Dr. Akin Seber / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 3, Issue No. 2, 159 - 160
Risk-Including Probability Measure and Arbitrage Pricing Theory Ass. Prof. Dr. Akin Seber Department of Financial Economics and Faculty of Commercial Sciences Yeditepe University, Istanbul, Turkey aseber@yeditepe.edu.tr
Keywords - risk neutral probability; binomial tree model; capital asset pricing model; arbitrage pricing theory; financial derivatives pricing.
I.
INTRODUCTION
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In the binomial-tree model the percentage change in the risky asset price is u if the up outcome materializes and d if the down outcome materializes, whereas the risk-free return is rF. The risk-neutral probability is the probability that gives the same risk-free return rF for the risky security with inherent risk of u and d values. The values would be defined as p* u + (1 – p*) d = rF and therefore p* = (rF – d) / (u – d).
For simplification, we will omit detailed analysis of well-known financial theorems that can be found in any finance theory textbook. The first reference we give for the paper is among these textbooks where a very solid mathematical approach to the financial theorems referred to in this paper is given, so any interested reader may refer to it. The second reference we give in our paper is our previous paper where we have used various historical time-series models for prediction of implied future parameter values. This paper has several references about the details of various timeseries approaches, and again interested readers may refer to them.
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Abstract — We propose 2 different risk-including probability measures to be used in arbitrage pricing theory. The use of these probabilities in place of the risk-neutral probability is shown to change the fundamental theorem of asset pricing which is used in financial derivatives pricing.
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The arbitrage pricing theorem is based on the idea that future risky security prices should be forecasted in such a way as to allow for no arbitrage opportunity. Meanwhile, the Fundamental Theorem of Asset Pricing (FTAP) states that if there exists a risk-neutral probability satisfying Martingale Property (MP) for a risky asset, which predicts future prices of risky securities with risk-neutral probability, there will be no arbitrage opportunity.
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The problem with this model is that the risky asset value for next period is predicted with risk-neutral probability. Actually, a better probability measure for predicting next period prices would take into account the risk inherent in the risky asset. So, our aim in this paper is to try and find out if there is any candidate for a better riskincluding probability measure and analyze the implications it would have. In this paper, we propose 2 such probability measures based on the Capital Asset Pricing Model (CAPM) whereas the volatility and mean estimates in the model can be based either on past period estimates or future predictions. Since risky security prices and derivative security prices are predicted based on the Fundamental Theorem of Asset Pricing, a change in the Theorem would also change the prediction for the prices of these assets.
ISSN: 2230-7818
II.
METHOD
First, we propose two risk-including probability measures in this paper. Both of these measures are based on the Capital Asset Pricing Model (CAPM), one based on the Capital Market Line (CML), the second based on the Security Market Line (SML). According to the CAPM, every rational investor will choose his portfolio on the tangent line to the Markowitz bullet constructed from the risk-return profile of the portfolios for different combination of securities existing in the market. The y-intercept of the tangency point of the CML is risk-free interest rate (rF, 0) and the tangency point to the Markowitz bullet is (rM, σM). Therefore, a prediction for the probability-measure for predicting risky-security prices based on the CML would have expected return rRCML equal to rF + (µM – rF) σR / σM. For the SML, βR of the risky asset or portfolio is found from covariance and variances σRM / σM2, and the expected return for the risky-portfolio rRSML is found from rF + (µM – rF) βR where µM stands for the expected return for the market portfolio. In both of these analysis, the market portfolio M with return µM and risk σM contain all risky securities with weights equal to their relative share in the whole market. In practice, the market portfolio is approximated by a suitable stock exchange index. As a result, the 2 risk-including probability measures to be used in arbitrage pricing models would be found by replacing rF with rRCML or rRSML in the equations used for risk-neutral probability estimation. The resulting risk-including probability measures will be represented by pRCML and pRSML instead of p*. In order to use these new probability measures we need to estimate σR, σM, βR (or additionally we only need the
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Dr. Akin Seber / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 3, Issue No. 2, 159 - 160
III.
APPLICATION
The next question we need to ask is how we would use these risk-including probability measures in our financial theorems. For explanatory purposes we will focus on forward / future price determination, Cox-Ross–Rubinstein option pricing formula, and the Fundamental Theorem of Asset Pricing. The basic idea in all of these models is that we need to predict the Future values with the risk-including probability measures and returns, whereas we need to do the discounting with the risk-free interest rate for Present Value (PV) analysis.
where Φ (m, N, p) = ∑
pk (1 – p)N–k is the cumulative
binomial distribution with N trials and probability p of success in each trial. The letter m stands for the least number of successes where S(0) (1 + u)m (1 + d)N-m becomes greater than the option exercise price X, and q = p* (1 + u) / (1 + rF). Similarly, in the FTAP, compounding is made with p* for forecasting financial asset price of n +1 with asset value known at time n. Since the PV of the of the asset from time n and n +1 will be made with rF, the Martingale Property will no longer hold if we change the compounding probability from p* to pRCML or pRSML. This introduces a change in FTAP since the Martingale Property no longer holds. IV.
CONCLUSION
Including risk in probability estimations for future prices of risky securities is a more logical alternative to risk-neutral probability. In this paper, we used the CAPM for risk estimation to be included in the new probability measure. The result is that the financial derivatives prices are different than the estimations with risk-neutral probability.
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For the Forward / Futures price determination, the relevant formula is F (t, T) = f (t, T) = S (t) e rF (T – t), where t is the starting and T is maturity time, and F, f, S stand for forward, futures and stock prices relatively. We need to replace rF with the new risk-including returns rRCML and rRSML for compounding. However, since the discounting still will be made with rF, the current price of the forward and futures contract will no longer be zero, but positive: F (0, T) = S (0) e ( rRCML - rF ) T and similarly for rRSML.
whereas the discounting factor rF stays the same. As a result, for example, the price of a European call may be more and put option less than the ones estimated with risk-neutral probability if [S(0) / X] (1 + u) (1 + rF)N-1 > 1, when Φ (m, N, p) / p < 0, and vice versa. The relevant equations for European call and put options are as follows: CE (0) = S (0) [1 – Φ (m – 1, N, q)] – (1 + rF) - N X [1 – Φ (m -1, N, p*)] and PE (0) = - S (0) Φ (m – 1, N, q) + (1 + rF) - N X Φ (m – 1, N, p*)
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correlation coefficient ρRM) and µM. For the estimation of the parameters for the new probability measures, we propose two different approaches: One based on the past values of these estimates, and the other based on future predictions. Past values are based on the definition of these variables, and the future predictions will be based on various time-series approaches. For an example of these time-series models, we refer to our previous paper where the historical volatility models MA, EWMA, AGARCH (1,1), GARCH (1,1), IGARCH (1,1), and GJR (1,1) are used for estimation of implied volatility, where implied volatility is the market expectation of future volatility estimated from the BlackScholes option pricing formula.
[1]
[2]
M. Capinski and T. Zastawniak, Mathematics for Finance, Springer, 2003. B. Akcay, O. Yorukoglu and A. Seber, “Comparison of Historical Volatility Models with Implied Volatility in Financial Markets in Normal and Crisis Periods”, paper presented at Maramara University Finance Seminar, Istanbul, 2010.
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For the Cox-Ross-Rubenstein option pricing formula, we have to change the p* in the Binomial Distribution estimation with the new risk-including probability measures
REFERENCES
ISSN: 2230-7818
@ 2011 http://www.ijaest.iserp.org. All rights Reserved.
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