Pricing index options by static hedging under finite liquidity
PPricing index options by static hedging underfinite liquidity
John Armstrong, Teemu Pennanen, Udomsak Rakwongwan ∗ Department of Mathematics,King’s College London,Strand, London, WC2R 2LS, United KingdomMarch 8, 2018
Abstract
We develop a model for indifference pricing in derivatives marketswhere price quotes have bid-ask spreads and finite quantities. Themodel quantifies the dependence of the prices and hedging portfolioson an investors beliefs, risk preferences and financial position as wellas on the price quotes. Computational techniques of convex optimi-sation allow for fast computation of the hedging portfolios and pricesas well as sensitivities with respect to various model parameters. Weillustrate the techniques by pricing and hedging of exotic derivativeson S&P index using call and put options, forward contracts and cashas the hedging instruments. The optimized static hedges provide goodapproximations of the options payouts and the spreads between indif-ference selling and buying prices are quite narrow as compared withthe spread between super- and subhedging prices.
In incomplete markets, the prices of financial products offered by an agentdepend on subjective factors such as views on the future development of theunderlying risk factors, risk preferences, the financial position as well as the ∗ [email protected] a r X i v : . [ q -f i n . P R ] M a r rading expertise of the agent. A agent’s prices also depend on the pricesat which the agent can trade other financial products since that affects thecosts of (partial) hedging when selling a product.The indifference pricing principle provides a consistent way to incorporatethe above factors into a pricing model. A classical reference on indifferencepricing of contingent claims under transaction costs is [6]. In the insurancesector, where market completeness would be quite an unrealistic assumption,indifference pricing seems to have longer history; see e.g. [2]. A more recentaccount with further references can be found in [3].Indifference pricing builds on an optimal investment model that describesthe relevant sector of financial markets as well as the agent’s financial posi-tion, views and risk preferences. Realistic models are often difficult to solvemuch like the investment problem they describe. This paper develops a com-putational framework for indifference pricing of European style options on theS&P500 index. Instead of the usual dynamic trading of the index and a cash-account, we take index options as the hedging instruments. For the ease ofimplementation, we consider only buy-and-hold strategies in the options butwe take actual market quotes as the trading costs. For the nearest maturities,there are some 200 strikes with fairly liquid quotes. This results in a convexstochastic optimization problem with one-dimensional uncertainty but over400 decision variables. The model is solved numerically using discretizationand an interior point solver for convex optimization. The indifference pricesfor a given payout are found within seconds so it is easy to study the effectof an agent’s views, risk preferences and financial position on the indifferenceprices.Much like the Breeden–Litzenberger formula, indifference pricing pro-vides automatic calibration to quoted call option prices. While the Breeden–Litzenberger formula provides only a heuristic approximation in real marketswith only a finite number of strikes and finite liquidity, the indifference ap-proach finds the best static hedge given the quotes, the agents views andpreferences. Moreover, the indifference approach gives explicit control of thehedging error in incomplete markets. Unlike the Breeden-Litzenberger for-mula would suggest, we find that in the presence of bid-ask spreads, the opti-mal hedges are often quite compressed portfolios of options taking positionsonly in few of the strikes. This is a significant benefit when implementing thehedges in practice. While the Breeden–Litzenberger formula applies only tooptions whose payouts are differences of convex functions of the underlying,the indifference pricing applies just as well to discontinuous payoffs such asdigital options. 2 The market
We study contingent exchange traded claims with common maturity T and payouts that only depend on the value of the S&P500 index at T . Thisincludes put and call options, forward contracts and cash. In general, lendingand borrowing rates for cash are different, so the payout on cash dependsnonlinearly on the position taken. Similarly, the forward rates available inthe market depend on whether one takes a long or short position. For theoptions, on the other hand, the payout per unit held is independent of theposition. The payoffs for holding x ∈ R units of an asset are given in Table 1. Asset Payoff with x units heldCash min { e r a T x, e r b T x } Forward min { ( X T − K a ) x, ( X T − K b ) x } Call max { ( X T − K ) , } x Put max { ( K − X T ) , } x Table 1: The payoffs of holding x units of the assets. Here r a and r b and theborrowing and lending rates, respectively, X T is the value of the underlyingat maturity, K a and K b are forward prices for long and short positions,respectively and K is the strike price of an optionWhile the option payoffs are linear in the position, the cost of entering aposition depends nonlinearly on the units x . For a long position x >
0, onepays the ask-price while for short position, one gets the bid-price. The costof buying x units of cash is simply x while for the forward, the cost is zero.For each contract, the market quotes come with finite quantities. Forthe nearest maturity, one can find quotes for some 400 options on S&P500.Table 2 gives an example of quotes available on the 8 April 2016 at 14:55:00for contracts expiring on 17 June 2016. Ticker Type Bid quantity Bid price Ask price Ask quantityESM6 Index Forward 258 2048.75 2049 377SPX US 6/17/2016 C2095 Index Call 623 26.90 28.20 506SPX US 6/17/2016 P2095 Index Put 27 72.60 74.70 22
Table 2: Market quotes on 8 April 2016 at 14:55:00 for the forward, a calland a put option maturing 17 June 2016. For the forward, the bid and askprice quotes are the forward prices for entering a short or a long position,respectively. The data was extracted from Bloomberg.3
The portfolio optimisation model
For given initial wealth and quotes on cash, forward and the options, ouraim is to find a portfolio with optimal net payoff at maturity. In general, thepayoff will depend on the value of the underlying at maturity so the opti-mality will depend on our risk preferences concerning the uncertain payoffs.The optimality of a portfolio also depends on our financial position whichmay involve uncertain cash-flows at time T .We will denote our initial wealth by w ∈ R and assume that our financialposition obligates us to pay c units of cash at time T . The collection of alltraded assets (cash, forward, options) is denoted by J . The cost of buying x j units of asset j ∈ J is given 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 . If j is cash, we simply have s jb = s ja = 1 while for the forward contract s jb = s ja = 0. The finite quantitiesfor the best quotes mean that there are upper and lower bounds q ja and q jb ,respectively, on the position x j one can take in asset j at the best availablequotes. For example, the quotes for the forward contract in Table 2 meanthat q ja = 377 while q jb = − x j units of asset j ∈ J by P j ( x j ).The functions P j are given in Table 1. We model the value X T of theunderlying at maturity as a random variable so that, in the case of forwardsand the options, P j ( x j ) will be random as well. We will assume that ourfinancial before the trade obligates us to deliver a random amount c of cashat maturity.Modelling our risk preferences with expected utility, the portfolio opti-mization problem can be written asminimize Ev ( c − (cid:88) j ∈ J P j ( x j )) over x ∈ D subject to (cid:88) j ∈ J S j ( x j ) ≤ w, (P)where D := (cid:89) j ∈ J [ q jb , q ja ]is the set of feasible portfolios, E denotes the expectation and v ( c ) := − u ( − c )with u being the utility function. In the terminology of [5], v : R → R is a4 oss function . The argument of v is the unhedged part of the claim c . Besidesthe available quantities, one could also include various margin requirementsin the constraints.It is clear that problem (P) is highly subjective. Its optimum value andsolutions depend on our • financial position described by the initial cash w and liability c , • views on the underlying X T described by the probabilistic model, • our risk preferences described by the loss function v .The dependence will be studied numerically in the following sections. In pric-ing of contingent claims, the subjective factors will be reflected in the pricesat which we are willing to trade the claims. The subjectivity is the drivingforce behind trading in practice but it is neglected e.g. by the traditional riskneutral pricing models.Another important feature of (P) is that it is a convex optimization prob-lem as soon as the loss function v is convex. The convexity simply meansthat we are risk averse. Convexity is crucial in numerical solution of (P) aswell as in the mathematical analysis of the indifference prices based on theoptimum value of (P). The first challenge in the numerical solution of problem (P) is that theobjective is given in terms of an integral which, in general, does not allowfor closed form expressions that could be treated by numerical optimizationroutines. However, in applications where the liability c only depends onthe the value of the underlying at maturity, the integral is one-dimensionalwhich can be treated fairly easily with integration quadratures. This will bethe case in the applications below where we study pricing and hedging ofclaims contingent on the underlying price at maturity. We will approximatethe expectation by Gauss-Legendre quadrature which results in an objectivegiven as a finite sum of convex functions of the portfolio vector x .We will reformulate the budget constraint as two linear inequality con-straints by writing the position in each asset as the sum of the long and shortposition. That is, x j = x j + − x j − , where both x j + and x j − are constrained tobe positive. This results in an inequality constrained convex optimizationproblem with the objective and constraints represented by smooth functions.The problem has 884 variables and 1769 constraints.5he resulting problem is solved with the interior-point solver of MOSEK [1]which is suitable for large-scale convex optimisation problems. To set up aninstance of the optimization problem in MATLAB takes on average 11.20seconds and its solution with MOSEK, 4.30 seconds on a PC with Intel(R)Core(TM) i5-4690 CPU @ 3.50GHz processor and 8.00 GB memory. We used quotes for S&P500 index options with maturity 17 June 2016.The quotes were obtained from Bloomberg on 8 April 2016 at 2:55:00PMwhen the value of S&P500 index was 2056 .
32. The available quantities atthe best quotes are given in terms of lot sizes which are 50 for forwardsand 100 for options. The lending and borrowing rates are 0 . . σ and degrees offreedom ν estimated from 25 years of historical daily data. The mean µ wasset to zero. The effect of varying the parameters will be studied later on. µ σ ν v ( c ) = e λc/w , where w is the initial wealth and λ > w used in the exampleswas w = 100 , Figure 1 illustrates the optimized portfolios obtained with two differentrisk aversions, λ = 2 (blue line) and λ = 6 (red line). The bottom panelsrepresent the optimal portfolios with the bars corresponding to the optimal6ositions in the assets. The top left plots the corresponding payoffs as func-tions of the index at maturity and the top right plots the kernel densityestimates (computed using 10,000,000 simulated values of the index at ma-turity) of the payoff distributions. As expected, higher risk aversion resultsin a payoff distribution with a thinner left tail. Increasing the risk aversionalso results in reduced quantities in the optimal portfolio compared with theportfolio of a less risk averse agent. Calls
Strikes -101 Q u a n t i t y ( c on t r ac t s ) Puts
Cash Forwards
S&P 500 index -10123456 P ay o ff risk aversion = 0.00002risk aversion = 0.00006 -0.5 0 0.5 1 1.5 2 2.5 S&P 500 index -4 risk aversion = 0.00002risk aversion = 0.00006 Calls
Forwards Cash
012 10 Strikes -505 Q u a n t i t y ( c on t r ac t s ) Puts
Figure 1: The optimal portfolios obtained with risk aversions λ = 2 and λ =6, respectively (bottom), the payoffs of the optimal portfolios as functionsof the index at maturity (top-left) and the kernel-density estimates of thepayoff distributions of the optimal portfolios (top-right)An interesting feature of the optimal portfolios is that they are sparse inthat our of more than 400 quoted options, the optimal portfolio has nonzeropositions in less than 10 options. This is explained by the spreads betweenthe quotes bid- and ask-prices. To illustrate this further, we repeated theoptimization with risk aversion λ = 2 by optimizing two variants of theproblem. In the first one, we increased the bid-ask spread by adding a 10%transaction cost on all trades and in the second, we set both the bid- andask-prices equal to mid-prices. The results are illustrated in Figure 2. The7ddition of the transaction cost made the optimal portfolio only slightlysparser while removal of the bid-ask spread had a dramatic effect by giving aportfolio that takes large positions in almost all the quoted options. For manyoptions, it was optimal to take maximal positions allowed by the availablebid/ask quantities. Calls
Strikes -200002000 Q u a n t i t y ( c on t r ac t s ) Puts
Cash Forwards
S&P 500 index -505101520 P ay o ff S&P 500 index -10123456 P ay o ff Calls
Strikes -2-101 Q u a n t i t y ( c on t r ac t s ) Puts
Cash
012 10 Forwards
Figure 2: The payoffs and optimal portfolios when an additional 10% trans-action cost is added to all trades (left) and when the bid-ask spread is ignoredby setting both bid- and ask-prices equal to the mid-price (right)To study the effect of views on the optimal portfolio, we reoptimized theportfolio after changing the parameters of the underlying t-distribution. Therisk aversion was kept at λ = 2. Figure 3 plots the payouts of the optimalportfolios in three cases. The first one is the base case already presented inFigure 1. The second if obtained by increasing the scale parameter σ to 0 . ν to 20. As expected,increasing σ results in a portfolio that gives higher payouts further in thetails (a straddle) while ν = 20 gives essentially a Gaussian distribution withthinner tails so the optimal portfolio has higher payouts near the median atthe expense of lower payoffs in the tails.8ase case -2.1499 σ = 0 .
40 -3.5121 ν = 20 -2.2339Table 4: Logarithms of the objective values corresponding to the three dif-ferent models of the underlying S&P 500 index -8-6-4-202468 P ay o ff nu = 4.83548, sigma = 0.0553835nu = 20, sigma = 0.0553835nu = 4.83548, sigma = 0.40 S&P 500 index nu = 4.83548,sigma = 0.0553835nu = 20,sigma = 0.0553835nu = 4.83548,sigma = 0.40
Figure 3: Distributions of the underlying (bottom) and optimal payoffs (top)in the base case (solid line), ν = 20 (dotted) and σ = 0 .
40 (dashed). All othermodel parameters were unchanged. 9 .150.125
Sigma Mu D i s u t ili t y -3.4-3.2-3-2.8-2.6-2.4 Figure 4: The entropic risk of the optimal portfolios as a function of themean µ and volatility σ when ν = ∞ The logarithms of the objective values obtained with the three models ofthe underlying in Figure 3 are given in Table 4. The logarithm of the expectedexponential utility is known as the entropic risk measure ; see e.g. [5]. We seethat the highest objective value is obtained with in the base case where themodel parameters are estimated from historical data. An explanation of thiscould be that the option prices used in the model correspond to the marketparticipants’ views of the future behaviour of the underlying. If we use amodel that is “inconsistent” with these prices, the option prices appear tooffer profitable trading opportunities.To explore this phenomenon more systematically, we repeated the opti-mization in the Gaussian case with ν = ∞ and the mean µ and volatility σ ranging over intervals. Figure 4 plots the corresponding logarithmic ob-jective value, i.e. the entropic risk measure as a function of µ and σ . Therisk seems to be concave as a function of ( µ , σ ) with the maximum around( µ, σ ) = ( − . , . − . We will denote the optimum value of (P) by ϕ ( w, c ) := inf { Ev ( c − (cid:88) j ∈ J P j ( x j )) | x ∈ D, (cid:88) j ∈ J S j ( x j ) ≤ w } . w, ¯ c ), the indifference price for sellinga claim c is given by π s ( ¯ w, ¯ c ; c ) := inf { w | 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 (P).Analogously, the indifference price for buying c is given by π b ( ¯ w, ¯ c ; c ) := sup { w | ϕ ( ¯ w − w, ¯ c − c ) ≤ ϕ ( ¯ w, ¯ c ) } . We have π b ( ¯ w, ¯ c ; c ) ≤ π s ( ¯ w, ¯ c ; c )as soon as π s ( ¯ w, ¯ c ; 0) = 0. Indeed, it is easily checked that the function c (cid:55)→ π s ( ¯ w, ¯ c ; c ) is convex so π s ( ¯ w, ¯ c ; 0) ≤ π s ( ¯ w, ¯ c ; c ) + 12 π s ( ¯ w, ¯ c ; − c )while π s ( ¯ w, ¯ c ; − c ) = − π b ( ¯ w, ¯ c ; c ), by definition.We will compare the indifference prices with the super- and subhedgingcosts defined for a claim c by π sup ( c ) := inf { (cid:88) j ∈ J S j ( x j ) | x ∈ D, (cid:88) j ∈ J P j ( x j ) − c ≥ P - a.s. } ,π inf ( c ) := sup {− (cid:88) j ∈ J S j ( x j ) | x ∈ D, (cid:88) j ∈ J P j ( x j ) + c ≥ P - a.s. } . The superhedging cost is the least cost of a superhedging portfolio while thesubhedging cost is the greatest revenue one could get by entering positionthat superhedges the negative of c . Whereas the indifference prices of a claimdepend on our financial position, views and risk preferences described by( w, c ), P and v , respectively, the super- and subhedging costs are independentof such subjective factors. In complete markets, the sub- and superhedgingcosts are equal for all claims c but, in general, the super- and subhedgingcosts are too wide apart to be considered as competitive quotes for a claim.Recall that if c : R + → R is the difference of convex functions, then itsright-derivative is of bounded variation and we have c ( X T ) = c (0) + c (cid:48) (0) X T + (cid:90) ∞ ( X T − K ) + dc (cid:48) ( K ) . This might suggest that the payout c could be replicated by a buy-and-holdportfolio of c (0) units of a zero-coupon bond, c (cid:48) (0) units of the underlying11nd a continuum of call options weighted according to the Borel-measureassociated with the BV function c (cid:48) . Even if one could buy and sell optionswith arbitrary strikes, it is not quite realistic to trade a continuum of them.Nevertheless, assuming that quotes for all strikes exist, the replication costof c would become c (0) P T + c (cid:48) (0) X + (cid:90) ∞ C ( K ) a dc (cid:48) + ( K ) − (cid:90) ∞ C ( K ) b dc (cid:48)− ( K ) , where c (cid:48) + and c (cid:48)− denote the positive and negative variations, respectively, of c (cid:48) and C ( K ) b and C ( K ) a denote the bid- and ask-prices of a call with strike K . The above formulas could be used to design approximate replicationstrategies given the finite number of quotes in real markets. We will findout that the hedges optimized for indifference pricing look quite differentfrom what the above replication approach would suggest. Instead of aimingfor approximate replication, indifference pricing optimizes the portfolios tothe given quotes, risk preferences and the given probabilistic description ofthe underlying. The definitions of the indifference prices involve the optimum value func-tion ϕ of problem (P) which can rarely be evaluated exactly. The definitionsstill make sense, however, if we replace the optimum value by the best valuewe are able to find numerically. Besides the financial position, future viewsand risk preferences of an agent, the indifference prices then also depend onthe agents’ expertise in portfolio optimization. In computations below, wewill replace ϕ by the approximate value we find with the numerical techniquesdescribed in Section 4.1. The evaluation of the indifference prices then comedown to a one-dimensional search over w . This can be done numerically bya line-search algorithm.The computation of the super- and subhedging costs come down to solv-ing linear programming problems where the constraints require the terminalposition of the agent to be nonnegative in every scenario; see [7]. In the con-text of put and call options, the constraint can be written in terms of finitelymany linear inequality constraints since we know that the net position willbe linear between consecutive strike prices.12 .2 Pricing exotic options We illustrate indifference pricing using the optimization model of Sec-tion 3 in the pricing of three “exotic” options namely, a digital option withpayoff c ( X T ) = (cid:40) ,
000 if X T ≥ K, X T < K a “quadratic forward” with c ( X T ) = | X T − K | and a “log-forward” with c ( X T ) = 100 ,
000 ln(
K/X T ), all with strike K = 2050. Log-forwards havebeen used in the hedging of variance swaps; see e.g. [4]. To compare witha simper option, we also price a European call option with the same strike.To make the last case nontrivial, we remove the call from the set of hedginginstruments.We compute the indifference selling prices assuming that ¯ w = 100 , c = 0, that is, assuming the agent has initial position consisting only of100,000 units of cash. The indifference prices together with the super- andsubhedging costs are given in Table 5. Superhedging is imposed on the in-terval [100,5000]. Clearly, superhedging the quadratic and log-forwards withthe given hedging instruments against all positive values of X T is impossible.The numbers reported in Table 5 are the costs of hedging over the interval[100,5000]. Claim subhedging buying price selling price superhedgingcall 51.2333 51.7338 51.7399 53.0483digital call 5280.00 6082.35 6160.65 6885.71quadratic forward 20383.68 20979.84 22044.92 24542.01log-forward 322.28 358.49 404.67 499.69
Table 5: Indifference prices, together with super- and subhedging costs.Figures 5–8 illustrate the corresponding hedging strategies. Each figuregives the optimal portfolio before and after selling the option together withthe payout of the“hedging portfolio” as a function of the underlying at ma-turity. The hedging portfolio is defined as the difference x − ¯ x , where ¯ x and x are the optimal portfolios before and after the sale of the option.13
600 1700 1800 1900 2000 2100 220001020
Calls
Strikes -4-202 Q u a n t i t y ( c on t r ac t s ) PutsCash Forwards
Calls
Forwards Cash Strikes -4-202 Q u a n t i t y ( c on t r ac t s ) Puts
S&P 500 index -20000200040006000800010000120001400016000 P ay o ff ( do ll a r s ) the payoff of the hedging portfoliothe payoff of one contract of a calloption with strike 2050 Figure 5: Optimal portfolios before (bottom left) and after (bottom right) thesale of a call option. The top panel gives the payoff of the hedging portfolio(solid line) together with the payoff of the claim being priced (dotted line).
Calls
Strikes -505 Q u a n t i t y ( c on t r ac t s ) Puts
Forwards Cash
012 10 S&P 500 index -4000-2000020004000600080001000012000 P ay o ff ( do ll a r s ) the payoff of the hedging portfoliothe payoff of a digital option with strike 2050 Calls
Forwards Cash
012 10 Strikes -505 Q u a n t i t y ( c on t r ac t s ) Puts
Figure 6: Optimal portfolios before (bottom left) and after (bottom right)the sale of a digital option. The top panel gives the payoff of the hedgingportfolio (solid line) together with the payoff of the claim being priced (dottedline). 14
000 1200 1400 1600 1800 2000 2200 2400 2600-20020
Calls
Strikes -1001020 Q u a n t i t y ( c on t r ac t s ) Puts
Cash
012 10 Forwards
S&P 500 index -202468101214 P ay o ff ( do ll a r s ) the payoff of the hedging portfoliothe payoff of one contract of a square forward with strike 2050 Calls
Forwards Cash
012 10 Strikes -505 Q u a n t i t y ( c on t r ac t s ) Puts
Figure 7: Optimal portfolios before (bottom left) and after (bottom right)the sale of a quadratic forward. The top panel gives the payoff of the hedgingportfolio (solid line) together with the payoff of the claim being priced (dottedline).
Calls
Strikes -505 Q u a n t i t y ( c on t r ac t s ) Puts
Forwards Cash
012 10 S&P 500 index -4-2024681012 P ay o ff ( do ll a r s ) the payoff of the hedging portfoliograph of a negative log return Calls
Forwards Cash
012 10 Strikes -505 Q u a n t i t y ( c on t r ac t s ) Puts
Figure 8: Optimal portfolios before (bottom left) and after (bottom right) thesale of a log-forward. The top panel gives the payoff of the hedging portfolio(solid line) together with the payoff of the claim being priced (dotted line).15 .3 Sensitivities
This section studies the sensitivities of the indifference prices with respectto some of the model parameters. Figure 9 plots indifference prices of a calloption with strike 2000 as functions of the “volatility” σ (Since we model theunderlying with the t-distributions, the variance of the log-price is σ ν/ ( ν − σ is close to its historicalestimate of 0 . Sigma P r i ce ( do ll a r s ) indifference prices for buying and sellingbest bid and ask pricessuper and sub hedging costs Figure 9: Indifference prices as functions of volatility. The dotted lines givethe best bid and ask quotes while the dashed lines give the super- and sub-hedging costs.Figure 10 plots the indifference prices as functions of the risk aversion. Asthe risk aversion increases, the gap between the indifference prices widens.The indifference price for selling a call option is more sensitive to the riskaversion. This seems quite natural as shorting a call results in unboundeddownside risk unless the call is superhedged.16
Risk aversion -5 P r i ce ( do ll a r s ) indifference prices for buying and sellingbest bid and ask pricessuper and sub hedging costs Figure 10: Indifference prices of a call option with strike 2000 as functionsof risk aversions.Figure 11 illustrates the dependence of the indifference prices on anagent’s initial position. While in earlier cases, the agent’s initial positionwas assumed to consist only of cash, in this case, we consider an agent withboth cash and call options of the same type as the one being priced. Fig-ure 11 plots the indifference prices as functions of the number of call optionsthe agent holds before the trade. As one might expect, an agent who alreadyhas exposure to the option would assign a higher price to the option. Aseller would increase her exposure to the option payout while for a buyer,the option would be a natural hedge and thus worth paying a higher pricefor. 17
30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30
Initial position in a call option (contracts) P r i ce ( do ll a r s ) indifference prices for buyingand sellingbest bid and ask pricessuper and sub hedging costs Figure 11: Indifference prices of a call option with strike 2000 as functionsof initial position in the same callTo illustrate the nonlinearity of the indifference prices as functions of theclaim, we computed the prices for different multiples M of the call. Figure 12plots the indifference prices per option as functions of the multiplier M . Thefigure plots the indifference prices also in a market model where the bestquotes are assumed to come with unlimited quantities. As the multiplier M increases, the quantity constraints become binding thus worsening the prices. quantity (contracts) p r i ce p e r on e op t i on ( do ll a r s ) indifference prices for buying without quantity constraintsindifference prices for buying with quantity constraints quantity (contracts) p r i ce p e r on e op t i on ( do ll a r s ) indifference prices for sellingwithout quantity constraintsindifference prices for sellingwith quantity constraints Figure 12: Indifference prices of a call option per unit as a function of thequantity traded. Buying price on the left and selling price on the right. Thesolid line gives the prices when quantity constraints are ignored18
Further developments
The developed indifference pricing framework should be taken merely asan illustration of the computational techniques that are available for port-folio optimization. The presented model could be extended in various waysin practice. For example, it would be straightforward to include margin re-quirements as portfolio constraints in the model, as long as the requirementsare given as explicit convex constraints on the portfolio. One could alsostudy options with different maturities by including the relevant maturitiesin the underlying probabilistic model. Such a multiperiod model, could alsoincorporate dynamic trading strategies of the underlying and cash.
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