Massimo Sapienza (2003)
Do Real Options perform better than Net Present Value? Testing in an artificial financial market
Journal of Artificial Societies and Social Simulation
vol. 6, no. 3
<http://jasss.soc.surrey.ac.uk/6/3/4.html>
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Received: 10-Nov-2002 Accepted: 5-Apr-2003 Published: 30-Jun-2003
where V(d_{t}) is the NPV value of the project already in place and F(d_{t}) is the value of the growth option that the firm can exercise when the cash flow is above its trigger value d*.
with is the cash flow's average rate of growth. is the scale multiplicative factor of the cash flow (under our assumptions equal to the dividend). It indicates how big the impact of new investments on cash flow will be when the expansion option is exercised. I is the exercise cost of the option, the value of the investments required to expand the scale. We assume and I to be common knowledge.
where is the agent's degree of relative risk aversion. The risky asset's demand is a function of the expected excess return E[ p_{t+1}+d_{t+1}] which the trader can obtain by investing in this asset rather than the riskless one. The model is closed by determining the clearing price p by setting the demand equal to the supply (i.e.. by assuming one share supplied per agent):
This assumption is perfectly in line with an NPV valuation process of firms.
Agent | Rationality |
Random | None |
Market Imitating | Imitation |
Stop Loss | Rule of thum |
Forecasting | ANN based forecasting |
Cross Targets | Cognitive abilities |
Table 3. Main simulation parameters |
Table 4. Classifier system's parameters |
where specificity is the relative specificity of the rule: the ratio between the number of # in both the action and the conditioning part and the length of the rule. The purpose of this cost is to make sure that each bit is actually serving a useful purpose in terms of a forecasting rule; s_{j,t-1} is the strength of rules j at time t-1, W is a random real number between 0 and 1. Bid_{j,t} is the amount the auction's winner actually has to pay.
Figure 1. Actual and REE prices in the HOCA Slow learning case |
Figure 2. Actual and REE prices in the HOCA fast learning case |
Table 5. Data on prices - HOCA |
Table 6. Data on returns - HOCA |
Table 7. Trading volume - HOCA |
Figure 3. Volume in the HOCA slow learning case |
Figure 4. Volume in the HOCA fast learning case |
Figure 5. Percentages of the conditioning part's bit set - HOCA slow learning case |
Table 8 |
Table 9. Da on prices-HECA |
Table 10. Da on returns-HECA |
Figure 6. Volume in then HECA slow learning case |
Figure 7. Volume in then HECA fast learning case |
Table 11. Trading volume-HECA |
Figure 8. Actual and REE prices in the HECA slow learning case |
Figure 9. Actual and REE pices in the HECA fast learning case |
Figure 10. CS agents performance |
Figure 11. Well performing agents in the HECA fast learning case |
Figure 12. Percentages of the conditioning part's bit set - HECA fast learning case |
^{2}See among others: Arthur et al. (1997), Chen , Yeh (2001), Tay and Linn (2001), Terna (2001).
^{3}See: Arifovic (1996), Arifovic (2000).
^{4}See: Le Baron et al. (1999), Lawenz and Westerhoff (2000)
^{5}Minar, Burkhart, Langton and Askenazi (1996) present the software. For a detailed user's guide see Johnson and Lancaster (1999). A tutorial by Stefansson is also contained in Luna and Stefansson (2000). Complete software documentation, source code and much other interesting material is available at: <http://www.swarm.org>.
^{6}In the time frame of the model, dividends are payed every 30 cycles; Each cycle represents a trading day. Dividends are obviously distributed to traders proportionally to their holdings.
^{7}In the version of SUM presented by Terna (2001) any kind of fundamentals is totally absent. The risky asset does not entitle the agent to earn any dividend, and consequently it's value is purely bubbly. The Santa Fe Stock market assumes that the dividend process is a mean reverting process. No assumptions are made about the relationship between the firm's cash flow and the dividends payed
^{8}We could also relax this assumption by considering the firm as a portfolio of different projects, but for the sake of simplicity we prefer to maintain this mono-project scheme.
^{9}By calculating the expansion option's value we obtain the following results:
^{10}See for instance Arthur et al. (1997), Le Baron et al. (1999), Tay and Linn (2001), Chen and Yeh (2001).
^{11}With σ=20% and λ=0.5 we would need a risk less interest rate greater than 40% to obtain a real value for the REE price. However we should point out that there is practically no evidence available to calibrate the parameter λ. The value of 0.5 has been practically just assumed. With λ in the range [ 0.01,0.1] we obtain values of the interest rate highly similar to those actually observed.
^{12} In the following exposition is instead the standard deviation of the dividends' process, the geometric Brownian motion.
^{13}Here is a strong difference with respect to the Santa Fe stock exchange market. In that context, agents are heterogenous in their learning history and in their knowledge but they share the same learning mechanisms. In that context all the agents have rationality empowered by a classifiers system.
^{14}Finally we have to add that our analysis is restricted to just one example of trading mechanism present in the financial markets. A different analysis on the impact of alternative trading schemes is left for future research.
^{15}More specifically in the simulation the following historical estimators for volatility are employed:
^{16}As an exemplification let consider the following conditions: [1 # # # # # # # # # # 1 1 # #] recognizes the states where the current price times the interest rate divided by the current dividend is below a quarter AND the dividend estimated volatility is greater than 30%.
^{17}The simulation code is available on-line at: <http://web.tiscali.it/msapienza/acepaper.htm>
^{18}To provide Monte Carlo-like evidence of the stochastic robustness of the results, running a sample of simulations under different random seeds would be desirable. Unfortunately this interesting exercise has not been done systematically. It is our aim to complete this research work in the near future. The 4 simulations illustrated in the text are based on different random seeds to allow some kind of statistical heterogeneity. In any event, two main elements avoid any inconsistency caused by the "real randomness" of the four simulations: the growth option continuous appearance mechanism which makes new options appear once the old one are exercised, and the initial calibration of the growth option which is "out-of-the money". These two elements make any slight difference in the dividend process almost irrelevant so as to determine which trading strategy is performing better.
^{19} In the original ASM (artificial stock market) configuration the initial rules' databases were composed by 100 rules per agent.
^{20}This algorithm described in Goldberg (1989) is quite standard in classifiers systems applications. It avoids the eccessive determinism which would be caused by a selection process only based on fitness measures. ASM does not exploit the advantages of such a selection process and relies on a purely deterministic mechanism. We consider this point as a significant improvement of our model.
^{21} More than two years.
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