Chapter 14: The Black-Scholes model

Question 1

The Black-Scholes market

Answer 1

The Black-Scholes market contains two assets, cash and stock

• both grow exponentially in value but cash earns interest at a deterministic rate, whereas the value of stock fluctuates

Question 2

The Black-Scholes market

STOCK

Answer 2

S_t= value of a unit of stock at time t
given by Geometric Brownian motion, the solution to
dS_t = μS_t.dt+ σS_t dB_t

initial value S_0

we know the solution:

S_t= S_0 exp( (μ-(σ^2/2))t + σB_t)

*sometimes work in stochastic diff - to use itos or may use solution

Question 3

The Black-Scholes market

CASH

Answer 3

if we hold x units of cash, for a time interval of length t.,
it grows to xe^{rt}.

ie earning interest in continuous time at rate r

C_t=value of unit of cash at time t is

dCt = rCt dt.
initial condition C0 = 1, and (unique) solution Ct = e^{rt}

Question 4

The Black-Scholes market restrictions

Answer 4

r≥,0
S_0 ≥ 0 (stock price)
μ∈R
σ strictly bigger than 0 (o/w stock price isn’t random, difficulties with arbitrage)

Question 5

in continuous time:

Answer 5

*We write the initial value of the stock as S0. We will use s as a variable. Note that this is different to our use of s in discrete time, in which we set s = S0

Question 6

filtered space generated by the Brownian motion Bt

Answer 6

As usual, we work over a filtered space (Ω,F,(Ft),P) where the filtration Ft is generated by the Brownian motion Bt, that is
Ft = σ(Bu ; u ≤ t).

Here, Bt is the same Brownian motion that drives the random stock price. Within the Black-Scholes model, Bt is the only source of randomness that we’ll need

Question 7

DEF 14.1.2 portfolio for Black Scholes model

Answer 7

A portfolio is a pair h = (x,y) where x ∈ R denotes an amount of cash and y ∈R denotes a number of units of stock. The value of this portfolio at time t is V h t = xCt + ySt.

*differs to binomial model as ash is an asset which changes value over time

Question 8

DEF 14.1.3
Black scholes process
portfolio strategy

Answer 8

*we continuously swap assets with time

A portfolio strategy is a pair of continuous stochastic processes (h_t) = (x_t,y_t) where both x_t and y_t are adapted to the filtration Ft. The value of (h_t) at time t is V^h _t = x_tC_t + y_tS_t.

(amount of cash at time t * value of cash + amount stock at time t * value stock)

*theyre adapted ie depend on info

(random values in capsvs deterministic)

Question 9

amounts of assets in BS

Answer 9

The amounts of cash xt and stock yt that we hold at a given time are stochastic processes. (Just like in discrete time.)

Question 10

DEF 14.1.4 self-financing

Answer 10

after time 0 don’t take out/input any value

A portfolio strategy (h_t) is said to be self-financing if

satisfies stochastic diff eq:

dV^h_t = x_t dC_t + y_t dS_t

*recall Y_tdX_t and ito int imply it represents Y_t (X_{t+δ} -X_t)
(Y_t decision and () random effect)

so it represents:
V^h_{t+δ}- V^h_t
= x_t (C_{t+δ}-C_t) + y_t(S_{t+δ}-S_t)

ie
change in value of portfolio strategy h is entirely explained by the change In value of C and S (during t to t+δ)

ie self financing

Question 11

DEF 14.1.5 arbitrage possibility in BS

Answer 11

We say that a self-financing portfolio strategy (h_t) is an arbitrage possibility if

V^h_0 = 0 (initial value 0 ie balance)

P[V^h_t ≥ 0] = 1 (chance that doesn’t lose is 1

P[V^h_t > 0] > 1 (chance gains value us greater than 0)

ie certain not to lose , make money for free

Question 12

DEF 14.1.6 Contingent claim BS

Answer 12

A contingent claim with date of exercise T is any random variable of the form X = Φ(S_T),
where Φ : [0,∞) → [0,∞) is a deterministic function.

(any function of stock price at time T, any contract that’s based on S_T)

We say that a portfolio strategy h_t = (xt,yt) hedges (or replicates) X if

(ht) is self-financing and
V^h_T = X.

(value at time contract expires = value of contingent claim)

Question 13

DEF 14.2.2

The risk-netural world in continuous time

Answer 13

The risk-netural world Q is the probability measure under which S_t evolves according to the SDE
dS_t = rS_t dt + σS_t dB_t.

Here, Bt has the same distribution (i.e. Brownian motion) under Q as in the ‘real’ world P.

*to transfer from world P to Q we replaced µ with r.

(here r is the drift and σ volatility)

Question 14

we’ll tend to write X (instead of our usual Φ(ST)) for contingent claims.

Answer 14

we’ll tend to write X (instead of our usual Φ(ST)) for contingent claims.

Question 15

LEMMA 14.2.4
specific case
when there exists a replicating portfolio

(X= Φ(S_T)

Answer 15

Let X ∈ F_T be a contingent claim in the Black-Scholes market with exercise time T. Suppose that the stochastic process
M_t = e^{−rT}E^Q [X|F_t] exists *

and has the stochastic differential
dM_t = f_t dZ_t

where Z_t = e^{−rt}St, **

for some (continuous, adapated) stochastic process f_t.

Then there exists a replicating portfolio strategy for X.

• assuming this exists is equiv to X is in L^1 ie E^Q[|X|] less than infinity

**Zt related to stock price

Question 16

LEMMA 14.2.4
specific case
when there exists a replicating portfolio:

Suppose that the stochastic process
M_t = e^{−rT}E^Q [X|F_t] exists

and has the stochastic differential
dM_t = f_t dZ_t

where Z_t = e^{−rt}St,

Answer 16

Don’t memorize the proof
don’t memorize the lemma
but hint: proof uses itos formula, we need to know how to use this..

WE PROVE THIS BY USING ITOS FORMULA

Want (h_t) st:
1) V^h_T = X (right value)

2) self financing: dV^h_T= x_t dCt + y_t dSt

we can use portfolio strategy :
x_t =(M_t −e^{−rt}f_tS_t)

y_t = f_t

x and y are continuous adaped processes as M_t , S_t and f_t adapted. So we have a portfolio stategy. We check 1) and 2) hold

1)
C_t=e^{rt}

V^h_t = xtCt + ytSt =
e^{rt}M_t −ftSt + ftSt
= e^{rt}M_t

Hence at time T
V^h_T= e^{rt}M_T
=e^{rT}[ e^{−rT}E^Q [X|F_t]]

(X∈mF_T)
=X
i.e. (h_t) replicates X
----
2)
check dV^h_T=  x_t dCt + y_t dSt:

we calculate itos using our calculations: need dV^h_t so need dM_t so need dZ_t where Z_t= e^{-rt}S_t

applications of itos formula with f(t,x)=e^{-rt}

dZt =(−rS_te^{−rt} + µS_te^{−rt} + 0)dt +
(σSt_e^{−rt}) dBt
= e^{−rt}S_t(µ−r)dt + σe^{−rt}S_t dBt

We have:
dM_t= f_t dZt

=e^{-rt}f_tS_t(µ-r).dt + σe^{−rt}f_tS_t dB

So by applying itos formula:

(NOT SUBSTITUTING)

dV^h_t= e^{rt}M_t

=(re^{rt}M_t+e^{-rt}f_tS_t(µ-r)e^{rt}+0).dt
+
e^{rt} σe^{−rt}f_tS_t dBt
= (re^{rt}Mt+ftSt(µ-r))dt + σftStdBt

=(re^{rt}M_t - rf_TS_t)dt + f_t[µStdt + σS_tdBt]

=
(M_t- e^{-rt}f_tS_t)[re^{rt}dt]+ (f_t)[µStdt + σS_tdBt]

= ()[dC_t]+()[dS_t]

(using definitions of dCt and dSt)

dV h t = xt dCt + yt dSt and, as required, we have checkedcond.
i.e. (h_t) is self-financing

Question 17

THM 14.2.5

contingent claim in BS market and replicating portfolio

Answer 17

Let X ∈F_T be a contingent claim in the Black-Scholes market with exercise time T.

Suppose that E^Q[|X|] < ∞.

Then there exists a replicating portfolio strategy for X.

Question 18

PROOF
Let X ∈F_T be a contingent claim in the Black-Scholes market with exercise time T.

Suppose that E^Q[|X|] < ∞.

Then there exists a replicating portfolio strategy for X.
THM 14.2.5

PROOF

Answer 18

PROOF:
Define M_t= e^{-rT}E^Q[X|F_t].

(Since E^Q[|X|] less than infinity the conditional exp is well defined;
as X is in L^1 our Mt exists)

Since e^{-rT} is constant M_t is a martingale.

using thm 12.3.3 in the risk neutral world Q, under Q there exists process g_t st
dMt = gt dBt

Let Z_t=e^{-rt}St. Recall in Q:
dSt=rSt dt+σSt dBt,

in Q apply ito’s formula to Zt:
dZt =(−re^{−rt}St + rSte^{−rt} + 0)dt + σe^{−rt}St dBt
= σZt dBt.

rearranging:
(1 /σZt) dZt = dBt

into dMt = gt dBt:

dM_t= (g_t/σZt) dZt

we have conditions for lemma 14.2.4
(ft=gt/σZt, adapted and continuous etc)
Then
there exists a replicating portfolio strategy for X

Question 19

NOTES THM 14.2.5:

Answer 19

It asserts that (essentially, all) contingent claims can be replicated, but it doesn’t tell us how to find a replicating portfolio. The root cause of this issue is that we used the martingale representation theorem – which told us the process g existed, but couldn’t give us an explicit formula for g. Without calculating g, we can’t calculate xt,yt either.

Of course, to trade in a real market, we would need a replicating portfolio ht = (xt,yt), or at the very least a way to numerically estimate one. With this in mind, in the next section, we will show how to find a replicating portfolio explicitly.

The argument we will use to do so relies on already knowing that a replicating portfolio exists; for this reason, we needed to prove Theorem 14.2.5 first.

Another issue is that we have not addressed is to ask if our replicating portfolio is unique. Potentially, two different replicating portfolios could exist. We will show in the next section that the replicating portfolio is, in fact, unique.

Question 20

The Black-Scholes equation

Answer 20

• look at how a replicating portfolio can be found explicitly for given contingent claim Φ(S_T)
• assume BS market Is arbitrage-free

Let Φ(ST) be a contingent claim in the Black-Scholes market (with parameters r,µ,σ) (r risk free rat, ,µ drift ,σ volatility) , and suppose that F(t,s) is a (suitably differentiable) function such that
F(t,St)
(deterministic function in time and value taken by stock price)

denotes the value of the contingent claim Φ(ST)
at time t ∈ [0,T].

Then, for all s > 0 and t ∈ [0,T] we have
(F solves the following)

∂F/∂t (t,s) + rs ∂F/∂s(t,s) + 1/2s^2σ^2∂^2F/∂s^2(t,s)− rF(t,s) = 0 *

F(T,s) = Φ(s) **

**boundary condition

Question 21

Theorem 14.3.1

risk neutral valuation formula

Answer 21

Let Φ(S_T) be a contingent claim such that EQ[Φ(S_T)] < ∞.

Then a replicating portfolio for Φ(ST) is given by

x_t = (1 /rCt) [∂F/∂t(t,St) +
1/2σ^2S^2_t ∂^2F/∂s^2(t,St)]

y_t =∂F/∂s(t,St)

where F(t,s) is the solution to * and **

The value of this portfolio at time t is equal
F(t,St) = e−r(T−t)E^Q[Φ(S_T)|F_t] ***

and in particular its value at time 0 is
e^{−rT}E^Q[Φ(S_T)].

• ** risk neutral valuation formula

Question 22

NOTES ABOUT:
Theorem 14.3.1

risk neutral valuation formula

Answer 22

The formulae given for xt,yt are a big improvement on Theorem 14.2.5, because a computer can simulate solutions to (14.10), as well as their partial derivatives, without (much) difficulty. This allows computation of the replicating portfolio in real-time. You are not expected to memorize these formulae for ht = (xt,yt). The other formula in Theorem 14.3.1 can be found on the formula sheet, in Appendix E. Of course, (14.13) is known as the risk-neutral valuation formula, in direct analogy to the discreet time version we found in Proposition

Question 23

PROOF:
Theorem 14.3.1

risk neutral valuation formula

F(t,s) is a sol to
∂F/∂t (t, s) + rs ∂F/∂s (t, s) +
(1/2)s^2 σ^2 ∂^2F/∂s^2(t, s) − rF(t, s) = 0, (*) BS eq

F(T, s) = Φ(s) (**)

and self-replicating replicating portfolio given by
x_t = 1/rCt
(∂F/∂t (t, St) + 1/2 σ^2S^2_t ∂^2F/∂s^2(t, St))

y_t =
∂F/∂s (t, St)

The value of Φ(S_T) at time t is
F(t,S_t)=

Answer 23

PROOF:

(**)boundary condition

To avoid arbitrage, at time T value of contingent claim must be equal to price
Φ(S_T) = F(T,S_T).

S_T may take any positive value so replace with general s greater than 0. ie as holds for every possible value
(we don’t worry about s less than 0 because S_t is always greater than 0. function Φ isn’t defined for these values )

So if h_t = (x_t, y_t) is a self replicating portfolio for our contingent claim, Then

value of portfolio = value of contingent claim

x_tC_t +y_t S_t = V^h_t = F(t,S_t)

so

dV^h_t=dF(t,s_t)
by itos and lemma 11.4.5 to equate dt and dBt coeffs

(h_t) is self financing so dV^h_t= x_t dC_t+y_tdS_t

we have dC_t=rC_t dt
dS_t= μS_t dt + σS_t dB_t
(ito process F is μS_t and G is σS_t )

substitute
dV^h_t= (x_t r C_t + y_t μS_t).dt
+ y_tσS_t dB_t

By itos formula
dF(t,S_t)=
( ∂F/∂t + ( μS_t ) ∂F/∂s+ (1/2)(sσ)^2 ∂^2F/∂s^2)dt +
(sσ)∂F/∂s dB_t
where we evaluate partial derivatives at (t,S_t)
From the dB_t coeff

y_t= ∂F/∂t
from the dt coefficients:

∂F/∂t + μS_t ∂F/∂s (t, s) +
(1/2)s^2 σ^2 ∂^2F/∂s^2
= x_t r C_t + y_t μS_t

gives
∂F/∂t + (1/2)s^2 σ^2 ∂^2F/∂s^2 = x_t r C_t

x_t =
(∂F/∂t (t, St) + 1/2 σ^2S^2_t ∂^2F/∂s^2(t, St))/rC_t

use V^h_t = F(t,S_t) to substitute in for C_t gives

∂F/∂t + 1/2 σ^2S^2_t ∂^2F/∂s^2
= rF-rS_t ∂F/∂S (**)

we evaluate at (t,S_T)
since S_t taks full range of values on (0, infinity) can replace s_t(0, infinity) for S_t

Apply Feynman-katz to **, lemma 13.13 α(t, s) = rs and β(r, s) = σs
sol to BS eq
F(t, s) = e^{−r(T −t)}E^Q_{t,s}[Φ(ST )].
value at time t of the replicating portfolio is

F(t, St) = e^{−r(T −t)}E^Q_{t,S_t}[Φ(S_T )]
by lemma 13.2.1 markov property

E^Q_{t,St}[Φ(S_T )] = E^Q[Φ(S_T )| F_t].
Setting t = 0, and recalling that F0 is the trivial σ-field (containing no information) we have initial value
E^Q[Φ(S_T )| F_0] = E^Q[Φ(ST )],
as required.

Question 24

Corollary 14.3.3

replicating portfolio given in Theorem 14.3.1

Answer 24

The replicating portfolio given in Theorem 14.3.1 for Φ(ST ) is unique.

Proof: The calculations in the proof of Theorem 14.3.1 showed that any replicating portfolio
(consisting solely of stocks and cash) could be written in terms of the solution to the BlackScholes equation, as in the statement of Theorem 14.3.1. Since the Black-Scholes equation has
a unique solution, this implies the replicating portfolio is also unique.

Question 25

Remark:he Feynman-Kac

Answer 25

the Feynman-Kac formula is important to mathematical
finance; it is the key to establishing the risk-neutral valuation formula. This formula links arbitrage free pricing theory to Brownian motion, through the stochastic differential equation
dSt = µSt dt + σSt dBt.

Question 26

Lemma 14.4.1

martingale in risk neutral world

Answer 26

The stochastic process
X_t= S_t/C_t
= e^{-rt}S_t

is a martingale in the risk-neutral world Q.

Question 27

proof:

Lemma 14.4.1
The stochastic process
X_t= S_t/C_t
= e^{-rt}S_t

is a martingale in the risk-neutral world Q.

Answer 27

C_t=e^{rt}
X_t=e^{-rt}S_t and in the risk-neutral world Q the process St satisfies dSt = rSt dt + σSt dBt so by Ito’s formula we have
dXt =
(−re^{−rt}S_t + rS_te^{−rt} + 0)dt + σS_te^{−rt} dBt
= σS_te^{−rt} dBt

By Theorem 11.2.1, we have that Xt is a martingale (under Q)

Question 28

PROPOSITION 14.4.2

value of contract and stock

Answer 28

Let Φ(S_T ) be a contingent claim and let Πt denote the price of this contingent claim at time t. Then
Πt/Ct is a martingale in the risk-neutral world

• C_t is a factor of interest
• martingale=fair game

so in Q, after factor out interest, buying/selling any contract is a fair game ie MARKETS ACT AS ARBITRAGE FREE

Question 29

proof

PROPOSITION 14.4.2

Let Φ(S_T ) be a contingent claim and let Πt denote the price of this contingent claim at time t. Then
Πt/Ct is a martingale in the risk-neutral world
Q

Answer 29

contract Φ(S_T )=S_T
value at time t Πt=S_t

Set X_t= Πt/Ct = e^{-rt}Π
dXt using itos:
If X_t is to be a martingale should be dXt=(…)dBt.

To avoid arbitrage Πt is equal to the value of the replicating portfolio for Φ(S_T) at time t. Thm 14.3.1 gives Πt=F(t,S_t) *

in risk neutral world Q, we have
dS_t=rS_t.dt + dt + σSt dBt
By Ito’s formula
(applied in the risk neutral world!), we have

dΠt =
(∂F/∂t + rSt ∂F/∂x +(1/2)σ^2S^2_t ∂^2F/∂x^2)dt + σS_t ∂F/∂x dB_t

for (t,S_t)
F satisfies BS PDE so
dΠt = rF dt + σSt ∂F/∂x dB_t
using * again

dΠt = rΠt dt + σSt ∂F/∂x dBt

which represents Πt as an Ito process.
We have Xt = e^{−rt}Πt

Using Ito’s formula again, this gives us
dXt =
(−re^{−rt}Πt + rΠte^{−rt} + 0)dt + σSt ∂F/∂x e^{−rt} dBt
= σSt ∂F/∂x e^{−rt} dBt

Hence, by Theorem 11.2.1, Xt
is a martingale (under Q)

Question 30

Risk neutral valuations summary

Answer 30

given a contingent claim Φ(S_T) , its value at time t in [0,T] is given by
F(t,S_t) = e^{-r(T-t)}E^Q[ Φ(S_T) | F_t]

here dS_t=rS_t dt + σSt dBt

S_T=S_t exp(
(r −(1/2)σ^2)(T − t) + σ(B_T − B_t))
=S_te^Z

(note B_T-B_t N(0,T-t))

where Z ∼ N(r −(1/2)σ^2)(T − t), σ^2(T − t)]
= N[u, v^2]

is independent of Ft

Here, we use u and v

Question 31

EXAMPLE:
We look to find the price, at time t ∈ [0, T], of the contingent claim
Φ(S_T ) =
2S_T + 1

Answer 31

We have contract and value of contract at time T, 2 units of stock and amount of cash with value 1 (not 1 unit of cash C_t, changes value)

The value at time t of this contingent claim is

e^{−r(T −t)}E^Q[2ST + 1 | Ft]

by S_T=S_te^z,
e^{−r(T −t)}E^Q[2ST + 1 | Ft]=

(S_t is F_t measurable)
e^{−r(T −t)}E^Q[2S_te^Z + 1 | Ft]

= e^{−r(T −t)}(2StE^Q[e^Z| Ft]+ 1)
(e^Z is indep of F_t)

= e^{−r(T −t)}(2StE^Q[e^Z]+ 1)
(Z~N(u,v^2) so E(e^Z)=c^{u + (v^2/2)})

= e^{−r(T −t)}(2S_te^{(r−(1/2)σ^2)(T −t)+ (1/2)σ^2(T −t) + 1

= e^{−r(T −t)}(2Ste^{r(T −t)} + 1)
= 2S_t + e^−r(T -t)
portfolio value at time t

*at t=0 2S_0 + e^{-rT}
2 units of stock + how much cash needed at t=0 to earn interest and have 1 in cash at time T

Question 32

Note compute explicit pricing formula

Answer 32

Using the (14.19) 
F(t, St) = e^{−r(T −t)}E^Q[Φ(ST )| Ft]  
in combination with either (14.20) or (14.21) 

S_T=S_t exp(
(r −(1/2)σ^2)(T − t) + σ(B_T − B_t))
=S_te^Z

is usually the best way to
compute explicit pricing formula

Question 33

Example: Consider the case of a call option(cont time) with strike
price K and exercise date T

contingent claim:
Φ(ST ) = max(S_T − K, 0).

Answer 33

F(t, St) = e^{−r(T −t)}E^Q[max(ST − K, 0)| Ft]

write s=S_t, (se^z is S_T into contingent claim)

(f_Z(z).dz’s z~N(u,v^2))

F(t, St) = 
 e^{−r(T −t)}
∫_{ -∞,∞}  [Φ(se^z)f_Z(z)] dz
=  e^{−r(T −t)} (
∫_{ -∞, log(K/s)} [Φ(se^z) f_Z(z)] dz + ∫_{ log(K/s),∞} [(se^z - K) f_Z(z)] dz
=
e^{−r(T −t)} (0 +
∫_{ log(K/s),∞} [(se^z - K) (1/root(2πv))e^{(z-u)^2/2v^2}] dz

= (split the integral into the cases se^Z < K and se^Z ≥ K)

e^{−r(T −t)} (

∫_{ log(K/s),∞} [(K) (1/root(2πv))e^{(z-u)^2/2v^2}] dz
=A+B

B term: want in terms of N(z)
= (1/√2π) ∫_{−∞,z} [e^{−x^2}dx

y = −z,

B= (K/√(2πv)) e^{−r(T −t)}
∫_{− log(K/s), −∞} [  e^{−(−y−u)^2/2v^2) (−1)] dy
=
(K/√(2πv)) e^{−r(T −t)}
∫_{−∞, log(s/K)}
[e^{−(y+u)^2/2v^2}] dy

sub x= (y+u)/v:

B=
(K/√(2πv)) e^{−r(T −t)}
∫_{−∞, (1/v)(log(s/K)+u)}
[e^{−x^2/2}v] dx
=
Ke^{−r(T −t)}
∫_{−∞, (1/v)(log(s/K)+u)}
[1/√(2π) e^{− x^2/2}] dx

=Ke^{-r(t-t)} N[ (1/v) {log(s/K)+ u}]

A term: e^z factor, complete the square inside the exponential term before x sub

….
A =sN((1/v) {log(s/K) +u+v^2})

sub u and v and recalling that we use the shorthand s = St, we obtain the following formula for price, at time t ∈ [0, T], of a European call option with strike price K and exercise date T.

F(t, St) = St N [d1] − Ke^{−r(T −t)}N[d2]

where
d1 = 1/(σ√(T − t)) {log(St/K)+
(r + (1/2)σ^2)(T − t)}

d2 =
1/(σ√(T − t)) {log(St/K)+
(r - (1/2)σ^2)(T − t)}