Deep Equal Risk Pricing of Financial Derivatives with Multiple Hedging Instruments
DDeep Equal Risk Pricing of Financial Derivatives withMultiple Hedging Instruments
Alexandre Carbonneau ∗ a and Fr´ed´eric Godin ba,b Concordia University, Department of Mathematics and Statistics, Montr´eal, Canada
February 26, 2021
Abstract
This paper studies the equal risk pricing (ERP) framework for the valuation of Europeanfinancial derivatives. This option pricing approach is consistent with global trading strategiesby setting the premium as the value such that the residual hedging risk of the long and shortpositions in the option are equal under optimal hedging. The ERP setup of Marzban et al.(2020) is considered where residual hedging risk is quantified with convex risk measures.The main objective of this paper is to assess through extensive numerical experimentsthe impact of including options as hedging instruments within the ERP framework. Thereinforcement learning procedure developed in Carbonneau and Godin (2020), which relieson the deep hedging algorithm of Buehler et al. (2019b), is applied to numerically solve theglobal hedging problems by representing trading policies with neural networks. Among otherfindings, numerical results indicate that in the presence of jump risk, hedging long-term putswith shorter-term options entails a significant decrease of both equal risk prices and marketincompleteness as compared to trading only the stock. Monte Carlo experiments demonstratethe potential of ERP as a fair valuation approach providing prices consistent with observablemarket prices. Analyses exhibit the ability of ERP to span a large interval of prices throughthe choice of convex risk measures which is close to encompass the variance-optimal premium.
Keywords:
Equal risk pricing, Deep hedging, Convex risk measure, Reinforcement learning. ∗ Corresponding author.
Email addresses: [email protected] (Alexandre Carbonneau), [email protected](Fr´ed´eric Godin). a r X i v : . [ q -f i n . C P ] F e b Introduction
In the famous setup of Black and Scholes (1973) and Merton (1973), every contingent claim canbe perfectly replicated through continuous trading in the underlying stock and a risk-free asset.These markets are said to be complete , and derivatives are redundant securities with a uniquearbitrage-free price equal to the initial value of the replicating portfolio. However, the giganticsize of the derivatives market demonstrates unequivalently that options are non-redundant andprovide additional value above the exclusive trading of the underlying asset from the standpointof speculation, risk management and arbitraging (Hull, 2003). Such value-added of derivatives inthe real world stems from market incompleteness which arises from several stylized features ofmarket dynamics such as discrete-time trading, equity risk (e.g. jump and volatility risks) andmarket impact (e.g. trading costs and imperfect liquidity). In contrast to the complete marketparadigm, in incomplete markets, the price of a derivative cannot be uniquely specified by ano-arbitrage argument since perfect replication is not always possible.The problem of determining the value of a derivative is intrinsically intertwined with its corre-sponding hedging strategy. On the spectrum of derivative valuation procedures in incompletemarkets, one extreme possibility is the so-called super-hedging strategy , where the derivativepremium is set as the value such that the residual hedging risk of the seller is nullified. However,the super-hedging premium is in general very large and is thus most often deemed impractical(Gushchin and Mordecki, 2002). On the other hand, a more reasonable and practical derivativepremium entails that some level of risk cannot be hedged away and is thus intrinsic to thecontingent claim. An additional layer of complexity to the hedging problem in incomplete marketsis in selecting not only the sequence of investments in trading instruments, but also the category of hedging instruments in the design of optimal hedges. Indeed, some categories of instrumentsare more effective to mitigate certain risk factors than others. For instance, it is well-known thatin the presence of random jumps, option hedges are much more effective than trading exclusivelythe underlying stock due to the convex property of derivatives prices (see, for instance, Colemanet al. (2007) and Carbonneau (2020)). More generally, the use of option hedges dampens tail risk1temming from different risk factors (e.g. jump and volatility risks). The focus of this paper liesprecisely on studying a derivative valuation approach called equal risk pricing (ERP) for pricingEuropean derivatives consistently with optimal hedging strategies trading in various categories ofhedging instruments (e.g. vanilla calls and puts as well as the underlying stock).The ERP framework introduced by Guo and Zhu (2017) determines the equal risk price (i.e.the premium) of a financial derivative as the value such that the long and short positions inthe contingent claim have the same residual hedging risk under optimal trading strategies. Animportant application of ERP in the latter paper is for pricing derivatives in the presence ofshort-selling restrictions for the underlying stock. Various studies have since extended thisapproach: Ma et al. (2019) provide Hamilton-Jacobi-Bellman equations for the optimizationproblem and establish additional analytical pricing formulas for equal risk prices, He and Zhu(2020) generalize the problem of pricing derivatives with short-selling restriction for the underlyingby allowing for short trades in a correlated asset and Alfeus et al. (2019) perform an empiricalstudy of equal risk prices when short selling is banned. One crucial pitfall of the Guo and Zhu(2017) framework considered in all of the aforementioned papers is that the optimization problemrequired to be solved for the computation of equal risk prices is very complex. Consequently,closed-form solutions are restricted to very specific setups (e.g. Black-Scholes market) and nonumerical scheme has been proposed to account for more realistic market assumptions.Marzban et al. (2020) recently extended the ERP framework by considering the use of convexrisk measures under the physical measure to quantify residual hedging risk. A major benefit ofthe ERP setup of the latter paper is that it does not require the specification of an equivalentmartingale measure (EMM), which is arbitrary in incomplete markets since there is an infiniteset of EMMs (Harrison and Pliska, 1981). Also, using convex measures to quantify residual riskis shown in Marzban et al. (2020) to significantly reduce the complexity of computing equal riskprices; the optimization problem essentially boils down to solving two distinct non-quadraticglobal hedging problems, one for the long and one for the short position in the option. Dynamicprogramming equations are provided in Marzban et al. (2020) for the aforementioned globalhedging problems. However, it is well-known that traditional dynamics programming procedures2re prone to the curse of dimensionality when the state and action spaces gets too large (Powell,2009). The main objective of this current paper consists in studying the impact of trading differentand possibly multiple hedging instruments on the ERP framework, which thus necessitates largeaction spaces. Furthermore, a specific focus of this study is on assessing the interplay betweendifferent equity risk factors (e.g. jump and volatility risks) and the use of options as hedginginstruments. Consequently, large state spaces are also required to model the dynamics of theunderlying stock and to characterize the physical measure dynamics of the implied volatility ofoptions used as hedging instruments. A feasible numerical procedure in high-dimensional stateand action spaces is therefore essential to this paper.Carbonneau and Godin (2020) expanded upon the work of Marzban et al. (2020) by developing atractable solution with reinforcement learning to compute equal risk prices in high-dimensionalstate and action spaces. The approach of the foremost study relies on the deep hedging algorithmof Buehler et al. (2019b) to represent the long and short optimal trading policy with two distinctneural networks. One of the most important benefits of parameterizing trading policies as neuralnetworks is that the computational complexity increases marginally with the dimension of thestate and action spaces. Carbonneau and Godin (2020) also introduce novel (cid:15) -completenessmetrics to quantify the level of market incompleteness which will be used throughout this currentstudy. Several papers have studied different aspects of the class of deep hedging algorithms:Buehler et al. (2019a) extend upon the work of Buehler et al. (2019b) by hedging path-dependentcontingent claims with neural networks, Carbonneau (2020) presents an extensive benchmarkingof global policies parameterized with neural networks to mitigate the risk exposure of verylong-term contingent claims, Cao et al. (2020) show that the deep hedging algorithm providesgood approximations of optimal initial capital investments for variance-optimal hedging problemsand Horvath et al. (2021) deep hedge in a non-Markovian framework with rough volatility modelsfor risky assets.The main objective of this paper consist in the assessment of the impact of using multiple hedginginstruments on the ERP framework through exhaustive numerical experiments. To the best of theauthors’ knowledge, this is the first study within the ERP literature that considers trades involving3ptions in the design of optimal hedges. The performance of these numerical experiments heavilyrelies on the use of reinforcement learning procedures to train neural networks representing tradingpolicies and would be hardly reachable with other numerical methods. The first key contributionof this paper consists in providing a broad analysis of the impact of jump and volatility riskson equal risk prices and on our (cid:15) -completeness metrics. These assessments expand upon thework of Carbonneau and Godin (2020) in two ways. First, the latter paper conducted sensitivityanalyses of the ERP framework under different risky assets dynamics by trading exclusively withthe underlying stock, not with options. However, the use of options as hedging instruments in thepresence of such risk factors allows for the mitigation of some portion of unattainable residual riskwhen trading exclusively with the stock. Second, this current paper examines the sensitivity ofequal risk prices and residual hedging risk to different levels of jump and volatility risks througha range of empirically plausible model parameters for asset prices dynamics (e.g. frequent smalljumps and rare extreme jumps). The motivation is to provide new qualitative insights into theinterrelation of different stylized features of jump and volatility risks on the ERP framework thatare more extensive than in previous work studies. The main conclusions of these experiments ofpricing 1-year European puts are summarized below.1) In the presence of downward jump risk, numerical values indicate that hedging withoptions entails significant reduction of both equal risk prices and on the level of marketincompleteness as compared to hedging solely with the underlying stock. The latter stemsfrom the fact that while the residual hedging risk of both the long and short positions in1-year puts decreases when short-term option trades are used for mitigating the presence ofjump risk, a larger decrease is observed for the short position due to jump risk dynamicsentailing predominantly negative jumps. These results further demonstrate that options arenon-redundant securities as it is the case in the Black-Scholes world.2) In the presence of volatility risk, numerical experiments demonstrate that while the use ofoptions as hedging instruments can entail smaller derivative premiums, the impact can alsobe marginal and is highly sensitive to the moneyness level of the put option being priced aswell as to the maturity of the traded options. This observation stems from the fact that4ontrarily to jump risk, volatility risk impacts both upside and downside risk. Thus, the useof option hedges does not necessarily benefit more the short position with a larger decreaseof residual hedging risk as observed in the presence of jump risk.3) The average price level of short-term options (i.e. average implied volatility level) used ashedging instruments is effectively reflected into the equal risk price of longer-term options.This demonstrates the potential of the ERP framework as a fair valuation approach consistentwith observable market prices, which could be used, for instance, to price over-the-counteror long-term less liquid derivatives with short-term highly liquid options.The last contribution of this paper is in benchmarking equal risk prices to derivative premiumsobtained with variance-optimal hedging (Schweizer, 1995). Variance-optimal hedging proceduressolve jointly for the initial capital investment and a self-financing strategy minimizing the expectedsquared hedging error. The optimized initial capital investment can be viewed as the productioncost of the derivative, since the resulting dynamic trading strategy replicates the derivative’spayoff as closely as possible in a quadratic sense. The main motivation for these experiments isthe popularity of variance-optimal hedging procedures in the literature for pricing derivatives.Furthermore, while these two derivative valuation procedures are both consistent with optimaltrading criteria, the underlying global hedging problem of each approach treats hedging shortfallthrough a radically different scope. Indeed, equal risk prices obtained under the ConditionalValue-at-Risk measure with large confidence level values, as considered in this paper, are theresult of joint optimizations over hedging decisions to minimize tail risk of hedging shortfallswhich penalize mainly (and most often exclusively) hedging losses, not gains. Conversely, variance-optimal procedures penalize equally hedging gains and losses, not solely losses. This benchmarkingof equal risk prices to variance-optimal premiums highlights the flexibility of ERP procedures forderivatives valuation through the choice of convex risk measure. Indeed, numerical values showthat the range of equal risk prices obtained with several convex measures can be very large and isclose to encompass the variance-optimal premium. Note that derivatives premiums prescribed by variance-optimal procedures coincide with the risk-neutral priceobtained under the so-called variance-optimal martingale measure (Schweizer, 1996).
This section details the equal risk pricing (ERP) framework considered in this paper, which isan extension of the derivative valuation scheme introduced in Marzban et al. (2020) with theaddition of multiple hedging instruments.
The financial market is in discrete-time with a finite time horizon of T years and N + 1 observationdates characterized by the set T := { t n : t n = n ∆ N , n = 0 , . . . , N } where ∆ N := T /N . Theprobability space (Ω , P , F ) is equipped with the filtration F := {F n } Nn =0 satisfying the usualconditions, where F n contains all information available to market participants at time t n . Assume F = F N . P is referred to as the physical probability measure. On each observation date, a totalof D + 2 financial securities can be traded on the market, which includes a risk-free asset, anon-dividend paying stock and D standard European calls and puts on the latter stock whosematurity dates fall within T . Let { B n } Nn =0 be the price process of the risk-free asset, where B n := e rt n for n = 0 , . . . , N with r ∈ R being the annualized continuously compounded risk-freerate. The definition of the price process for the risky securities is now outlined. Since some of thetradable options can mature before the final time horizon T , the set of options that can be tradedat the beginning of two different observation periods could differ. To reflect this modeling featureand properly represent gains of trading strategies, two different stochastic processes are defined,namely the price of tradable assets at the beginning and at the end of each period. First, let { ¯ S ( b ) n } Nn =0 be the beginning-of-period risky price process whose element ¯ S ( b ) n contains the time- t n price of all risky assets traded at time t n . More precisely, ¯ S ( b ) n := [ S (0 ,b ) n , . . . , S ( D,b ) n ] with S (0 ,b ) n and S ( j,b ) n respectively being the time- t n price of the underlying stock and of the j th option that canbe traded at time t n for j = 1 , . . . , D . Similarly, let { ¯ S ( e ) n } N − n =0 be the end-of-period risky price rocess where ¯ S ( e ) n := [ S (0 ,e ) n , . . . , S ( D,e ) n ] with S (0 ,e ) n and S ( j,e ) n respectively being the time t n +1 priceof the underlying stock and j th option that can be traded at time t n . Since the underlying asset isdenoted as the risky asset with index 0, S (0 ,e ) n = S (0 ,b ) n +1 for n = 0 , . . . , N −
1. Also, if the j th optionthat can be traded at t n matures at time t n +1 , then S ( j,e ) n is the payoff of that option. In thatcase, S ( j,b ) n +1 is the price of a new contract with the same characteristics in terms of payoff function,moneyness level and time-to-maturity. Otherwise, S ( j,b ) n +1 = S ( j,e ) n holds for all time steps and allrisky assets (i.e. for j = 0 , . . . , D and n = 0 , . . . , N − This paper studies the problem of pricing a simple European-type derivative providing a time- T payoff denoted by Φ( S (0 ,b ) N ) ≥ For such purposes, the equal risk pricing scheme is considered,which entails optimizing two distinct self-financing dynamic trading strategies separately forboth the long and short positions on the derivative, and then determining the premium whichequates the residual hedging risk of the two hedged positions. The mathematical formalismused for trading strategies in the current study is now outlined. A trading strategy { δ n } Nn =0 is an F -predictable process where δ n := [ δ (0) n , . . . , δ ( D ) n , δ ( B ) n ] with δ ( B ) n and δ ( j ) n , j = 0 , . . . , D ,respectively denoting the number of shares of the risk-free asset and the j th risky asset tradedat time t n − held in the hedging portfolio throughout the period ( t n − , t n ], except for the case n = 0 which represents the hedging portfolio composition exactly at time t . The notation δ (0: D ) n := [ δ (0) n , . . . , δ ( D ) n ] is used to define the vector containing exclusively positions in the riskyassets. Furthermore, the initial capital investment of the trading strategy is always assumedto be completely invested in the risk-free asset, i.e. δ ( B )0 is the initial investment amount and Note that the optimization procedure for global policies described in Section 3 can naturally be generalizedfor the case of rebalancing multiple times option contracts prior to their expiry. The derivative valuation approach presented in this paper can easily be adapted for European options whosepayoff is of the form Φ( S (0 ,b ) N , Z N ) ≥ { Z n } Nn =0 as some F -adapted potentially multidimensional randomprocess encompassing the path-dependence property of the payoff function. For examples of such exotic derivatives,the reader is referred to Carbonneau and Godin (2020). A process X = { X n } Nn =0 is said to be F -predictable if X is F -measurable and X n is F n − -measurable for n = 1 , . . . , N . (0: D )0 := [0 , . . . , { V δn } Nn =0 be the hedging portfolio value process associated with the trading strategy δ , where V δn is the time- t n portfolio value prior to rebalancing with V δ := δ ( B )0 and V δn := δ (0: D ) n • ¯ S ( e ) n − + δ ( B ) n B n , n = 1 , . . . , N, (2.1)where • is the dot product operator. Furthermore, denote as { G δn } Nn =0 the discounted gain processassociated with δ where G δn is the time- t n discounted gain prior to rebalancing with G δ := 0 and G δn := n X k =1 δ (0: D ) k • ( B − k ¯ S ( e ) k − − B − k − ¯ S ( b ) k − ) , n = 1 , . . . , N. (2.2)The trading strategies considered in this paper are always self-financing : they require no cashinfusion nor withdrawal at intermediate times except possibly at the initialization of the strategy.More formally, a trading strategy is said to be self-financing if it is predictable and if the followingequality holds P -a.s. for n = 0 , . . . , N − δ (0: D ) n +1 • ¯ S ( b ) n + δ ( B ) n +1 B n = V δn . (2.3)Lastly, denote Π as the set of accessible trading strategies, which includes all trading strategiesthat are self-financing and sufficiently well-behaved. Remark 2.1.
It can be shown that δ ∈ Π is self-financing if and only if V δn = B n ( V δ + G δn ) holds P -a.s. for n = 0 , . . . , N ; see for instance Lamberton and Lapeyre (2011). The latterrepresentation of portfolio values implies the following useful recursive equation (2.4) to compute For X := [ X , . . . , X K ] and Y := [ Y , . . . , Y K ], X • Y := P Kj =1 X j Y j . δn for n = 1 , . . . , N given V δ : V δn = B n ( V δ + G δn )= B n ( V δ + G δn − + δ (0: D ) n • ( B − n ¯ S ( e ) n − − B − n − ¯ S ( b ) n − ))= B n B n − V δn − + δ (0: D ) n • ( ¯ S ( e ) n − − B n B n − ¯ S ( b ) n − )= e r ∆ N V δn − + δ (0: D ) n • ( ¯ S ( e ) n − − e r ∆ N ¯ S ( b ) n − ) . (2.4) The financial market setting considered in this paper implies incompleteness stemming fromdiscrete-time trading and equity risk factors (e.g. jump risk and volatility risk). For the hedger,these many sources of incompleteness entail that most contingent claims are not attainablethrough dynamic hedging. Following the work of Marzban et al. (2020) and Carbonneau andGodin (2020), this study quantifies the level of residual hedging risk with convex risk measures asdefined in F¨ollmer and Schied (2002).
Definition 2.1. (Convex risk measure) For a set of random variables X representing liabilitiesand X , X ∈ X , ρ : X → R is a convex risk measure if it satisfies the following properties:0) Normalized: ρ (0) = 0 (empty portfolio has no risk).1) Monotonicity: X ≤ X = ⇒ ρ ( X ) ≤ ρ ( X ) (larger liability is riskier).2) Translation invariance: for c ∈ R and X ∈ X , ρ ( X + c ) = ρ ( X ) + c (borrowing amount c increases the risk by that amount).3) Convexity: for c ∈ [0 , , ρ ( cX + (1 − c ) X ) ≤ cρ ( X ) + (1 − c ) ρ ( X ) (diversification doesnot increase risk). The hedging problem underlying the ERP framework is now formally defined.
Definition 2.2. (Long- and short-sided risk) For a given convex risk measure ρ , define (cid:15) ( L ) ( V ) and (cid:15) ( S ) ( V ) respectively as the measured risk exposure of a long and short position in Φ under he optimal hedge if the value of the initial hedging portfolio is V ∈ R : (cid:15) ( L ) ( V ) := min δ ∈ Π ρ (cid:16) − Φ( S (0 ,b ) N ) − B N ( V + G δN ) (cid:17) , (2.5) (cid:15) ( S ) ( V ) := min δ ∈ Π ρ (cid:16) Φ( S (0 ,b ) N ) − B N ( V + G δN ) (cid:17) . (2.6) Remark 2.2.
As noted in Carbonneau and Godin (2020), an assumption implicit to Definition 2.2is that the minimum in (2.5) or (2.6) is indeed attained by some trading strategy, i.e. that theinfimum is in fact a minimum. Note that the same risk measure ρ is used for both the long and short positions global hedgingproblems. The rationale for this choice is threefold. First, considering the same convex risk measurefor long and short positions is in line with the trading activities of some market participants thatboth buy and sell options with no directional view of the market. One example of such participantis a market maker of derivatives which typically expects to make a profit on bid-ask spreads, notby speculating (Basak and Chabakauri, 2012). Another motivation for using the same convexmeasure is for cases where a price quote must be given prior to knowing if the derivative is beingpurchased or sold. For instance, a client asks his broker to provide a quote for a derivative withoutrevealing his intention of buying or selling the option. A similar argument is made in Bertsimaset al. (2001) to motivate the use of a quadratic loss function for hedging shortfalls, which entailsthe same derivative price for the long and short position. Lastly, as shown in Carbonneau andGodin (2020), using the same risk measure for both positions guarantees, under some specificconditions, that the ERP derivative premium is arbitrage-free. It is interesting to note that the translation invariance property of ρ entails that the optimalstrategies solving (2.5)-(2.6), denoted respectively by δ ( L ) and δ ( S ) , are invariant to the initial Nevertheless, the authors want to emphasize that the numerical scheme developed in Section 3 for the globalhedging problems (2.5) and (2.6) could easily be extended to include two distinct convex measures respectively forthe long and short position hedges (see Remark 3 . V . The latter significantly enhances the tractability of the solution: δ ( L ) := arg min δ ∈ Π ρ (cid:16) − Φ( S (0 ,b ) N ) − B N ( V + G δN ) (cid:17) = arg min δ ∈ Π ρ (cid:16) − Φ( S (0 ,b ) N ) − B N G δN (cid:17) , (2.7) δ ( S ) := arg min δ ∈ Π ρ (cid:16) Φ( S (0 ,b ) N ) − B N ( V + G δN ) (cid:17) = arg min δ ∈ Π ρ (cid:16) Φ( S (0 ,b ) N ) − B N G δN (cid:17) . (2.8)Based on the aforementioned global hedging problems, the equal risk price of a derivative isdefined as the initial hedging portfolio value equating the measured risk exposures for both thelong and short positions. Definition 2.3. (Equal risk price) The equal risk price C ? of Φ is defined as the real number C such that (cid:15) ( L ) ( − C ) = (cid:15) ( S ) ( C ) . (2.9)As shown for instance in Marzban et al. (2020), equal risk prices have the following representationwhich is used throughout the rest of the paper: C ? = (cid:15) ( S ) (0) − (cid:15) ( L ) (0)2 B N . (2.10)Carbonneau and Godin (2020) introduced the market incompleteness metric (cid:15) ? defined as thelevel of residual risk faced by the hedgers of Φ if the hedged derivative price is set to C ? andoptimal trading strategies are used by both the long and short position hedgers: (cid:15) ? := (cid:15) ( L ) ( − C ? ) = (cid:15) ( S ) ( C ? ) = (cid:15) ( L ) (0) + (cid:15) ( S ) (0)2 . (2.11)Consistently with the terminology of Carbonneau and Godin (2020), (cid:15) ? and (cid:15) ? /C ? are referredrespectively as the measured residual risk exposure per derivative contract and per dollar invested. The last equality of (2.11) can easily be obtained with the translation invariance property of ρ , see equation(8) of Carbonneau and Godin (2020) for the details. (cid:15) ? -metrics will be extensively studied in numerical experiments conducted in Section 4to assess, for instance, the impact of the use of options as hedging instruments on the level ofmarket incompleteness. The problem of solving the ERP framework, that is evaluating equal risk prices and (cid:15) -completenessmeasures, boils down to the computation of the measured risk exposures (cid:15) ( S ) (0) and (cid:15) ( L ) (0). Thissection presents a reinforcement learning method to compute such quantities. The approach wasfirst proposed in Carbonneau and Godin (2020) and relies on approximating optimal tradingstrategies with the deep hedging algorithm of Buehler et al. (2019b) through the representationof the long and short global trading policy with two distinct neural networks. In its essence,neural networks are a class of composite functions mapping feature vectors (i.e. input vectors) to output vectors through multiple hidden layers , with the latter being functions applying successiveaffine and nonlinear transformations to input vectors. In this paper, the type of neural networkconsidered to represent global hedging policies is the long short-term memory (LSTM, Hochreiterand Schmidhuber (1997)). LSTMs belong to the class of recurrent neural networks (RNNs,Rumelhart et al. (1986)), which have self-connections in hidden layers: the output of the time- t n hidden layer is a function of both the time- t n feature vector as well as the output of the time- t n − hidden layer. The periodic computation of long short-term memory neural networks is done withso-called LSTM cells , which are similar to but more complex than the typical hidden layer ofRNNs. LSTMs have recently been applied with success to approximate global hedging policiesin several studies: Buehler et al. (2019a), Cao et al. (2020) and Carbonneau (2020). Additionalremarks are made in subsequent sections to motivate this choice of neural networks for the specificsetup of this paper.
The following formally defines the architecture of long-short term memory neural networks. Forconvenience, a very similar notation for neural networks as the one of Carbonneau (2020) is used.Note that the time steps of the feature and output vectors coincide with the set of financial12arket trading dates T . For additional general information about LSTMs, the reader is referredto Chapter 10 .
10 of Goodfellow et al. (2016) and the many references therein.
Definition 3.1. (LSTM) For
H, d , . . . , d H +1 ∈ N , let F θ : R N × d → R N × d H +1 be an LSTMwhich maps the sequence of feature vectors { X n } N − n =0 to output vectors { Y n } N − n =0 where X n ∈ R d and Y n ∈ R d H +1 for n = 0 , . . . , N − . The computation of Y n , the subset of outputs of F θ associated with time t n , is achieved through H LSTM cells, each of which outputs a vector of d j neurons denoted as h ( j ) n ∈ R d j × for j = 1 , . . . , H . More precisely, the computation applied by the j th LSTM cell for the time- t n output is as follows: i ( j ) n = sigm ( U ( j ) i h ( j − n + W ( j ) i h ( j ) n − + b ( j ) i ) ,f ( j ) n = sigm ( U ( j ) f h ( j − n + W ( j ) f h ( j ) n − + b ( j ) f ) ,o ( j ) n = sigm ( U ( j ) o h ( j − n + W ( j ) o h ( j ) n − + b ( j ) o ) ,c ( j ) n = f ( j ) n (cid:12) c ( j ) n − + i ( j ) n (cid:12) tanh ( U ( j ) c h ( j − n + W ( j ) c h ( j ) n − + b ( j ) c ) ,h ( j ) n = o ( j ) n (cid:12) tanh ( c ( j ) n ) , (3.1) where (cid:12) denotes the Hadamard product (the element-wise product), sigm ( · ) and tanh ( · ) are thesigmoid and hyperbolic tangent functions applied element-wise to each scalar given as input and • U ( j ) i , U ( j ) f , U ( j ) o , U ( j ) c ∈ R d j × d j − , W ( j ) i , W ( j ) f , W ( j ) o , W ( j ) c ∈ R d j × d j and b ( j ) i , b ( j ) f , b ( j ) o , b ( j ) c ∈ R d j × for j = 1 , . . . , H .At each time-step, the input of the first LSTM cell is the feature vector ( i.e. h (0) n := X n ) and thefinal output is an affine transformation of the output of the last LSTM cell: Y n = W y h ( H ) n + b y , n = 0 , . . . , N − , (3.2) At time 0 (i.e. n = 0), the computation of the LSTM cells is the same as in (3.1) with h ( j ) − and c ( j ) − defined asvectors of zeros of dimension d j for j = 1 , . . . , H . For X := [ X , . . . , X K ], sigm( X ) := h e − X , . . . , e − XK i and tanh( X ) := h e X − e − X e X + e − X , . . . , e XK − e − XK e XK + e − XK i . here W y ∈ R d H +1 × d H and b y ∈ R d H +1 × . Lastly, the set of trainable parameters denoted as θ consists of all weight matrices and bias vectors: θ := n { U ( j ) i , U ( j ) f , U ( j ) o , U ( j ) c , W ( j ) i , W ( j ) f , W ( j ) o , W ( j ) c , b ( j ) i , b ( j ) f , b ( j ) o , b ( j ) c } Hj =1 , W y , b y o . (3.3)In this study, the computation of hedging positions is done through the mapping of a sequence ofrelevant financial market observations into the periodic number of shares held in each hedginginstrument with an LSTM. One of the main objectives of this paper is to analyze the impactof including vanilla options as hedging instruments on the ERP framework. For the numericalexperiments conducted in the subsequent Section 4, the hedging instruments consist of either onlythe underlying asset (without options) or exclusively options (without the underlying asset). Thecase of using both the stock and options is not considered since the options can always replicatepositions in the underlying asset with calls and puts by relying on the put-call parity. In whatfollows, let { X n } N − n =0 and { Y n } N − n =0 be respectively the sequence of feature vectors and outputvectors of an LSTM as in Definition 3.1. When hedging only with the underlying, the time- t n feature vector considered is X n = [log( S (0 ,b ) n /K ) , V δn , ϕ n ] , n = 0 , . . . , N − , (3.4)where K is the strike price of Φ and { ϕ n } N − n =0 is a sequence of additional relevant state variablesassociated with the dynamics of asset prices. For instance, if the underlying log-returns aremodeled with a GARCH process, it is well-known that the bivariate process of the underlyingprice and the GARCH volatility has the Markov property under P with respect to the marketfiltration F . The time- t n volatility of the GARCH process is thus added to the feature vectorsthrough ϕ n . Furthermore, in that same case where the underlying stock is considered as the onlyhedging instrument, the output vectors of the LSTM consist of the number of underlying asset The use of log( S (0 ,b ) n /K ) instead of S (0 ,b ) n in feature vectors was found to improve the learning speed of theneural networks (i.e. time taken to find a good set of trainable parameters). Note that log transformation forrisky asset prices was also considered in Carbonneau (2020), Buehler et al. (2019b) and Buehler et al. (2019a). Y n = δ (0) n +1 for n = 0 , . . . , N − implied volatilities (IVs) of such options denoted as { IV n } N − n =0 are added to feature vectors with IV n encompassingevery implied volatilities needed to price the D options used for hedging: X n = [log( S (0 ,b ) n /K ) , V δn , ϕ n , IV n ] , n = 0 , . . . , N − . (3.5)In that case, the output vectors are the number of option contracts held in the portfolio for thevarious time steps: Y n = [ δ (1) n +1 , . . . , δ ( D ) n +1 ] for n = 0 , . . . , N −
1. Recall that when options are usedas hedging instruments, δ (0) n +1 = 0 for n = 0 , . . . , N − Remark 3.1.
Although the portfolio value V δn is in theory a redundant feature in the context ofLSTMs since it can be retrieved as a function of previous times inputs and outputs of the neuralnetwork (see (2.4) ), incorporating it to feature vectors was found to significantly improve uponthe hedging effectiveness of the LSTMs in the numerical experiments conducted in Section 4. To numerically solve the underlying global hedging problems of the ERP framework, Carbonneauand Godin (2020) propose to use two distinct neural networks denoted as F ( L ) θ and F ( S ) θ toapproximate the global trading policies of respectively the long and short positions in Φ. This isthe approach considered in the current paper. As illustrated below, the procedure consists insolving the alternative problems of optimizing the neural networks trainable parameters so as tominimize the corresponding hedging shortfall: (cid:15) ( L ) ( V ) ≈ min θ ∈ R q ρ (cid:16) − Φ( S (0 ,b ) N ) − B N ( V + G δ ( L ,θ ) N ) (cid:17) , (3.6) (cid:15) ( S ) ( V ) ≈ min θ ∈ R q ρ (cid:16) Φ( S (0 ,b ) N ) − B N ( V + G δ ( S ,θ ) N ) (cid:17) , (3.7) Note that the bijection relation between implied volatilities and option prices entails that either values couldtheoretically be used in feature vectors as one is simply a nonlinear transformation of the other. δ ( L ,θ ) and δ ( S ,θ ) are to be understood respectively as the output sequences of F ( L ) θ and F ( S ) θ , and q ∈ N is the total number of trainable parameters of F ( L ) θ and F ( S ) θ . The approximatedmeasured risk exposures obtained through (3.6) and (3.7) are subsequently used to computeequal risk prices and (cid:15) -completeness measures with (2.10) and (2.11). One implicit assumptionassociated with (3.6) and (3.7) is that the architecture of all neural networks in terms of thenumber of LSTM cells and neurons per cell is always fixed; the hyperparameter tuning step ofthe optimization problem is not considered in this paper. Section 3.3 that follows presents theprocedure considered in this study to optimize the trainable parameters of the LSTMs. Remark 3.2.
Carbonneau and Godin (2020) show that when relying on feedforward neuralnetworks (FFNNs ) instead of LSTMs, the alternative problems (3.6) - (3.7) allow for arbitrarilyprecise approximations of the measured risk exposures (2.5) - (2.6) due to results from Buehleret al. (2019b). Despite this theoretical ability of FFNNs to approximate arbitrarily well globalhedging policies in such context, the authors of the current paper found that LSTMs are ableto learn significantly better trading policies than FFNNs in the numerical experiments carriedout in Section 4, which motivates their use over FFNNs. The theoretical justifications forthe outperformance of LSTMs over FFNNs in the financial market settings of this paper areout-of-scope and are left-out as interesting potential future research. The numerical scheme to optimize the trainable parameters of neural networks as entailed by theglobal hedging optimization problems (3.6)-(3.7) is now described. The procedure first proposedin Buehler et al. (2019b) uses minibatch stochastic gradient descent (SGD) to approximate thegradient of the cost function with Monte Carlo sampling. For convenience, the notation used forthe optimization procedure is similar to the one from Carbonneau and Godin (2020). Withoutloss of generality, the numerical procedure is only presented for the short measured risk exposure;the corresponding procedure for the long position is simply obtained through modifying theobjective function (3.8) that follows. Let J : R q → R be the cost function to be minimized for FFNNs are another class of neural networks which map input vectors into output vectors, in contrast toLSTMs which map input vector sequences to output vector sequences. θ is the set of trainable parameters of F ( S ) θ : J ( θ ) := ρ (cid:16) Φ( S (0 ,b ) N ) − B N G δ ( S ,θ ) N (cid:17) , θ ∈ R q . (3.8)A typical stochastic gradient descent procedure entails adapting the trainable parameters iterativelyand incrementally in the opposite direction of the cost function gradient with respect to θ : θ j +1 = θ j − η j ∇ θ J ( θ j ) , (3.9)where θ is the initial values for the trainable parameters, η j is a small deterministic positive realvalue commonly called the learning rate and ∇ θ denotes the gradient operator. In the currentstudy, the Glorot uniform initialization of Glorot and Bengio (2010) is always used to selectinitial parameters in θ . Since closed-form solutions for the gradient of the cost function withrespect to the trainable parameters are unavailable in the general market setting considered inthis work, the approach relies instead on Monte Carlo sampling to provide an estimate. Thus, let B j := { π i,j } N batch i =1 be a minibatch of simulated hedging errors of size N batch ∈ N where π i,j is the i th simulated hedging error when θ = θ j : π i,j := Φ( S (0 ,b ) N,i ) − B N G δ ( S ,θj ) N,i , (3.10)where S (0 ,b ) N,i and G δ ( S ,θj ) N,i are the i th random realization among the minibatch of the terminalunderlying asset price and discounted hedging portfolio gains, respectively. Furthermore, denoteˆ ρ : R N batch → R as the empirical estimator of ρ (Φ( S (0 ,b ) N ) − B N G δ ( S ,θ ) N ) evaluated with minibatchesof hedging errors. Minibatch SGD consists in approximating the gradient of the cost function ∇ θ J ( θ j ) with ∇ θ ˆ ρ ( B j ) in the update rule for trainable parameters: θ j +1 = θ j − η j ∇ θ ˆ ρ ( B j ) . (3.11) Minimizing J with respect to θ corresponds to the alternative problem (3.7) with zero initial capital. Recallthat (cid:15) ( L ) (0) and (cid:15) ( S ) (0) are required for the computation of C ? and (cid:15) ? . Consequently, hedging portfolio valuesused in LSTM feature vectors are equal to hedging gains, i.e. V δn = B n G δn . , the CVaR has the representationCVaR α ( X ) := E [ X | X ≥ VaR α ( X )] , α ∈ (0 , , (3.12)where VaR α ( X ) := min { x : P ( X ≤ x ) ≥ α } is the Value-at-Risk (VaR) with confidence level α of the liability X . Let { π [ i ] ,j } N batch i =1 be the order statistics (i.e. values sorted by increasing order)of B j . For ˜ N := d αN batch e where d x e is the ceiling function (i.e. the smallest integer greater orequal to x ), the empirical estimator of the CVaR used in this study is from the work of Honget al. (2014) and has the representationVaR V α ( B j ) := π [ ˜ N ] ,j , CVaR V α ( B j ) := VaR V α ( B j ) + 1(1 − α ) N batch N batch X i =1 max( π i,j − VaR V α ( B j ) , . The gradient of the empirical estimator of the Conditional Value-at-Risk with respect to thetrainable parameters (i.e. ∇ θ CVaR V α ( B j )) required for the update rule (3.11) can be computedexactly without discretization or other numerical approximations. Such computations canbe implemented with modern deep learning libraries such as Tensorflow (Abadi et al., 2016).Furthermore, algorithms which dynamically adapt the learning rate η j in (3.11) such as Adam (Kingma and Ba, 2014) have been shown to improve upon the effectiveness of SGD proceduresfor neural networks. For all numerical experiments conducted in Section 4, an implementation ofTensorflow with the Adam algorithm is used to optimize neural networks; the reader is referredto the online Github repository for samples of codes in Python. Also, Appendix B presents apseudo-code of the training procedure for F ( S ) θ . In Section 4, the only dynamics considered for the risky assets produce integrable and absolutely continuoushedging errors. github.com/alexandrecarbonneau. Numerical experiments
This section performs various numerical experimentations of the ERP approach for derivativesvaluation. The main goal is to study the impact of including options as hedging instruments onequal risk prices and on the level of market incompleteness. A special case assessed throughoutthis section is the trading of short-term vanilla options for the pricing and hedging of longer-termderivatives. The conduction of these experiments heavily relies on the neural network schemedescribed in Section 3 to solve the underlying global hedging problems of the ERP framework.Such exhaustive numerical study would have been hardly accessible with traditional methods(e.g. conventional dynamic programming algorithms) due to the high-dimensional continuousstate and action spaces of the hedging problem stemming from the use of multiple short-termoptions as hedging instruments and from the asset price dynamics considered. As a result, theuse of neural networks enables us to provide novel qualitative insights into the ERP framework.The analysis begins in Section 4.2 and Section 4.3 with the assessment of the sensitivity of equalrisk prices and residual hedging risk to the presence of two salient equity stylized features: jumpand volatility risks. The impact of the choice of convex risk measure on the ERP frameworkwhen trading exclusively options is examined in Section 4.4. Lastly, Section 4.5 presents thebenchmarking of equal risk prices to derivative premiums obtained with variance-optimal hedging.The specific financial market setup and asset dynamics models considered for all numericalexperiments are described in Section 4.1 that follows.
For the rest of the paper, the derivative to price is a European vanilla put option of payofffunction Φ( S (0 ,b ) N ) = max( K − S (0 ,b ) N ,
0) with K = 90 ,
100 and 110 corresponding respectively toan out-of-the-money (OTM), an at-the-money (ATM) and an in-the-money (ITM) option. Thematurity of the derivative is set to 1 year (i.e. T = 1) with 252 days. The annualized continuouslycompounded risk-free rate is r = 0 .
03. In addition to the risk-free asset, the hedging instrumentsconsist of either only the underlying stock, or exclusively shorter-term ATM European calls andputs. When hedging is performed with the underlying stock, daily and monthly rebalancing are19onsidered, corresponding to respectively N = 252 and N = 12 trading periods per year. Whenhedging with options, all options are assumed to have a single-period time-to-maturity, i.e. theyare traded once and held until expiration. We consider either 1-month or 3-months maturitiesATM calls and puts as hedging instruments, which respectively entails N = 12 or N = 4. Lessfrequent rebalancing when hedging with options rather than only with the underlying stock isconsistent with market practices; such hedging instruments are commonly embedded in semi-statictype of trading strategies, see for instance Carr and Wu (2014). Lastly, note that daily variationsfor the underlying log-returns and implied volatilities are always considered throughout the restof the paper, even with non-daily rebalancing periods (i.e. when hedging with the underlyingstock on a monthly basis or with 1-month and 3-months maturities options) by aggregating dailyvariations over the rebalancing period. The asset price dynamics models considered in stochastic simulations are now introduced. Tocharacterize jump risk, the Merton jump-diffusion model (MJD, Merton (1976)) is considered.Furthermore, the impact of volatility risk is assessed with the GJR-GARCH model of Glostenet al. (1993). Several sets of parameters are tested for each model to conduct a sensitivity analysisand highlight the impact of various model features on both equal risk prices and residual hedgingrisk.Denote y n := log( S (0 ,b ) n /S (0 ,b ) n − ) as the periodic underlying stock log-return between the tradingperiods t n − and t n . Since our modeling framework assumes daily variations for asset prices andpossibly non-daily rebalancing, let { ˜ y j,n } Mj =1 be the M daily stock log-returns in the time interval20 t n − , t n ] such that y n = M X j =1 ˜ y j,n , n = 1 , . . . , N, N × M = 252 , (4.1)where N corresponds to the number of trading dates to hedge the 1 year maturity derivative Φand M to the number of days between two trading dates. Thus, daily stock hedges correspondsto the case of N = 252 and M = 1, monthly stock and 1-month option hedges to N = 12 and M = 21, and 3-months option hedges to N = 4 and M = 63.The asset price dynamics are now formally defined for the daily log-returns. For the rest of thesection, let { (cid:15) j,n } M,Nj =1 ,n =1 be a sequence of independent standardized Gaussian random variableswhere the subsequence { (cid:15) j,n } Mj =1 will be used to model the M daily innovations of log-returns inthe time interval [ t n − , t n ]. The Merton-jump diffusion dynamics expands upon the ideal market conditions of the Black-Scholes model by incorporating random Gaussian jumps along stock paths. Let { N j,n } M,Nj =0 ,n =1 be a discrete-time sampling from a Poisson process of intensity parameter λ > { N j,n } Mj =0 corresponds to the M + 1 daily values of the Poisson process occurringduring the time interval [ t n − , t n ]. N , := 0 is the initial value of the process and N ,n +1 := N M,n for n = 1 , . . . , N −
1. Furthermore, denote { ξ k } ∞ k =1 as a sequence of random Gaussian variablescorresponding to the jumps of mean µ J and variance σ J . { N j,n } M,Nj =0 ,n =1 , { ξ k } ∞ k =1 and { (cid:15) j,n } N,Mn =1 ,j =1 are independent. For n = 1 , . . . , N and j = 1 , . . . , M , the daily log-return dynamics can be For completeness, let { ˜ S (0 ,b ) j,n } M,Nj =0 ,n =1 be the daily underlying stock prices where { ˜ S (0 ,b ) j,n } Mj =0 corresponds tothe M + 1 daily prices during the period [ t n − , t n ]. Also, let G := {G j,n } M,Nj =0 ,n =1 be a filtration satisfying theusual conditions with G j,n containing all information available to market participants at the j th day of the timeperiod [ t n − , t n ]. The filtration used to optimize trading strategies F with time steps t , t , . . . , t N is a subset of G by construction. However, since the risky asset dynamics considered in this paper have the Markov property,optimizing trading strategies with the filtration F or G results in the same trading policy. ˜ y j,n = 1252 (cid:18) ν − λ (cid:16) e µ J + σ J / − (cid:17) − σ (cid:19) + σ r (cid:15) j,n + N j,n X k = N j − ,n +1 ξ k , (4.2)where { ν, µ J , σ J , λ, σ } are the model parameters with { ν, λ, σ } being on a yearly scale, ν ∈ R and σ >
0. Furthermore, since { S (0 ,b ) n } Nn =0 has the Markov property with respect to the filtration F generated by the trading dates observations, no additional state associated to the risky assetdynamics is required to be added to the feature vectors of neural networks (i.e. ϕ n = 0 for alltime steps n in (3.4) and (3.5)). GARCH processes also expand upon the Black-Scholes ideal framework by exhibiting well-knownempirical features of risky assets such as time-varying volatility, volatility clustering and theleverage effect (i.e. negative correlation between underlying returns and its volatility). Dailylog-returns modeled with a GJR-GARCH(1,1) dynamics have the representation˜ y j,n = µ + ˜ σ j,n (cid:15) j,n , ˜ σ j +1 ,n = ω + υ ˜ σ j,n ( | (cid:15) j,n | − γ(cid:15) j,n ) + β ˜ σ j,n , (4.3)where { ˜ σ j,n } M +1 ,Nj =1 ,n =1 are the daily conditional variances of log-returns. More precisely, { ˜ σ j,n } M +1 j =1 are the M + 1 daily conditional variances in the time interval [ t n − , t n ]. Also, ˜ σ ,n +1 := ˜ σ M +1 ,n for n = 1 , . . . , N − . Model parameters consist of { µ, ω, υ, γ, β } with { ω, υ, β } being positive realvalues and γ, µ ∈ R . Note that if the starting value of the GARCH process ˜ σ , is deterministic,then { ˜ σ j,n } M +1 ,Nj =1 ,n =1 can be computed recursively with the observed daily log-returns. In this paper,˜ σ , is set as the stationary variance: ˜ σ , := ω − υ (1+ γ ) − β . Also, contrarily to the MJD model, theGJR-GARCH(1,1) requires adding at each trading time t n the current stochastic volatility value This paper adopts the convention that if N j,n = N j − ,n , i.e. that no jumps occurred on that day, then: N j,n X k = N j − ,n +1 ξ k = 0 .
22o the feature vectors of the neural networks, i.e. ϕ n = ˜ σ ,n +1 for n = 0 , . . . , N − This work proposes to model the daily variations of the logarithm of ATM implied volatilitieswith 1-month and 3-months maturities as a discrete-time version of the Ornstein-Uhlenbeck (OU)process. The choice of an OU type of dynamics for IVs is motivated by the work of Cont andDa Fonseca (2002) which shows that for S&P 500 index options, the first principal component ofthe daily variations of the logarithm of the IV surface accounts for the majority of its variance andcan be interpreted as a level effect. Also, this first principal component can be well representedby a low-order autoregressive (AR) model. The OU dynamics considered in this study thereforehas the representation of an AR model of order 1.The dynamics for the daily evolution of IVs is now formally defined. For convenience, this paperassumes that 1-month and 3-months IVs are the same. Using a similar notation as for dailylog-returns, let { f IV j,n } M,Nj =0 ,n =1 be the daily ATM IV process for both 1-month and 3-monthsmaturities where { f IV j,n } Mj =0 are the M +1 daily observations during the time interval [ t n − , t n ] with f IV ,n +1 := f IV M,n for n = 1 , . . . , N −
1. Furthermore, let { Z j,n } M,Nj =1 ,n =1 be an additional sequence ofindependent standardized Gaussian random variables characterizing shocks in the IV dynamics. Inorder to incorporate the stylized feature of strong negative correlation between implied volatilitiesand asset returns (Cont and Da Fonseca (2002)), the modeling framework assumes that the dailyinnovations of log-returns and IVs are correlated with parameter % := corr ( (cid:15) j,n , Z j,n ) set at − . n = 1 , . . . , N and j = 0 , . . . , M −
1: log f IV j +1 ,n = log f IV j,n + κ ( ϑ − log f IV j,n ) + σ IV Z j +1 ,n , (4.4) It is important to note that implied volatilities are used strictly for pricing options used as hedging instruments.They are not used to price the derivative Φ. It is worth highlighting that since trading strategies allow for the use of either 1-month or 3-months maturitiesATM calls and puts, but not both maturities within the same strategy, 1-month and 3-months IVs are never usedat the same time. { κ, ϑ, σ IV } are the model parameters with κ, ϑ ∈ R and σ IV >
0. The initial value ofthe process is set as log f IV , = ϑ . Also, recall that when trading options, their correspondingimplied volatilities at each trading date are added to the feature vectors of neural networks, i.e. IV n − = f IV ,n in (3.5) for n = 1 , . . . , N .The pricing of calls and puts used as hedging instruments is done with the well-known Black-Scholes formula hereby stated with the annual volatility term set at the implied volatility value.For the underlying price S , implied volatility IV , strike price K and time-to-maturity ∆ T , theBlack-Scholes pricing formulas for calls and puts are respectively C ( S, IV, ∆ T, K ) := S N ( d ) − e − r ∆ T K N ( d ) , (4.5) P ( S, IV, ∆ T, K ) := e − r ∆ T K N ( − d ) − S N ( − d ) , (4.6)where N ( · ) denotes the standard normal cumulative distribution function and d := log( SK ) + ( r + IV )∆ TIV √ ∆ T , d := d − IV √ ∆ T .
The set of hyperparameters for the LSTMs are two LSTM cells (i.e. H = 2) and 24 neurons percell (i.e. d j = 24 for j = 1 , ,
000 paths is used to optimize the trainableparameters with a total of 50 epochs and a minibatch size of 1 ,
000 sampled exclusively from thetraining set. The deep learning library Tensorflow (Abadi et al., 2016) is used to implement thestochastic gradient descent procedure with the Adam optimizer of Kingma and Ba (2014) with alearning rate hyperparameter value of 0 . /
6. All numerical results presented throughout thissection are computed based on a test set (i.e. out-of-sample dataset) of 100 ,
000 paths. Lastly,unless specified otherwise, the convex risk measure chosen for all experiments is the CVaR withconfidence level α = 0 .
95. Sensitivity analyses of equal risk prices and residual hedging risk with One epoch consists of a complete iteration of SGD on the training set. For a training set of 400 ,
000 paths anda minibatch of size 1 , This section examines the sensitivity of the ERP solution to equity jump risk. The analysis iscarried out by considering three different sets of parameters for the MJD dynamics which inducedifferent levels of jump frequency and severity. While maintaining empirical plausibility, thisis done by modifying the intensity parameter λ controlling the expected frequency of jumps aswell as parameters µ J and σ J controlling the severity component of jumps. In order to betterisolate the impact of different stylized features of jump risk on the ERP framework, the diffusionparameter is fixed for all three sets of parameters. Also, the parameters { λ, µ J , σ J , ν } are chosensuch that the yearly expected value and standard deviation of log-returns are respectively 10%and 15% for all three cases. To facilitate the analysis, the three sets of parameters are referredto as scenario 1, scenario 2 and scenario 3 for jump risk. Model parameter values for the threescenarios are presented in Table 1. Scenario 1 represents relatively smaller but more frequent Table 1:
Parameters of the Merton jump-diffusion model for the three scenarios. ν σ λ µ J σ J Scenario 1 0 . . − .
05 0 . . . . − .
10 0 . . . . − .
20 0 . ν , σ and λ are on an annual basis.jumps with on average one jump per year of mean −
5% and standard deviation 5%. Scenario 2entails more severe, but less frequent jumps with on average one jump every four years of mean −
10% and standard deviation 10%. Lastly, scenario 3 depicts the most extreme case with rarebut very severe jumps with on average one jump every twelve and a half years of mean − ϑ is set at the logarithm The parameter σ in (4.2) corresponds to the diffusion parameter of the MJD dynamics.
25f the yearly standard deviation of log-returns with ϑ = log 0 .
15, and other parameters are chosenin an ad hoc fashion so as to produce reasonable values for implied volatilities.
Table 2:
Parameters of the log-AR(1) model for the evolution of implied volatilities. κ ϑ σ IV % .
15 log(0 .
15) 0 . − . Table 3 presents equal risk prices C ? and residual hedging risk (cid:15) ? across the three scenarios of jumpparameters and different trading instruments. Numerical values indicate that in the presence ofjump risk, hedging with options entails significant reduction of both equal risk prices and marketincompleteness as compared to hedging solely with the underlying stock across all moneynesslevels and jump risk scenarios. The reduction in hedging residual risk by trading options isobtained despite less frequent rebalancing than when only the stock is used. These results addadditional evidence that options are indeed non-redundant as prescribed by the Black-Scholesworld: the equal risk pricing framework dictates that hedging with options in the presence ofjump risk can significantly impact both derivative premiums and hedging risk as quantified byour incompleteness metrics.The relative reduction achieved in C ? with 1-month and 3-months options as compared to hedgingwith the stock is most important for OTM puts, followed by ATM and ITM contracts. Forinstance, the relative reduction obtained with 3-months options hedging over daily stock hedgingranges across the three jump risk scenarios between 8% to 29% for OTM, 6% to 22% for ATMand 4% to 13% for ITM puts. This reduction in C ? when using options as hedging instrumentscan be explained by the following observations. As pointed out in Carbonneau and Godin (2020),the fact that a put option payoff is bounded below at zero entails that the short position hedgingerror has a thicker right tail than the long position hedging error. Also, it is widely documented inthe literature that hedging jump risk with options significantly dampens tail risk as compared to If C ? (daily stock) and C ? (3-months options) are equal risk prices obtained respectively by hedging with thestock on a daily basis and with 3-months options, the relative reduction is computed as 1 − C ? (3-months options) C ? (daily stock) forall examples. able 3: Sensitivity analysis of equal risk prices C ? and residual hedging risk (cid:15) ? to jump risk forOTM ( K = 90), ATM ( K = 100) and ITM ( K = 110) put options of maturity T = 1.OTM ATM ITMJump Scenario (1) (2) (3) (1) (2) (3) (1) (2) (3) C ? Daily stock 1 .
89 2 .
58 3 .
36 5 .
21 6 .
01 6 .
81 10 .
81 11 .
68 12 . .
97 2 .
60 3 .
31 5 .
04 5 .
77 6 .
38 10 .
73 11 .
44 11 . .
82 2 .
24 2 .
55 4 .
99 5 .
36 5 .
60 10 .
48 10 .
86 10 . .
74 2 .
08 2 .
39 4 .
87 5 .
12 5 .
28 10 .
43 10 .
51 10 . (cid:15) ? Daily stock 1 .
09 1 .
98 2 .
67 1 .
76 2 .
74 3 .
54 1 .
82 2 .
78 3 . .
82 2 .
52 3 .
26 3 .
00 3 .
88 4 .
57 3 .
07 3 .
91 4 . .
76 1 .
18 1 .
52 1 .
14 1 .
53 1 .
78 1 .
17 1 .
56 1 . .
03 1 .
37 1 .
68 1 .
59 1 .
82 2 .
02 1 .
70 1 .
79 1 . (cid:15) ? /C ? Daily stock 0 .
58 0 .
77 0 .
79 0 .
34 0 .
46 0 .
52 0 .
17 0 .
24 0 . .
92 0 .
97 0 .
99 0 .
60 0 .
67 0 .
72 0 .
29 0 .
34 0 . .
42 0 .
53 0 .
60 0 .
23 0 .
28 0 .
32 0 .
11 0 .
14 0 . .
59 0 .
66 0 .
70 0 .
33 0 .
36 0 .
38 0 .
16 0 .
17 0 . ,
000 independent paths generated from the MertonJump-Diffusion model for the underlying (see Section 4.1.2 for model description). Three differentsets of parameters values are considered with λ = { , . , . } , µ J = {− . , − . , − . } and σ J = { . , . , . } respectively for jump scenario 1, 2, and 3 (see Table 1 for all parametersvalues). Hedging instruments : daily or monthly rebalancing with the underlying stock and1-month or 3-months options with ATM calls and puts. Options used as hedging instrumentsare priced with implied volatility modeled with a log-AR(1) dynamics (see Section 4.1.4 formodel description and Table 2 for parameters values). The training of neural networks is done asdescribed in Section 4.1.5. The confidence level of the CVaR measure is α = 0 . Consequently, the choice of trading options to mitigate jump risk reduces the measured riskexposure of both the long and short positions, but the thicker right tail for the short positionhedging error entails a larger decrease for the latter than for the long position. In such situations,the ERP framework dictates that the long position should be compensated with a lower derivativepremium C ? to equalize residual hedging risk of both positions.Moreover, values for both (cid:15) ? -metrics indicate that in the presence of jump risk, the use of optionscontributes significantly to the reduction of market incompleteness as both the long and shortposition hedges achieve risk reduction when compared to trading only with the stock. The latterconclusion is in itself not novel, and is widely documented in the literature (see, for instance,Cont and Tankov (2003) and the many references therein). Indeed, this is a consequence ofthe well-known convex property of put option prices, which implies that hedging random jumpssolely with the underlying stock is ineffective. Our (cid:15) ? -metrics have the advantage of allowing fora precise quantification of such reduction in residual hedging risk achieved through the use ofoptions as hedging instruments.The sensitivity of equal risk prices and residual hedging risk across the three jump risk scenarios foreach set of hedging instruments is now examined. Numerical results presented in Table 3 indicatethat for a fixed set of hedging instruments, both the equal risk price and the level of incompletenessincreases with the severity of jumps across all moneyness levels. Indeed, the relative increase ofequal risk prices observed under scenario 3 as compared to scenario 1 respectively for OTM, ATMand ITM puts is 78% ,
31% and 12% with the daily stock, 68%, 27% and 11% with the monthlystock, 40%, 12% and 3% with 1-month options and 38%, 8% and 1% with 3-months options. Similar observations can be made for both incompleteness metrics: increases in jump severityleads to larger (cid:15) ? and (cid:15) ? /C ? . This positive association between both equal risk prices and the Horvath et al. (2021) deep hedge derivatives under a rough Bergomi volatility model by trading the underlyingstock and a variance swap. The latter paper shows that this dynamics exhibits jump-like behaviour when discretized.As results presented in this current paper highlights the fact that global hedging jump risk with option hedges isvery effective, deep hedging with options could also potentially be effective under such rough volatility models. For a fixed hedging instrument and moneyness level, if C ? (scenario 1) and C ? (scenario 3) are respectively theequal risk price obtained under jump risk scenario 1 and 3, the relative increase is computed as C ? (scenario 3) C ? (scenario 1) − Having examined the impact of jump risk on the ERP framework, the impact of volatility risk isnow studied. In the same spirit as analyses done for jump risk, three different sets of parametersare considered for the GARCH dynamics which imply annualized stationary (expected) volatilitiesof 10% ,
15% and 20%. The three sets of parameters are presented in Table 4. Note that everyparameter is fixed for all three sets, except for the level parameter ω , which is adjusted to attainthe wanted stationary volatility. The value of the drift parameter µ is set such that the yearlyexpected value of log-returns is 10%. Also, values for { υ, γ, β } are inspired from parametersestimated with maximum likelihood on a time series of daily log-returns on the S&P 500 indexfor the period 1986-12-31 to 2010-04-01 used in Carbonneau and Godin (2020). The same setupis considered as in Section 4.2 in terms of the derivative to be priced (1-year maturity Europeanputs) and for the choice of hedging instruments (underlying stock traded on a daily or monthlybasis and 1-month or 3-months maturities ATM calls and puts). The same parameters as in thestudy of jump risk conducted in Section 4.2 are used for { κ, σ IV , % } of the log-AR(1) dynamics for The annualized stationary volatility with 252 days per year is computed as s ω − υ (1 + γ ) − β . κ = 0 . , σ IV = 0 .
06 and % = − . ϑ , which is set to be in line with the underlying GARCH process aslog(0 . , log(0 .
15) and log(0 .
20) when the stationary volatility is 10%, 15% and 20%, respectively.It is worth highlighting that the choice of modeling implied volatilities for short-term options withhigher and smaller average levels enables us to assess the impact of larger and smaller averagecosts for trading options on the equal risk price and residual hedging risk of longer-term options.
Table 4:
Parameters of the GJR-GARCH model for 10% ,
15% and 20% stationary yearlyvolatilities. Stationary volatility µ ω υ γ β
10% 3 . . .
05 0 . . . . .
05 0 . . . . .
05 0 . . Table 5 presents equal risk prices C ? and (cid:15) ? -metrics for put options of 1 year maturity acrossthe three sets of volatility risk parameters and hedging instruments. Numerical results indicatethat in the presence of volatility risk, the use of options as hedging instruments can reduce C ? as compared to daily stock hedging. However, this impact on C ? when trading options can bemarginal and is highly sensitive to the moneyness level of the put option being priced as wellas to the maturity of the traded options. Furthermore, the impact on C ? of the use of optionswithin hedges tends to diminish when traded options are more costly (i.e. as the average levelof implied and GARCH volatility increases). Indeed, the relative reduction in equal risk pricesachieved with 1-month options hedging as compared to daily stock hedging with 10% ,
15% and20% stationary volatility is respectively 44% ,
26% and 15% for OTM puts, 12% ,
9% and 5% forATM and 1% ,
2% and 1% for ITM options. However, the relative reduction in C ? with 3-monthsoption hedges as compared to using the stock on a daily basis is overall much more marginal,with the notable exceptions of OTM and ATM puts with 10% stationary volatility which achieve If C ? (daily stock) and C ? (1-month options) are respectively the equal risk price obtained by hedging withthe stock on a daily basis and with 1-month options, the relative reduction is computed as 1 − C ? (1-month options) C ? (daily stock) for all examples. able 5: Sensitivity analysis of equal risk prices C ? and residual hedging risk (cid:15) ? to volatility riskfor OTM ( K = 90), ATM ( K = 100) and ITM ( K = 110) put options of maturity T = 1.OTM ATM ITMStationary volatility 10% 15% 20% 10% 15% 20% 10% 15% 20% C ? Daily stock 1 .
01 2 .
35 3 .
85 3 .
23 5 .
36 7 .
24 8 .
56 10 .
55 12 . .
17 2 .
65 4 .
23 3 .
37 5 .
44 7 .
58 8 .
85 10 .
82 12 . .
56 1 .
74 3 .
27 2 .
86 4 .
87 6 .
89 8 .
46 10 .
32 12 . .
76 2 .
07 3 .
65 3 .
01 5 .
08 7 .
16 8 .
51 10 .
38 12 . (cid:15) ? Daily stock 0 .
77 1 .
51 2 .
21 1 .
28 2 .
12 2 .
67 1 .
12 1 .
98 2 . .
15 2 .
44 3 .
67 2 .
15 3 .
41 4 .
62 1 .
92 3 .
29 4 . .
26 0 .
65 1 .
06 0 .
59 1 .
00 1 .
36 0 .
62 1 .
03 1 . .
59 1 .
32 2 .
02 1 .
10 1 .
77 2 .
41 1 .
04 1 .
68 2 . (cid:15) ? /C ? Daily stock 0 .
77 0 .
64 0 .
57 0 .
40 0 .
40 0 .
37 0 .
13 0 .
19 0 . .
99 0 .
92 0 .
87 0 .
64 0 .
63 0 .
61 0 .
22 0 .
30 0 . .
45 0 .
37 0 .
32 0 .
21 0 .
20 0 .
20 0 .
07 0 .
10 0 . .
77 0 .
64 0 .
55 0 .
36 0 .
35 0 .
34 0 .
12 0 .
16 0 . ,
000 independent paths generated from the GJR-GARCH(1,1) model for the underlying with three sets of parameters implying stationary yearlyvolatilities of 10% ,
15% and 20% (see Section 4.1.3 for model description and Table 4 for parametersvalues).
Hedging instruments : daily or monthly rebalancing with the underlying stock and 1-month or 3-months options with ATM calls and puts. Options used as hedging instrumentsare priced with implied volatility modeled as a log-AR(1) dynamics with κ = 0 . , σ IV = 0 . % = − . ϑ set to log(0 . , log(0 .
15) and log(0 .
20) when the GARCHstationary volatility is 10% ,
15% and 20%, respectively (see Section 4.1.4 for the log-AR(1) modeldescription). The training of neural networks is done as described in Section 4.1.5. The confidencelevel of the CVaR measure is α = 0 .
95. 31espectively 25% and 7% reduction as well as for the OTM moneyness under 15% stationaryvolatility with a 12% reduction. Also, as expected, values presented in Table 5 confirm that thelevel of market incompleteness as measured by the (cid:15) ? metric has a positive relationship with theaverage level of stationary volatility for all hedging instruments.The previously described observations about the impact of option hedges on both equal risk pricesand residual hedging risk all stem from the realized reduction in measured risk exposure by thelong and short positions. However, contrarily to results obtained with jump risk, the reduction inmeasured risk exposure when hedging volatility risk with options can be very similar for boththe long and short positions, whereas with jump risk, the reduction is asymmetric by alwaysfavoring the short position with a larger reduction. The latter can be explained by the fact thatvolatility risk impacts both upside and downside risk, while the impact of jump risk dynamicsconsidered in this paper is very asymmetric by entailing significantly more weight on the right(resp. left) tail of the short (resp. long) hedging error with predominantly negative jumps. Valuespresented in Table 5 confirm this analysis of the interrelation between volatility risk and thechoice of hedging instruments. For instance, for ITM puts, the measured risk exposure of the longand short positions decreases by a similar amount when trading 1-month or 3-months optionsas compared to daily stock hedges, which explains the significant decrease in (cid:15) ? , but also theinsensitivity of C ? to the choice of hedging instruments and rebalancing frequency. On the otherhand, for OTM puts, 1-month and 3-months option hedges results in larger decreases of measuredrisk exposure for the short position than for the long position, which explains the reduction in C ? and (cid:15) ? as compared to daily stock hedges.Lastly, it is very interesting to observe that the average price level of short-term options usedas hedging instruments is effectively reflected into the equal risk price of longer-term options.Indeed, numerical results for C ? presented in Table 5 highlight the fact that higher hedgingoptions implied volatilities for 1-month and 3-months ATM calls and puts leads to higher equalrisk prices for 1-year maturity puts. Furthermore, to isolate the idiosyncratic contribution ofthe variations of option prices used as hedging instruments on the equal risk price from theimpact of the stationarity volatility of the GARCH process, the authors also tested fixing the32tationarity volatility of the GARCH process to 15% and setting the long-run parameter of theIV process to 14% and 16%. These results presented in the Supplementary Material, TableSM2, confirm that higher implied volatilities for options used as hedging instruments leads tohigher equal risk prices. All of these benchmarking results demonstrate the potential of the ERPframework as a fair valuation approach consistent with observable market prices. For instance,the ERP framework could be used to price and optimally hedge over-the-counter derivativeswith vanilla options. An additional potential application is the marking-to-market of less liquidlong-term derivatives (e.g. Long-Term Equity AnticiPation Securities (LEAPS)) consistently withhighly liquid shorter-term option hedges. The ERP framework could also be used for the fairvaluation of segregated funds guarantees, which are equivalent to very long-term (up to 40 years)derivatives sold by insurers. Indeed, International Financial Reporting Standards 17 (IFRS 17,IASB (2017)) mandates a market consistent valuation of options embedded in segregated fundsguarantees with readily available observable market prices at the measurement date. The ERPframework could potentially be applied to price such very long-term options consistently withshorter-term implied volatility surface dynamics, with the latter being much less challenging tocalibrate due to the higher liquidity of short-term options. α This section conducts sensitivity analyses with respect to the choice of convex risk measure onthe ERP framework when trading exclusively options. Similarly to the work of Carbonneau andGodin (2020), values for equal risk prices and (cid:15) ? -metrics are examined across the confidence levels0 . , .
95 and 0 .
99 for the CVaR α measure. As argued in the latter paper, higher confidencelevels corresponds to more risk averse agents by concentrating more relative weight on losses of Note that Carbonneau (2020) demonstrates the potential of the deep hedging algorithm for global hedginglong-term lookback options embedded in segregated funds guarantees with multiple hedging instruments. It isalso worth highlighting that Barigou et al. (2020) developed a pricing scheme consistent with local non-quadratichedging procedures for insurance liabilities which relies on neural networks. In the context of segregated funds, the short position of the embedded option is assumed to be held by aninsurance company who has to provide a quote and mitigate its risk exposure. The long position is held by anunsophisticated investor who will not be hedging his risk exposure. Nevertheless, as IFRS 17 mandates the use ofa fair valuation approach for embedded options consistent with observable market prices, the ERP frameworkcould potentially be used in this context. C ? and (cid:15) ? metrics. The objective of this section is to assess if this finding isrobust to the use of short-term option hedges instead of the underlying stock. For each confidencelevel, the authors of the current paper computed both equal risk prices and residual hedging riskobtained by trading 3-months ATM calls and puts with the same setup as in Section 4.2 andSection 4.3, i.e. for all three jump and volatility scenarios of parameters. Overall, the mainconclusions are found to be qualitatively similar for all of the different setups. Thus, to savespace, values for equal risk prices and residual hedging risk are only reported under the MJDdynamics with jump risk scenario 2 by trading 3-months options; these results are presented inTable 6. The interested reader in numerical results obtained under jump risk scenarios 1 and 3 aswell as under the three sets of volatility risk parameters is referred to Table SM3 and Table SM4of the Supplementary Material.Numerical values reported in Table 6 indicate that with option hedges, an increase in the confidencelevel parameter of the CVaR α measure leads to larger equal risk prices C ? and residual hedgingrisk (cid:15) ? across all examples. These results confirm that the finding of Carbonneau and Godin (2020)with respect to the sensitivity of C ? and (cid:15) ? to the risk aversion of the hedger is robust to usingexclusively options as hedging instruments. Furthermore, values for equal risk prices C ? showa largest increase when using CVaR . and CVaR . as compared to CVaR . for OTM puts,followed by ATM and ITM moneyness levels; the same conclusion was observed in Carbonneauand Godin (2020) when trading the underlying stock. The increase in C ? with the risk aversionlevel of the hedger stems from the thicker right tail of the short position hedging error thanfor the long position hedging error. The latter observation is consistent with previous analyses:while option hedges are more effective than stock hedges in the presence of equity jump risk as Unreported tests performed by the authors show that values lower than 0 .
90 for the confidence level of CVaR α with 1-month and 3-months option hedges lead to trading policies with significantly larger tail risk in a way whichwould deem such policies as inadmissible by hedgers. Using the CVaR . measure with 1-month options alsoresulted in trading policies with significantly larger tail risk. However, this large increase in tail risk was notobserved with the CVaR . and CVaR . measures when trading 1-month options, nor with CVaR . , CVaR . and CVaR . when trading 3-months options. These observations motivated the choice of performing sensitivityanalysis for CVaR α with α = 0 . , .
95 and 0 .
99 exclusively when trading 3-months options. able 6: Sensitivity analysis of equal risk prices C ? and residual hedging risk (cid:15) ? for OTM( K = 90), ATM ( K = 100) and ITM ( K = 110) put options of maturity T = 1 under the MJDdynamics with jump risk scenario 2. C ? (cid:15) ? (cid:15) ? /C ? Moneyness OTM ATM ITM OTM ATM ITM OTM ATM ITMCVaR . .
86 4 .
93 10 .
40 0 .
99 1 .
43 1 .
50 0 .
53 0 .
29 0 . .
12% 4% 1% 39% 28% 20% 24% 23% 18%CVaR .
40% 10% 4% 116% 76% 65% 54% 60% 59%Notes: Results are computed based on 100 ,
000 independent paths generated from the MertonJump-Diffusion model for the underlying (see Section 4.1.2 for model description) with parameters ν = 0 . , σ = 0 . , λ = 0 . , µ J = − .
10 and σ J = 0 .
10 corresponding to jump risk scenario2 of Table 1. Hedging instruments consist of 3-months ATM calls and puts priced with impliedvolatility modeled with a log-AR(1) dynamics (see Section 4.1.4 for model description and Table 2for parameters values). The training of neural networks is done as described in Section 4.1.5.Values for the CVaR . and CVaR . measures are expressed relative to CVaR . (% increase).demonstrated in Section 4.2, their inclusion within hedging portfolios does not fully mitigate theasymmetry in tail risk of the residual hedging error. This section presents the benchmarking of equal risk prices to derivative premiums obtainedwith variance-optimal hedging procedures (VO, Schweizer (1995)), also commonly called globalquadratic hedging . Variance-optimal hedging solves jointly for the initial capital investment and aself-financing strategy minimizing the expected value of the squared hedging error:min δ ∈ Π ,V ∈ R E (cid:20)(cid:16) Φ( S (0 ,b ) N ) − B N ( V + G δN ) (cid:17) (cid:21) . (4.7)The optimized initial capital investment denoted hereafter as C ( V O )0 can be viewed as the productioncost of Φ, since the resulting dynamic trading strategy replicates the derivative’s payoff as closelyas possible in a quadratic sense. The optimization problem (4.7) can also be solved in a similarfashion as the non-quadratic global hedging problems embedded in the ERP framework, but withtwo distinctions: the initial capital investment is treated as an additional trainable parameterand a single neural network is considered since the optimal trading strategy is the same for the35ong and short position due to the quadratic penalty. The reader is referred to Appendix A fora complete description of the numerical scheme for variance-optimal hedging implemented in thisstudy.The setup considered for the examination of this benchmarking is the same as in Section 4.2 withthe MJD dynamics under the three jump risk scenarios, with the exception of the confidencelevel of the CVaR α measure, which is studied at first with α = 0 .
95 fixed as in Section 4.2and Section 4.3, and subsequently across α = 0 . , .
95 and 0 .
99 as in Section 4.4. Note thatthe authors also conducted the same experiments under the setup of Section 4.3 with volatilityrisk, and found that the main qualitative conclusions are very similar. The reader is referred toTable SM6 and Table SM8 of the Supplementary Material for the benchmarking of ERP to VOprocedures in the presence of volatility risk.
Table 7 presents benchmarking results of equal risk prices C ? to variance-optimal prices C ( V O )0 under the MJD dynamics with the CVaR . measure. Numerical experiments show that C ? isat least larger than C ( V O )0 for all examples, but the relative increase is always smaller and lesssensitive to jump severity when trading options. Furthermore, the relative increase in derivativepremiums observed with the ERP framework over VO hedging is the largest for OTM puts,followed by ATM and ITM options across all jump risk scenarios and hedging instruments. Forinstance, the relative increase in C ? as compared to C ( V O )0 when trading the daily stock rangesfrom jump scenario 1 to scenario 3 between 17% to 75% for OTM puts, 11% to 42% for ATMand 6% to 20% for ITM options. On the other hand, the relative increase in C ? as comparedto C ( V O )0 is much less sensitive to jump severity when trading 1-month options by ranging fromscenario 1 to scenario 3 between 6% to 10% for OTM puts, 3% to 6% for ATM and 2% to 4% forITM. Based on these results, we can assert that although both derivative valuation schemes areconsistent with optimal trading criteria, the choice of hedging instrument and pricing procedure Cao et al. (2020) showed that the deep hedging algorithm for variance-optimal hedging problems providesgood approximations of optimal initial capital investments by comparing the optimized values to known formulas. The relative increase is computed as C ? C ( V O )0 − able 7: Equal risk prices C ? and variance-optimal (VO) prices C ( V O )0 with jump risk for OTM( K = 90), ATM ( K = 100) and ITM ( K = 110) put options of maturity T = 1.OTM ATM ITMJump Scenario (1) (2) (3) (1) (2) (3) (1) (2) (3) C ( V O )0 Daily stock 1 .
62 1 .
77 1 .
92 4 .
71 4 .
79 4 .
80 10 .
20 10 .
18 10 . .
55 1 .
73 1 .
86 4 .
62 4 .
72 4 .
72 10 .
14 10 .
11 10 . .
71 2 .
04 2 .
38 4 .
82 5 .
11 5 .
31 10 .
27 10 .
45 10 . .
58 1 .
83 2 .
08 4 .
64 4 .
82 4 .
97 10 .
11 10 .
15 10 . C ? Daily stock 17% 45% 75% 11% 25% 42% 6% 15% 20%Monthly stock 27% 51% 78% 9% 22% 35% 6% 13% 18%1-month options 6% 10% 7% 3% 5% 6% 2% 4% 3%3-months options 10% 14% 15% 5% 6% 6% 3% 4% 4%Notes: Results are computed based on 100 ,
000 independent paths generated from the MertonJump-Diffusion model for the underlying (see Section 4.1.2 for model description). Three differentsets of parameters values are considered with λ = { , . , . } , µ J = {− . , − . , − . } and σ J = { . , . , . } respectively for the jump scenario 1, 2, and 3 (see Table 1 for allparameters values). Hedging instruments : daily or monthly rebalancing with the underlying stockand 1-month or 3-months options with ATM calls and puts. Options used as hedging instrumentsare priced with implied volatility modeled with a log-AR(1) dynamics (see Section 4.1.4 for modeldescription and Table 2 for parameters values). The training of neural networks for ERP and VOhedging is done as described in Section 4.1.5 and Appendix A, respectively. The confidence levelof the CVaR measure is α = 0 . C ? are expressed relative to C ( V O )0 (% increase).(hence implicitly of the treatment of hedging gains and losses) has a material impact on resultingderivative premiums and must thus be carefully chosen.This smaller disparity between equal risk and variance-optimal prices with option hedges is inline with previous analyses: in the presence of jump or volatility risk, hedging with optionsentails significant reduction of the level market incompleteness as compared to trading solely theunderlying stock. In such cases, premiums obtained with both derivative valuation approaches37 able 8: Sensitivity analysis of equal risk prices C ? with CVaR . , CVaR . and CVaR . measures to variance-optimal (VO) prices C ( V O )0 under jump risk for OTM ( K = 90), ATM( K = 100) and ITM ( K = 110) put options of maturity T = 1.OTM ATM ITMJump Scenario (1) (2) (3) (1) (2) (3) (1) (2) (3) C ( V O )0 .
58 1 .
83 2 .
08 4 .
64 4 .
82 4 .
97 10 .
11 10 .
15 10 . C ? (CVaR . ) 3% 2% 0% 2% 2% 2% 2% 2% 2% C ? (CVaR . ) 10% 14% 15% 5% 6% 6% 3% 4% 4% C ? (CVaR . ) 32% 43% 55% 10% 12% 16% 6% 6% 8%Notes: Results are computed based on 100 ,
000 independent paths generated from the MertonJump-Diffusion model for the underlying (see Section 4.1.2 for model description). Three differentsets of parameters values are considered with λ = { , . , . } , µ J = {− . , − . , − . } and σ J = { . , . , . } respectively for jump scenario 1, 2, and 3 (see Table 1 for all parametersvalues). Hedging instruments consist of 3-months ATM calls and puts priced with impliedvolatility modeled as a log-AR(1) dynamics (see Section 4.1.4 for model description and Table 2for parameters values). The training of neural networks for ERP and VO hedging is doneas described in Section 4.1.5 and Appendix A, respectively. C ? with CVaR . , CVaR . andCVaR . are expressed relative to C ( V O )0 (% increase).should be closer with the limiting case of being the same in a complete market. These observationsexpand upon the work of Carbonneau and Godin (2020), which shows that equal risk prices ofputs obtained by hedging solely with the underlying stock are always larger than risk-neutralprices computed under convential change of measures. Indeed, benchmarking results presented inthis current paper provide important novel insights into this price inflation phenomenon observedwith the ERP framework: the disparity between equal risk and variance-optimal prices is alwayssignificantly smaller and less sensitive to stylized features of risky assets (e.g. jump or volatilityrisk) when option hedges are considered instead of trading exclusively the underlying stock.Moreover, Table 8 presents benchmarking results of C ? to C ( V O )0 with CVaR . , CVaR . andCVaR . measures with 3-months option hedges. Values presented in this benchmarking demon- To further illustrate this phenomenon, the authors also performed the same benchmarking with the Black-Scholes dynamics under which market incompleteness solely stems from discrete-time trading. The latter resultsare presented in the Supplementary Material. Numerical values show that under the Black-Scholes dynamics,trading the underlying stock on a daily basis leads for most combinations of moneyness level and yearly volatilityto the closest derivative premiums between ERP and VO procedures as compared to the other hedging instruments(see Table SM5). Also, as expected under the Black-Scholes dynamics, daily stock hedging entails the smallestlevel of residual hedging risk across the different hedging instruments (see Table SM1). . measure, we observe that C ? values are very close to C ( V O )0 where the relative difference rangesbetween 0% and 3% across all moneynesses and jump risk scenarios. On the other hand, opti-mizing trading policies with more risk averse agents, i.e. with CVaR . or CVaR . , providesa very wide range of derivative premiums with the ERP framework, especially for the OTMmoneyness level. It is very interesting to note that this added flexibility of ERP procedures forpricing derivatives does not come at the expense of less effective hedging policies. Indeed, amajor drawback of variance-optimal hedging lies in penalizing equally gains and losses through aquadratic penalty for hedging shortfalls. Conversely, the long and short trading policies solvingthe non-quadratic global hedging problems of the ERP framework are optimized to minimize aloss function which is possibly more in line with the financial objectives of the hedger by mainly(and most often exclusively) penalizing hedging losses, not gains. This paper studies the equal risk pricing (ERP) framework for pricing and hedging Europeanderivatives in discrete-time with multiple hedging instruments. The ERP approach sets derivativeprices as the value such that the optimally hedged residual risk of the long and short positionsin the contingent claim are equal. The ERP setup of Marzban et al. (2020) is considered whereresidual hedging risk is quantified through convex measures. The main objective of this currentpaper is in assessing the impact of including options within hedges on the equal risk price C ? and on the level of market incompleteness quantified by our (cid:15) ? -metrics. A specific focus is onthe examination of the interplay between different stylized features of equity jump and volatilityrisks and the use of options as hedging instruments within the ERP framework. The numericalscheme of Carbonneau and Godin (2020), which relies on the deep hedging algorithm of Buehleret al. (2019b), is used to solve the embedded global hedging problems of the ERP frameworkthrough the representation of the long and short trading policies with two distinct long-shortterm memory (LSTM) neural networks. 39ensitivity analyses with Monte Carlo simulations are performed under several empiricallyplausible sets of parameters for the jump and volatility risk models in order to highlight theimpact of different stylized features of the models on C ? and (cid:15) ? . Numerical values indicate thatin the presence of jump risk, hedging with options entails a significant reduction of both equalrisk prices and market incompleteness as compared to hedging solely with the underlying stock.The latter stems from the fact that using options as hedging instruments rather than only theunderlying stock shrinks the asymmetry of tail risk, which tends to both shrink option prices andreduce market incompleteness. On the other hand, in the presence of volatility risk, while optionhedges can reduce equal risk prices as compared to stock hedges, the impact can be marginal andis highly sensitive to the moneyness level of the put option being priced as well as to the maturityof traded options. This can be explained by the fact that while the impact of jump risk dynamicsconsidered in this paper is asymmetric by entailing significantly more weight on the right (resp.left) tail of the short (resp. long) hedging error through predominantly negative jumps, volatilityrisk impacts both upside and downside risk. Furthermore, additional experiments conductedshow that the average price level of short-term options used as hedging instruments is effectivelyreflected into the equal risk price of longer-term options. The latter highlights the potential ofthe ERP framework as a fair valuation approach providing prices consistent with observablemarket prices. Thus, ERP could be applied for instance in the context of pricing over-the-counterderivatives with vanilla calls and puts hedges or pricing less liquid long-term derivatives (e.g.LEAPS contracts) with shorter-term liquid options.Moreover, the benchmarking of equal risk prices to variance-optimal derivative premiums C ( V O )0 is performed. The deep hedging algorithm is also used as the numerical scheme to solve thevariance-optimal hedging problems. Numerical results show that while C ? tends to be larger than C ( V O )0 , trading options entails much smaller disparity between equal risk and variance-optimalprices as compared to trading only the underlying stock in the presence of jump or volatility risk.The latter is due to the market incompleteness being significantly smaller when option hedgesare used to mitigate jump and volatility risks. Furthermore, additional experiments conducteddemonstrate the ability of ERP to span a large interval of prices through the choice of convex40isk measures, which is close to encompass the variance-optimal premium. Alexandre Carbonneau gratefully acknowledges financial support from the Fonds de recherche duQu´ebec - Nature et technologies (FRQNT, grant number 205683) and The Montreal Exchange.Fr´ed´eric Godin gratefully acknowledges financial support from Natural Sciences and EngineeringResearch Council of Canada (NSERC, grant number RGPIN-2017-06837).
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A Variance-optimal hedging
Denote J ( V O ) : R q × R → R as the cost function to be minimized for variance-optimal procedures: J ( V O ) ( θ, V ) := E (cid:20)(cid:16) Φ( S (0 ,b ) N ) − B N ( V + G δ θ N ) (cid:17) (cid:21) , ( θ, V ) ∈ R q × R , (A.1)where θ is the set of trainable parameters of the LSTM F θ , V is the initial capital investmentand δ θ is to be understood as the output sequence of F θ . Let ˜ θ := { θ, V } be the augmented setof trainable parameters which includes the initial portfolio value. Minibatch SGD with MonteCarlo sampling can naturally also be used to minimize (A.1) jointly for the trainable parametersand the initial capital investment by updating iteratively the augmented set ˜ θ :˜ θ j +1 = ˜ θ j − η j ∇ ˜ θ ˆ J ( V O ) ( B j , V ,j ) , (A.2)44here ˜ θ := { θ , V , } is the initial set and ˆ J ( V O ) ( B j , V ,j ) is the empirical estimator of J ( V O ) ( θ, V )evaluated with the minibatch of hedging errors B j = { Φ( S (0 ,b ) N,i ) − B N ( V ,j + G δ θj N,i ) } N batch i =1 when˜ θ = ˜ θ j (i.e. θ = θ j and V = V ,j ):ˆ J ( V O ) ( B j , V ,j ) := 1 N batch N batch X i =1 (cid:16) Φ( S (0 ,b ) N,i ) − B N ( V ,j + G δ θj N,i ) (cid:17) . (A.3) B Pseudo-code deep hedging
Algorithm 1 presents the pseudo-code to perform a one-step update of the trainable parametersas in (3.11) for the global hedging problems of the ERP framework, i.e. updating θ j to θ j +1 . Forconvenience, the pseudo-code is presented for the case of trading exclusively the underlying stockand for the short position trading policy, but it is trivial to generalize to the case of trading otherhedging instruments (e.g. short-term options) and for the long position trading policy. Note thatthe pseudo-code is described for the MJD dynamics, but it can be generalized to the GARCHdynamics by sampling log-returns from (4.3) in line (6), and adding the stochastic volatilitiesto feature vectors as described in Section 4.1.3. Furthermore, the pseudo-code can also easilybe extended to variance-optimal hedging by updating the augmented set ˜ θ j to ˜ θ j +1 with (A.2)instead of θ j to θ j +1 in line (17) and by adapting the empirical cost function in line (15) to (A.3).Lastly, recall that a GitHub repository with samples of codes in Python for the training procedureof neural networks is available online: github.com/alexandrecarbonneau. The implementationreplicates results of Table 3 with jump risk scenario 2, and can easily be adapted to reproduce allresults presented in Section 4. As described in Section 4.1.4, an implied volatility dynamics is considered to price options used as hedginginstruments. In numerical experiments of Section 4, V , is set at the price obtained with the time-0 impliedvolatility. The authors also tested the naive initialization scheme V , = 0 as a robustness test, and found that theresulting variance-optimal premiums were marginally affected by this choice. Also, the Glorot uniform initializationof Glorot and Bengio (2010) is used to select θ . lgorithm 1 Pseudo-code short trading policy with stock hedges under the MJD modelInput: θ j Output: θ j +1 for i = 1 , . . . , N batch do . Loop over each path of minibatch X ,i = [log( S (0 ,b )0 ,i /K ) , V δ ,i ] . Time-0 feature vector of F ( S ) θ with V δ ,i = 0 for n = 0 , . . . , N − do Y n,i ← time- t n output of LSTM F ( S ) θ with θ = θ j δ (0) n +1 ,i = Y n,i y n +1 ,i ∼ (4.2) . Sample next log-return S (0 ,b ) n +1 ,i = S (0 ,b ) n,i e y n +1 ,i V δn +1 ,i = e r ∆ N V δn,i + δ (0) n +1 ,i ( S (0 ,b ) n +1 ,i − e r ∆ N S (0 ,b ) n,i ) . See (2.4) for details X n +1 ,i = [log( S (0 ,b ) n +1 ,i /K ) , V δn +1 ,i ] . Time- t n +1 feature vector for F ( S ) θ end for Φ( S (0 ,b ) N,i ) = max( K − S (0 ,b ) N,i , π i,j = Φ( S (0 ,b ) N,i ) − V δN,i end for VaR V α = π [ ˜ N ] ,j . ˜ N th ordered hedging error with ˜ N := d αN batch e CVaR V α = VaR V α + − α ) N batch P N batch i =1 max( π i,j − VaR V α , η j ← Adam algorithm θ j +1 = θ j − η j ∇ θ CVaR V α . ∇ θ CVaR V α computed with TensorflowNotes: Subscript i represents the i th simulated path among the minibatch of size N batch . Also,the time-0 feature vector is fixed for all paths, i.e. S (0 ,b )0 ,i = S (0 ,b )0 and V δ ,i = V δ = 0.46 upplementary Material: Deep Equal Risk Pricing ofFinancial Derivatives with Multiple Hedging Instruments Alexandre Carbonneau ∗ a and Fr´ed´eric Godin ba,b Concordia University, Department of Mathematics and Statistics, Montr´eal, Canada
February 26, 2021
This supplementary material presents additional numerical experiments of the ERP framework.Section A1 presents values for C ? and (cid:15) ? under the Black-Scholes model with the same setupas in Table 3 and Table 5. Section A2 presents the sensitivity analysis of C ? and (cid:15) ? to impliedvolatility risk with a similar setup as in Table 5. Section A3 presents the corresponding sensitivityanalysis of Table 6 under jump risk scenario 1 and 3 as well as under the three sets of parametersfor volatility risk. Section A4 presents the benchmarking of C ? to C ( V O )0 under the BSM andGARCH dynamics with the same setup as in Table 7 and Table 8. A1 Sensitivity of equal risk pricing under Black-Scholes
The Black-Scholes dynamics models log-returns as independent and identically distributed Gaus-sian random variables of daily periodic mean ( µ − σ ) and variance σ . Daily log-returnsunder this model have the representation˜ y j,n = 1252 (cid:18) µ − σ (cid:19) + σ r (cid:15) j,n , (A1.1)where µ and σ are the parameters on a yearly scale. Under the BSM, feature vectors of the neuralnetworks do not require the inclusion of additional state variables (i.e. ϕ n = 0 for all time steps n ∗ Corresponding author
Email addresses: [email protected] (Alexandre Carbonneau), [email protected](Fr´ed´eric Godin). a r X i v : . [ q -f i n . C P ] F e b n (3 .
4) and (3 . C ? and (cid:15) ? -metrics under the BSMwith µ = 0 . σ = 0 . , .
15 and 0 . A2 Sensitivity of equal risk pricing to implied volatility
Table SM2 presents the sensitivity analysis of the ERP framework to the long-run impliedvolatility when trading 1-month or 3-months options. These options are priced with impliedvolatility modeled as a log-AR(1) dynamics with parameters of Table 2 for { κ, σ IV , % } (i.e. κ = 0 . , σ IV = 0 .
06 and % = − . ϑ set to log(0 . , log(0 .
15) andlog(0 . A3 Additional sensitivity analysis convex risk measure
Table SM3 presents the sensitivity analysis of the ERP framework to the choice of convex riskmeasure when trading 3-months options under jump risk scenario 1 and 3 of Table 1. Table SM4presents this sensitivity analysis under volatility risk with the three sets of parameters describedin Table 4.
A4 Benchmarking of equal risk prices to variance-optimal premiums
Table SM5 and Table SM6 present the benchmarking of C ? with the CVaR . risk measure to C ( V O )0 with the same setup as Table 7 under the Black-Scholes and GARCH dynamics, respectively.Furthermore, Table SM7 and Table SM8 perform the benchmarking of C ? to C ( V O )0 with theCVaR . , CVaR . and CVaR . measures and 3-months option hedges under the BSM andGARCH dynamics, respectively. The latter experiments are performed with the same setup asTable 8 under the MJD dynamics. 2 able SM1: Equal risk prices C ? and residual hedging risk (cid:15) ? under the Black-Scholes modelfor OTM ( K = 90), ATM ( K = 100) and ITM ( K = 110) put options of maturity T = 1.OTM ATM ITMAnnual volatility ( σ ) 10% 15% 20% 10% 15% 20% 10% 15% 20% C ? Daily stock 0 .
43 1 .
48 2 .
83 2 .
65 4 .
60 6 .
68 8 .
37 10 .
19 12 . .
49 1 .
67 3 .
15 2 .
71 4 .
70 6 .
72 8 .
56 10 .
42 12 . .
44 1 .
56 3 .
00 2 .
74 4 .
70 6 .
71 8 .
48 10 .
30 12 . .
44 1 .
59 3 .
08 2 .
81 4 .
80 6 .
82 8 .
58 10 .
47 12 . (cid:15) ? Daily stock 0 .
19 0 .
41 0 .
65 0 .
38 0 .
64 1 .
00 0 .
39 0 .
67 0 . .
50 1 .
52 2 .
59 1 .
54 2 .
53 3 .
48 1 .
51 2 .
59 3 . .
19 0 .
57 0 .
94 0 .
57 0 .
91 1 .
30 0 .
64 1 .
01 1 . .
29 0 .
93 1 .
55 0 .
96 1 .
55 2 .
11 1 .
09 1 .
76 2 . (cid:15) ? /C ? Daily stock 0 .
44 0 .
28 0 .
23 0 .
14 0 .
14 0 .
15 0 .
05 0 .
07 0 . .
01 0 .
91 0 .
82 0 .
57 0 .
54 0 .
52 0 .
18 0 .
25 0 . .
44 0 .
37 0 .
31 0 .
21 0 .
19 0 .
19 0 .
07 0 .
10 0 . .
67 0 .
59 0 .
50 0 .
34 0 .
32 0 .
31 0 .
13 0 .
17 0 . ,
000 independent paths generated from the Black-Scholes model for the underlying with µ = 0 . σ = 0 . , .
15 and 0 .
20 (see (A1.1) for modeldescription).
Hedging instruments : daily or monthly rebalancing with the underlying stock and1-month or 3-months options with ATM calls and puts. Options used as hedging instruments arepriced with implied volatility modeled as a log-AR(1) dynamics with κ = 0 . , σ IV = 0 .
06 and % = − . ϑ set to log(0 . , log(0 .
15) and log(0 .
20) when σ = 0 . , .
15 and 0 . . . . .
5. The confidence level of the CVaR measure is α = 0 . able SM2: Sensitivity analysis of equal risk prices C ? and residual hedging risk (cid:15) ? to impliedvolatility (IV) risk for OTM ( K = 90), ATM ( K = 100) and ITM ( K = 110) put options ofmaturity T = 1. OTM ATM ITMLong-run IV 14% 15% 16% 14% 15% 16% 14% 15% 16% C ? .
52 1 .
74 1 .
98 4 .
52 4 .
87 5 .
20 10 .
01 10 .
32 10 . .
86 2 .
07 2 .
28 4 .
75 5 .
08 5 .
39 10 .
08 10 .
38 10 . (cid:15) ? .
61 0 .
65 0 .
66 0 .
96 1 .
00 0 .
96 0 .
99 1 .
03 1 . .
28 1 .
32 1 .
36 1 .
72 1 .
77 1 .
78 1 .
63 1 .
68 1 . (cid:15) ? /C ? .
40 0 .
37 0 .
34 0 .
21 0 .
20 0 .
18 0 .
10 0 .
10 0 . .
69 0 .
64 0 .
60 0 .
36 0 .
35 0 .
33 0 .
16 0 .
16 0 . ,
000 independent paths generated from the GJR-GARCH(1,1) model for the underlying with µ = 3 . , ω = 1 . υ = 0 . γ = 0 . β = 0 .
91 which entails stationary yearly volatility of 15% (see Section 4 . . κ = 0 . , σ IV = 0 .
06 and % = − . ϑ set tolog(0 . , log(0 .
15) or log(0 .
16) (see Section 4 . . . .
5. The confidence level of theCVaR measure is α = 0 .
95. 4 able SM3:
Sensitivity analysis of equal risk prices C ? and residual hedging risk (cid:15) ? for OTM( K = 90), ATM ( K = 100) and ITM ( K = 110) put options of maturity T = 1 under the MJDdynamics with jump risk scenario 1 and scenario 3. C ? (cid:15) ? (cid:15) ? /C ? Moneyness OTM ATM ITM OTM ATM ITM OTM ATM ITM
Jump risk scenario . .
63 4 .
76 10 .
32 0 .
78 1 .
28 1 .
42 0 .
48 0 .
27 0 . .
6% 2% 1% 32% 24% 20% 24% 21% 19%CVaR .
28% 7% 4% 103% 70% 65% 59% 58% 58%
Jump risk scenario . .
07 5 .
08 10 .
44 1 .
21 1 .
61 1 .
57 0 .
58 0 .
32 0 . .
15% 4% 1% 39% 25% 20% 21% 21% 19%CVaR .
56% 13% 5% 128% 76% 68% 47% 56% 60%Notes: Results are computed based on 100 ,
000 independent paths generated from the MertonJump-Diffusion model for the underlying (see Section 4 . . . . . .
5. Values for the CVaR . and CVaR . measuresare expressed relative to CVaR . (% increase).5 able SM4: Sensitivity analysis of equal risk prices C ? and residual hedging risk (cid:15) ? for OTM( K = 90), ATM ( K = 100) and ITM ( K = 110) put options of maturity T = 1 under the GARCHdynamics. C ? (cid:15) ? (cid:15) ? /C ? Moneyness OTM ATM ITM OTM ATM ITM OTM ATM ITM
Stationary volatility: . .
65 2 .
89 8 .
42 0 .
40 0 .
83 0 .
86 0 .
61 0 .
29 0 . .
17% 4% 1% 46% 32% 21% 25% 27% 20%CVaR .
70% 13% 3% 159% 94% 66% 52% 72% 60%
Stationary volatility: . .
88 4 .
91 10 .
28 0 .
97 1 .
39 1 .
40 0 .
51 0 .
28 0 . .
10% 4% 1% 36% 27% 21% 24% 23% 19%CVaR .
38% 11% 4% 122% 81% 69% 61% 63% 63%
Stationary volatility: . .
40 6 .
95 12 .
27 1 .
54 1 .
93 1 .
89 0 .
45 0 .
28 0 . .
7% 3% 1% 31% 25% 22% 22% 21% 20%CVaR .
28% 10% 5% 103% 77% 71% 58% 61% 64%Notes: Results are computed based on 100 ,
000 independent paths generated from the GJR-GARCH(1,1) model for the underlying with three sets of parameters implying stationary yearlyvolatilities of 10% ,
15% and 20% (see Section 4 . . κ = 0 . , σ IV = 0 .
06 and % = − . ϑ set to log(0 . , log(0 .
15) and log(0 .
20) when the GARCH stationary volatility is 10% ,
15% and20%, respectively (see Section 4 . . . .
5. Values for the CVaR . and CVaR . measuresare expressed relative to CVaR . (% increase).6 able SM5: Equal risk prices C ? and variance-optimal (VO) prices C ( V O )0 under the Black-Scholes model for OTM ( K = 90), ATM ( K = 100) and ITM ( K = 110) put options of maturity T = 1. OTM ATM ITMAnnual volatility ( σ ) 10% 15% 20% 10% 15% 20% 10% 15% 20% C ( V O )0 Daily stock 0 .
38 1 .
39 2 .
77 2 .
62 4 .
53 6 .
46 8 .
34 10 .
12 12 . .
35 1 .
36 2 .
73 2 .
55 4 .
47 6 .
39 8 .
27 10 .
07 11 . .
42 1 .
47 2 .
84 2 .
67 4 .
58 6 .
52 8 .
35 10 .
13 12 . .
39 1 .
44 2 .
84 2 .
65 4 .
56 6 .
50 8 .
32 10 .
12 12 . C ? Daily stock 12% 6% 2% 1% 2% 3% 0% 1% 0%Monthly stock 40% 23% 15% 6% 5% 5% 3% 4% 3%1-month options 5% 6% 6% 3% 3% 3% 2% 2% 2%3-months options 11% 10% 9% 6% 5% 5% 3% 3% 4%Notes: Results are computed based on 100 ,
000 independent paths generated from the Black-Scholes model for the underlying with µ = 0 . σ = 0 . , .
15 and 0 .
20 (see (A1.1) for modeldescription).
Hedging instruments : daily or monthly rebalancing with the underlying stock and1-month or 3-months options with ATM calls and puts. Options used as hedging instruments arepriced with implied volatility modeled as a log-AR(1) dynamics with κ = 0 . , σ IV = 0 .
06 and % = − . ϑ set to log(0 . , log(0 .
15) and log(0 .
20) when σ = 0 . , .
15 and 0 . . . . . α = 0 . C ? are expressed relative to C ( V O )0 (%increase). 7 able SM6: Equal risk prices C ? and variance-optimal (VO) prices C ( V O )0 with volatility riskfor OTM ( K = 90), ATM ( K = 100) and ITM ( K = 110) put options of maturity T = 1.OTM ATM ITMStationary volatility 10% 15% 20% 10% 15% 20% 10% 15% 20% C ( V O )0 Daily stock 0 .
78 1 .
88 3 .
24 2 .
94 4 .
75 6 .
60 8 .
29 9 .
98 11 . .
81 1 .
91 3 .
27 2 .
97 4 .
80 6 .
62 8 .
30 10 .
01 11 . .
50 1 .
60 3 .
01 2 .
75 4 .
66 6 .
60 8 .
32 10 .
10 12 . .
61 1 .
74 3 .
19 2 .
79 4 .
72 6 .
68 8 .
23 10 .
05 11 . C ? Daily stock 29% 25% 19% 10% 13% 10% 3% 6% 6%Monthly stock 45% 39% 30% 13% 13% 14% 7% 8% 9%1-month options 12% 9% 9% 4% 4% 4% 2% 2% 3%3-months options 24% 19% 14% 8% 8% 7% 3% 3% 4%Notes: Results are computed based on 100 ,
000 independent paths generated from the GJR-GARCH(1,1) model for the underlying with three sets of parameters implying stationary yearlyvolatilities of 10% ,
15% and 20% (see Section 4 . . Hedging instruments : daily or monthly rebalancing with the underlying stock and 1-month or 3-months options with ATM calls and puts. Options used as hedging instruments arepriced with implied volatility modeled as a log-AR(1) dynamics with κ = 0 . , σ IV = 0 .
06 and % = − . ϑ set to log(0 . , log(0 .
15) and log(0 .
20) when the GARCH stationaryvolatility is 10% ,
15% and 20%, respectively (see Section 4 . . . . α = 0 . C ? areexpressed relative to C ( V O )0 (% increase). 8 able SM7: Sensitivity analysis of equal risk prices C ? with CVaR . , CVaR . and CVaR . measures to variance-optimal (VO) prices C ( V O )0 under the Black-Scholes model for OTM ( K = 90),ATM ( K = 100) and ITM ( K = 110) put options of maturity T = 1.OTM ATM ITMAnnual volatility ( σ ) 10% 15% 20% 10% 15% 20% 10% 15% 20% C ( V O )0 .
39 1 .
44 2 .
84 2 .
65 4 .
56 6 .
50 8 .
32 10 .
12 12 . C ? (CVaR . ) −
2% 4% 4% 3% 3% 3% 3% 2% 2% C ? (CVaR . ) 11% 10% 9% 6% 5% 5% 3% 3% 4% C ? (CVaR . ) 44% 26% 22% 12% 13% 9% 6% 7% 6%Notes: Results are computed based on 100 ,
000 independent paths generated from the Black-Scholes model for the underlying with µ = 0 . σ = 0 . , .
15 and 0 .
20 (see (A1.1) for modeldescription). Hedging instruments consist of 3-months ATM calls and puts priced with impliedvolatility modeled as a log-AR(1) dynamics with κ = 0 . , σ IV = 0 .
06 and % = − . ϑ set to log(0 . , log(0 .
15) and log(0 .
20) when σ = 0 . , .
15 and 0 .
20, respectively(see Section 4 . . . . C ? withCVaR . , CVaR . and CVaR . are expressed relative to C ( V O )0 (% increase). Table SM8:
Sensitivity analysis of equal risk prices C ? with CVaR . , CVaR . and CVaR . measures to variance-optimal (VO) prices C ( V O )0 under volatility risk for OTM ( K = 90), ATM( K = 100) and ITM ( K = 110) put options of maturity T = 1.OTM ATM ITMStationary volatility 10% 15% 20% 10% 15% 20% 10% 15% 20% C ( V O )0 .
61 1 .
74 3 .
19 2 .
79 4 .
72 6 .
68 8 .
23 10 .
05 11 . C ? (CVaR . ) 6% 8% 6% 4% 4% 4% 2% 2% 2% C ? (CVaR . ) 24% 19% 14% 8% 8% 7% 3% 3% 4% C ? (CVaR . ) 81% 49% 37% 17% 16% 15% 6% 6% 7%Notes: Results are computed based on 100 ,
000 independent paths generated from the GJR-GARCH(1,1) model for the underlying with three sets of parameters implying stationary yearlyvolatilities of 10% ,
15% and 20% (see Section 4 . . κ = 0 . , σ IV = 0 .
06 and % = − . ϑ setto log(0 . , log(0 .
15) and log(0 .
20) when the GARCH stationary volatility is 10% ,
15% and 20%,respectively (see Section 4 . . . . C ? with CVaR . , CVaR . and CVaR . are expressed relative to C ( V O )0)0