Options constitute integral part of modern financial trades, and are priced according to the risk associated with buying or selling certain asset in future. Financial literature mostly concentrates on risk-neutral methods of pricing options such as Black- Scholes model. However, using trading agents with utility function to determine the option’s potential payoff is an emerging field in option pricing theory. In this paper, we use one of such methodologies developed by Othman and Sandholm to design portfolioholding agents that are endowed with popular option portfolios such as bullish spread, bearish spread, butterfly spread, straddle, etc to price options. Agents use their portfolios to evaluate how buying or selling certain option would change their current payoff structure. We also develop a multi-unit direct double auction which preserves the atomicity of orders at the expense of budget balance. Agents are simulated in this mechanism and the emerging prices are compared to risk-neutral prices under different market conditions. Through an appropriate allocation of option portfolios to trading agents, we can simulate market conditions where the population of agents are bearish, bullish, neutral or non-neutral in their beliefs

Abdullaev, S.

McBurney, P.

Musial-Gabrys, Katarzyna

Bournemouth University Research Online

Pricing Options with Portfolio-Holding Trading Agents inDirect Double AuctionSarvar Abdullaev1 and Peter McBurney2 and Katarzyna Musial3Abstract. Options constitute integral part of modern ﬁnancialtrades, and are priced according to the risk associated with buyingor selling certain asset in future. Financial literature mostly con-centrates on risk-neutral methods of pricing options such as Black-Scholes model. However, it is an emerging ﬁeld in option pricingtheory to use trading agents with utility functions to determine theoption’s potential payoff for the agent. In this paper, we use one ofsuch methodologies developed by Othman and Sandholm to designportfolio-holding agents that are endowed with popular option port-folios such as bullish spread, butterﬂy spread, straddle, etc to priceoptions. Agents use their portfolios to evaluate how buying or sell-ing certain option would change their current payoff structure, andform their orders based on this information. We also simulate theseagents in a multi-unit direct double auction. The emerging prices arecompared to risk-neutral prices under different market conditions.Through an appropriate endowment of option portfolios to agents,we can also mimic market conditions where the population of agentsare bearish, bullish, neutral or non-neutral in their beliefs.1 IntroductionOption is the type of ﬁnancial derivative that enables its holder (i.e.owner) to buy or sell speciﬁed assets at certain future price to writer(i.e. issuer) of the option. Holder of the option buys for an additionalcost (i.e option premium) determined by the market or the writer ofthe option. On the other hand, the writer of the option sells by takingfuture obligation to trade assets if holder chooses to exercise his rightto buy or sell. Option contract must specify the underlying asset to betraded, its volume, strike price and expiration date. European optionscan be exercised only on their maturity date, while American optionson any date until expiration. We will use only European options inthe scope of this paper.Traders can take different positions with options of different mon-eyness and create option portfolios which can align with their fore-cast and at same time limit their loss in case if their forecast isnot true. Cohen counts more than 40 option portfolios and classi-ﬁes them based on their market direction (i.e. bullishness or bearish-ness), volatility level, riskiness and gain [2]. Let us consider, butterﬂyspread. This type of spread involves taking positions in options withthree different strike prices. In butterﬂy call spread, trader has an es-timate that the price is not going to change sharply, so he wants tostay neutral. He buys 2 call options: one ITM with low K1 and oneOTM with high K3. At the same time, he sells 2 ATM calls with K2,where K2 is halfway between the range of K1 and K3. This spread1 King’s College London, UK, email: sarvar.abdullaev@kcl.ac.uk2 King’s College London, UK, email: peter.mcburney@kcl.ac.uk3 Bournemouth University, UK, email: kmusialgabrys@bournemouth.ac.ukleads to a proﬁt if the asset price will not go far from its current spotprice. It will incur in ﬁxed loss if the asset price changes sharply ineither directions. Butterﬂy spread can be created using put options aswell.Pricing options is mostly based on Black-Scholes framework [1]which values options from the perspective of no arbitrage assump-tion. There is an analytical solution for ﬁnding the risk-neutral valueof the option. However, in agent-oriented approach, traders can havedifferent assumptions and pricing strategies (not necessarily risk-neutral) based on their private utility functions. Gerber and Pafumdescribed risk-averse traders based on an exponential utility func-tion which could produce a bid-ask spread around risk-neutral optionprices [3]. We will use Othman and Sandholm’s indifferent optionpricing method which takes into account agent’s inventory [7].In this paper, we study how option prices may differ from the risk-neutral prices if the traders come to the market already endowed withsome option portfolio. We developed an agent-based system whichuses direct double auction mechanism to run option traders. We usedinventory-based Logarithmic Market Scoring Rule (LMSR) optiontrader developed by Othman and Sandholm [7] to enable the optionpricing based on the payoff structure of their current portfolio. Weendow the LMSR traders with commonly used option portfolios suchas bullish spread, bearish spread, butterﬂy spread etc. and run themin our proposed mechanism. This allows us to set up different sen-timent in the market such as more bullish traders (or more bearishtraders) and observe the resulted option prices in comparison to risk-neutral price. We explain our key ﬁndings from this experiment.2 Portfolio-holding Trading AgentIn prediction markets, the agents are allowed to change the marketmaker’s payoff structure for a corresponding payment. For example,if market maker is accepting bets for teams A and B on a footballmatch, and his current payoff structure is (300,200) meaning thatthe aggregator has to pay $300 in total if team A wins, and $200if team B wins. However one would like to bet on team A, and heexpects to receive $50 if his bet is achieved. The aggregator changeshis payoff structure to (350,200) by accepting the bet, and he alsoneeds to decide how he can charge the client for accepting his bet.The most commonmethod for evaluating the cost of accepting the betin prediction markets, LMSR [4] and it is deﬁned as a cost functionfor the vector of payoffs x = {x1, x2, . . . , xn} on the probabilityspace of events Ω = {ω1, ω2, . . . , ωn}:C(x) = b log(∑iexp(xi/b))(1)ECAI 2016G.A. Kaminka et al. (Eds.)© 2016 The Authors and IOS Press.This article is published online with Open Access by IOS Press and distributed under the termsof the Creative Commons Attribution Non-Commercial License 4.0 (CC BY-NC 4.0).doi:10.3233/978-1-61499-672-9-17541754where b > 0 is a liquidity parameter. The larger values of b producetighter bid/ask spreads, but may also incur larger worst-case lossescapped by b log(n) [8]. The agent who wishes to change the payofffrom x to y has to pay the difference between the costsC(y)−C(x).In our above example, given that b = 100, the aggregator acceptingbets must charge the client C((350, 200)) − C((300, 200)) ≈ $39for the bet.The same principle can be used for the option trader who holdsa certain portfolio of options that generate certain payoff for differ-ent asset price outcomes in future. The agent can virtually simulatebuying or selling particular type of option and compute the changesit makes to its payoff structure. For example, let agent take butterﬂycall spread with buying ITM call at strike K1 = 80 and OTM callat K3 = 120, and selling 2 ATM calls at K2 = 100. We can com-pute his discounted payoffs for the range of possible prices wherethe asset price can end up at time T . Let this payoff structure be x.Trader feels bullish and wants to buy one more call option at strikeK4 = 130, so his bid for buying an OTM option atK4 = 130 can becomputed from the difference of his payoff structures C(y)−C(x),and in our particular case, it is $1.42 given that the liquidity param-eter is b = 2500. We can also compute the Black-Scholes price ofsuch option given parameters T = 1, r = 0.05, S0 = 100, σ = 0.02and it is $2.52. This would mean that the trader places a bid for givenOTM option less than its risk-neutral value.3 Experimental ResultsWe have simulated the asset prices using the Geometric BrownianMotion with a calibrated parameters according to the historic dataof NASDAQ-100 index in 2014. The daily mean drift is computedas μ = 0.0007, and and the volatility is σ = 0.0089. The ini-tial asset price is the same as NASDAQ-100 on 2 January 2014,S0 = $3563.57, and it is ST = $3597.59 at the end of the year.We also analyse option with strike $3563.00 which expires in oneyear. We use only call options for the simulation, because put pricescan be directly computed from the call price using put-call parityrelationship. Mechanism simulates 365 trading days going up to thepoint when the option expires. Because mechanism is direct, it clearsorders in one round and moves to the next trading day. Every trad-ing day, the traders are re-instantiated with the same distribution ofportfolios so they do not remember their previous choices.LMSR traders create a positive bid-ask spread, which forbids themfrom trading if the market is uniformly populated with LMSR tradersholding the same portfolio. Therefore LMSR trader should be alsosimulated in mixed groups each holding different set of portfolios,and thus produce different prices. It is also important to note thatLMSR trader is deterministic in their pricing, because the only factorwhich affects their pricing decision is their portfolio and ﬁxed rangeof events horizon that they use to compute their ﬁnal payoff. There-fore two LMSR traders holding the same portfolio produce same bidsor same asks. To make market more heterogeneous, I use multipleoption portfolios to mimic neutral, non-neutral, bullish and bearishtraders. Some of the option portfolios used are bearish spread, bullishspread, butterﬂy spread, ladder, strangle straddle, strip, etc [5]. Afterrunning several experiments with LMSR traders, we found out thatliquidity b = 100 provides reasonable range of bids and asks whichare likely to produce trades in the market. LMSR trader picks ran-dom quantities between -2000 and 2000 while submitting orders, soagent’s decision to buy or sell is uniformly distributed. The negativequantities stand for asks, and the positive ones are bids.Figure 1 shows the trade between neutral and non-neutral portfo-lio holders. It can be seen that the prices are volatile around Black-Scholes prices. This is explained using the deterministic nature ofLMSR traders. The whole market consists of 2 neutral LMSR tradersand 2 non-neutral LMSR traders who output all together 8 differ-ent pricing quotes, 4 for bids and 4 for asks. Because mechanismhas very few choices to determine the clearing price among mostlyhomogeneous quotes, the option price for each trading day differssigniﬁcantly. In the market of bearish traders as shown in Figure 2,the call options are initially underpriced, as they are considered lessproﬁtable for the traders expecting the drop in asset prices. Howeverthe prices cross the risk-neutral price only after option lives the halfof its lifespan. This is because the payoff from the option becomesmore certain, as the asset prices continue to grow defying the bearishbelief of the trader.Figure 1. Option prices with neutraland non-neutral LMSR tradersFigure 2. Option prices with morebearish LMSR traders4 Concluding RemarksIn this paper, we simulated LMSR-based option trading agents indirect double auction mechanism and were able to observe optionprices under different population of traders with bearish, bullish, neu-tral and non-neutral portfolios. Our simulation results have shownthat pricing option via double auctions is a valid technique as theobtained prices were close to risk-neutral solution if the market wastruly populated with traders having different beliefs. Moreover, wesaw that the neutral and the non-neutral traders create volatility astheir portfolios consist of opposite payoff structures. Also in morebearish market population, the calls were initially underpriced un-til the option reached half-way its maturity. This approach can befurther improved using other incentive compatible mechanisms suchas McAfee’s double auction [6] in future researches. Also the traderscan be more sophisticated in making buying/selling decisions insteadof randomly choosing either action.REFERENCES[1] Fischer Black and Myron Scholes, ‘The pricing of options and corporateliabilities’, The journal of political economy, 637–654, (1973).[2] G. Cohen, The Bible of Options Strategies: The Deﬁnitive Guide forPractical Trading Strategies, Prentice Hall, 2005.[3] Hans U Gerber and Ge´rard Pafum, ‘Utility functions: from risk theory toﬁnance’, North American Actuarial Journal, 2(3), 74–91, (1998).[4] R. Hanson, ‘Logarithmic market scoring rules for modular combinato-rial information aggregation’, Journal of Prediction Markets, 1(1), 1–15,(2007).[5] J. C. Hull, Options, Futures and Other Derivatives, Prentice Hall, 5 edn.,2001.[6] R PrestonMcAfee, ‘A dominant strategy double auction’, Journal of eco-nomic Theory, 56(2), 434–450, (1992).[7] A. Othman and T. Sandholm, ‘Inventory-Based Versus Prior-Based Op-tions Trading Agents’, Algorithmic Finance, 1(2), 95–121, (2012).[8] David M Pennock and Rahul Sami, ‘Computational aspects of predictionmarkets’, Algorithmic game theory, 651–674, (2007).S. Abdullaev et al. / Pricing Options with Portfolio-Holding Trading Agents in Direct Double Auction 1755

Pricing Options with Portfolio-based Option Trading Agents in Direct Double Auction

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Pricing Options with Portfolio-based Option Trading Agents in Direct Double Auction

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