τ z e − rs f π ( ˜ Z z ,πs , t − s ) ds + e − r ( t ∧ τ z ) g ( ˜ Z z ,πt ∧ τ z , t ∧ τ z ) (cid:19) − E (cid:18) (cid:90) t ∧ τ z e − rs f π ( ˜ Z z ,πs , t − s ) ds + e − r ( t ∧ τ z ) g ( ˜ Z z ,πt ∧ τ z , t ∧ τ z ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) E (cid:18) (cid:90) ρ e − rs (cid:8) f π ( ˜ Z z ,πs , t − s ) − f π ( ˜ Z z ,πs , t − s ) (cid:9) ds (cid:12)(cid:12)(cid:12)(cid:12) I ρ ≤ ρ δ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) E (cid:18) (cid:90) tρ e − rs (cid:8) f π ( ˜ Z z ,πs , t − s ) − f π ( ˜ Z z ,πs , t − s ) (cid:9) ds + e − rt (cid:8) g ( ˜ Z z ,πt , t ) − g ( ˜ Z z ,πt , t ) (cid:9)(cid:12)(cid:12)(cid:12)(cid:12) I ρ ≤ ρ δ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) E (cid:18) (cid:90) t e − rs (cid:8) f π ( ˜ Z z ,πs , t − s ) − f π ( ˜ Z z ,πs , t − s ) (cid:9) ds + e − rt (cid:8) g ( ˜ Z z ,πt , t ) − g ( ˜ Z z ,πt , t ) (cid:9)(cid:12)(cid:12)(cid:12)(cid:12) I ρ >ρ δ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = B + B + B .(A.3)For B , since f π is Lipschitz continuous, we can take δ ∈ (0 , δ ) small enough such that(A.4) B < C (cid:107) ˜ Z z ,πs − ˜ Z z ,πs (cid:107) < (cid:15)/ MARKET DRIVER VOLATILITY MODEL VIA PIA 23
For B , due to the definition of ˜ Z , the processes ˜ Z z ,πt and ˜ Z z ,πt take the same values in thistime frame in consideration, so B = 0.Due to the boundedness of f π and g , the last term B could be bounded by some constantmultiplied by P ( ρ > ρ δ ). If we denote by ρ δ ( Y ) and ρ ( Y ) the first hitting times of the levels δ (cid:48) and 0 respectively for any process Y , we have from Lemma A.3 P ( ρ δ < ρ ) ≤ P (cid:18) ρ δ ( G τ ) < ρ ( G τ ) (cid:19) ≤ P (cid:18) ρ δ (cid:18) G ν (cid:19) < ρ (cid:18) G ν (cid:19)(cid:19) .(A.5)Using the scale property of the squared Bessel process we get P (cid:18) ρ δ (cid:18) G ν (cid:19) < ρ (cid:18) G ν (cid:19)(cid:19) = P (cid:18) ρ δ (cid:18) ν G (cid:19) < ρ (cid:18) ν G (cid:19)(cid:19) = P ( ρ ν δ ( G ) < ρ ( G )).(A.6)Recall that the scale function of the Bessel process with dimension 1+ (cid:15) is given by s ( z ) := z − (cid:15) ,and that the process G starts at (cid:107) z − z (cid:107) < ˆ δ . Hence we obtain(A.7) P ( ρ ν δ ( G ) < ρ ( G )) = s ( (cid:107) z − z (cid:107) ) − s (0) s ( ν δ ) ≤ (cid:18) ˆ δν δ (cid:19) − (cid:15) .We set δ = δ and take ˆ δ ∈ (0 , δ ) small enough so that(A.8) 2 C (cid:18) ˆ δν δ (cid:19) − (cid:15) < (cid:15) (cid:107) z − z (cid:107) < ˆ δ implies | V g, E ,π ( z , t ) − V g, E ,π ( z , t ) | < (cid:15) , so we have uniform continuity of V g, E ,π ( · , t ). ♦ Lemma A.5.
For every Lipschitz Markov policy π , the function V g, E ,π is continuous.Proof. If we proved the continuity of V g, E ,π with respect to t for fixed z , the statement is provedusing the triangle inequality and Lemma A.4. Therefore, we prove the continuity in t with fixed z . Due to Corollary A.2, we have V g, E ,π ( z, t + δ ) − V g, E ,π ( z, t ) = E (cid:18) (cid:90) δ e − rs f π ( Z z,πs , t − s ) ds (cid:19) + e − rδ E (cid:18) V g, E ,π ( Z z,πδ , t ) − e rδ V g, E ,π ( z, t ) (cid:19) .(A.9)Applying Lemma A.4, we obtain(A.10) | V g, E ,π ( z, t + δ ) − V g, E ,π ( z, t ) | ≤ Cδ + C (cid:48) E ( (cid:107) Z z,πδ − z (cid:107) ). The SDE for Z z,πδ yields Z z,πδ − z = (cid:90) δ µ π ( Z z,πs , s ) ds + (cid:90) δ σ π ( Z z,πs , s ) dW s .(A.11)Therefore, we have | V g, E ,π ( z, t + δ ) − V g, E ,π ( z, t ) | ≤ Cδ + C (cid:48) E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) δ µ π ( Z z,πs , s ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + C (cid:48)(cid:48) E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) δ σ π ( Z z,πs , s ) dW s (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) .(A.12)The second term on RHS can be bounded by some multiple of δ as µ π is bounded. For the lastterm, using Jensen’s inequality and Burkholder-Davis-Gundy inequality, E (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) δ σ π ( Z z,πs , s ) dW s (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ (cid:18) E (cid:18) (cid:90) δ σ π ( Z z,πs , s ) dW s (cid:19) (cid:19) (cid:46) (cid:18) E (cid:18) (cid:90) δ σ π ( Z z,πδ , s ) ds (cid:19)(cid:19) .(A.13)This proves the continuity of V g, E ,π with respect to t with fixed z . Therefore, the continuity of V g, E ,π is proved. ♦ University of Warwick, Coventry, United Kingdom
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