Lecture notes, lectures 1-10 PDF

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The Black Scholes model

1

The model

The Black-Scholes model for a stock price process is defined as 1

S0 e(µ− 2 σ

=

St

2

)t+σW t

.

(1)

where Wt is a standard Brownian motion. From Ito’s lemma, we have seen that St satisfies the SDE =

dSt

St [µdt + σdWt ]

which is known as Geometric Brownian motion. µ is the drift of the process which describes the overall upward/downward trend, and σ is the volatility, which describes its variability.

2

The stock price distribution under the Black-Scholes model • Re-arranging (1) we see that log St − log S0 Thus we can calculate the distribution of log log

= St S0

log

St S0

=

1 (µ − σ 2 )t + σWt . 2

as

1 ∼ N (µ − σ 2 )t, σ 2 t) 2

St S0

or equivalently log St ∼ N (log S0 + (µ − 21 σ 2 )t, σ 2 t). • We compute P(ST > K) as follows: ST K > log ) S0 S0 log SST − (µ − 21 σ 2 )T log SK0 − (µ − 21 σ 2 )T 0 P( √ > √ ) σ T σ T log SK0 − (µ − 21 σ 2 )T P(Z > √ ) σ T c Φ (z )

P(ST > K) = P(log = = = log

K

2 −(µ− 12 σ )T

S0 √ where z = ∫ ∞ 1 −z2 /2 σ T √ e dz. x 2π

, because Z =

log

ST S0

2 −(µ− 12 σ )T √ σ T

is a standard N (0, 1) random variable, and Φc (x) =

• Example: set S = 1, K = 1.1, σ = .1, τ = .25, µ = .05. Then we have z = 1.681203596 and P(ST > K ) = Φc (z) = 0.046361687633 (CHECK!). Use Normsdist(.) to calculate Φ and Φc in Excel.

2

Figure 1: Here we have plotted the stock price density fSt (S) for σ = .1 and t = 1.

• Note that =

P(St ≤ S)

P(log St ≤ log S)

=

F (log S )

where F is the distribution function of log S. Differentiating both sides with respect to S and using the chain rule, we see that the density fSt (S) of St is given by fSt (S)

=

d P(St ≤ S) = dS

1 ′ F (log S) S

=

1 fX (x) S t

where x = log S and fXt (x) is the density of Xt which is given by 1 2 )t)2

fXt (x)

(x−x0 −(µ− σ 2 1 2σ2 t √ e− 2 2πσ t

=

(2)

so fSt (S) =

1σ [log S−log S0 −(µ− 2 1 2σ2 t √ e− S 2πσ 2 t

2 )t]2

St has what is known as a lognormal distribution. Note the presence of the defined for S > 0 because the stock price cannot go negative. • fSt (S) is a probability density function and thus must integrate to 1, i.e. ∫ ∞ fSt (S)dS = 1 . 0

. 1 S

pre-factor, and this pdf is only

3

Figure 2: Here we have plotted the “hockey stick” payoff max(S − K, 0).

3

The Black-Scholes PDE

we obtain the celebrated Black-Scholes partial differential equation 1 ft + rSfS + σ 2 fSS S 2 2

=

rf .

Note (as for the binomial model) that µ has disappeared, because we are computing f using a hedging argument with S .

4

The Black-Scholes formula • To solve the Black-Scholes PDE ft + rSfS +

1 2 2 σ S fSS 2

=

rf ,

we have to specify a boundary condition for the PDE at the time of maturity t = T .

(3)

4

• For a European call option, the boundary condition at t = T is f (S, T ) = max(S − K, 0), because the call option is worth max(S − K, 0) at expiry. • It can be shown that there is a unique solution to the Black-Scholes PDE with this boundary condition, given by the (Nobel prize winning) Black-Scholes formula C(S, τ ) =

SΦ(d 1 ) − K e−rτ Φ(d 2 ) ,

where τ = T − t and

where Φ(x) =

∫x

−∞

−z 2 /2

e√ 2π

d1

=

d2

=

S K

+ (r + 12 σ 2 )τ √ , σ τ √ d1 − σ τ log

is the standard cumulative Normal distribution function.

• If the Stock also pays a continuous dividend q, then the BS PDE becomes ft + (r − q)SfS +

1 2 2 σ S fSS 2

=

rf ,

(4)

and the formula has to be adjusted as follows C(S, t) = where d 1 =

4.1

S +(r−q+ 1 2 log K σ )τ 2 √ σ τ

SΦ(d 1 ) − Ke−rτ Φ(d 2 ) ,

√ , d 2 = d 1 − σ τ . We will not consider dividends on this course.

Numerical example

Assume current stock price is 1, volatility is .10 and interest rate is .05. Price a call option with strike 1.1 with maturity 1: take S = 1, K = 1.1, σ = .1, τ = 1, r = .05. Plugging these numbers into the BS formula we obtain d1

=

d2

=

−0.403101798043249 ,

−0.503101798043249

and the call price C

=

0.021739503382137.

(the Excel sheet “BlackScholesModel.xls” on the course website implements this formula in Visual Basic.)

5

The Greeks

We can compute partial derivatives of the BS formula with respect to each of the parameters ∆ = Γ

=

Λ =

∂C = Φ(d 1 ) > 0, ∂S 2 ∂ C n(d 1 ) ∂∆ √ > 0, = = ∂S ∂S 2 Sσ τ √ ∂C = Sn(d 1 ) τ > 0 ∂σ

2

(5)

where n(z) = √12π e−z /2 is the standard Normal density. ∆, Γ and Λ are known as the Delta, Gamma and Vega respectively of the option. The proof of these expressions for the Greeks are very tedious and not examinable. • Delta measures the responsiveness of the call option price to small changes in the underlying stock price. • Vega measures the responsiveness of the call option price to small changes in the volatility. • Gamma measures the responsiveness of the Delta to small changes in the underlying stock price.

5

5.1

Numerical example

For the example above, we obtain ∆ = 0.343435379743, Γ = 3.677756432729 and Λ = 0.367811626137.

5.2

Calculating Greeks using finite differences

• We can also calculate Greeks numerically (rather than using the exact formulae in (5)). e.g. choose ∆S > 0 small. Then we can approximate ∆ as 1 ∆ ≈ (C (S + ∆S, K, σ, T , r) − C (S, K, σ, T, r)). ∆S • We get a better approximation if we use the following two-sided formula:

1 (C(S + ∆S, K, σ, T , r) − C(S − ∆S, K, σ, T, r)). 2∆S This is just the usual way in which we use finite differences to approximate a true derivative, by making a small change to one variable. ∆

≈

• Similarly, we can approximate Λ as

1 (C(S, K, σ + ∆σ, T , r) − C(S, K, σ − ∆σ, T, r)) 2∆σ and for Γ (which is a second derivative), we use Λ

Γ

≈

≈

1 (C(S + ∆S, K, σ, T , r) − 2C (S, K, σ, T, r) + C (S − ∆S, K, σ, T , r)). (∆S)2

• These finite difference approximations converge to the true answers in (5) as ∆S or ∆σ tend to zero. • Recall Taylor’s theorem for a differentiable function of one variable: 1 f (x + ∆x) ≈ f (x) + f ′ (x)∆x + f ′′(x)(∆x)2 . 2 We can do the same thing for the Black-Scholes formula: ∂C 1 ∂2C (∆S)2 ∆S + 2 ∂S 2 ∂S 1 = C(S) + ∆ · ∆S + Γ(∆S)2 . 2 This is the practical financial use of Delta and Gamma - we can approximate the new value of the Black Scholes call price if the stock price moves from S to S + ∆S, if ∆S is small. We obtain a cruder approximation if we ignore the Gamma term Γ. C(S + ∆S, K, σ, τ, r) ≈

6

C(S, K, σ, τ, r) +

Obtaining the Black-Scholes formula via risk neutral valuation • It turns out that C(S, τ ) (the price of a call option under the Black-Scholes model) also has a probabilistic representation C(S, τ) =

e−rτ E(max(ST − K, 0) ,

where S follows the process dSt = St [rdt + σdWt ]

(6)

(this will be proved using the Feynman-Kac formula in the next set of notes). In words: the call price is the discounted expected value of max(ST − K, 0) in the risk-neutral world where the drift is r − q. We refer to this world as the risk neutral measure. • Note that (6) this is the same as the Black-Scholes SDE dSt = St [µdt + σdW ], but the real-world µ has been replaced by r − q .

6

6.1

Put-Call parity

By considering both cases ST > K and ST < K we see that max(ST − K, 0) + K

= max(K − ST , 0) + ST .

(7)

A portfolio of 1 call option and K e−rT dollars will be equal in value to the left hand side at T . Similarly, a portfolio of 1 put option and S0 shares will be equal in value to the right hand side at T . Thus we obtain the put-call parity: C + Ke−rT

6.2

=

P + S0 .

Implied volatility

√ ∂C The Vega ∂σ = Sn(d 1 ) τ of a call option under Black-Scholes is positive, so C is montonically increasing as a function of σ. Thus, given an observed call price C obs in the market, we can extract the unique σ value consistent with this price by solving C(S, K, σˆ , τ, r) = C obs . This σ is known as the implied volatility of the option, and is a very important concept in practice.

(8)

Brownian motion and Ito’s lemma

1

Brownian motion

A stochastic process (Wt )t≥0 is said to be a standard one-dimensional Brownian motion if it satisfies the following four properties: • W0 = 0 . • W has independent increments, i.e. Wt2 − Wt1 , ... , Wtn − Wtn−1

(1)

are independent for all 0 ≤ t1 < t2 < ... < tn . • The increments are Normally distributed: Wt − Ws ∼ N (0, t − s) for all 0 ≤ s ≤ t. • Wt is continuous as a function of t almost surely (i.e. with probability one). Remark 1.1 For this course, the third property is the most important to remember. Setting s = 0, we see that Bt ∼ N (0, t) and thus (from lecture 1) we know that P(Bt > x) = where

∫∞ z

1 2 √1 e− 2 z dz. 2π

x Φc ( √ ) t

(2)

We will use (2) many times.

It is not immediately obvious that we can construct a process with these properties. The following theorem answers this question. Theorem 1.1 (Wiener 1923) Brownian motion exists. Remark 1.2 We can prove that Wt is not differentiable with respect to t at all times almost surely, i.e. with probability 1.

1.1

Simulating Brownian motion

• We can simulate Brownian motion as follows: fix a small step size ∆ > 0. Then √ W(n+1)∆t = Wn∆t + ∆t Zn where Zn ∼ N (0, 1) is a sequence of i.i.d. (independent and identically distributed) standard N (0, 1) random variables. • This process starts at zero, has independent increments and we see that W(n+1)∆t − Wn∆t ∼ N (0, ∆t), which is consistent with the property that Wt − Ws ∼ N (0, t) (recall that for any random variable X, we have that Var(aX) = a2 Var(X)). • The simulation gives a piecewise linear approximation to the true Brownian motion at the times ∆t, 2∆t, ... (see Excel sheet on the course webpage).

2

2

Ito’s lemma •

•

.

The next theorem (Ito’s lemma) is hugely important and fundamental to the whole course. Theorem 2.1 (Ito’s lemma). Let f (x, t) be twice differentiable in x and once differentiable in t. Then ∫ t ∫ t 1 fx (Ws , s)dWs + f (Wt , t) = f (0, 0) + [ft (Ws , s) + fxx (Ws , s)]ds . 2 0 0 Remark 2.2 It is customary to write (3) in differential form as df which is really just a lazy shorthand for (3).

=

1 ft dt + fx dWt + fxx dt . 2

(3)

3

3

The stochastic integral

.

4

Examples from past exam questions

1. Let f (x, t) = x2 − t so f (Wt , t) = Wt2 − t. Then fx = 2x, fxx = 2 and ft = −1. Thus from Ito’s lemma we have 1 df (Wt , t) = ft dt + fx dWt + fxx dt 2 1 = 2Wt dWt + · 2dt − dt 2 = 2Wt dWt . Writing this in integrated form we have Wt2 − t 1

2

1

2. Let f (x, t) = S0 e(µ− 2 σ )t+σx , so f (Wt , t) = S0 e(µ− 2 σ ft = (µ − 12 σ 2 )f . Thus from Ito’s lemma we have dSt

∫

= 2

t

2Ws dWs . 0

)t+σWt

and let St = f (Wt , t). Then fx = σf , fxx = σ 2 f and

1 ft dt + fx dWt + fxx dt 2 1 2 1 = σSt dWt + σ 2 St dt + (µ − σ )St dt 2 2 = µSt dt + σSt dWt .

=

This is the famous Black-Scholes model for a stock price process. µ describes the overall trend of the process, i.e. its 1 2 tendency to go up or down in the long run (it can be shown that E(St ) = S0 e(µ+ 2 σ )t , and σ is the volatility which control the variability of the stock price. Note that St is always positive. 3. Let T > 0 be fixed, and f (x, t) = (1 − Tt )x. Then fx = (1 − Tt ), fxx = 0 and ft = − T1 x. Thus from Ito’s lemma we have 1 t t d[(1 − )Wt ] = (1 − )dWt − Wt dt . T T T Integrating from t = 0 to t = T we find that (1 −

t )Wt |t=T t=0 T

= 0−0

=

∫

0

T

(1 −

This identity will be very useful later on for pricing Asian options.

1 t )dWt − T T

∫

T

Wt dt . 0

Continuous time martingales

1 A continuous time martingale (Xt )t≥0 is a stochastic process which satisfies (i) E(Xt | Fs ) =

Xs

(1)

where Fs is (informally) the information available at time s. For our purposes on this course, Fs will just mean all the information contained in the historical sample path of the process from time 0 to time s (this concept is made rigorous using the concept of filtrations in stochastic analysis, which is beyond the scope of this course). (ii) E(|Xt |) < ∞ for all t.

1.1

Example: Brownian motion

• For standard Brownian motion, we know that Wt − Ws ∼ N (0, t − s). Thus E(Wt | Fs ) = E(Wt − Ws + Ws | Fs ) = =

Ws + E(Wt − Ws | Fs ) Ws .

• We now need to prove that E|Wt | < ∞. We know that ∫ ∞ 1 −x2 /2t e dx |x| √ E(|Wt |) = 2πt −∞ But whether this is infinite or not depends only on how the integrand behaves near x = ∞, so it suffices to consider ∫ ∞ 2 1 e−x /2t dx |x| √ 2πt c for some c > 1, because the integrand is also an even function. Then ∫ ∞ ∫ ∞ ∫ ∞ 2 x x2 −x2 /2t x2 −x2 /2t √ e−x /2t dx ≤ √ e dx ≤ √ e dx = E(W t2 ) = t < ∞ . 2πt 2πt 2πt c c −∞

2

1.2

Example 2: Wt2 − t

• Let Mt = Wt2 − t. Then E(Mt − Ms | Fs ) = = = = = = = = =

E(Wt2 − t − (Ws2 − s) | Fs ) E(W t2 − Ws2 | Fs ) − (t − s) E((Ws + Wt − Ws )2 − W 2s | Fs ) − (t − s) E(W s2 + 2Ws (Wt − Ws ) + (Wt − Ws )2 − W 2s | Fs ) − (t − s) E(2Ws (Wt − Ws ) + (Wt − Ws )2 | Fs ) − (t − s)

2Ws E(Wt − Ws ) | Fs ) + E((Wt − Ws )2 | Fs ) − (t − s)

2Ws E(Wt − Ws )) + E((Wt − Ws )2 ) − (t − s) 0 + t − s − (t − s) 0

(-12)

from the standard properties of Brownian motion. • We can also verify that E(|Mt |) < ∞, so Mt = Wt2 − t is a martingale.

1.3

Example 3: the Black-Scholes model with µ = 0 1

2

• Let St = S0 eσWt − 2 σ t . Then 1

E(St − Ss | Fs ) = E(Ss eσ(Wt −Ws )− 2 σ

2

(t−s)

− Ss | Fs ) .

1 2

where we have used that Ss = S0 eσWs −2 σ s . • But for any normal random variable X ∼ N (µ, ν 2 ), the moment generating function of X is given by E(epX ) = 1 2 2 eµp+ 2 ν p . • In our case, X = σ(Wt − Ws ) − 21 σ 2 (t − s) so µ = − 12 σ 2 (t − s) and ν 2 = σ 2 (t − s) and p = 1, so we see that 1 2

E(Ss eσ(Wt −Ws )−2 σ

(t−s)

1

| Fs ) = Ss eµp+ 2 ν =

Ss e

=

Ss .

2 2

p

− 21 σ 2 (t−s)+ 12 σ 2 (t−s)

• Moreover, setting s = 0 we see that E(|St |) = E(St ) = S0 < ∞, so St is a martingale.

The Feynman-Kac formula and the forward/backward Kolmogorov equations

The Feynman-Kac formula Consider a solution to the PDE 1 ft + µ(x)fx + σ(x)2 fxx 2

=

V (x)f

subject to the terminal condition f (x, T ) = ϕ(x), with µ, σ, V bounded, and assume |fx | is also bounded. Theorem 0.1 The Feynman-Kac formula states that f (x, t) has the stochastic representation f (x, t) = E(e−

∫T t

V (X s )ds

ϕ(XT ) | Xt = x)

where Xt is a stochastic process which satisfies the SDE dXt with Xt = x.

=

µ(Xt )dt + σ(Xt )dWt

2

Remark 0.1 Applying the Feynman-Kac formula to the Black-Scholes PDE 1 ft + rSfS + S 2 σ 2 fSS 2

=

rf ,

subject to f (S, T ) = max(S − K, 0) (note that V is just r here), we recover the probabilistic representation for the price of a call option under the Black-Scholes model f (S, t) =

e−r(T −t) EQ (max(ST − K, 0) | St = S)

where St satisfies the SDE dSt = St [rdt + σdW ] under the probability measure Q. Example. Price a digital call option under the Black-Scholes model which pays 1 if ST > K and zero otherwise. Solution: By the same hedging argument as for European options, the price f (S, t) of the digital call satisfies also satisfies the Black-Scholes PDE in (5), but with boundary condition f (S, T ) = 1S>K . Then from the Feynman-Kac formula, f (S, t) has the probabilistic representation f (S, t) = =

e−r(T −t) EQ (1ST >K | St = S)

e−r(T −t) Q(ST > K | St = S)

where (again) St satisfies the SDE dSt = St [rdt + σdW ] under the probability measure Q, and Q(A) denotes the probability of an event A under the probability measure Q, and we have used that for any random variable X , P(X > K) = E(1X >K ). If we now let t = 0, then we can compute Q(ST > K ) similar to before as: K ST > log ) S0 S0 log SK0 − (r − 12 σ 2 )T log SS0T − (r − 12 σ 2 )T √ > √ Q( ) σ T σ T log SK0 − (r − 12 σ 2 )T Q(Z > √ ) σ T c Φ (z )

Q(ST > K) = Q(log = = = where z =

log

K S0

−(r−21σ 2 )T √ σ T

(note that we have now just replaced µ with r).

The forward and backward Kolmogorov equations • Consider again the stochastic differential equation dXt

=

µ(Xt )dt + σ(Xt )dWt .

Then if µ, σ are sufficiently well behaved, then Xt has a unique strong solution and a probability density which we denote by p(t, x; T , y) (this is the density at time T in the future as a function of y, when the process X starts at time t at the level x). • Then p = p(t, x; T , y) satisfies the backward Kolmogorov equation in t and x: ∂2p ∂p ∂p 1 + µ(x) + σ(x)2 2 2 ∂t ∂x ∂x

=

0

(5)

with boundary condition p(T , x; T , y) = δ(x − y), where δ(x) denotes the Dirac delta function. This is a partial differential equation. We refer to t and x as the backward variables. ∫∞ −x2 • δ(x) is +∞ if x = 0 and zero otherwise, but −∞ δ(x)dx = 1. We can approximate δ(x) with δϵ (x) = √ 1 e 2ϵ , 2πϵ i.e. a Normal density with mean 0 and variance ϵ small. Informally, we can view δ(x) is the derivative of the Heaviside step function H (x) which is 1 for x > 0 and zero otherwise. H (x) is the distribution function of a random variable which is just constant and equal to zero.

3

• If t = T , then XT is known to be x, so there is no randomness, and the “density” of XT is given by δ(y − x). • Similarly, p = p(t, x; T , y) satisfies the forward Kolmogorov equation in T and y: ∂p ∂T

=

−

∂2 1 ∂ (µ(y)p) + ( σ(y)2 p) ∂y ∂y2 2

(6)

with the same bound...