Recipes for hedging exotics with illiquid vanillas
aa r X i v : . [ q -f i n . T R ] M a y Recipes for hedging exotics with illiquid vanillas ∗Joaquin Fernandez-Tapia † , Olivier Guéant ‡ Abstract
In this paper, we address the question of the optimal Delta and Vega hedging ofa book of exotic options when there are execution costs associated with the trading ofvanilla options. In a framework where exotic options are priced using a market model(e.g. a local volatility model recalibrated continuously to vanilla option prices) and vanillaoptions prices are driven by a stochastic volatility model, we show that, using simpleapproximations, the optimal dynamic Delta and Vega hedging strategies can be computedeasily using variational techniques.
The classical theory to price and manage derivatives contracts is based on the simplifyingassumption of a frictionless market in which traded assets are liquid and agents incur notrading costs. Although this assumption is not realistic, it has led to very powerful pricingmodels that have been used on trading floors for more than 40 years. In spite of their success,classical pricing models need to be amended when reality is too far from classical modellingassumptions, in particular when there are transaction costs (e.g. a large bid-ask spread) orwhen the liquidity of the underlying assets is limited.Classical option pricing models have been improved to account for transaction costs. One ofthe first models to include transaction costs is that of Leland [24] who proposed an amendmentto the seminal model of Black and Scholes through a change in the volatility parameter totake account of both the transaction costs and the frequency of hedging. Several incompletemarket approaches have been discussed in the literature for taking account of transactioncosts. The super-replication approach, for instance, has been shown to be of no help to dealwith transaction costs (see [14, 22, 34]). Most authors introduced therefore utility functions ∗ The authors would like to thank the Research Initiative “Modélisation des marchés actions, obligationset dérivés” financed by HSBC France under the aegis of the Europlace Institute of Finance for their sup-port regarding an early version of the paper. The authors would also like to thank Bastien Baldacci (EcolePolytechnique), Philippe Bergault (Université Paris 1 Panthéon-Sorbonne), François Bouscarle (HSBC), Ar-naud Gocsei (HSBC), Nicolas Grandchamp des Raux (HSBC), Greg Molin (HSBC), Jean Nguyen (HSBC),and Jiang Pu (Institut Europlace de Finance) for the discussions they had on the topic. The readers shouldnevertheless be aware that the views, thoughts, and opinions expressed in the text belong solely to the authors. † Institut Europlace de Finance. 28, place de la Bourse, 75002 Paris, France. ‡ Université Paris 1 Panthéon-Sorbonne. Centre d’Economie de la Sorbonne. 106, boulevard de l’Hôpital,75013 Paris, France. Corresponding author. email: [email protected]
1o tackle the questions of the pricing and hedging of contingent claims in presence of trans-action costs. Interesting examples include the paper of Barles and Soner [6] who obtainedan elegant formula for the price using a modification of the implied volatility, the paper ofConstantinides and Zariphopoulou [12] who obtained bounds on option prices or the paperof Cvitanic and Karatzas [14]. Alternative approaches include also quantile hedging or theminimization of classical risk measures (see [15] and [16] for an introduction to these incom-plete market methods).In addition to the literature dealing with transaction costs, there is an interesting literaturedealing with execution costs (and market impact). The classical paper by Çetin, Jarrow,and Protter [8] (see also [5] and [9, 10]) belongs to this category, although the authors donot phrase their approach in these terms. Their trader is not price-taker and the price shepays depends on the quantity she trades. Although it is very interesting, the main drawbackof this framework is that it leads to prices identical to those of the Black-Scholes model.Çetin, Soner, and Touzi improved this approach in [11] by adding a restriction to the spaceof admissible strategies (see also [25]), and they obtained positive liquidity costs and pricesthat depart from those of Black and Scholes. By considering absolutely continuous hedg-ing strategies, a recent literature, inspired by the literature on optimal execution (see [1, 2]and [17]), obtained new results for the Delta hedging of options when liquidity has to be takeninto account. Articles in this category include that of Rogers and Singh [29], and the papersof Almgren and Li [3] and Guéant and Pu [19] motivated by the observations of saw-toothpatterns on stocks the day of expiry of some options (see [23]). All the previously discussed papers deal with Delta hedging when the underlying asset(s)is/are illiquid. When it comes to exotic equity derivatives, the hedging process does not onlyinvolve stocks or futures but rather stocks or futures and options, typically vanilla options.In this paper, we address the question of the optimal Delta and Vega hedging of a book ofexotic derivatives under the assumption that the underlying asset is liquid but that tradingvanilla options is costly. More precisely, we consider that exotic derivatives are valued usinga market model and that vanilla options, whose price dynamics is driven by a stochasticvolatility model, can be traded, with execution costs, using absolutely continuous strategiesas in the literature on optimal execution.In a mean-variance setting, using simple approximations, we manage to write the optimalhedging strategy of the trader as the solution of a deterministic variational problem involvingthree kinds of terms: terms to penalize fast execution (execution costs), terms modeling theVega risk associated with the portfolio, and terms to profit from the trader’s view on themarket. When execution costs are quadratic as in the original paper of Almgren and Chriss(see [2]), our main result is a closed-form representation of the optimal hedging strategy intwo different problems: one in which the portfolio is progressively hedged in the stochasticvolatility model and another one in which we impose a complete unwinding of the risk in themarket model while hedging in the stochastic volatility model. See also [27], [32], and [33] for the feedback effect of hedging on option prices.
We consider a probability space (Ω , F , P (cid:1) with a filtration ( F t ) t ∈ R + satisfying the usualconditions. Throughout the paper, we assume that all stochastic processes are defined on (cid:0) Ω , F , ( F t ) t ∈ R + , P (cid:1) . We consider an asset whose price dynamics is described by a one-factor stochastic volatilitymodel of the form ( dS t = µ t S t dt + √ ν t S t dW S, P t dν t = a P ( t, ν t ) dt + ξ √ ν t dW ν, P t , where ξ ∈ R + ∗ , ( W S, P t , W ν, P t ) t ∈ R + is a couple of Brownian motions with quadratic covariationgiven by ρ = d h W S, P ,W ν, P i t dt ∈ ( − , µ t ) t ∈ R + andthe function a P are such that the processes are well defined (in particular, we assume thatthe process ( ν t ) t ∈ R + stays positive almost surely). Remark 1.
A classical example for the function a P is that of the Heston model. In that case, a P : ( t, ν ) κ P ( θ P − ν ) where κ P , θ P ∈ R + satisfy the Feller condition κ P θ P > ξ (see [20]). Assuming interest rates are equal to 0, we introduce a risk-neutral / pricing probabilitymeasure Q equivalent to P under which the price and volatility processes become ( dS t = √ ν t S t dW S, Q t dν t = a Q ( t, ν t ) dt + ξ √ ν t dW ν, Q t , where ( W S, Q t , W ν, Q t ) t ∈ R + is another couple of Brownian motions, this time under Q , withquadratic covariation given by ρ = d h W S, Q ,W ν, Q i t dt ∈ ( − , a Q is such that theprocesses are well defined.We consider a set of N ≥ P t at time t is assumed to be a function Π of the time, the priceof the underlying asset, and the price of the N vanilla options.3enoting the price of the i -th vanila option at time t by O it , we know that O it = Ω i ( t, S t , ν t )where Ω i is solution of the following partial differential equation (PDE):0 = ∂ Ω i ∂t ( t, S, ν ) + ∂ Ω i ∂ν ( t, S, ν ) a Q ( t, ν )+ 12 ∂ Ω i ∂S ( t, S, ν ) νS + 12 ∂ Ω i ∂ν ( t, S, ν ) ξ ν + ∂ Ω i ∂ν∂S ( t, S, ν ) ρξνS, for ( t, S, ν ) ∈ [0 , T i ) × R +2 where T i is the maturity of the i -th option.Therefore, for i ∈ { , . . . , N } and t < T i , the dynamics of the i -th vanilla option is givenby dO it = ∂ Ω i ∂t ( t, S t , ν t ) dt + ∂ Ω i ∂S ( t, S t , ν t ) dS t + ∂ Ω i ∂ν ( t, S t , ν t ) dν t + 12 ∂ Ω i ∂S ( t, S t , ν t ) d h S, S i t + 12 ∂ Ω i ∂ν ( t, S t , ν t ) d h ν, ν i t + ∂ Ω i ∂ν∂S ( t, S t , ν t ) d h S, ν i t = ∂ Ω i ∂t ( t, S t , ν t ) dt + ∂ Ω i ∂S ( t, S t , ν t ) µ t S t dt + ∂ Ω i ∂S ( t, S t , ν t ) √ ν t S t dW S, P t + ∂ Ω i ∂ν ( t, S t , ν t ) a P ( t, ν t ) dt + ∂ Ω i ∂ν ( t, S t , ν t ) ξ √ ν t dW ν, P t + 12 ∂ Ω i ∂S ( t, S t , ν t ) ν t S t dt + 12 ∂ Ω i ∂ν ( t, S t , ν t ) ξ ν t dt + ∂ Ω i ∂ν∂S ( t, S t , ν t ) ρξν t S t dt = ∂ Ω i ∂S ( t, S t , ν t ) µ t S t + ∂ Ω i ∂ν ( t, S t , ν t ) (cid:16) a P ( t, ν t ) − a Q ( t, ν t ) (cid:17)! dt + ∂ Ω i ∂S ( t, S t , ν t ) √ ν t S t dW S, P t + ∂ Ω i ∂ν ( t, S t , ν t ) ξ √ ν t dW ν, P t . The resulting dynamics for the value of the book of exotic options is dP t = ∂ Π ∂S ( t, S t , O t ) + X i ∂ Π ∂O i ( t, S t , O t ) ∂ Ω i ∂S ( t, S t , ν t ) ! √ ν t S t dW S, P t + X i ∂ Π ∂O i ( t, S t , O t ) ∂ Ω i ∂ν ( t, S t , ν t ) ξ √ ν t dW ν, P t + ∂ Π ∂t ( t, S t , O t ) dt + ∂ Π ∂S ( t, S t , O t ) µ t S t dt + X i ∂ Π ∂O i ( t, S t , O t ) ∂ Ω i ∂S ( t, S t , ν t ) µ t S t + ∂ Ω i ∂ν ( t, S t , ν t ) (cid:16) a P ( t, ν t ) − a Q ( t, ν t ) (cid:17)! dt + 12 ∂ Π ∂S ( t, S t , O t ) ν t S t dt + 12 X i X i ′ ∂ Π ∂O i ∂O i ′ ( t, S t , O t ) ∂ Ω i ∂ν ( t, S t , ν t ) ∂ Ω i ′ ∂ν ( t, S t , ν t ) ξ ν t dt + X i ∂ Π ∂O i ∂S ( t, S t , O t ) ∂ Ω i ∂S ( t, S t , ν t ) ν t S t + ρ ∂ Ω i ∂ν ( t, S t , ν t ) ξν t S t ! dt. .2 Trading strategies and objective function We now consider a trader in charge of hedging the book of exotic options over a short period oftime [0 , T ] where T is significantly smaller than min i T i . For that purpose, she can trade theunderlying asset with no friction but has to pay execution costs to trade the vanilla options.In what follows, we denote by q St the number of underlying assets held at time t and by q it the number of i -th vanilla options held in the portfolio. For i ∈ { , . . . , N } , we assume that dq it = v it dt and that the running costs paid (in addition to the MtM value) to trade the i -thoption at velocity v it is given by L i ( v it ) where L i satisfies the classical assumptions of executioncosts functions:• L i (0) = 0,• L i is increasing on R + and decreasing on R − ,• L i is strictly convex,• L i is asymptotically superlinear, that is lim ρ → + ∞ L i ( ρ ) ρ = + ∞ . The dynamics of the trader’s PnL is therefore d PnL t = q St dS t + X i q it dO it − X i L i ( v it ) dt + dP t = q St µ t S t dt + X i q it ∂ Ω i ∂S ( t, S t , ν t ) µ t S t + ∂ Ω i ∂ν ( t, S t , ν t ) (cid:16) a P ( t, ν t ) − a Q ( t, ν t ) (cid:17)! dt − X i L i ( v it ) dt + (cid:18) q St + ∂ Π ∂S ( t, S t , O t ) (cid:19) + X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂S ( t, S t , ν t ) ! √ ν t S t dW S, P t + X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) ξ √ ν t dW ν, P t + ∂ Π ∂t ( t, S t , O t ) dt + ∂ Π ∂S ( t, S t , O t ) µ t S t dt + X i ∂ Π ∂O i ( t, S t , O t ) ∂ Ω i ∂S ( t, S t , ν t ) µ t S t + ∂ Ω i ∂ν ( t, S t , ν t ) (cid:16) a P ( t, ν t ) − a Q ( t, ν t ) (cid:17)! dt + 12 ∂ Π ∂S ( t, S t , O t ) ν t S t dt + 12 X i X i ′ ∂ Π ∂O i ∂O i ′ ( t, S t , O t ) ∂ Ω i ∂ν ( t, S t , ν t ) ∂ Ω i ′ ∂ν ( t, S t , ν t ) ξ ν t dt + X i ∂ Π ∂O i ∂S ( t, S t , O t ) ∂ Ω i ∂S ( t, S t , ν t ) ν t S t + ρ ∂ Ω i ∂ν ( t, S t , ν t ) ξν t S t ! dt. (1)Ideally, we would like to maximize an objective function of the form E [PnL T ] − γ V [PnL T ]over a constrained set of trading strategies. 5ssuming that the trading strategies are such that the local martingales involved in (1) aremartingales, we have that the part of E [PnL T ] that depends on q S and ( q i ) i is: E "Z T q St µ t S t + X i q it ∂ Ω i ∂S ( t, S t , ν t ) µ t S t + ∂ Ω i ∂ν ( t, S t , ν t ) (cid:16) a P ( t, ν t ) − a Q ( t, ν t ) (cid:17)!! dt − E "Z T X i L i ( v it ) dt . When it comes to the variance of the PnL, the computation is however far more cumbersome.For that reason, we consider an expansion of V [PnL T ] for T small: V [PnL T ]= V "Z T (cid:18) q St + ∂ Π ∂S ( t, S t , O t ) (cid:19) + X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂S ( t, S t , ν t ) ! √ ν t S t dW S, P t + Z T X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) ξ √ ν t dW ν, P t + o ( T )= E Z T (cid:18) q St + ∂ Π ∂S ( t, S t , O t ) (cid:19) + X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂S ( t, S t , ν t ) ! ν t S t dt + E Z T X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) ! ξ ν t dt + 2 ρ E "Z T (cid:18) q St + ∂ Π ∂S ( t, S t , O t ) (cid:19) + X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂S ( t, S t , ν t ) ! × X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) ! ξν t S t dt + o ( T ) . Therefore, to the first order in T , we can approximate our problem by the minimization, over q S and ( v i ) i in a set to be specified, of E " − Z T q St µ t S t + X i q it ∂ Ω i ∂S ( t, S t , ν t ) µ t S t + ∂ Ω i ∂ν ( t, S t , ν t ) (cid:16) a P ( t, ν t ) − a Q ( t, ν t ) (cid:17)!! dt + E "Z T X i L i ( v it ) dt + 12 γ E Z T (cid:18) q St + ∂ Π ∂S ( t, S t , O t ) (cid:19) + X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂S ( t, S t , ν t ) ! ν t S t dt + E Z T X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) ! ξ ν t dt + 2 ρ E "Z T (cid:18) q St + ∂ Π ∂S ( t, S t , O t ) (cid:19) + X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂S ( t, S t , ν t ) ! × X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) ! ξν t S t dt . u t = (cid:18) q St + ∂ Π ∂S ( t, S t , O t ) (cid:19) + X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂S ( t, S t , ν t ) ! √ ν t S t , our problem boils down to minimizing over u and ( v i ) i in a set to be specified the expres-sion E "Z T − µ t u t √ ν t + µ t S t ∂ Π ∂S ( t, S t , O t ) + X i ∂ Π ∂O i ( t, S t , O t ) ∂ Ω i ∂S ( t, S t , ν t ) !! dt + E "Z T − X i q it ∂ Ω i ∂ν ( t, S t , ν t ) (cid:16) a P ( t, ν t ) − a Q ( t, ν t ) (cid:17)! + X i L i ( v it ) ! dt + 12 γ E Z T u t + X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) ! ξ ν t + 2 ρu t X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) ξ √ ν t dt . If there is no constraint on the process ( u t ) t , and if we consider that the process ( q t ) t =( q t , . . . , q Nt ) t is given, then the optimal value of u t must verify u ∗ t = µ t γ √ ν t − ρξ √ ν t X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) , i.e. q S ∗ t = µ t γν t S t − ∂ Π ∂S ( t, S t , O t ) + X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂S ( t, S t , ν t ) + ρξS t ∂ Ω i ∂ν ( t, S t , ν t ) !! . This formula has two components: (i) the term µ t γν t S t that corresponds to the optimal numberof underlying assets to hold in a pure dynamic mean-variance portfolio choice problem à laMerton, given the trader’s view (see [26] for instance), and (ii) a Delta term that correspondsto the Delta of the portfolio in our model with a correction term taking account of the possi-bility to Delta-hedge part of the Vega in the stochastic volatility model whenever the vol-spotcorrelation parameter ρ is not equal to 0.Now, plugging u ∗ t in the optimization problem, we see that our objective function writes E "Z T µ t S t ∂ Π ∂S ( t, S t , O t ) + X i ∂ Π ∂O i ( t, S t , O t ) ∂ Ω i ∂S ( t, S t , ν t ) ! dt + E "Z T − X i q it ∂ Ω i ∂ν ( t, S t , ν t ) (cid:16) a P ( t, ν t ) − a Q ( t, ν t ) (cid:17)! + X i L i ( v it ) ! dt + E Z T − µ t γν t + 12 γ (1 − ρ ) X i q it + ∂ Π ∂O i ( t, S t , O t ) ∂ Ω i ∂ν ( t, S t , ν t ) !! ξ ν t + ρµ t ξ X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) dt . E "Z T − X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) (cid:16) a P ( t, ν t ) − a Q ( t, ν t ) (cid:17)! + X i L i ( v it ) ! dt + E Z T γ (1 − ρ ) X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) ! ξ ν t + ρµ t ξ X i (cid:18) q it + ∂ Π ∂O i ( t, S t , O t ) (cid:19) ∂ Ω i ∂ν ( t, S t , ν t ) dt . (2)where ( q t ) t = ( q t , . . . , q Nt ) t has to satisfy constraints to be specified. Although considering a short-time horizon simplifies the objective function, the problem re-mains difficult to address numerically because of its dimensionality. We consider thereforean approximation of the above optimization problem where the terms characterizing the dy-namics of the underlying asset and the Vega terms are freezed over [0 , T ]. More precisely, weconsider the following approximations: • The Sharpe ratio µ t √ ν t is approximated by a constant denoted by s .• The rescaled difference between the drifts of the volatility under P and Q , defined as a P ( t,ν t ) − a Q ( t,ν t ) √ ν t , is approximated by a constant denoted by ζ .• For each i ∈ { , . . . , N } , the Vega of the i -th option in the stochastic volatility model,i.e. ∂ Ω i ∂ √ ν ( t, S t , ν t ) = 2 √ ν t ∂ Ω i ∂ν ( t, S t , ν t ) ( i ∈ { , . . . , N } ), is approximated by a constantdenoted by V i SV .• The Vegas of the exotic portfolio in the market model (with respect the N impliedvolatilities) are constant and equal to V i MM ( i ∈ { , . . . , N } ). Therefore, for each i ∈ { , . . . , N } , ∂ Π ∂O i ( t, S t , O t ) is equal to V i MM V i BS : the ratio of the Vega of the book ofexotic options with respect to the implied volatility associated with the i -th vanilla op-tion in the market model and the Vega of that vanilla option in the Black-Scholes model. Similar approximations are used in the recently published paper [4] to address an option market makingproblem. E Z T X i L i ( v it ) + 18 γ (1 − ρ ) ξ X i q it + V i MM V i BS ! V i SV ! dt + E "Z T
12 ( ρ s ξ − ζ ) X i q it + V i MM V i BS ! V i SV dt . Focusing on deterministic strategies (that can be shown to be optimal by following the samereasoning as in [31] – see also [17]) our objective function (for minimizing) is in fact: Z T X i L i ( v it ) + 18 γ (1 − ρ ) ξ X i q it + V i MM V i BS ! V i SV ! dt + Z T
12 ( ρ s ξ − ζ ) X i q it + V i MM V i BS ! V i SV dt. In what follows, we consider two different problems corresponding to two different sets ofconstraints for the trading strategies. In the first problem, we simply impose the initial stateof the portfolio of vanilla options, i.e. q is given. In the second one, we additionally imposethat the portfolio of vanilla options at time T makes the portfolio hedged in the market model(up to the approximation of constant Vegas), i.e. q T = − v where v := (cid:18) V V , . . . , V N MM V N BS (cid:19) ′ . Inother words, the first problem is a Vega hedging problem in the stochastic volatility modelwhile the second is a Vega Bucket cancellation problem with Vega hedging in the stochasticvolatility model that corresponds better to the real problem faced by traders. Remark 2.
Because of the above approximations, our optimization problems are only mean-ingful over a short period of time. This may be regarded as a problem but it must be notedthat one can use the output of our models over a short period of time and then run the modelagain with updated values of s , ζ , and the Vegas. Although this approach is time-inconsistent,it is a classical practice in applied optimal control, when parameters are estimated online forinstance. Vega hedging in the stochastic volatility model
In the first problem we consider, wesimply impose the initial condition (i.e. q given). Therefore, the problem writesinf ( q t ) t ,q given Z T X i L i ( v it ) + 18 γ (1 − ρ ) ξ X i q it + V i MM V i BS ! V i SV ! dt + Z T
12 ( ρ s ξ − ζ ) X i q it + V i MM V i BS ! V i SV dt. t ∈ [0 , T ] q ∗ ( t ) = ( q ∗ ( t ) , . . . , q ∗ N ( t )) ′ characterized by the following Hamiltonian system: ( ˙ p ( t ) = γ (1 − ρ ) ξ V SV V ′ SV ( q ∗ ( t ) + v ) + ( ρ s ξ − ζ ) V SV , p ( T ) = 0 , ˙ q ∗ i ( t ) = H i ′ ( p i ( t )) , q ∗ (0) = q , (3)where the Hamiltonian functions ( H , . . . , H N ) are defined by ∀ i ∈ { , . . . , N } , H i ( z ) = sup v vz − L i ( v ) . In the case where execution cost functions are quadratic as in the original paper of Almgrenand Chriss [2], we can in fact solve in closed form the system (3). If indeed we have L i ( v ) = η i v for all i ∈ { , . . . , N } , then (3) is the linear system ( ˙ p ( t ) = γ (1 − ρ ) ξ V SV V ′ SV ( q ∗ ( t ) + v ) + ( ρ s ξ − ζ ) V SV , p ( T ) = 0 , ˙ q ∗ ( t ) = Λ p ( t ) , q ∗ (0) = q , or equivalently¨ q ∗ ( t ) = 18 γ (1 − ρ ) ξ Λ V SV V ′ SV q ∗ ( t ) + 18 γ (1 − ρ ) ξ Λ V SV V ′ SV v + 12 ( ρ s ξ − ζ ) Λ V SV (4)with boundary conditions q ∗ (0) = q and ˙ q ∗ ( T ) = 0, where Λ = η . . . η N .In the case of quadratic execution cost functions, the problem therefore boils to a linear or-dinary differential equation of order 2 that can be addressed using standard techniques.For solving this ordinary differential equation, it is interesting to notice that ∀ t ∈ [0 , T ] , ¨ q ∗ ( t ) ∈ span(Λ V SV ) . As ˙ q ∗ ( T ) = 0, we deduce that ∀ t ∈ [0 , T ] , ˙ q ∗ ( t ) ∈ span(Λ V SV ) . Since q ∗ (0) = q , there exists a C function α such that ∀ t ∈ [0 , T ] , q ∗ ( t ) = q + α ( t )Λ V SV .Using (4), we obtain¨ α ( t ) = 18 γ (1 − ρ ) ξ V ′ SV Λ V SV α ( t ) + 18 γ (1 − ρ ) ξ V ′ SV ( v + q ) + 12 ( ρ s ξ − ζ ) , with boundary conditions α (0) = 0 and ˙ α ( T ) = 0.10t is then straightforward to see that ∀ t ∈ [0 , T ] , α ( t ) = − V ′ SV ( v + q ) V ′ SV Λ V SV + ρ s ξ − ζ λ ! − cosh( √ λ ( T − t ))cosh( √ λT ) ! , (5)where λ = γ (1 − ρ ) ξ V ′ SV Λ V SV , and therefore ∀ t ∈ [0 , T ] , q ∗ ( t ) = q − V ′ SV ( v + q ) V ′ SV Λ V SV + ρ s ξ − ζ λ ! − cosh( √ λ ( T − t ))cosh( √ λT ) ! Λ V SV . This result deserves to be commented upon.First, in this optimization problem, the optimal Vega hedging strategy in the case of quadraticexecution costs always consists in trading the same basket of vanilla options. This result mayseem odd at first sight but one has to remember that it is based on a one-factor dynamicsfor the vanilla options and that, because there is no final constraint in this first problem,the trader only hedges her position in the stochastic volatility model. The weights of thebasket are given (up to a multiplicative constant) by the vector Λ V SV = (cid:18) V η , . . . , V N SV η N (cid:19) ′ : thehigher its Vega and the more liquid a vanilla option, the higher its weight in the basket. Itis in particular important to notice that the basket of vanilla options is independent of theportfolio of exotic options.Second, the volume we trade of that basket of vanilla options depends on the Vegas of theportfolio of exotic options (see (5)). For instance, in the case where the trader has no viewon the market (i.e. s = 0 and ζ = 0), whether the trader should buy or sell the basket ofvanilla options depends on the scalar product V ′ SV v of the Vegas of the vanilla options inthe stochastic volatility model and the sensitivities of the portfolio of exotic options to thedifferent vanilla options: if V ′ SV v is positive (resp. negative) the trader will sell (resp. buy)the basket of vanilla options in order to hedge the portfolio of exotic options. Interestingly,the composition of the portfolio of exotic options determines the level (i.e. the scale) of thetrading curve but not the shape (as a function of time) – which only depends on the constant λ .Third, the trader’s view on the market influences her strategy. Indeed, even if the portfoliois already hedged in the market model, i.e. q = − v , the trader trades whenever ρξ s − ζ isnot equal to 0. The trader’s view on the market impacts the trading strategy through twoeffects. A first effect is related to the trader’s view on the dynamics of the instantaneousvolatility in the stochastic volatility model: the larger ζ , i.e. the more upward the view of thetrader on the instantaneous volatility, the bigger the incentive to buy the basket of vanillaoptions. Since the Vegas of the vanilla options in the stochastic volatility model are positive,this means that the more upward the view of the trader on the instantaneous volatility, thebigger the incentive to buy vanilla options. This was expected. The second effect is moresubtle: the higher the product of the vol-spot correlation in the stochastic volatility modeland the view on the Sharpe ratio of the underlying asset, the smaller the incentive to buy thebasket of vanilla options (and the vanilla options themselves because of the sign the Vegasin the stochastic volatility model). This effect is due to the incentive of using the underlying11sset to partially hedge the Vega of the portfolio in the stochastic volatility model when thevol-spot correlation is not equal to nought. To better understand this effect, let us considerthe case where ρ and s are positive and let us consider the case ρ = s = 0 as a benchmark case.Would the trader buy more of the basket of vanilla options in the former case than in thelatter (benchmark) case, then, because the vol-spot correlation is positive, the trader wouldsell more of the underlying asset in order to Delta-hedge part of the Vega of the portfolio inthe stochastic volatility model, and this would result in an expected loss as the Sharpe ratiois positive. Subsequently, there is a reduced incentive to buy the basket of vanilla optionswhen ρ and s are positive. Vega Bucket cancellation with Vega hedging in the stochastic volatility model
Inpractice, the above problem does not lead to a complete cancellation of the Vega risk exposurein the market model. To reach this objective, we consider our second probleminf ( q t ) t ,q given ,q T = − v Z T X i L i ( v it ) + 18 γ (1 − ρ ) ξ X i q it + V i MM V i BS ! V i SV ! dt + Z T
12 ( ρ s ξ − ζ ) X i q it + V i MM V i BS ! V i SV dt in which the trader hedges her portfolio in the stochastic volatility model, which describes thedynamics of vanilla options, and has to reach at time T a position that hedges her portfolioin the market model.This problem is a problem of Bolza and, as above, there is a unique optimal trajectory t ∈ [0 , T ] q ∗ ( t ) = ( q ∗ ( t ) , . . . , q ∗ N ( t )) ′ characterized by the following Hamiltonian sys-tem: ( ˙ p ( t ) = γ (1 − ρ ) ξ V SV V ′ SV ( q ∗ ( t ) + v ) + ( ρ s ξ − ζ ) V SV ˙ q ∗ i ( t ) = H i ′ ( p i ( t )) , q ∗ (0) = q , q ∗ ( T ) = − v . (6)In the case where execution cost functions are quadratic as above, (6) is in fact the linearsystem ( ˙ p ( t ) = γ (1 − ρ ) ξ V SV V ′ SV ( q ∗ ( t ) + v ) + ( ρ s ξ − ζ ) V SV , ˙ q ∗ ( t ) = Λ p ( t ) , q ∗ (0) = q , q ∗ ( T ) = − v , or equivalently¨ q ∗ ( t ) = 18 γ (1 − ρ ) ξ Λ V SV V ′ SV q ∗ ( t ) + 18 γ (1 − ρ ) ξ Λ V SV V ′ SV v + 12 ( ρ s ξ − ζ ) Λ V SV (7)with boundary conditions q ∗ (0) = q and q ∗ ( T ) = − v .As above, in the case of quadratic execution cost functions, the problem therefore boils toa linear ordinary differential equation of order 2 that can be addressed using standard tech-niques. 12s above, we notice that ∀ t ∈ [0 , T ] , ¨ q ∗ ( t ) ∈ span(Λ V BS ) . Then, recalling that ∀ t ∈ [0 , T ] , q ∗ ( t ) = ( T − t ) q (0) + tq ( T ) T + Z t ¨ q ∗ ( s )( t − s ) ds − tT Z T ¨ q ∗ ( s )( T − s ) ds = (cid:18) − tT (cid:19) q − tT v + Z t ¨ q ∗ ( s )( t − s ) ds − tT Z T ¨ q ∗ ( s )( T − s ) ds, we clearly see that there exists a C function α such that ∀ t ∈ [0 , T ] , q ∗ ( t ) = (cid:18) − tT (cid:19) q − tT v + α ( t )Λ V BS . From (7), we deduce that ∀ t ∈ [0 , T ] , ¨ α ( t ) = λα ( t ) + 18 γ (1 − ρ ) ξ (cid:18) − tT (cid:19) V ′ SV ( v + q ) + 12 ( ρ s ξ − ζ ) , with boundary conditions α (0) = 0 and α ( T ) = 0.It is then straightforward to see that ∀ t ∈ [0 , T ] , α ( t ) = V ′ SV ( v + q ) V ′ SV Λ V SV sinh( √ λ ( T − t ))sinh( √ λT ) − (cid:18) − tT (cid:19)! + ρ s ξ − ζ λ sinh( √ λt ) + sinh( √ λ ( T − t ))sinh( √ λT ) − ! . The optimal hedging strategy in this second problem, which is more in line with the actualproblem faced by the trader, is then ∀ t ∈ [0 , T ] , q ∗ ( t ) = (cid:18) − tT (cid:19) q − tT v + V ′ SV ( v + q ) V ′ SV Λ V SV sinh( √ λ ( T − t ))sinh( √ λT ) − (cid:18) − tT (cid:19)! Λ V SV + ρ s ξ − ζ λ sinh( √ λt ) + sinh( √ λ ( T − t ))sinh( √ λT ) − ! Λ V SV . This strategy enables to hedge progressively – in fact linearly appears to be optimal – thebook of exotic options in the market model (this is the term (cid:0) − tT (cid:1) q − tT v that goes linearlyfrom q to − v as t goes from 0 to T ) while hedging the evolving portfolio in the stochasticvolatility model (this is the term V ′ SV ( v + q ) V ′ SV Λ V SV (cid:18) sinh( √ λ ( T − t ))sinh( √ λT ) − (cid:0) − tT (cid:1)(cid:19) Λ V SV ). As for our firstmodel, we notice that we use a unique basket of vanilla options for hedging in the stochasticvolatility model. The strategy also takes account of the view of the trader on the market asabove. Unlike in the first problem however, the strategy associated with the trader’s viewhas to be a round trip: this is the term ρ s ξ − ζ λ (cid:18) sinh( √ λt )+sinh( √ λ ( T − t ))sinh( √ λT ) − (cid:19) Λ V SV .13 onclusion In this paper, we built a framework in which exotic options are priced using a market modeland where the prices of vanilla options are driven by a stochastic volatility model. Using thisframework, we derived, after some simplifying approximations, the optimal hedging strategyassociated with a book of exotic options in presence of execution costs à la Almgren-Chrissto trade vanilla options. In the case of quadratic execution costs, we even showed that thehedging strategy could be computed in closed form.