Lecture 8: Finance: GBM model for share prices, futures and options, hedging, and the Black-Scholes...

Lecture 8: Finance: GBM model for share prices, futures and options, hedging, and

the Black-Scholes equation

Outline:• GBM as a model of share prices• futures and options, hedging• derivation of the Black-Scholes equation

Geometric Brownian motion as a model of share prices

Empirical facts:


Empirical facts:

On average, stocks appreciate in value at an approximately constant rate:


Empirical facts:


€

dSdt

= r S


Empirical facts:


Stock return (δs/s) fluctuations have a very short correlation time

€

dSdt

= r S


Empirical facts:



€

dSdt

= r S

dSdt

= rS + σSξ (t); ξ (t)ξ ( ′ t ) = δ(t − ′ t )


Empirical facts:



This is GBM (that we already studied).

€

dSdt

= r S

dSdt

= rS + σSξ (t); ξ (t)ξ ( ′ t ) = δ(t − ′ t )


Empirical facts:



This is GBM (that we already studied). σ is called the volatility.

€

dSdt

= r S

dSdt

= rS + σSξ (t); ξ (t)ξ ( ′ t ) = δ(t − ′ t )

derivative securitiesFuture: A contract to buy a fixed number of shares at a particular price X at a particular time T in the future.


Option: A contract giving the buyer the option of buying (“call option”) or selling (“put option”) a fixed number of shares ata particular price X at a particular time T in the future (“European” option)


Option: A contract giving the buyer the option of buying (“call option”) or selling (“put option”) a fixed number of shares ata particular price X at a particular time T in the future (“European” option) or any time up to that particular time (“American” option).


Option: A contract giving the buyer the option of buying (“call option”) or selling (“put option”) a fixed number of shares ata particular price X at a particular time T in the future (“European” option) or any time up to that particular time (“American” option). (The buyer need not exercise the option.)


Option: A contract giving the buyer the option of buying (“call option”) or selling (“put option”) a fixed number of shares ata particular price X at a particular time T in the future (“European” option) or any time up to that particular time (“American” option). (The buyer need not exercise the option.) Options are insurance.



A derivative has a price. Our aim here is to determine the fair price.



A derivative has a price. Our aim here is to determine the fair price.

Since a derivative security has a price, one can also buy or sell a future or option on it (derivative of a derivative).

Fair price of a future(?)

What is the fair price of a future (per share of a stock S)?



A possible answer: S will fluctuate according to GBM. Wehave already calculated the distribution of prices at T:




€

P(S,t | S0,0) = 1S 2πσ 2t

exp −log S /S0( ) − r −σ 2 /2( )t( )

2

2σ 2t

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥




€

P(S,t | S0,0) = 1S 2πσ 2t

exp −log S /S0( ) − r −σ 2 /2( )t( )

2

2σ 2t

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

The average value of S at time T is

€

S(T) = SP(S, t | S0,0)dS0

∞∫




€

P(S,t | S0,0) = 1S 2πσ 2t

exp −log S /S0( ) − r −σ 2 /2( )t( )

2

2σ 2t

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

The average value of S at time T is

€

S(T) = SP(S, t | S0,0)dS0

∞∫Shouldn’t this be the fair price of the future?

Hedging

No!

Hedging

No! This argument ignores the possibility of hedging.

Hedging


The seller could just buy a share of S herself at t = 0 and sellit to the buyer at maturity (T). This is riskless for the seller.Cost to the seller: She had to spend some of her own money (=S0)on that share at t = 0. That money could (risklessly) have beeninvested in T-bills, which give a rate r0. By buying the share, she has lost the chance to see her original cash become S0exp(r0T).

Hedging


The seller could just buy a share of S herself at t = 0 and sellit to the buyer at maturity (T). This is riskless for the seller.Cost to the seller: She had to spend some of her own money (=S0)on that share at t = 0. That money could (risklessly) have beeninvested in T-bills, which give a rate r0. By buying the share, she has lost the chance to see her original cash become S0exp(r0T).Therefore, the fair price of the future contract is

S0exp(r0T)

Hedging


The seller could just buy a share of S herself at t = 0 and sellit to the buyer at maturity (T). This is riskless for the seller.Cost to the seller: She had to spend some of her own money (=S0)on that share at t = 0. That money could (risklessly) have beeninvested in T-bills, which give a rate r0. By buying the share, she has lost the chance to see her original cash become S0exp(r0T).Therefore, the fair price of the future contract is

S0exp(r0T)

(Any other price would allow arbitrage).

Options: call

European call option: What is the fair price to charge for theoption to buy a share at a strike price X at time T?

Options: call


If S(T) > X, the buyer will certainly take advantage of the option.(He could immediately resell it and gain the difference.)

Options: call


If S(T) > X, the buyer will certainly take advantage of the option.(He could immediately resell it and gain the difference.)If S(T) < X, the buyer will not exercise the option (he can get it for less than X on the market).

Options: call


If S(T) > X, the buyer will certainly take advantage of the option.(He could immediately resell it and gain the difference.)If S(T) < X, the buyer will not exercise the option (he can get it for less than X on the market).

This means a net cost to the seller of

€

S(T) − X( )Θ S(T) − X( )

Options: put

European put option: What is the fair price to charge for the option to sell a share at a strike price X at time T?

Options: put


If S(T) < X, the buyer of the option will certainly take advantage .

Options: put


If S(T) < X, the buyer of the option will certainly take advantage .(He could then immediately buy it on the market instead and gain the difference.)

Options: put


If S(T) < X, the buyer of the option will certainly take advantage .(He could then immediately buy it on the market instead and gain the difference.)If S(T) > X, the buyer will not exercise the option (he can sell it for more than X on the market).

Options: put


If S(T) < X, the buyer of the option will certainly take advantage .(He could then immediately buy it on the market instead and gain the difference.)If S(T) > X, the buyer will not exercise the option (he can sell it for more than X on the market).

This means a net cost to the seller of

€

X − S(T)( )Θ X − S(T)( )

The strategy to find the right price at no risk to the seller

(European call option) Let f(t) be the value of the option at t, 0 < t ≤ T.



If we were already at time T (now we know S(T)), the fair priceis just .

€

f (T) = S(T) − X( )Θ S(T) − X( )




Suppose we are at time T – dt (the day before maturity), and S(T - dt) > X. The seller needs this much stock in order to be able to be able to sell it to the buyer.

€

f (T) = S(T) − X( )Θ S(T) − X( )




Suppose we are at time T – dt (the day before maturity), and S(T - dt) > X. The seller needs this much stock in order to be able to be able to sell it to the buyer. If, on the other hand, if S(T - dt) < X, she need do nothing.

€

f (T) = S(T) − X( )Θ S(T) − X( )




Suppose we are at time T – dt (the day before maturity), and S(T - dt) > X. The seller needs this much stock in order to be able to be able to sell it to the buyer. If, on the other hand, if S(T - dt) < X, she need do nothing.These two possibilities are described by: She should be holding shares.

€

f (T) = S(T) − X( )Θ S(T) − X( )

€

∂f /∂S

hedging strategy (continued)

Farther back in time: Always hold shares. If S changes,buy shares. (Sell if .)

€

∂f /∂S

€

∂f /∂S( )dS

€

∂f /∂S < 0



Formally: f depends on S, which varies randomly in timeaccording to

€

∂f /∂S

€

∂f /∂S( )dS

€

dS = rSdt + σSdW (t)€

∂f /∂S < 0




Use Ito’s lemma to see how f varies:

€

∂f /∂S

€

∂f /∂S( )dS

€

dS = rSdt + σSdW (t)€

∂f /∂S < 0





€

∂f /∂S

€

∂f /∂S( )dS

€

dS = rSdt + σSdW (t)

df = ∂f∂t

+ ∂f∂S

rS + 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt + ∂f

∂SσSdW (t)

€

∂f /∂S < 0





€

∂f /∂S

€

∂f /∂S( )dS

€


df = ∂f∂t

+ ∂f∂S

rS + 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt + ∂f

∂SσSdW (t)

€

∂f /∂S < 0

The seller’s portfolio is

€

Π=−f + ∂f∂S

S





€

∂f /∂S

€

∂f /∂S( )dS

€


df = ∂f∂t

+ ∂f∂S

rS + 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt + ∂f

∂SσSdW (t)

€

∂f /∂S < 0

The seller’s portfolio is

so it varies according to

€

Π=−f + ∂f∂S

S

dΠ = −df + ∂f∂S

dS

deriving the Black-Scholes equation

€


dS


€


dS

= − ∂f∂t

+ ∂f∂S

rS + 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt − ∂f

∂SσSdW (t) + ∂f

∂SrSdt + σSdW (t)( )


€


dS

= − ∂f∂t

+ ∂f∂S

rS + 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt − ∂f



= − ∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt


€


dS

= − ∂f∂t

+ ∂f∂S

rS + 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt − ∂f



= − ∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt

Because of the hedging strategy, the noise has cancelled out!


€


dS

= − ∂f∂t

+ ∂f∂S

rS + 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt − ∂f



= − ∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt

Because of the hedging strategy, the noise has cancelled out!Therefore the risk has been eliminated.


€


dS

= − ∂f∂t

+ ∂f∂S

rS + 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt − ∂f



= − ∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt


And because of that (as in the case of futures), the portfolio hasto earn the T-bill rate r0:


€


dS

= − ∂f∂t

+ ∂f∂S

rS + 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt − ∂f



= − ∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt


And because of that (as in the case of futures), the portfolio hasto earn the T-bill rate r0:

€

dΠ = r0Πdt = r0 − f + S ∂f∂S

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt

Black-Scholes equation

Combining the two expressions for dΠ,

€

−∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt = r0 − f + S ∂f

∂S ⎛ ⎝ ⎜

⎞ ⎠ ⎟dt



€

−∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt = r0 − f + S ∂f

∂S ⎛ ⎝ ⎜

⎞ ⎠ ⎟dt

⇒ ∂f∂t

+ r0S∂f∂S

+ 12 σ 2S2 ∂ 2 f

∂S2 = r0 f



€

−∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt = r0 − f + S ∂f

∂S ⎛ ⎝ ⎜

⎞ ⎠ ⎟dt

⇒ ∂f∂t

+ r0S∂f∂S

+ 12 σ 2S2 ∂ 2 f

∂S2 = r0 f




€

−∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt = r0 − f + S ∂f

∂S ⎛ ⎝ ⎜

⎞ ⎠ ⎟dt

⇒ ∂f∂t

+ r0S∂f∂S

+ 12 σ 2S2 ∂ 2 f

∂S2 = r0 f


This looks “Fokker-Planck-ish”.



€

−∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt = r0 − f + S ∂f

∂S ⎛ ⎝ ⎜

⎞ ⎠ ⎟dt

⇒ ∂f∂t

+ r0S∂f∂S

+ 12 σ 2S2 ∂ 2 f

∂S2 = r0 f


This looks “Fokker-Planck-ish”. It is in fact (except for the r0fterm on the right-had side) and a flip of the time direction,the adjoint (backward) FP equation (or backward Kolmogorovequation).



€

−∂f∂t

+ 12 σ 2S2 ∂ 2 f

∂S 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt = r0 − f + S ∂f

∂S ⎛ ⎝ ⎜

⎞ ⎠ ⎟dt

⇒ ∂f∂t

+ r0S∂f∂S

+ 12 σ 2S2 ∂ 2 f

∂S2 = r0 f


This looks “Fokker-Planck-ish”. It is in fact (except for the r0fterm on the right-had side) and a flip of the time direction,the adjoint (backward) FP equation (or backward Kolmogorovequation). (The S’s occur in front of the derivative operatorsrather than after them.)

a digression on adjoint equations

Recall for a Markov process

€

Pn (t +1) = Tn ′ n ′ n

∑ P ′ n (t)



What about calculating Pn(t -1), given Pn(t)?

€

Pn (t +1) = Tn ′ n ′ n

∑ P ′ n (t)



What about calculating Pn(t -1), given Pn(t)?governed by the transpose (adjoint) matrix Tn’n:

€

Pn (t +1) = Tn ′ n ′ n

∑ P ′ n (t)

€

Pn (t −1) = T ′ n n′ n

∑ P ′ n (t)




€

Pn (t +1) = Tn ′ n ′ n

∑ P ′ n (t)

€

Pn (t −1) = T ′ n n′ n

∑ P ′ n (t)

For master equation, we had the forward equation

€

dPm

dt= WmnPn −WnmPm( )

n≠m∑




€

Pn (t +1) = Tn ′ n ′ n

∑ P ′ n (t)

€

Pn (t −1) = T ′ n n′ n

∑ P ′ n (t)

For master equation, we had the forward equation

€

dPm

dt= WmnPn −WnmPm( )

n≠m∑

− dPm

dt= WnmPn −WmnPm( )

n≠m∑

=> backward equation is

some facts about adjoint matrices

Suppose

€

T = T1T2


Suppose

Then

€

T = T1T2

˜ T = ˜ T 2 ˜ T 1


Suppose

Then

For differential operators, fx€

T = T1T2

˜ T = ˜ T 2 ˜ T 1

€

Uf[ ](x) ≡ ddx

a(x) f (x)( )

˜ U f[ ](x) = a(x) − ddx

⎛ ⎝ ⎜

⎞ ⎠ ⎟ f (x)

adjoint Fokker-Planck operator

FP operator:

€

Lf[ ](x) = − ∂∂x

u(x) f (x)( ) + ∂ 2

∂x 2 D(x) f (x)( )


FP operator:

backward FP operator:

€

Lf[ ](x) = − ∂∂x

u(x) f (x)( ) + ∂ 2

∂x 2 D(x) f (x)( )

˜ L f[ ](x) = u(x) ∂∂x

f (x) + D(x) ∂ 2

∂x 2 f (x)


FP operator:


The solution of the FP equation

€

Lf[ ](x) = − ∂∂x

u(x) f (x)( ) + ∂ 2

∂x 2 D(x) f (x)( )

˜ L f[ ](x) = u(x) ∂∂x

f (x) + D(x) ∂ 2

∂x 2 f (x)

€

∂f∂t

= L(x) f[ ](x) = δ(x − x0)δ(t − t0)


FP operator:



€

Lf[ ](x) = − ∂∂x

u(x) f (x)( ) + ∂ 2

∂x 2 D(x) f (x)( )

˜ L f[ ](x) = u(x) ∂∂x

f (x) + D(x) ∂ 2

∂x 2 f (x)

The solution of the adjoint (or backward) FP equation

€

∂f∂t

= L(x) f[ ](x) = δ(x − x0)δ(t − t0)

∂f∂t0

= ˜ L (x0) f[ ](x0) = δ(x − x0)δ(t − t0)


FP operator:



€

Lf[ ](x) = − ∂∂x

u(x) f (x)( ) + ∂ 2

∂x 2 D(x) f (x)( )

˜ L f[ ](x) = u(x) ∂∂x

f (x) + D(x) ∂ 2

∂x 2 f (x)

The solution of the adjoint (or backward) FP equation

describes the change of the probability density with the final time

€

∂f∂t

= L(x) f[ ](x) = δ(x − x0)δ(t − t0)

∂f∂t0

= ˜ L (x0) f[ ](x0) = δ(x − x0)δ(t − t0)

back to Black-Scholes

€

∂f∂t

+ r0S∂f∂S

+ 12 σ 2S2 ∂ 2 f

∂S 2 = r0 f

The BS equation:


€

∂f∂t

+ r0S∂f∂S

+ 12 σ 2S2 ∂ 2 f

∂S 2 = r0 f

f (S,T) = max(S − X,0)

The BS equation:

Boundary condition:


€

∂f∂t

+ r0S∂f∂S

+ 12 σ 2S2 ∂ 2 f

∂S 2 = r0 f

f (S,T) = max(S − X,0)First, get rid of the r0f term by defining

€

f = ger0t

The BS equation:

Boundary condition:


€

∂f∂t

+ r0S∂f∂S

+ 12 σ 2S2 ∂ 2 f

∂S 2 = r0 f


€

f = ger0t

€

∂g∂t

+ r0S∂g∂S

+ 12 σ 2S2 ∂ 2g

∂S2 = 0

The BS equation:

Boundary condition:


€

∂f∂t

+ r0S∂f∂S

+ 12 σ 2S2 ∂ 2 f

∂S 2 = r0 f


€

f = ger0t

€

∂g∂t

+ r0S∂g∂S

+ 12 σ 2S2 ∂ 2g

∂S2 = 0

The BS equation:

Boundary condition:

The boundary condition is now

€

g(S,T) = e−r0T max(S − X,0)

solving Black-Scholes

Define y = log S. Then

€

∂g∂S

= 1S

∂g∂y



€

∂g∂S

= 1S

∂g∂y

∂ 2g∂S2 = ∂

∂S1S

∂g∂y

⎛ ⎝ ⎜

⎞ ⎠ ⎟= − 1

S2∂g∂y

+ 1S

∂∂S

∂g∂y ⎛ ⎝ ⎜

⎞ ⎠ ⎟= 1

S2∂g∂y

+ 1S2

∂2g∂y 2



€

∂g∂S

= 1S

∂g∂y

∂ 2g∂S2 = ∂

∂S1S

∂g∂y

⎛ ⎝ ⎜

⎞ ⎠ ⎟= − 1

S2∂g∂y

+ 1S

∂∂S

∂g∂y ⎛ ⎝ ⎜

⎞ ⎠ ⎟= 1

S2∂g∂y

+ 1S2

∂2g∂y 2

€

⇒ −∂g∂t

= r0 − 12 σ 2( )

∂g∂y

+ 12 σ 2 ∂ 2g

∂y 2



€

∂g∂S

= 1S

∂g∂y

∂ 2g∂S2 = ∂

∂S1S

∂g∂y

⎛ ⎝ ⎜

⎞ ⎠ ⎟= − 1

S2∂g∂y

+ 1S

∂∂S

∂g∂y ⎛ ⎝ ⎜

⎞ ⎠ ⎟= 1

S2∂g∂y

+ 1S2

∂2g∂y 2

€

⇒ −∂g∂t

= r0 − 12 σ 2( )

∂g∂y

+ 12 σ 2 ∂ 2g

∂y 2

Except for the sign of t, this is a standard FP equation with constantdrift and diffusion constants, so define τ = T – t, h(τ) = g(T - τ)



€

∂g∂S

= 1S

∂g∂y

∂ 2g∂S2 = ∂

∂S1S

∂g∂y

⎛ ⎝ ⎜

⎞ ⎠ ⎟= − 1

S2∂g∂y

+ 1S

∂∂S

∂g∂y ⎛ ⎝ ⎜

⎞ ⎠ ⎟= 1

S2∂g∂y

+ 1S2

∂2g∂y 2

€

⇒ −∂g∂t

= r0 − 12 σ 2( )

∂g∂y

+ 12 σ 2 ∂ 2g

∂y 2


€

⇒∂h∂τ

= r0 − 12 σ 2( )

∂h∂y

+ 12 σ 2 ∂ 2h

∂y 2



€

∂g∂S

= 1S

∂g∂y

∂ 2g∂S2 = ∂

∂S1S

∂g∂y

⎛ ⎝ ⎜

⎞ ⎠ ⎟= − 1

S2∂g∂y

+ 1S

∂∂S

∂g∂y ⎛ ⎝ ⎜

⎞ ⎠ ⎟= 1

S2∂g∂y

+ 1S2

∂2g∂y 2

€

⇒ −∂g∂t

= r0 − 12 σ 2( )

∂g∂y

+ 12 σ 2 ∂ 2g

∂y 2


€

⇒∂h∂τ

= r0 − 12 σ 2( )

∂h∂y

+ 12 σ 2 ∂ 2h

∂y 2

with boundary condition

€

h(y,0) = e−r0T max(ey,X) = e−r0T (ey − X)Θ(ey − X )

solving Black-Scholes (2)

Green’s function: If the initial condition werethe solution would be

€

h(y,0) = δ(y − y0)



€

h(y,0) = δ(y − y0)

€

G(y,τ | y0,0) = 12πσ 2τ

exp −y − y0 + (r0 − 1

2 σ 2)τ( )2

2σ 2τ

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥



€

h(y,0) = δ(y − y0)

€

G(y,τ | y0,0) = 12πσ 2τ

exp −y − y0 + (r0 − 1

2 σ 2)τ( )2

2σ 2τ

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

note sign



€

h(y,0) = δ(y − y0)

€

G(y,τ | y0,0) = 12πσ 2τ

exp −y − y0 + (r0 − 1

2 σ 2)τ( )2

2σ 2τ

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

For general h(y,0),

€

h(y,τ ) = dy0G(y,τ | y0,0)h(y0,0)∫

note sign



€

h(y,0) = δ(y − y0)

€

G(y,τ | y0,0) = 12πσ 2τ

exp −y − y0 + (r0 − 1

2 σ 2)τ( )2

2σ 2τ

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

For general h(y,0),

Here€

h(y,τ ) = dy0G(y,τ | y0,0)h(y0,0)∫

€

h(y,T) = g(y,0) = e−r0T dy0

ey0 − X( )2πσ 2Tlog X

∞∫ exp −y − y0 + (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

note sign



€

h(y,0) = δ(y − y0)

€

G(y,τ | y0,0) = 12πσ 2τ

exp −y − y0 + (r0 − 1

2 σ 2)τ( )2

2σ 2τ

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

For general h(y,0),

Here€

h(y,τ ) = dy0G(y,τ | y0,0)h(y0,0)∫

€

h(y,T) = g(y,0) = e−r0T dy0

ey0 − X( )2πσ 2Tlog X

∞∫ exp −y − y0 + (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= I1 + I2

note sign

doing the integrals:

€

I2 = −Xe−r0T dy0

2πσ 2Tlog X

∞∫ exp −y0 − y − (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€


2πσ 2Tlog X

∞∫ exp −y0 − y − (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= −Xe−r0T du2πσ 2Tlog X −y−(r0 − 1

2σ2 )T

∞∫ exp − u2

2σ 2T ⎡ ⎣ ⎢

⎤ ⎦ ⎥=


€


2πσ 2Tlog X

∞∫ exp −y0 − y − (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


2σ2 )T

∞∫ exp − u2

2σ 2T

⎡ ⎣ ⎢

⎤ ⎦ ⎥=

= −Xe−r0T dz2π

logX −y−(r0 − 12σ 2 )T

σ T

∞∫ e− 12 z2


€


2πσ 2Tlog X

∞∫ exp −y0 − y − (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


2σ2 )T

∞∫ exp − u2

2σ 2T

⎡ ⎣ ⎢

⎤ ⎦ ⎥=

= −Xe−r0T dz2π

logX −y−(r0 − 12σ 2 )T

σ T

∞∫ e− 12 z2

= −Xe−r0T Hlog X − y − (r0 − 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟


€


2πσ 2Tlog X

∞∫ exp −y0 − y − (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


2σ2 )T

∞∫ exp − u2

2σ 2T

⎡ ⎣ ⎢

⎤ ⎦ ⎥=

= −Xe−r0T dz2π

logX −y−(r0 − 12σ 2 )T

σ T

∞∫ e− 12 z 2

= −Xe−r0T Hlog X − y − (r0 − 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟ H(x) ≡ dz

2πx

∞∫ e− 12 z 2

doing the integrals (2):

€

I1 = e−r0T dy0ey0

2πσ 2Tlog X

∞∫ exp −y − y0 + (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€

I1 = e−r0T dy0ey0

2πσ 2Tlog X

∞∫ exp −y − y0 + (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

ˆ y = y + (r0 − 12 σ 2)T


€

I1 = e−r0T dy0ey0

2πσ 2Tlog X

∞∫ exp −y − y0 + (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

ˆ y = y + (r0 − 12 σ 2)T

I1 = e−r0T dy0

2πσ 2Tlog X

∞∫ exp −y0 − ˆ y ( )

2 − 2(y0 − ˆ y )σ 2T − 2 ˆ y σ 2T2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€

I1 = e−r0T dy0ey0

2πσ 2Tlog X

∞∫ exp −y − y0 + (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

ˆ y = y + (r0 − 12 σ 2)T

I1 = e−r0T dy0

2πσ 2Tlog X

∞∫ exp −y0 − ˆ y ( )

2 − 2(y0 − ˆ y )σ 2T − 2 ˆ y σ 2T2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= e−r0T + ˆ y dy0

2πσ 2Tlog X

∞∫ exp −y0 − ˆ y ( )

2 − 2(y0 − ˆ y )σ 2T + (σ 2T)2

2σ 2T+ 1

2 σ 2T ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€

I1 = e−r0T dy0ey0

2πσ 2Tlog X

∞∫ exp −y − y0 + (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

ˆ y = y + (r0 − 12 σ 2)T

I1 = e−r0T dy0

2πσ 2Tlog X

∞∫ exp −y0 − ˆ y ( )

2 − 2(y0 − ˆ y )σ 2T − 2 ˆ y σ 2T2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


2πσ 2Tlog X

∞∫ exp −y0 − ˆ y ( )

2 − 2(y0 − ˆ y )σ 2T + (σ 2T)2

2σ 2T+ 1

2 σ 2T ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= e−r0T + ˆ y + 12σ 2T dy0

2πσ 2Tlog X

∞∫ exp −y0 − ˆ y −σ 2T( )

2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€

I1 = e−r0T dy0ey0

2πσ 2Tlog X

∞∫ exp −y − y0 + (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

ˆ y = y + (r0 − 12 σ 2)T

I1 = e−r0T dy0

2πσ 2Tlog X

∞∫ exp −y0 − ˆ y ( )

2 − 2(y0 − ˆ y )σ 2T − 2 ˆ y σ 2T2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


2πσ 2Tlog X

∞∫ exp −y0 − ˆ y ( )

2 − 2(y0 − ˆ y )σ 2T + (σ 2T)2

2σ 2T+ 1

2 σ 2T ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= e−r0T + ˆ y + 12σ 2T dy0

2πσ 2Tlog X

∞∫ exp −y0 − ˆ y −σ 2T( )

2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= S du0

2πσ 2TlogX −y−(r0 + 12σ 2 )T

∞∫ exp − u2

2σ 2T

⎡ ⎣ ⎢

⎤ ⎦ ⎥


€

I1 = e−r0T dy0ey0

2πσ 2Tlog X

∞∫ exp −y − y0 + (r0 − 1

2 σ 2)T( )2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

ˆ y = y + (r0 − 12 σ 2)T

I1 = e−r0T dy0

2πσ 2Tlog X

∞∫ exp −y0 − ˆ y ( )

2 − 2(y0 − ˆ y )σ 2T − 2 ˆ y σ 2T2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


2πσ 2Tlog X

∞∫ exp −y0 − ˆ y ( )

2 − 2(y0 − ˆ y )σ 2T + (σ 2T)2

2σ 2T+ 1

2 σ 2T ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= e−r0T + ˆ y + 12σ 2T dy0

2πσ 2Tlog X

∞∫ exp −y0 − ˆ y −σ 2T( )

2

2σ 2T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= S du0

2πσ 2TlogX −y−(r0 + 12σ 2 )T

∞∫ exp − u2

2σ 2T

⎡ ⎣ ⎢

⎤ ⎦ ⎥

= SHlog(X /S) − (r0 + 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟

result:

So the price of the European call option is

result:


€

f (S,0) = SHlog(X /S) − (r0 + 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟− Xe−r0T H

log(X /S) − (r0 − 12 σ 2)T

σ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟

result:


€

f (S,0) = SHlog(X /S) − (r0 + 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟− Xe−r0T H

log(X /S) − (r0 − 12 σ 2)T

σ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟

in the limit S -> ∞, H( ) -> 1

result:


€

f (S,0) = SHlog(X /S) − (r0 + 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟− Xe−r0T H

log(X /S) − (r0 − 12 σ 2)T

σ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟


€

f (S,0) → S − Xe−r0T

result:


€

f (S,0) = SHlog(X /S) − (r0 + 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟− Xe−r0T H

log(X /S) − (r0 − 12 σ 2)T

σ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟


€

f (S,0) → S − Xe−r0T

effectively a future

result:


€

f (S,0) = SHlog(X /S) − (r0 + 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟− Xe−r0T H

log(X /S) − (r0 − 12 σ 2)T

σ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟


€

f (S,0) → S − Xe−r0T


in the limit σ -> 0

result:


€

f (S,0) = SHlog(X /S) − (r0 + 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟− Xe−r0T H

log(X /S) − (r0 − 12 σ 2)T

σ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟


€

f (S,0) → S − Xe−r0T



€

f (S,0) → max S − Xe−r0T ,0( )

result:


€

f (S,0) = SHlog(X /S) − (r0 + 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟− Xe−r0T H

log(X /S) − (r0 − 12 σ 2)T

σ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟


€

f (S,0) → S − Xe−r0T



€

f (S,0) → max S − Xe−r0T ,0( )

“at the money” (X = S), short maturity (small T):

result:


€

f (S,0) = SHlog(X /S) − (r0 + 1

2 σ 2)Tσ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟− Xe−r0T H

log(X /S) − (r0 − 12 σ 2)T

σ T

⎛ ⎝ ⎜

⎞ ⎠ ⎟


€

f (S,0) → S − Xe−r0T



€

f (S,0) → max S − Xe−r0T ,0( )

“at the money” (X = S), short maturity (small T):

€

f (S,0) ≈ σ T2π

Date post:	17-Jan-2018
Category:	Documents
Upload:	christiana-matthews
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Lecture 8: Finance: GBM model for share prices, futures and options, hedging, and the Black-Scholes...

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