Black-Scholes Model

Chapter 5

Black-Scholes Model

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)√/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. √ Let us now see if the h-day volatility increases in proportion to h.√ The last column of Table 5.1 shows the h-day volatilities divided by h. There, we can see that they remain roughly constant.√ In Figure 5.9 are plotted h-day volatilities and a graph y = 0.016436 h that best ﬁts the data. In either case, we√ can see that the h-day volatilities scales roughly in proportion to h. √ 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

√ 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 Itˆoformula 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 {∅, Ω},

(ii) Ft1 ⊂ Ft2 if t1 ≤ t2.

For any t ∈ [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 ∈ 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 (continuous 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 constant. 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. Utilizing the standard procedure in Girsanov theorem, define the exponential martingale Mt by

dMt = −γMtdWt

M0 = 1, for some constant γ to be determined later. First, deﬁne the measure Qt on Ft by dQt = MtdP.

Then let Q = QT . We learned that

Qt = Q |Ft .

The Girsanov theorem says that the new stochastic process Wft de- ﬁned by dWft = dWt + γdt 5.3. RISK-NEUTRAL VALUATION PRINCIPLE IN THE BLACK-SCHOLES MODEL 165

is a Brownian motion with respect to Q(= QT ). Now

∗ ∗ dSt = St [(µ − r)dt + σdWt] ∗ h i = St (µ − r − σγ)dt + σdWft .

If we deﬁne γ by µ − r γ = , (5.3) σ ∗ St now satisﬁes

∗ ∗ dSt = σSt dWft. (5.4)

∗ Namely, St is a Q-martingale. The constant γ deﬁned in (5.3) is called the “market price of risk.” Although a misnomer, it nonetheless plays a very important role in ﬁnance. It is also useful to notice that (5.4) is equivalent to

dSt = St(rdt + σdWft). (5.5) In ﬁnance jargon, (5.5) is to be interpreted as saying that in the risk- neutral (martingale) world, the mean instantaneous return of any risky asset should be exactly the same as that of riskless asset.

5.3 Risk-Neutral Valuation Principle in the Black-Scholes Model

In Chapter 2, we have established the risk-neutral valuation principle for the discrete model. The same risk neutral valuation principle holds for continuous model. In this section, we establish it for the Black-Scholes model. However any alert reader will notice that this derivation works without any significant modification for any continuous market model with stochastic volatility. Definition 5.2. A European option or a European contingent claim with expiry (time) at t = T is an FT -random variable. Let us now suppose X is a European option with expiry t = T . The risk-neutral valuation principle we are trying to derive here has the same form as the one given for discrete case.

5.3.0.1 • Construction of Portfolio

Let Q be the martingale measure obtained in section 5.2. Deﬁne Vt by X Vt = BtEQ | Ft BT 5.3. RISK-NEUTRAL VALUATION PRINCIPLE IN THE BLACK-SCHOLES MODEL 166 and ∗ Vt Vt = . Bt ∗ Thus Vt is obviously a Q-martingale. Then by the Martingale Rep- 1 resentation theorem, there exists αt ∈ Ft such that

∗ dVt = αtdWft. Combining this with (5.4), and setting

αt ζt = ∗ , (5.6) σSt it is easy to see that ∗ ∗ dVt = ζtdSt . It should be noted that the denominator in (5.6) never vanishes as ∗ σ > 0 and St > 0. Deﬁne ξt by ∗ ∗ ξt = Vt − ζtSt . (5.7) Thus we have

Vt = ξtBt + ζtSt. (5.8)

Namely, if one construct a portfolio of ξt units of riskless bond and ζt shares of stock at each time t, then Vt must be the one that represents the value of this portfolio. We will show below that this portfolio is self-ﬁnancing.

• Continuous Trading and Self-Financing In the Black-Scholes model, we assume the portfolio is continuously changed, i.e. traded (rebalanced, to use ﬁnance jargon). This continuous trading assumption is not of course realistic but can be en- visioned as a “limit” of frequent trading.

Deﬁnition 5.3. A portfolio (ξt, ζt) that consists of ξt units of riskless bond and ζt shares of stocks is called self-ﬁnancing, if

dVt = ξtdBt + ζtdSt, where Vt is the values of the portfolio at time t given by

Vt = ξtBt + ζtSt. (5.9)

This deﬁnition comes as a “limit” of the discrete counterpart. (See especially Deﬁnition 2.39.)

1 In fact, αt ∈ Ft−. But since Ft− = Ft, αt ∈ Ft. 5.3. RISK-NEUTRAL VALUATION PRINCIPLE IN THE BLACK-SCHOLES MODEL 167

Theorem 5.4. The portfolio (ξt, ζt) is self-ﬁnancing if and only if

∗ ∗ dVt = ζtdSt .

Proof. First note that

∗ dVt = d(BtVt ) ∗ ∗ = Vt dBt + BtdVt , (5.10) by Itˆo’sformula. On the other hand, using (5.9), we have

∗ ∗ ξtdBt + ζtdSt = (Vt − ζtSt )dBt + ζtdSt ∗ −2 −1 = Vt dBt + ζtBt −Bt StdBt + Bt dSt ∗ −1 = Vt dBt + ζtBtd(Bt St) ∗ ∗ = Vt dBt + ζtBtdSt . (5.11)

Note that the self-ﬁnancing condition is equivalent to the LHS of (5.10) being equal to the LHS of (5.11), which in turn is equivalent ∗ ∗ to dVt = ζtdSt by looking at the right hand sides of (5.10) and (5.11)

Remark 5.5. Theorem 5.4 is the continuous counter-part of Propo- sition 2.40. It also proves that the portfolio constructed in “Con- structing Portfolio” subsection is self-ﬁnancing.

• Replicating Portfolio

Deﬁnition 5.6. A portfolio consisting of ξt units of riskless bond and ζt shares of stock is said to replicate a European option X ∈ Ft if ξT BT + ζT ST = X as random variables.

• Risk-Neutral Valuation Principle We are now ready to state and prove the risk-neutral valuation principle in the Black-Scholes model setting. Suppose (ξt, ζt) is a portfolio constructed above using the martingale representation theorem as in “Construction of Portfolio” subsection above. Then, in particular, ∗ the discounted portfolio value Vt satisﬁes

∗ ∗ dVt = ζtdSt , 5.4. THE BLACK-SCHOLES FORMULA 168 which then implies that this portfolio is self-ﬁnancing by Theorem 5.4. Now note that since X and BT ∈ FT , X VT = BT EQ | FT BT = X, which means that this portfolio replicates X. Therefore the price of X at time t must be X Vt = BtEQ | Ft . BT For, otherwise, there must exist an arbitrage at time t involving the portfolio (ξt, ζt) and the option itself X. We summarize our result as follows:

Theorem 5.7. (Risk-Neutral Valuation Principle) The price Vt at time t of the European option X ∈ FT is X Vt = BtEQ | Ft . BT

5.4 The Black-Scholes Formula

The above risk-neutral valuation principle gives a method of comput- ing the price of any European option. However, in practice, it is not easy to carry out actual closed-form computation. In this section, we derive the celebrated formula called the Black-Scholes formula for pricing the European call or put options. A European call option is a contract to pay at expiry, say at time T , the amount that is the diﬀerence between the preset value, say K, called the strike price, and the stock price ST as long as ST is greater than K; and zero, otherwise. (In this chapter all options are European.) Thus the call option can be succinctly expressed as

+ X = (ST − K) , where a+ means max(a, 0).

Let us now ﬁnd the formula for the price of X at time 0, i.e., we are seeking a closed-form formula for V0, −rT + V0 = e EQ (ST − K) −rT = e EQ (ST − K) · 1{ST >K} −rT −rT = e EQ ST · 1{ST >K} − Ke EQ 1{ST >K} = I1 − I2, 5.4. THE BLACK-SCHOLES FORMULA 169

where 1{ST >K} is the indicator (characteristic) function of the set {ST > K}.

Compute I2: First, note that 1 2 ST = S0exp r − σ T + σWfT > K 2 is equivalent to √ 1 2 L log(S0/K) + r − σ T > −σWfT = −σ T Wf1, 2 where the symbol L above the equal sign denotes the equality in law Ω, or equivalently in distribution. Then

1 2 ! log(S0/K) + (r − 2 σ )T P rob(ST > K) = P rob −Wf1 < √ . σ T

Thus, since the distribution of −Wf1 is the standard normal distribution N(0, 1) of mean 0 and variance 1,

1 2 ! log(S0/K) + (r − 2 σ )T EQ 1{S >K} = P rob Z < √ , T σ T where Z is a standard Gaussian random variable with mean 0 and variance 1. Therefore EQ 1{ST >K} = N(d2), where d 1 Z 2 N(d) = √ e−x /2dx 2π −∞ and 1 2 log(S0/K) + (r − 2 σ )T d2 = √ . σ T Thus −rT I2 = Ke N(d2). (5.12)

Compute I1: ST I1 = EQ · 1{ST >K} BT ∗ = EQ ST · 1{ST >K} Z ∗ = ST dQ. {ST >K} 5.4. THE BLACK-SCHOLES FORMULA 170

Recall ∗ ∗ dSt = σSt dWft. ∗ Thus S∗ is an exponential martingale, and so is St . In view of the t S0 ∗ ∗ facts that St is an exponential martingale and St = 1, define a S0 S0 t=0 new measure Pe by first defining Pe t by ∗ St dPe t = dQ S0 and then setting Pe = Pe T . Thus by the machinery of Girsanov theorem, there exists Wft, which is a Pe-Brownian motion such that

dWft = dWft − σdt.

Thus

Wft = Wft − σt. (5.13)

Now from the fact that

dSt = St(rdt + σdWft), we have 1 2 ST = S0exp r − σ T + σWft 2 1 2 = S0exp r + σ T + σWft 2 by (5.13). Hence ST > K if and only if

1 2 f log(S0/K) + r + 2 σ T WfT d1 = √ > − √ = Z, σ T T 5.4. THE BLACK-SCHOLES FORMULA 171 where Z is a standard Gaussian random variable of mean 0 and variance 1. Therefore ST EQ · 1{ST >K} BT ∗ = EQ ST · 1{ST >K} ∗ ST = S0EQ · 1{ST >K} S0 Z ∗ ST = S0 dQ {ST >K} S0 Z = S0 dPe {ST >K} Z = S0 dPe {Z

−rT V0 = S0N(d1) − Ke N(d2). In general, by shifting time and conditioning everything at time t, one can obtain the following:

Theorem 5.8. The price Vt at time t of the European call option + X = (ST − K) is given by

−r(T −t) Vt = StN(d1) − Ke N(d2), where d 1 Z 2 N(d) = √ e−x /2dx, 2π −∞ 1 2 log(St/K) + (r + 2 σ )(T − t) d1 = √ , σ T − t 1 2 log(St/K) + (r − 2 σ )(T − t) d2 = √ . σ T − t

Note also that Theorem 5.8 provides a valuable information on the replicating portfolio. Namely,

−rT ξt = −Ke N(d2),

ζt = N(d1).

The negative sign of ξt the short position of the riskless bond; the positive sign of ζt indicates the long position of the stock, etc. 5.5. THE BLACK-SCHOLES PARTIAL DIFFERENTIAL EQUATION 172

5.5 The Black-Scholes Partial Diﬀerential Equa- tion

The original derivation of the Black-Scholes formula Black and Sc- holes gave was via the so-called Black-Scholes partial diﬀerential equation. In this section, we present two derivations of the celebrated Black-Scholes PDE. The ﬁrst one is via the martingale framework and the second without it.

5.5.1 First Derivation of the Black-Scholes PDE We use the notations and facts from the previous sections. Recall that in the risk-neutral world, i.e., with respect to the martingale measure, the discounted stock price process satisﬁes

∗ ∗ dSt = σSt dWft, which is equivalent to

dSt = St(rdt + σdWft).

Let (ξt, ζt) be the self-ﬁnancing portfolio consisting of ξt units of riskless bond and ζt shares of stock that replicates the call option

+ X = (ST − K) with the expiry T and the strick price K. The existence of such portfolio was already veriﬁed in section 5.3, and the discounted value ∗ Vt of this portfolio is seen to satisfy

∗ ∗ dVt = ζtdSt ∗ = σζtSt dWft. (5.14)

Let us now assume that Vt can be expressed as

Vt = C(t, St) (5.15) for some smooth deterministic function C(t, S) of two variables t and S. At this juncture, we do not yet know such function exists. But we will later show that such function indeed exists. Deﬁne C∗ = ∗ −rt C (t, St) = C(t, St)/Bt = e C. Then,

dC∗ = d(e−rtC) = e−rtdC − re−rtCdt ∂C ∂C 1 ∂2C = e−rt dt + dS + (dS )2 − re−rtCdt. ∂t ∂S t 2 ∂S2 t 5.5. THE BLACK-SCHOLES PARTIAL DIFFERENTIAL EQUATION 173

The last equality is due to Itˆo’sformula. It is to be noted that ∂C ∂C = (t, S ) means that ﬁrst one takes the partial derivative ∂S ∂S t ∂C (t, S) of the deterministic function C(t, S) of two variables t and ∂S S, and then replace the variable S with St; the similar interpretation ∂C ∂2C applies to the and . Collecting terms and using (5.14), we ∂t ∂S2 have ∂C ∂C 1 ∂2C dC∗ = e−rt + rS + σ2S2 − rC dt ∂t t ∂S 2 t ∂S2

−rt ∂C +σe St dWft. ∂S

Thus utilizing (5.14) the necessary and suﬃcient condition for C(t, St) to be equal to Vt is that

∂C ∂C 1 ∂2C + rS + σ2S2 − rC = 0 (5.16) ∂t ∂S 2 ∂S2 t=t,S=St and ∂C σe−rtS = σζ S∗. (5.17) t ∂S t t Therefore by (5.17), we must have

∂C ∂C ζ = = (t, S ) (5.18) t ∂S ∂S t and (5.16) says that the deterministic function C(t, S) of two variables t and S must satisﬁes the following partial diﬀerential equation

∂C ∂C 1 ∂2C + rS + σ2S2 − rC = 0. (5.19) ∂t ∂S 2 ∂S2 The reason for the validity of this partial diﬀerential equation is that St can take any positive value therefore (5.19) must hold for any t ≥ 0 and S. (In fact it suﬃces to assume S > 0. But it is a moot point here.) Now it is easy to see that the replication condition is

C(T,S) = (S − K)+. (5.20)

(5.18) is also very useful in that it gives a method or formula to calculate the number of shares of stock the replication portfolio is to have. 5.5. THE BLACK-SCHOLES PARTIAL DIFFERENTIAL EQUATION 174

The promised proof of the existence of C(t, S) satisfying (5.15) can now be seen very easily. Namely, suppose C(t, S) is a deterministic function of two variables satisfying (5.19) and (5.20). Then by backtracking the arguments presented above, (5.15) can be easily seen to be satisﬁed by such C(t, S). Thus the proof boils down to the existence of the solution of (5.19) and (5.20), which will be shown in the subsequent sections.

5.5.2 Second Derivation of the Black-Scholes PDE The derivation given above is the most succinct and conceptually cleanest derivation of the Black-Scholes PDE. The reader is advised to get familiar with this type of argument. However, it presup- poses some knowledge of the risk-neutral valuation theory utilizing the martingale methodology. There is a way, however, to derive it directly without resorting to the martingale measure Q and Q- Brownian motion Wft. This also is the way Black and Scholes origi- nally derived it. Let us present it here.

As before, the stock price process is assumed to satisfy

dSt = St(µdt + σdWt), where Wt is a Brownian motion with respect to some Wiener measure P , usually called the underlying (physical) measure.

+ Let X = (ST − K) be a European call option, and let (ξt, ζt) be a self-ﬁnancing portfolio consisting of ξt units of riskless bond and ζt shares of stock that replicates X. Of course, it is far from trivial that such portfolio exists. The existence of such portfolio is part of the derivation process of the Black-Scholes PDE given in this subsection.

Let us for now assume such portfolio exists, and let

Vt = ξtBt + ζtSt (5.21) be the value at time t of such portfolio. Then the self-ﬁnancing condition implies that dVt = ξtdBt + ζtdSt. Therefore,

dVt = ξtdBt + ζtdSt rt = re ξtdt + ζtSt [µdt + σdWt] rt = re ξt + µζtSt dt + σζtStdWt. (5.22)

As before, let us assume that Vt can be expressed as

Vt = C(t, St) 5.5. THE BLACK-SCHOLES PARTIAL DIFFERENTIAL EQUATION 175 for some smooth function C(t, S) of two variables t and S. Then by Itˆo’sformula,

∂C ∂C 1 ∂2C dC = dt + dS + (dS )2 ∂t ∂S t 2 ∂S2 t ∂C ∂C 1 ∂2C = + µS + σ2S2 dt ∂t t ∂S 2 t ∂S2 ∂C +σS dW . (5.23) t ∂S t

Equating (5.22) and (5.23) and collecting the coeﬃcients of dWt, we have ∂C ζ = (t, S ) (5.24) t ∂S t and ∂C ∂C 1 ∂2C rertξ + µζ S = + µS + σ2S2 . (5.25) t t t ∂t t ∂S 2 t ∂S2 Since ∂C V = C,V = ertξ + ζ S and ζ = , t t t t t t ∂S plugging them (5.21), we have

∂C ξ = e−rt C(t, S ) − S (t, S ) . (5.26) t t t ∂S t

Plugging (5.24) and (5.26) into (5.25) and simplifying, we have

∂C ∂C 1 ∂2C + rS + σ2S2 − rC = 0, (5.27) ∂t ∂S 2 ∂S2 t=t,S=St which is exactly what we have derived before. Therefore the Black- Scholes PDE (5.19) must also hold. Note that in the Black-Scholes PDE, the drift coeﬃcient µ dis- appears and in its stead the riskless interest rate r appears. By the same token, the physical measure P plays no role but the martingale measure Q is what matters.

Let us now give an argument for the promised existence of the self-ﬁnancing portfolio. Let C be the solution of the Black-Scholes PDE (5.19) satisfying the condition

C(T,S) = (S − K)+. (5.28) 5.6. SOLUTION OF THE BLACK-SCHOLES PDE 176

As suggested by (5.24), deﬁne ζt by ∂C ζ = (t, S ), (5.29) t ∂S t and ξt, as in (5.26), by

∂C ξ = e−rt C(t, S ) − S (t, S ) . (5.30) t t t ∂S t

Construct a portfolio consisting of ξt units of riskless bond and ζt shares of stock. Then it is trivial to see that

rt Vt = e ξt + ζtSt = C

= C(t, St). (5.31)

Let us now prove the self-ﬁnancing property of this portfolio. Rewrit- ing the Black-Scholes PDE by

∂C ∂C 1 ∂2C ∂C + µS + σ2S2 = (µ − r)S + rC ∂t t ∂S 2 t ∂S2 t ∂S and plugging this into (5.23), we have

dVt = dC(t, St) ∂C ∂C = (µ − r)S + rC dt + σS dW . t ∂S t ∂S t

∂C Using ζ = and plugging (5.31) into the above equation, we then t ∂S have

dVt = dC rt = (µ − r)ζtSt + r(ξte + ζtSt) dt + σζtStdWt rt = rξte dt + ζtSt [µdt + σdWt]

= ξtdBt + ζtdSt, which is the self-ﬁnancing condition we were after. The replicating condition is trivially satisﬁed by (5.28).

5.6 Solution of the Black-Scholes PDE

Now solve the Black-Scholes PDE (5.19) subject to the condition (5.20). Let S0 be the constant that represents the known stock price 5.6. SOLUTION OF THE BLACK-SCHOLES PDE 177

at time t = 0. Deﬁne z = log (S/S0) and Ce(t, z) = C(t, S). Then

∂Ce ∂C = S , ∂z ∂S ∂2Ce ∂2C ∂C = S2 + S . ∂z2 ∂S2 ∂S Therefore, Black-Scholes equation becomes

∂Ce 1 ∂Ce 1 ∂2Ce + r − σ2 + σ2 − rCe = 0. (5.32) ∂t 2 ∂z 2 ∂z2

Now let x = Az +Bt, where A and B are constant to be determined. Let

v(t, x) = Ce(t, z) = C(t, S).

Then

∂Ce ∂v ∂v ∂x ∂v ∂v = + = + B , ∂t ∂t ∂x ∂t ∂t ∂x ∂Ce ∂v ∂t ∂v ∂x ∂v = + = A , ∂z ∂t ∂z ∂x ∂z ∂x ∂2Ce ∂2v = A2 . ∂z2 ∂x2 Therefore (5.32) becomes

∂v 1 ∂v 1 ∂2v + B + r − σ2 A + σ2A2 − rv = 0. (5.33) ∂t 2 ∂x 2 ∂x2

Let us choose A and B so that 1 B + r − σ2 A = 0 and σA = 1, 2 i.e.,

1 1 1 A = and B = − r − σ2 . (5.34) σ σ 2

Then (5.33) becomes

∂v 1 ∂2v + − rv = 0. (5.35) ∂t 2 ∂x2 Deﬁne

u(t, x) = e−rtv(t, x), 5.6. SOLUTION OF THE BLACK-SCHOLES PDE 178 then (5.35) becomes

∂u 1 ∂2u + = 0. (5.36) ∂t 2 ∂x2 If we replace of A and B in x by (5.34), then 1 1 1 x = z − r − σ2 t, σ σ 2 and so 1 z = r − σ2 t + σx. 2 Back to the ﬁrst change of variable, we have

z S = S0e 1 = S exp r − σ2 t + σx . (5.37) 0 2

In conclusion, if u is deﬁned by

u(t, x) = e−rtv(t, x) = e−rtC(t, S), u satisﬁes the following heat equation initial boundary problem

 2  ∂u 1 ∂ u  + 2 = 0 ∂t 2 ∂x (5.38)   u(T, x) = e−rT C(T,S), where x and S are related by 1 S = S exp r − σ2 t + σx . 0 2 The equation in (5.38) is the so-called heat equation, but with the time seemingly ﬂowing backward. However, if we make one more substitution τ = T − t for time, (5.38) becomes an equation for u(τ, x) satisfying

 2  ∂u 1 ∂ u  = 2 ∂τ 2 ∂x (5.39)   u(0, x) = e−rT C(T,S).

This is the usual form of the initial value problem of the heat equation. In other words, (5.38) is the correct, well-posed heat equation 5.6. SOLUTION OF THE BLACK-SCHOLES PDE 179 because the initial condition u(T, x) = e−rT C(T,S) is posed at a future time while we are interested in the solution at times before that ﬁxed future time. Incidentally, in ﬁnance literature, (5.38) is called the boundary value problem. It is a well known fact from the theory of parabolic (heat) equa- tions the solution u(t, x) of (5.38) can be written as

Z ∞ (x−y)2 1 − u(t, x) = p e 2(T −t) u(T, y)dy. (5.40) −∞ 2π(T − t) Here the function 2 1 − (x−y) e 2(T −t) p2π(T − t) is called the heat kernel (for (5.38)) and its probabilistic meaning is that it represents the probability density function for a Brownian particle (motion) to move from y to x in time T − t. Let us now derive the value of the European call option. Since the stock price at time t = 0 is S0, it must be C(0,S0). It is easily seen from (5.37) that this case correspond to the case in which t = 0 and x = 0. Namely, by (5.40) the options value at time t = 0 is nothing but

Z ∞ 2 1 − x u(0, 0) = √ e 2T u(T, x)dx. (5.41) −∞ 2πT We now derive the Black-Scholes formula for the call option. Note that −rT + u(T, x) = e (ST − K) . By (5.41),

Z ∞ 2 1 − x −rT + u(0, 0) = √ e 2T e (ST − K) dx, −∞ 2πT where at T , ST and x are related by 1 S = S exp r − σ2 T + σx . T 0 2 One can compute u(0, 0) as follows:

Z 2 −rT 1 − x 1 2 u(0, 0) = e √ e 2T S0 exp r − σ T + σx − K dx D 2πT 2 Z 2 −rT 1 − x 1 2 = e √ e 2T S0 exp r − σ T + σx dx D 2πT 2 Z 2 −rT 1 − x −Ke √ e 2T dx D 2πT −rT = I1 − Ke I2, 5.6. SOLUTION OF THE BLACK-SCHOLES PDE 180 where n 1 o D = x : S exp r − σ2 T + σx > K , 0 2 Z 2 −rT 1 − x 1 2 I1 = e √ e 2T S0 exp r − σ T + σx dx, D 2πT 2 Z 2 1 − x I2 = √ e 2T dx. D 2πT

Compute I2: x Put y = √ , then T n 1 o D = x : S exp r − σ2 T + σx > K 0 2 1 2 n log(S0/K) + (r − σ )T o = y : y > − √ 2 , σ T and

Z 2 1 − y I2 = √ e 2 dy D 2π Z ∞ 2 1 − y = √ e 2 dy log(S /K)+(r− 1 σ2)T − 0 √ 2 2π σ T 1 2 log(S0/K)+(r− 2 σ )T Z √ 2 σ T 1 − y = √ e 2 dy −∞ 2π 1 2 log(S0/K) + (r − σ )T = N √ 2 . σ T

Compute I1: x − σT Put y = √ , then T n 1 o D = x : S exp r − σ2 T + σx > K 0 2 1 2 n log(S0/K) + (r + σ )T o = y : y > − √ 2 , σ T 5.6. SOLUTION OF THE BLACK-SCHOLES PDE 181 and Z 1 1x − σT 2 I1 = S0 √ exp − √ dx D 2πT 2 T Z ∞ 2 1 − y = S √ e 2 dy 0 log(S /K)+(r+ 1 σ2)T − 0 √ 2 2π σ T 1 2 log(S0/K)+(r+ 2 σ )T Z √ 2 σ T 1 − y = S0 √ e 2 dy −∞ 2π 1 2 ! log(S0/K) + (r + 2 σ )T = S0N √ . σ T

Therefore, we can obtain the following:

−rT C(0,S0) = S0N(d1) − e KN(d2), (5.42) where

1 2 log(S0/K) + (r + 2 σ )T d1 = √ , σ T 1 2 log(S0/K) + (r − 2 σ )T d2 = √ . σ T

In general, if we know the value St at time t, then we can apply the same formula to get

−r(T −t) C(t, St) = StN(d1) − e KN(d2), (5.43) where T − t is the time to expiry from time t and

1 2 log(St/K) + (r + 2 σ )(T − t) d1 = √ , σ T − t 1 2 log(St/K) + (r − 2 σ )(T − t) d2 = √ . σ T − t

Note also that (5.43) provides a valuable information on the replicating portfolio. Namely,

 −rT  ξt = −Ke N(d2), (5.44)  ζt = N(d1).

The negative sign of ξt the short position of the riskless bond; the positive sign of ζt indicates the long position of the stock. 5.7. PUT-CALL PARITY 182

5.7 Put-Call Parity

Let X be the European call option with the expiry T and the strike price K, i.e. + X = (ST − K) as FT measurable random variables. Let Ct be its value at time t. Then the risk-neutral valuation principle, Theorem 5.7, says that X Ct = BtEQ | Ft , BT where Q is the martingale measure. Let Y be the European put option with the same expiry T and the same strike price K. Namely, + Y = (K − ST ) . Then these call and put options are intricately linked via a principle called the “put-call parity.” Let Pt be the value of Y at time t, then Theorem 5.7 says that Y Pt = BtEQ | Ft . BT Note now that CT − PT = X − Y = ST − K. Therefore, upon applying the risk-neutral valuation principle, The-

K ST

Figure 5.10: Payoﬀ of CT . orem 5.7, to the above, we have X − Y Ct − Pt = BtEQ | Ft BT ST K = BtEQ | Ft − BtEQ | Ft BT BT −r(T −t) = St − Ke . (5.45) 5.7. PUT-CALL PARITY 183

K ST

Figure 5.11: Payoﬀ of PT .

K ST

−K

Figure 5.12: Payoﬀ of CT − PT .

∗ The last equality is the consequence of the fact that St = St/Bt is rt a Q-martingale and that Bt = e . (5.45) combined with the Black- Scholes formula (5.43) gives the Black-Scholes formula for put option. From (5.45), we have

−r(T −t) Pt = Ct − St + Ke −r(T −t) = −St (1 − N(d1)) + Ke (1 − N(d2)) −r(T −t) = −StN(−d1) + Ke N(−d2) where 1 2 log(St/K) + (r + 2 σ )(T − t) d1 = √ , σ T − t 1 2 log(St/K) + (r − 2 σ )(T − t) d2 = √ . σ T − t 5.7. PUT-CALL PARITY 184

Here we used the fact that

1 − N(d) = N(−d) for the cumulative normal distribution N(d). This Black-Scholes formula also gives the replicating portfolio (ξt, ζt) for the put option that consists of ξt units of riskless bond and ζt shares of stock by

 −rT  ξt = Ke N(−d2), (5.46)  ζt = −N(−d1). EXERCISES 185

Exercises

5.1. Suppose u(x, t) is a C2 function satisfying

1 ∂2u ∂u (x, t) + (x, t) = 0, 2 ∂x2 ∂t for all x ∈ R and t ≥ 0. Let Wt be a Brownian motion. Show that Zt = u(Wt, t) is a martingale.

2 5.2. Let u(t, x) be a deterministic C function. Let Xt be a stochastic process satisfying dXt = adt + bdWt for some constants a and b. Find the partial diﬀerential equation u must satisfy in order for u(t, Xt) to become a martingale. 5.3. Let r be a positive constant representing the instantaneous riskless interest rate. Let St be the price process of a stock satisfying the following stochastic diﬀerential equation

dSt = St(µ1dt + σ1dWt), where Wt is a P -Brownian motion while µ1 and σ1 > 0 are constants. The market price of risk of St is a constant γ such that the discounted ∗ −rt stock price process St = e St become a martingale with respect to a new measure Q which is given by dQ = MT dP , where Mt is the exponential martingale satisfying dMt = −γMtdWt.

(a) Find γ in terms of r, µ1 and σ1.

(b) Let Ut be another stochastic process given by

dUt = Ut(µ2dt + σ2dWt)

where µ2 and σ2 > 0 are constants and Wt is the same Brown- ian motion that drives St. It is well known that the arbitrage- free condition for both securities must imply that there is a ∗ −rt ∗ −rt martingale measure Q that makes St = e St and Ut = e Ut simultaneously martingales. (Such Q can be found by the use of Girsanov theorem as described (a) above.) Assuming this fact, show that the market prices of risk for both securities are the same.