Optimal semi-static hedging in illiquid markets
OOptimal semi-static hedging in illiquid markets
Teemu Pennanen ∗ , Udomsak Rakwongwan † Department of Mathematics,King’s College London,Strand, London, WC2R 2LS, United KingdomAugust 5, 2020
Abstract
We study indifference pricing of exotic derivatives by using hedgingstrategies that take static positions in quoted derivatives but trade theunderlying and cash dynamically over time. We use real quotes that comewith bid-ask spreads and finite quantities. Galerkin method and integra-tion quadratures are used to approximate the hedging problem by a finitedimensional convex optimization problem which is solved by an interiorpoint method. The techniques are extended also to situations where theunderlying is subject to bid-ask spreads. As an illustration, we computeindifference prices of path-dependent options written on S&P500 index.Semi-static hedging improves considerably on the purely static optionsstrategy as well as dynamic trading without options. The indifferenceprices make good economic sense even in the presence of arbitrage oppor-tunities that are found when the underlying is assumed perfectly liquid.When transaction costs are introduced, the arbitrage opportunities vanishbut the indifference prices remain almost unchanged. ∗ [email protected] † [email protected] a r X i v : . [ q -f i n . P R ] A ug Introduction
Unlike in complete markets where derivative prices are uniquely determinedby replication arguments, in incomplete markets, quoted prices depend on sub-jective factors such as the agents’ financial positions, risk preferences and viewsconcerning future market developments. Such dependencies are consistently de-scribed by indifference pricing which can be viewed as an extension of replicationarguments to the incomplete markets; see e.g. [B¨uh70, HN89, Car09, IJS04] andthe references therein. Extensions to illiquid markets and the correspondingduality theory has been studied in [Pen14] and [PP18], respectively.This paper develops computational techniques for utility-based semi-statichedging with a finite number of derivatives whose quotes have bid-ask spreadsand finite quantities. The hedging strategies involve buy-and-hold positions inthe derivatives while the underlying and cash are traded dynamically. We usea Galerkin method to approximate the hedging problem by a finite-dimensionalconvex optimization problem which is then numerically solved by an interiorpoint method much like in [APR18] in a purely static setting. The approachextends with minor modifications to situations where the dynamically tradedunderlying is also subject to bid ask spreads.The techniques are illustrated by computing indifference prices of variouspath-dependent options (including knock-out, Asian and look-back options) onthe S&P500 index. As hedging instruments, we use exchange-traded puts andcalls on the index. For the nearest maturities, one can find hundreds of optionswith bid and/or ask quotes. We find that semi-static hedging significantly im-proves on the hedges obtained by purely static or purely dynamic strategies.The semi-static hedging strategies provide good approximations of the payoutsof the hedged derivatives and the corresponding spreads between seller’s andbuyer’s prices are considerably tighter than those obtained with purely staticor dynamic hedging. The computational approach applies to arbitrary utilityfunctions and stochastic models that allow for numerical sampling.Compared to the more traditional super/subhedging, indifference pricing isless sensitive to market imperfections and it makes good sense even in the pres-ence of arbitrage. This was found a useful feature as the quotes on exchangetraded options seem to often lead to arbitrage if the dynamically traded un-derlying is assumed perfectly liquid (as is the case in most models studied inthe literature). The arbitrage opportunities vanish when moderate transactioncosts on the underlying are introduced but the indifference prices remain almostunaffected.When the statically hedged options are discarded, the optimal investmentstrategy to maximize the expected exponential utility coincides with the clas-sical Merton strategy. More surprisingly, the corresponding hedging strategiesobtained with indifference pricing seem to coincide with the delta-hedging strate-gies for replication in complete market models.Semi-static hedging has been actively studied in the recent literature butmainly under the assumption of perfect liquidity for both the static and dynamicinstruments. Moreover, it is common to assume also that there exist quotes fora continuum of strikes as opposed to the finite number of strikes traded in realmarkets. Much of the research has focused on duality theory in a distributionallyroust superhedging; see e.g. [BHLP13]. Guo and Obloj [GOo19] developedcomputational techniques for the martingale optimal transport problems by2sing discretization and interior point methods much like we do below. Theirproblem can be viewed as the dual of a semi-static superhedging problem with acontinuum of strikes for statically held call options (which fixes the marginals ofthe martingale measures). Extensions of the duality theory of model-free semi-static superhedging to illiquid markets were given in the examples of [PP19]. Inthe computational studies below, we find that with real finite-liquidity quotes forfinitely many options, superhedging tends to give prices with very wide spreads.The present paper is closely related to [IS06] and [IJS08] that studied utilityindifference pricing under semi-static trading. While they studied duality andasymptotics of indifference prices in a perfectly liquid continuous-time model,we focus on real illiquid markets and compute prices and hedging portfoliosnumerically.The rest of this paper is organized as follows. Sections 2 and 3 describe theoptimal hedging model and the corresponding indifference prices, respectively.Section 4 presents the techniques employed in the numerical computation ofoptimal hedging and indifference pricing. Section 5 extends the techniques tomarkets with an illiquid underlying. Sections 6 and 7 present the numericalresults obtained with S&P500 derivatives.
Consider a finite set J of quoted derivatives whose payouts are determinedby the values of an underlying index X at times t = 0 , , . . . , T . We assumethat the derivatives are traded only at t = 0 and they are held to maturity. Theunderlying can be traded at any t = 0 , , . . . , T . The cost of buying x j units of j ∈ J is denoted by S j ( x j ) := (cid:40) s ja x j if x j ≥ ,s jb x j if x j ≤ , where s jb < s ja are the bid and ask prices of j . The quantities available at thebest bid and ask quotes will be denoted by q jb and q ja , respectively. This meansthat the position x j we take in asset j has to lie in the interval [ − q jb , q ja ]. Thepayoff of j ∈ J at time t will be denoted by P jt . We assume for now that theunderlying index is perfectly liquid and can be dynamically traded at price X t , t = 0 , . . . , T . More realistic markets will be considered in Section 5.Consider an agent with w ∈ R units of initial cash and the liability to deliver c t units of cash at t = 1 , . . . , T . In the applications below, c t will be the payoutof an exotic option to be priced. We model the price process X = ( X t ) Tt =1 ,the payouts p j = ( p jt ) Tt =1 and the liability c = ( c t ) Tt =1 as adapted stochasticprocesses in a filtered probability space (Ω , F , ( F t ) Tt =1 , P ). We will study theoptimal investment problemminimize Ev T (cid:88) t =1 [ c t − (cid:88) j ∈ J p jt x j ] − T − (cid:88) t =0 z t ∆ X t +1 over x ∈ D, z ∈ N subject to (cid:88) j ∈ J S j ( x j ) ≤ w, (SSP)3here D := (cid:89) j ∈ J [ − q jb , q ja ] , N is the linear space of adapted trading strategies z = ( z t ) T − t =1 , and v : R → R is a loss function describing the investor’s risk preferences; see e.g. [FS11,Section 4.9]. One may think of u ( c ) := − v ( − c ) as a utility function so v will beassumed nondecreasing and convex. The argument of v is the unhedged partof the claims ( c t ) Tt =1 , the last term being interpreted as the payout of a self-financing trading strategy in the underlying and cash. One could also includevarious margin requirements in the specification of the set D .It is clear that the optimum value and solutions of problem (SSP) dependon 1. the financial position described by the initial cash w and liability c ,2. the views concerning the future values of X , p and c described by theprobabilistic model,3. the risk preferences described by the loss function v all of which are subjective. The effect of these factors on the optimal hedgingstrategies and the associated prices of c will be studied below. It turns out that,if the claims c are replicable, then the prices will be unique and independent ofthe subjectivities; see Theorem 1 below.Another important feature of (SSP) is that it is a convex optimization prob-lem . Convexity is crucial in numerical solution of (SSP) as well as in the math-ematical analysis of the indifference prices. We shall denote the optimum value of (SSP) by ϕ ( w, c ) := inf x ∈ D, z ∈N (cid:26) Ev (cid:0) T (cid:88) t =1 [ c t − (cid:88) j ∈ J p jt x j ] − T − (cid:88) t =1 ∆ X t +1 z t (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J S j ( x j ) ≤ w (cid:27) . For an agent with financial position of ¯ w units of initial wealth and a liability ofdelivering a sequence ¯ c = (¯ c t ) Tt =1 of payments, the indifference price for sellinga claim c = ( c t ) Tt =1 is given by π s ( ¯ w, ¯ c ; c ) := inf { w ∈ R | ϕ ( ¯ w + w, ¯ c + c ) ≤ ϕ ( ¯ w, ¯ c ) } . This is the minimum price at which the agent could sell the claim c withoutworsening her financial position as measured by the optimum value of (SSP).Analogously, the indifference price for buying c is given by π b ( ¯ w, ¯ c ; c ) := sup { w ∈ R | ϕ ( ¯ w − w, ¯ c − c ) ≤ ϕ ( ¯ w, ¯ c ) } . We shall compare the indifference prices with the superhedging and sub-hedging costs defined by π sup ( c ) := inf x ∈ D, z ∈N (cid:26) (cid:88) j ∈ J S j ( x j ) (cid:12)(cid:12)(cid:12)(cid:12) T (cid:88) t =1 (cid:88) j ∈ J p jt x j + T − (cid:88) t =1 ∆ X t +1 z t − T (cid:88) t =1 c t ≥ P -a.s. (cid:27) , π inf ( c ) := sup x ∈ D, z ∈N (cid:26) − (cid:88) j ∈ J S j ( x j ) (cid:12)(cid:12)(cid:12)(cid:12) T (cid:88) t =1 (cid:88) j ∈ J p jt x j + T − (cid:88) t =1 ∆ X t +1 z t + T (cid:88) t =1 c t ≥ P -a.s. (cid:27) . The superhedging cost is the least cost of a superhedging portfolio while thesubhedging cost is the greatest revenue one could get by entering position thatsuperhedges the negative of c . Whereas the indifference prices of a claim dependon our financial position, views and risk preferences described by ( ¯ w, ¯ c ), P and v , respectively, the superhedging and subhedging costs are independent of suchsubjective factors.In situations where the quantities available at the best quotes are largeenough to be nonbinding, the indifference prices lie between the superhedgingand subhedging costs. Indeed, an application of [Pen14, Theorem 4.1] to thepresent situation gives the following. Theorem 1.
The function π s ( ¯ w, ¯ c ; · ) is convex, nondecreasing and π s ( ¯ w, ¯ c ; 0) ≤ . If there are no quantity constraints (or if they are not active), then π s ( ¯ w, ¯ c ; c ) ≤ π sup ( c ) . If in addition, π s ( ¯ w, ¯ c ; 0) = 0 , then π inf ( c ) ≤ π l ( ¯ w, ¯ c ; c ) ≤ π s ( ¯ w, ¯ c ; c ) ≤ π sup ( c ) ∀ c ∈ L with equalities throughout if s b = s a and c is replicable. As long as one can (numerically) compute the optimum values ϕ ( w, c ) forgiven ( w, c ), the indifference prices can be computed by a simple line searchwith respect to the price. This can, however, be computationally expensive. Ifcash is perfectly liquid and the interest rate is zero, the indifference prices canbe expressed in terms of two optimization problems as follows. Proposition 1.
If cash is perfectly liquid with zero interest rate, the indifferenceprices for buying and selling for an agent with exponential risk measure can beexpressed as, π b ( ¯ w, ¯ c ; c ) = ¯ wλ log (cid:18) ϕ ( ¯ w, ¯ c ) ϕ ( ¯ w, ¯ c − c ) (cid:19) ,π s ( ¯ w, ¯ c ; c ) = ¯ wλ log (cid:18) ϕ ( ¯ w, ¯ c + c ) ϕ ( ¯ w, ¯ c ) (cid:19) , where ¯ w is an initial wealth, and λ is a risk aversion factor.Proof. By definition, ϕ ( ¯ w + w, ¯ c + c ) = inf x ∈ D, z ∈N (cid:26) E exp( λ ¯ w ( T (cid:88) t =1 [¯ c t + c t − (cid:88) j ∈ J p jt x j ] − T − (cid:88) t =1 ∆ X t +1 z t )) (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J S j ( x j ) ≤ ¯ w + w (cid:27) , = inf x ∈ D, z ∈N (cid:26) E exp( λ ¯ w ( T (cid:88) t =1 [¯ c t + c t − (cid:88) j ∈ J p jt x j ] − w − T − (cid:88) t =1 ∆ X t +1 z t )) (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ∈ J S j ( x j ) ≤ ¯ w (cid:27) , = ϕ ( ¯ w, ¯ c + c ) exp( − λw ¯ w ) . π b ( ¯ w, ¯ c ; c ) = inf { w | ϕ ( ¯ w − w, ¯ c − c ) ≤ ϕ ( ¯ w, ¯ c ) } , = inf { w | ϕ ( ¯ w, ¯ c − c ) exp( λw ¯ w ) ≤ ϕ ( ¯ w, ¯ c ) } , = ¯ wλ log (cid:18) ϕ ( ¯ w, ¯ c ) ϕ ( ¯ w, ¯ c − c ) (cid:19) , and π s ( ¯ w, ¯ c ; c ) = inf { w | ϕ ( ¯ w + w, ¯ c + c ) ≤ ϕ ( ¯ w, ¯ c ) } , = inf { w | ϕ ( ¯ w, ¯ c + c ) exp( − λw ¯ w ) ≤ ϕ ( ¯ w, ¯ c ) } , = ¯ wλ log (cid:18) ϕ ( ¯ w, ¯ c + c ) ϕ ( ¯ w, ¯ c ) (cid:19) , which completes the proof. We assume from now on that the derivative and liability payouts p j and c ,respectively, are adapted to the filtration generated by the underlying X . Thisclearly holds when p j and c are contingent claims on X . To solve (SSP), we need to optimize the dynamic part z over the infinite-dimensional space N of adapted stochastic processes. We will employ the Galerkin method where one optimizes z only over the finite-dimensional sub-space N N ⊂ N spanned by the simple processes z s,n ∈ N of the form z s,nt ( ω ) := (cid:40) t = s , X t ( ω ) ∈ [ K n , K n +1 ) , , where s = 1 , , . . . , T − K n , n = 1 , , . . . , N s are the strikes of the quotedoptions with maturity s , while K = 0 and K N s +1 = + ∞ . The dimension ofthe linear span N N is thus N = (cid:81) T − s =1 ( N s + 1). Clearly, each z s,nt is adapted tothe filtration generated by X so indeed, N N ⊂ N . The linear span N N consistsof simple processes that that are constant between consecutive strikes. Since the filtration ( F t ) Tt =0 is generated by X , the Doob-Dynkin lemma im-plies that the random variables c t and p t are functions of X t = ( X , X , . . . , X t ).The objective of (SSP) can be written as Ef ( x, z ) = (cid:90) R T + f ( x, z ( X ) , X ) ϕ ( X ) dX, ϕ is the density function of X, and f ( x, z ( X ) , X ) := v T (cid:88) t =1 [ c t ( X t ) − (cid:88) j ∈ J p jt ( X t ) x j ] − T − (cid:88) t =1 ∆ X t +1 z t ( X t ) . We will approximate the multivariate integral by an integration quadrature: (cid:90) R T + f ( x, z ( X ) , X ) ϕ ( X ) dX ≈ M (cid:88) i =1 w i f ( x, z ( X ( i ) ) , X ( i ) ) ϕ ( X ( i ) ) , where M is the number of the quadrature points, X ( i ) are the quadrature pointsand w i are the corresponding weights.There are many possible choices for the integration quadrature. In thisstudy, we take X ( i ) = ( X ( i ) t ) Tt =1 where X ( i ) t are the strikes at maturity t . Thecorresponding weights w i will be the volumes of the hyper cubes defined bythe consecutive strikes. This choice of quadrature points yields fairly accurateresults because the portfolio payout depends linearly on the index value betweentwo strikes. In addition, the probability that the index is smaller than thesmallest strike or bigger than the biggest strike is very small. The approximate problem obtained with the Galerkin method and the in-tegration quadrature, problem (SSP) becomes a finite-dimensional convex opti-mization problem with finitely many constraints. It can be written asminimize M (cid:88) i =1 v T (cid:88) t =1 [ c t ( X t ) − (cid:88) j ∈ J p jt ( X t ) x j ] − T − (cid:88) t =1 ∆ X t +1 z t ( X t ) w i ϕ ( X i )over x ∈ D, z ∈ N N subject to (cid:88) j ∈ J S j ( x j ) ≤ w. This is a finite-dimensional convex optimization problem that can be solved nu-merically e.g. by interior-point methods. In this study, we use the exponentialutility so the problem can be written as a conic exponential optimization prob-lem and solved using the interior-point solver of MOSEK [ApS15]. Numericalresults are given in Section 6 below.
Up to now, we have assumed that the underlying index is a perfectly liquidasset that can bought and sold at price X . The same assumption is madein most of the literature on semi-static hedging but from a practical point ofview, this is not quite realistic. This section considers a more realistic variantof (SSP) where the index is subject to a transaction costs, or equivalently, aconstant bid-ask spread. More precisely, we assume that an agent needs to paya δ % transaction cost to buy or sell the index at time t = 1 , , . . . , T −
1, and7he index is liquid at T . The semi-static hedging problem can then be writtenas minimize Ev T (cid:88) t =1 [ c t − (cid:88) j ∈ J p jt x j ] + T (cid:88) t =1 S t (∆ z t ) over x ∈ D, z ∈ N subject to (cid:88) j ∈ J S j ( x j ) ≤ w, where ∆ z t := z t − z t − is the number of the unit of the underlying bought at t and S t (∆ z t ) := (cid:40) (1 + δ ) X t ∆ z t if ∆ z t ≥ , (1 − δ ) X t ∆ z t if ∆ z t ≤ . Here and in what follows, z − = z T = 0. Note that if δ = 0, T (cid:88) t =1 S t (∆ z t ) = T (cid:88) t =1 X t ∆ z t = − T − (cid:88) t =1 ∆ X t +1 z t , so the original model (SSP) is a special case of the above.To solve the problem numerically, we express the purchases ∆ z as∆ z t = ∆ z + t − ∆ z − t , where ∆ z + t , ∆ z − t ≥ S t (∆ z t ) = (1 + δ
100 ) X t ∆ z + t − (1 − δ
100 ) X t ∆ z − t . We then can apply the Galerkin method to ∆ z + t and ∆ z − t where the multipliersof the basis functions are restricted to be positive. The rest is similar to thenumerical solution of (SSP) described in Section 4. This section illustrates the presented models and techniques in the S&P500derivatives market with option quotes taken from Bloomberg. For the nearestmaturities, there are hundreds of exchange traded puts and calls whose quotescome with bid-ask spreads and finite quantities. The resulting optimization andpricing problems are then solved using the techniques described in Sections 4–5.We start by finding optimal portfolios in the quoted derivatives when as-suming that the liability c in Problem (SSP) is zero. We study the dependenceof the optimal solution on the risk preferences as well as on the distribution ofthe underlying, both of which are highly subjective components of the model.Building on the optimization model, we then compute indifference prices forpath-dependent derivatives namely a knock-out call option, an Asian call op-tion, look-back call options and a look-back digital option. We compare the op-timized portfolios and indifference prices obtained by semi-static hedging withdifferent transaction costs to those obtained by purely dynamic hedging withoutoptions. 8 .1 Quotes, views, and preferences We use quotes for S&P500 index options with maturities 21 April 2017 and19 May 2017. The strikes of the options range from 1500 to 2500. The quoteswere obtained from Bloomberg on 21 March 2017 at 3:00:00 PM when the valueof the S&P500 index was 2360. All quotes come with bid ask spreads and finitequantities. Table 1 gives an example of quotes on put and call options writtenon the S&P500 index available on 21 March 2017 at 15:00:00 expiring on 21April 2017 and 19 May 2017. The index value was 2,360 at the time. Thebid and ask prices shown in the table are per one option, whereas the availablequantities are given in terms of a lot size which is 100. For example, the cost ofbuying or selling a call option with strike 2300 expiring on 5/19/2017 are 81.80and 79.50, and there are 51 contracts (5100 options) and 48 contracts (4800options) available for buying and selling respectively.Call options are more liquid at lower strikes. One can find quotes for calloptions whose strikes are as low as 500, whereas the lowest strike for put optionsavailable in the market is 1,555. For the two nearest maturities, one can findquotes for 678 options whose strikes range from 1,500 to 2,500 with 5 dollarincrements.
Ticker Type Bid quantity Bid price Ask price Ask quantitySPX US 5/19/2017 C2300 Index Call 48 79.5 81.8 51SPX US 5/19/2017 P2300 Index Put 182 22.6 24 376SPX US 4/21/2017 C2370 Index Call 300 18.7 20.3 273SPX US 4/21/2017 P2370 Index Put 275 28.6 30.5 322
Table 1: Market quotes extracted from Bloomberg on 21 March 2017 at 15:00:00for put and call options expiring on 21 April 2017 and 19 May 2017.In the applications below, we assume zero interest on cash. In practice, theindex is not tradable, but one can trade exchange-traded funds, ETFs, whichare securities that track the index. An example of a fairly liquid ETF whichefficiently tracks the S&P500 index is the SPY is issued by State Street GlobalAdvisors.We model the logarithm of the S&P500 index by a variance gamma pro-cess, obtained by evaluating Brownian motion with drift θ and volatility σ ata random time change given by a gamma process with a unit mean rate anda variance rate ν ; see [MCC98] and [MS90]. The parameters θ and ν providecontrol over skewness and kurtosis, respectively. As a base case, we use theparameter values given in Table 2. The parameter θ is assumed to be zero,whereas the parameters σ , and ν are estimated using 10-year historical dailydata. The effect of varying the parameters will be studied later on. The initialwealth w is 100 ,
000 USD and for now, the claim c t is assumed to be zero forall t .As for risk preferences, we use exponential loss function v ( c ) = e λc/w , where w is the initial wealth and λ > θ σ ν λ and variancegamma parameters θ, σ , and ν used to model the index value. To simplify the presentation and to ease the extensive numerical computa-tions on a relatively modest computational setup, we will study a two-periodinstance of semi-static hedging and pricing. With the available set of quotesoptions and the numerical procedure described in Section 4, there are over 1700variables and 2700 constraints in the discretized optimization problem. In thequadrature approximation of the expected loss function there are over 160,000points on the grid; see Section 4.2. The interior point solver of MOSEK takeson average of 650.40 seconds on a PC with Intel(R) Core(TM) i5-4690 CPU @3.50GHz processor and 16.00 GB memory.Figure 1 represents the structure of the optimized semi-static strategy. Thebars represent the optimal positions in the options, whereas the line plots showthe positions in cash and the index taken at t = 1 as functions of X . Figure 2plots the payout of the portfolio as a function of X and X .The portfolio enjoys higher profit if the index values at the first and secondmaturities are close to each other, while its loss is greater elsewhere. Thismakes sense as it is unlikely that the index value at the second maturity greatlydeviates from that of the first maturity. Index value at T1 -1.4-1.3-1.2-1.1-1-0.9 I nd ex qu a n t i t y Calls T2
Puts T2
Strikes -100-50050 Q u a n t i t y ( c on t r ac t s ) Calls T1
Puts T1
Strikes -50050100 Q u a n t i t y ( c on t r ac t s ) Index value at T1 C as h qu a n t i t y cash quantity at T0 = -5.1028e+06 Figure 1: The structure of the optimized semi-static strategy where an indexvalue is modelled by symmetric variance gamma with parameters given in Table2. 10igure 2: The payout of the optimal portfolio by semi-static optimization as afunction of X and X . The grey horizontal plane represents the initial wealth.Figure 3 represents the structure of the optimized portfolio obtained withrisk aversion λ = 6. The other parameters are as in Table 2. The payout of theportfolio is plotted in Figure 4 (left) together with the payout of the optimalportfolio obtained with risk aversion λ = 2. The right plot of Figure 4 shows thekernel density estimates (using 1,000,000 out-of-sample simulated price paths)of the terminal wealth of the optimal portfolios obtained with risk aversion 2and 6. Index value at T1 -9000-8500-8000-7500-7000 I nd ex qu a n t i t y Calls T2
Puts T2
Strikes -100-50050 Q u a n t i t y ( c on t r ac t s ) Index value at T1 C as h qu a n t i t y cash quantity at T0 = -4.1043e+06 Calls T1
Puts T1
Strikes -50050 Q u a n t i t y ( c on t r ac t s ) Figure 3: The structure of the optimized semi-static strategy obtained whenthe risk aversion increased from 2 to 6.11
Terminal wealth K e r n a l d e n s i t y -5 risk aversion = 2risk aversion = 6 Figure 4: The payout as functions of X and X of the optimal portfoliosobtained with risk aversions 2 and 6 (left). The kernel density estimates of theterminal wealths of the optimal portfolios obtained with risk aversions 2 and 6using 1,000,000 out-of-sample simulated price paths (right).We see that the positions of the optimized portfolio obtained with risk aver-sion λ = 6 are smaller than the ones with risk aversion λ = 2. As expected, thepayout of the portfolio obtained with higher risk aversion is less variable. Ex-cept for the scale, the shapes of the two kernel density plots look fairly similar,both exhibiting profits in roughly the same area. This makes sense as changingrisk aversion does not change the view on the index value. base case λ = 6 σ = 0 . σ = 0 . Table 3: Logarithms of optimum objective values obtained with different param-eters in the semi-static optimization problem (SSP). As one of the parameterschanges, the others remain the same as in the base case.Figure 5 plots the optimal payouts obtained with σ = 0 .
08 (left) and σ = 0 . σ results in a portfolio that gives higherpayout further in the tails (a straddle). Table 3 gives the logarithms of theoptimum objective values when σ = 0 . σ = 0 .
08 and σ = 0 .
2. Notethat, since we use the exponential loss function, the logarithmic objective is the“entropic risk measure” which has units of cash. It can also be interpreted as the“certainty equivalent”. The highest objective value is obtained with the base-case parameters which are estimated from historical data. This may be thoughtof as consistency of the market quotes and the market participants’ views ofthe future behavior of the underlying. With a model that is inconsistent withthe “market views”, the available quotes may seem to offer profitable tradingopportunities. 12igure 5: The payout of the optimal portfolios by semi-static optimization asfunctions of X and X , obtained with σ = 0 .
08 (left) and σ = 0 . We found that, with the quotes obtained from Bloomberg, there exists anarbitrage opportunity if the index can be traded without transaction costs. Dueto the finite quantities at the best bid and ask quotes, however, the optimizationmodel admits a bounded solution so that the pricing and hedging problems stillmake economic sense.To identify an arbitrage portfolio, we add the constraint T (cid:88) t =1 (cid:88) j ∈ J p jt x j + T − (cid:88) t =1 ∆ X t +1 z t ≥ w P -a.s.to problem (SSP). This means that the portfolio payout is at least the initialwealth in all scenarios. In numerical computations, we impose the constraint onall quadrature points. As the payout is a linear function between the strikes, theconstraint will then hold everywhere. Figure 6 represents the structure of thearbitrage strategy and Figure 7 plots the corresponding payout. The solutionuses both static and dynamic trading and achieves a net payout that never fallsbelow the initial wealth but is likely to end up strictly higher. Index value at T1 -1.09-1.088-1.086-1.084-1.082-1.08-1.078 I nd ex qu a n t i t y Calls T2
Puts T2
Strikes -100-50050 Q u a n t i t y ( c on t r ac t s ) Calls T1
Puts T2
Strikes -50050100 Q u a n t i t y ( c on t r ac t s ) Index value at T1 C as h qu a n t i t y Figure 6: The structure of the arbitrage strategy13igure 7: The payout of the arbitrage portfolio as a function of X and X We will now study the effect of a bid-ask spread on the dynamically tradedunderlying. Figure 8 illustrates the structure of the optimal solution when theproportional transaction cost is δ = 0 .
1. The optimized options portfolio issparser than the one obtained with perfectly liquid underlying. In addition,the quantities traded are smaller in the options as well as the index. A kerneldensity plot of the net payout is given in Figure 9.
Index value at T1 -5000-4000-3000-2000-10000 I nd ex qu a n t i t y Calls T2
Puts T2
Strikes -20-10010 Q u a n t i t y ( c on t r ac t s ) Index value at T1 C as h qu a n t i t y cash quantity = 8.4781e+04 Calls T1
Puts T1
Strikes -10010 Q u a n t i t y ( c on t r ac t s ) Figure 8: The structure of the optimal portfolio by semi-static optimizationwith 0.1% transaction cost 14igure 9: The payout of the optimized portfolio by semi-static optimization asa function of X and X with 0.1% transaction cost. The grey horizontal planerepresents the initial wealth.To examine the effects of the transaction cost further, we study the payouts ofthe optimized portfolios for varying levels of the transaction costs. The left plotof Figure 10 shows the payout of the optimized portfolio with 0.1% transactioncost subtracted by the payout of the optimized portfolio with 1% transactioncost, whereas the right plot shows the payout of the optimized portfolio with0.1% transaction cost subtracted by the payout of the optimized portfolio with10% transaction cost. Figure 11 shows the optimal index quantities bought orsold at t = 1 obtained with different transaction costs as functions of X .We see that, for the index values up to 2500, which is the highest strikeamong the options available in the market, a higher transaction cost results inpayouts that tend to be higher when X and X are close to each other. As thetransaction cost increases, we invest less in the index at t = 1; see Figure 11.Figure 10: The payout of the optimized portfolio with 0.1% transaction costsubtracted by 1% transaction cost’s (left). The payout of the optimized portfoliowith 0.1% transaction cost subtracted by 10% transaction cost’s (right).15
500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500
Index value at T1 -5000-4000-3000-2000-100001000 I nd ex qu a n t i t y spread = 0.1%spread = 1%spread = 10% Figure 11: The optimal index quantities bought or sold at t = 1 by semi-staticoptimization as functions of X with different transaction costsTable 4 shows how the expected loss function value increases with the trans-action cost. transaction cost 0% 0.1% 1% 10%log objective -2.4404 -2.3845 -2.3559 -2.3136 Table 4: Logarithms of the optimum objective values in semi-static optimizationwith different transaction costs
To illustrate the benefits of employing buy-and-hold strategies in the quotedoptions, we will compare the results with a purely dynamic optimization modelwhere we are not allowed to trade the options. Other than that, the modelis identical with the ones studied above. The numerical optimization is donewith the Galerkin discretization and quadrature approximations as described inSection 4.Figure 12 plots the mark-to-market values of the optimal index holding attime t = 1 as functions of the underlying X for varying levels of transactioncosts δ . When there are no transaction costs, the amount of wealth investedin the underlying does not depend on the value of the underlying except forthe more extreme values. This is in line with the theory which says that withexponential utility, the amount of wealth invested in the risky assets is con-stant. The deviations at the extremes are due to discretization errors. Thecorresponding terminal wealth of the optimal index position as a function of X and X when there are no transaction costs is given in Figure 13. With highertransaction costs, the terminal wealth is lower as one would expect, and with0 .
20% transaction cost, the terminal wealth is constant as there is no tradingat all. 16igure 12: Mark-to-market value of the index at time t = 1 as a function of X obtained by dynamic optimization with different transaction costsFigure 13: The payout of the optimized portfolio by purely dynamic optimiza-tion when there are no transaction costs. The grey horizontal plane representsthe initial wealth. To illustrate numerical indifference pricing, we will consider a standard calloption and four path-dependent options namely, a knock-out call option withpayoff c ( X , X ) = (cid:40) ( X − K ) + if X < B, X ≥ B , (1)an Asian call option with payoff c ( X , X ) = (cid:18) X + X − K (cid:19) + , (2)a look-back call option with payoff c ( X , X ) = max t =1 , { ( X t − K ) + } , (3)and a look-back digital call option with payoff c ( X , X ) = (cid:40)
10 if X or X ≥ K, K = 2 ,
350 and barrier B = 2 , δ = 0 . , X is likely to deviate from X more than from X . The indifference prices for the look-back call option are between 0 and 10but closer to 10 as it is in the money.As reported in Section 6.3, there is arbitrage opportunity in the semi-staticmodel without transaction costs. Accordingly, the superhedging costs are belowthe subhedging costs. However, the existence of the arbitrage does not prohibitus from computing sensible indifference prices. One should note that in thepresence of arbitrage, the quantity constraints for the options are binding soTheorem 1 does not apply in the present situation. Adding a 0.1% transactioncost on the underlying removes the arbitrage and puts us back in the setting ofTheorem 1 in terms of the order of the four prices; see Table 6.As expected, adding transaction costs increases superhedging costs and low-ers the subhedging costs. Removing the statically traded options has a similareffect. This is simply because the construction of a superhedging strategy be-comes cheaper when trading costs are reduced. The same does not apply toindifference pricing because both sides of the indifference inequality increasewhen trading becomes more expensive.Without the statically traded options, the true superhedging cost is + ∞ forall but the digital option. Accordingly, the numerically computed superhedgingcosts in Table 7 would converge to infinity when the scenario grid is extendedfurther. Similarly, the true subhedging costs of all but the call and Asian optionare zero. claim subhedging buying price selling price superhedgingcall 52.9626 45.3296 45.3939 37.4974knock-out call 18.1167 22.4763 22.7125 18.6974Asian 38.9026 35.0562 35.1019 29.8066look-back call 53.6604 53.9293 54.0110 51.5058look-back digital 14.4026 7.6834 7.6966 0.6058 Table 5: Indifference prices, together with super- and subhedging costs by semi-static hedging without transaction costs on the underlying18 laim subhedging buying price selling price superhedgingcall 43.4250 44.5308 44.8265 45.8000knock-out call 4.8957 20.6770 21.1444 29.5397Asian 29.8327 35.0303 35.2427 38.9201look-back call 43.9763 53.7226 54.0400 61.1320look-back digital 5.2640 7.5498 7.5663 9.2374
Table 6: Indifference prices, together with super- and subhedging costs by semi-static hedging with 0.1% transaction cost on the underlying claim subhedging buying price selling price superhedgingcall 10.0000 49.9490 51.2605 442.0000knock-out call 0.0000 15.3326 16.5480 442.0000Asian 0.0000 41.1187 42.1857 442.0000look-back call 10.0000 60.4879 62.3530 485.2906look-back digital 0.1538 6.4321 6.4469 10.0000
Table 7: Indifference prices, together with super- and subhedging costs by two-period dynamic hedging without statically held optionsThe hedging portfolios, which are x − ¯ x and z − ¯ z where ¯ x and x are optionsportfolios, and ¯ z and z are index quantities before and after selling the options,as well as their payouts for each hedging will be shown in the later subsections. Figure 14 illustrates the hedging strategy for selling one contract of the calloption strike K = 2 , alls T2 Puts T2
Strikes -101 Q u a n t i t y ( c on t r ac t s ) Index value at T1 -101234 C as h qu a n t i t y cash quantity at T0 = -4.7095e+04 -500025000 25005000 P ay o ff X2 X1 Index value at T1 -40-35-30-25-20-15-10 I nd ex qu a n t i t y Calls T1
Puts T1
Strikes -0.200.20.4 Q u a n t i t y ( c on t r ac t s ) Figure 14: The hedging portfolio for selling one contract of a call option withstrike 2,350. -500025000 25005000 P ay o ff X2 X1 Knock-out call -125000 2500 P ay o ff X2 X1 Asian call -125000 2500110 P ay o ff X2 X1 Look-back call -50025000 2500500 P ay o ff X2 X1 Look-back digital call
Figure 15: The payouts of the hedging portfolios for selling one contract of aknockout call option, Asian call option, look-back call option, and look-backdigital option all with strike 2,350 and barrier 2,400.
Figure 16 illustrates the hedging strategy for selling one contract of the calloption strike K = 2 ,
350 when the underlying is subject to 0.1% transactioncost. The payouts of the hedging portfolios for the other options are shown inFigure 15. 20 P ay o ff X2 X1 Calls T1
Puts T1
Strikes -0.2-0.100.1 Q u a n t i t y ( c on t r ac t s ) Calls T2
Puts T2
Strikes -2-1012 Q u a n t i t y ( c on t r ac t s ) Index value at T1 -3-2.5-2-1.5 C as h qu a n t i t y cash quantity at T0 = 3.9929e+03 Index value at T1 I nd ex qu a n t i t y Figure 16: The hedging portfolio for selling one contract of a call option withstrike 2,350 with 0.01% transaction cost. -12500-0.50 2500 P ay o ff X2 X1 Knock-out call -5000250005000 2500 P ay o ff X2 X1 Asian call -50025000 2500500 P ay o ff X2 X1 Look-back digital call -5000250005000 2500 P ay o ff X2 X1 Look-back call
Figure 17: The payouts of the hedging portfolios for selling one contract of aknockout call option, Asian call option, look-back call option, and look-backdigital option all with strike 2,350 with 0.01% transaction cost.Despite having the transaction cost, the path-dependent options are stillhedged well. However, the lower the transaction cost, the better the hedge aswe can see from Figure 18 which shows the payouts of the hedging portfolios forselling the look-back call option with strike 2,350 when the transaction costs are1 and 10 percents. The semi-static hedging with the 10% transaction cost coin-cides with the static hedging as the underlying is not traded. Note that statichedging is a special case of semi-static hedging. We see that the 2-dimensional21hapes of the payout are identical for any given values of X or X -5000250005000 2500 P ay o ff X2 X1 -5000250005000 2500 P ay o ff X2 X1 Transaction cost = 1% Transaction cost = 10%
Figure 18: The payouts of hedging portfolios for selling one contract of a look-back call option with strike 2350 with 1% and 10% transaction costs.
Figure 19 illustrates the hedging strategy for selling one contract of the calloption strike K = 2 ,
350 by two-period dynamic hedging. The payouts of thehedging portfolios for the other options are shown in Figure 20. Only cash andthe underlying, allowed to be traded without transaction costs at t = 0 ,
1, arethe hedging instruments. We see that, without the call and put options as thehedging instruments, they badly hedge the options. However, hedging portfoliostend to be more profitable when X and X are close to each other which is anarea with higher probability of occurring. -22500-1 25000 P ay o ff X2 X1 Index value at T1 -60-40-200204060 I nd ex qu a n t i t y dynamic hedgingdelta hedging index quantity at T0 dynamic hedging = 54.93 delta hedging = 54.45 Figure 19: The quantity of the index bought at t = 1 as a function of X of thehedging portfolio for selling one contract of a call option with strike 2,350 bydynamic strategy (left) and its payout (right).22 P ay o ff X2 X1 Knock-out call -500025000 25005000 P ay o ff X2 X1 Asian call -5000250005000 2500 P ay o ff X2 X1 Look-back call -50025000 2500500 P ay o ff X2 X1 Look-back digital call
Figure 20: The payouts of the hedging portfolios for selling one contract of aknockout call option, Asian call option, look-back call option, and look-backdigital option all with strike 2,350 and barrier 2,400 by dynamic hedging.The quantity of trade in the index as shown in Figure 19 looks similar to theone of the delta hedging. This is very surprising because the dynamic hedging isimplemented in a two-period setting, whereas the delta hedging is a continuoustrading strategy. The indifference prices for buying and selling are 49.9490 and51.2605, whereas the Black-Scholes price with µ = 0 and σ = 0 . t = 1 of the hedgingportfolios for selling one contract of a call option with strike 2,350 as functionsof X obtained by two-period dynamic hedging with different transaction costs.We see that the quantities traded in the underlying at both t = 0 and t = 1decrease as the transaction cost increases. Except at the tails, which havelow probability of occurring, the strategies are to buy some underlying at thebeginning and buy more if it goes up or sell if it goes down. However, we seethat, for the 0 .
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