A new median-based formula for the Black-Scholes-Merton Theory
aa r X i v : . [ q -f i n . M F ] A p r A new median-based formula for theBlack-Scholes-Merton Theory
Takuya Okabe ∗ and Jin Yoshimura † ‡ § ∗ Graduate School of Integrated Science and Technology, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561, Japan, † Graduate School of Science and Technology, andDepartment of Mathematical Systems Engineering, Shizuoka University, Hamamatsu 432-8561, Japan, ‡ Department of Environmental and Forest Biology, State University ofNew York College of Environmental Science and Forestry, Syracuse, NY 13210 USA, and § Marine Biosystems Research Center, Chiba University, Uchiura, Kamogawa, Chiba299-5502, Japan
The Black-Scholes-Merton (BSM) theory for price variation has beenwell established in mathematical financial engineering. However, ithas been recognized that long-term outcomes in practice may di-vert from the Black-Scholes formula, which is the expected value ofthe stochastic process of price changes. While the expected valueis expected for the long-run average of infinite realizations of thesame stochastic process, it may give an erroneous picture of nearlyevery realization when the probability distribution is skewed, as is thecase for prices. Here we propose a new formula of the BSM theory,which is based on the median of the stochastic process. This formulamakes a more realistic prediction for the long-term outcomes thanthe current Black-Scholes formula. stock market — random walk — geometric Brownian motion — geometricfitness — multiplicative growth
The Black-Scholes-Merton (BSM) theory has been consid-ered the standard model of prices in finalcial markets [1, 2].The Black-Scholes (BS) formula gives the price of a Europeancall option, i.e., the right to buy a stock on a future day. Thisformula is derived under the assumptions of the BSM theory,e.g., a constant riskless rate, a geometric Brownian motionwith constant drift and volatility, no dividends, no arbitrageopportunity, no commissions and transactions costs, and africtionless market [3]. Although the BSM theory has beengenerally successful and widely accepted, some shortcomingsof the BS formula, including the long-term prediction, havebecome clear over the past decades. Although most of theshortcomings may originate from the assumptions of the BSMtheory, here we accept them valid as they have little to do withthe point we make. Instead, we cast doubt on practicality ofthe fundamental concept in probability theory, the expectedvalue. The expectation value may deviate from a middle ofthe distribution owing to its strong dependence on outliers,extremely large values of extremely small probabilities. In bi-ology, e.g., of population growth in stochastic environments, itis widely acknowledged that the use of the expected value cangive a completely erroneous picture of nearly every population[4]. Instead of the arithmetic mean (the expected value), thegometric mean of growth rates (geometric mean fitness) pro-vides a satisfactory picture of population growth [5, 6, 7, 8].For example, risk-spreading strategies in animals have beenunderstood in terms of geometric mean fitness [7, 9]. We thuspropose a better alternative to the current BS formula usingthe median of the distribution in the BSM theory.Suppose that population size S t at time t varies in the mul-tiplicative manner S t +1 = l t S t , where the rate l t varies ac-cording to a probability distribution [4, 5, 10]. If the ratesat different times are independent of each other, the expectedvalue of S t is given by S t = µ tl S , (1) where µ l = E [ l ] is the mean (expectation value) of l t . If themean is greater than unity ( µ l > S t blows up as t grows. However, this expectation may be subverted almost certainly. For example, when l t comprisestwo possibilities l = 0 . l = 1 . µ l = 1 . E [ S ] = 1 . S = 13781 S . As a matter of fact, however,the probablity that the final size S surpasses the originalvalue S is very small (Prob[ S > S ] = 0 . S ≃ S t > E [ S ]] = 0 .
022 and Prob[ S t < E [ S ]] = 0 . µ log l = E [log( l )], which is the log-arithm of the geometric mean of l t . Accordingly, a moresatisfactory solution to this problem is provided by the ge-ometric mean M = e µ log l ( S t = M t S ) than by the arith-metic mean µ l ( S t = µ tl S )[4, 11]. In the above example, µ log l = − . <
0. Hence, the most typical behavior isthe exponential decay S ≃ . S . Here we make theimportant point that the geometric-mean growth factor M isthe median of the probability distribution, i.e., M satisfiesProb[ S t > M ]=Prob[ S t < M ].The same remark applies to stock prices modelled by a log-normal distribution. In the BSM model, the market consistsof a risky asset (stock S ) and a riskless asset (bond B ). Theformer obeys the stochastic differential equation dS t = µS t dt + σS t dW t , (2) where dW t is a stochastic variable. The latter varies as B t = B e rt with B = 1 [3]. Since the logarithm of thestock price S t discounted by B t follows a normal distribution,the terminal stock price S T follows the log-normal distribu-tion f S T ( x ) with mean log S t + ( r − σ / τ and variance σ τ ,where τ = T − t is the time to maturity. For the strike price K , the BS formula for the price of a European call option is C BS ( S t , K, T ) = e − rτ Z ∞ K ( S T − K ) f S T ( x ) dx (3) = S t Φ( d ) − e − rτ K Φ( d ) , (4) where d = (cid:0) log ( S t /K ) + r + σ τ / (cid:1) / ( σ √ τ ) and d = d − σ √ τ and Φ( y ) = √ π R y −∞ e − t dt is the normal cumulativedistribution function. Instead of the expected value C BS inEq.(3), we propose to use the median (Fig.1) C ( S t , K, T ) = S t exp (cid:18) σ √ τ Φ − (cid:18) − Φ ( d )2 (cid:19) − σ τ (cid:19) − Ke − rτ , (5) where Φ − ( x ) is the inverse function of Φ( y ). The two for-mulae C BS and C give the same result in the shortest term,i.e., C BS = C at τ = 0. However, they make strikingly differ-ent long-term predictions (Fig. 2A). As the time τ increases, C BS increases continuously to the stock price S t , indepen-dently of K [1, 2]. This is not the case for C , which is notven a monotonic function. Accordingly, the S t -dependence of C is modified drastically (Fig.2B). The difference is becausethe probability above the mean of the log-normal distributiondrops exponentially as Φ ( − σ √ τ/
2) while the median of thelog-normal distribution is independent of τ .In probability theory, the expected value (mean, average)is known to be away from most typical outcomes when theprobability distribution has a large skew or includes a few ex-treme outliers. Here the median is much closer to these typicaloutcomes than the mean. Thus, the alternative median for-mula gives a better estimate for the call options when thereare outliers or large skews, e.g., the subprime mortgage crisisin 2007-2008 [3]. Recently, the log-normal distribution usedfor stock prices is questioned and the probability distributionof stock prices have been estimated from the past records us-ing, for example, the boot-strap model instead [3]. We shouldnote that the assumption of the lognormal distribution canbe replaced to such a practical distribution to gain the betterresults to avoid the problem of the abovementioned statisticaloutliers. S T f S T C ( Black - Scholes ) C ( median ) K Fig. 1.
The median C and the mean C BS (Black-Scholes). The probability dis-tribution of price S T . Two shaded regions have the same area. S t = 1 . , σ = 1 , τ = 1 , r = 0 and K = 0 . . C ( K = ) C BS ( K = ) C ( K = ) C BS ( K = ) σ τ C C ( τ = ) C BS ( τ = ) C ( τ = ) C BS ( τ = ) S t C S t S t - K Fig. 2.
The median C and the mean C BS (Black-Scholes). A (top) C and C BS for K = 0 . and 0.7 are plotted against σ √ τ ( S t = 1 . and r = 0 ). B (bottom) C and C BS for τ = 0 . and τ = 2 are plotted against S t ( r = 0 . , K = 1 and σ = 1 ). The BS formula C BS gives concave curves between S t and S t − K .
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