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Black-Scholes Model Chapter 5 Black-Scholes Model Copyright c 2008{2012 Hyeong In Choi, All rights reserved. 5.1 Modeling the Stock Price Process One of the basic building blocks of the Black-Scholes model is the stock price process. In particular, in the Black-Scholes world, the stock price process, denoted by St, is modeled as a geometric Brow- nian motion satisfying the following stochastic differential equation: dSt = St(µdt + σdWt); where µ and σ are constants called the drift and volatility, respec- tively. Also, σ is always assumed to be a positive constant. In order to see if such model is \reasonable," let us look at the price data of KOSPI 200 index. In particular Figure 5.1 depicts the KOSPI 200 index from January 2, 2001 till May 18, 2011. If it is reasonable to model it as a geometric Brownian motion, we must have ∆St ≈ µ∆t + σ∆Wt: St In particular if we let ∆t = h (so that ∆St = St+h − St and ∆Wt = Wt+h − Wt), the above approximate equality becomes St+h − St ≈ µh + σ(Wt+h − Wt): St 2 Since σ(Wt+h − Wt) ∼ N(0; σ h), the histogram of (St+h − St)p=St should resemble a normal distribution with standard deviation σ h. (It is called the h-day volatility.) To see it is indeed the case we have plotted the histograms for h = 1; 2; 3; 5; 7 (days) in Figures 5.2 up to 5.1. MODELING THE STOCK PRICE PROCESS 156 Figure 5.1: KOSPI 200 from 2001. 1. 2 to 2011. 5. 18. 5.7. In there, we have also computed the normal distributions that are closest to the given histograms. Such normal distributions are overlaid together in Figure 5.8. In there, we can see the standard deviations (h-day volatility) for each time interval h increases as shown in Table 5.1. p Let us now see if the h-day volatility increases in proportion to h.p The last column of Table 5.1 shows the h-day volatilities divided by h. There, we can see that they remain roughly constant.p In Figure 5.9 are plotted h-day volatilities and a graph y = 0:016436 h that best fits the data. In either case, wep can see that the h-day volatilities scales roughly in proportion to h. p day h-day volatility (h-day volatility)= day 1 0.016858 0.016858 2 0.023430 0.016568 3 0.029624 0.017103 5 0.036611 0.016373 7 0.040421 0.015278 Table 5.1: Standard Deviations (volatility) for the Normal Distribu- tions. Next, if St were to be modeled s satisfying ∆St ≈ µ∆t + σWt; St using the increment Brownian motion Wt, it is not enough to check 5.1. MODELING THE STOCK PRICE PROCESS 157 Figure 5.2: 1-day percentage changes of KOSPI 200. Figure 5.3: 1-day percentage changes of KOSPI 200 with Normal Distribution Approximation. 5.1. MODELING THE STOCK PRICE PROCESS 158 Figure 5.4: 2-day percentage changes of KOSPI 200 with Normal Distribution Approximation. Figure 5.5: 3-day percentage changes of KOSPI 200 with Normal Distribution Approximation. 5.1. MODELING THE STOCK PRICE PROCESS 159 Figure 5.6: 5-day percentage changes of KOSPI 200 with Normal Distribution Approximation. Figure 5.7: 7-day percentage changes of KOSPI 200 with Normal Distribution Approximation. 5.1. MODELING THE STOCK PRICE PROCESS 160 Figure 5.8: Overlaid normal distributions. Figure 5.9: Change of the h-day Volatilities. 5.1. MODELING THE STOCK PRICE PROCESS 161 p that the standard deviation of ∆St=St scales like h. We also need to check the independence of increment. In order to do that, let us go back to the stochastic differential equation for St and rewrite d log St using the It^oformula as: 1 d log S = (µ − σ2)dt + σdW : t 2 t Therefore we can write 1 ∆ log S ≈ µ − σ2 ∆t + σ∆W : (5.1) t 2 t On the other hand 1 log S = log S + µ − σ2 t + σW : (5.2) t 0 2 t Therefore from (5.1) and (5.2), log St and ∆ log St should exhibit \independent" behavior. For that purpose, the following lemma comes in handy. Lemma 5.1. Let X and Y be both Gaussian random variable. Then X and Y are independent if and only if Cor(X; Y ) = 0, where Cor(X; Y ) is the correlation given by X − m Y − m Cor(X; Y ) = E X Y ; σX σY 2 2 where mX = E[X] and σX = E[(X − mX ) ] and mY and σY are defined similarly. From the data we can compute Cor(log St; ∆ log St), which turns out to be −0:023749. This correlation between log St and ∆ log St is quite small; therefore it is reasonable to presume it to be zero, which then implies by Lemma 5.1 that log St and ∆ log St are independent. The h-day volatility scaling and the independence expounded above indicate that it is \reasonable" to model St as a geometric Brownian motion. It should be noted that we can use ∆St=St and ∆ log St inter- changeably. The reason is as follows. First St+h ∆ log St = log St+h − log St = log ; St But S S − S log t+h = log 1 + t+h t : St St 5.2. BLACK-SCHOLES MARKET 162 By the Taylor expansion, log(1 + x) = x + O(x2). Therefore S S − S log t+h ≈ t+h t : St St In other words, it is reasonable to say that ∆ log St ≈ ∆St=St. A remark on the drift term is in order. The h-day drift terms are listed as Table 5.2. day(h) h-day mean 1 0.0007 2 0.0014 3 0.0021 5 0.0035 7 0.0046 Table 5.2: h-day mean (h-day drift terms). It confirms the expected pattern that h-day mean is supposed to be proportional to h. However, it is a happy accident of data. In many cases, h-day mean will change quite a bit so that they gener- ally do not exhibit such pattern. Estimating the drift term is not a statistically robust process. So any estimate of the drift is not a reliable data. However, as we shall see below, the drift term plays no role in the Black-Scholes model. So it really does not matter how one sets it. 5.2 Black-Scholes Market Black-Scholes model is the simplest yet most widely used continuous model in finance. Even with its many shortcomings, its importance can not be too emphasized. Anyone who is interested in gaining deeper understanding of mathematical finance must be thoroughly familiar with it. Let us now spell out the details of this model. • Time The interval is [0;T ], where t = 0 denotes the present. • Probability space The underlying probability space is denoted by (Ω; F;P ). 5.2. BLACK-SCHOLES MARKET 163 • Information Structure The information structure is given by a filtration (Ft)t≥0 of sub σ- fields of F such that (i) F0 is the trivial σ-field f;; Ωg; (ii) Ft1 ⊂ Ft2 if t1 ≤ t2. For any t 2 [0;T ], we define Ft− to be the σ-field generated by all Fs,(s < t) and Ft+ the σ-field that is the intersection of Fs for all s > t. We then assume that Ft− = Ft = Ft+: • Brownian motion The Brownian motion with respect to the measure P is denoted by Wt. It is assumed that Wt 2 Ft: • Bank account (riskless bond) process The bank account (riskless bond) process Bt is assumed to satisfy the following ordinary differential equation dBt = rBtdt B0 = 1 for a fixed positive constant r which is usually called the riskless (instantaneous) interest rate. It is trivial to see that rt Bt = e ; which is the result of the continuous compounding with interest rate r. • Time scale It is customary to use the time scale in such a way that one year is given duration (time span) 1. Since we only count the trading days, one year is considered to have roughly 250 trading days, which means that one trading day is given 1=250 (year). By the same token, the interest rate is always quoted as the annualize interest rate. However, in reality, banks do not compound continuously; and if the compounding is to be done daily, banks do compound even for holidays. So reconciling the differences between the theory (con- tinuous compounding) and the practice (daily, monthly or quarterly compounding) is a subtle matter. 5.2. BLACK-SCHOLES MARKET 164 • Stock price process We assume there is only one risky asset, which we call the stock. Its price process is denoted by St. Following the motivation given in Sec- tion 5.1, it is assumed to satisfy the following stochastic differential equation dSt = St(µdt + σdWt); where µ is a constant while we always assume σ is a positive con- stant. Thus St is the so-called geometric Brownian motion studied in Example 4.25 of Chapter 4. In particular, we know that 1 S = S exp µ − σ2 t + σW : t 0 2 t • Discounted stock price process ∗ Define the discounted stock price process St by ∗ St −rt St = = e St: Bt Then it is easily seen that ∗ ∗ dSt = St [(µ − r)dt + σdWt] : The standard procedure of the martingale approach in finance is to find a new measure Q, called the martingale measure or risk- ∗ neutral measure, with respect to which St becomes a martingale.
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