Chapter 14: The Black-Scholes model Flashcards
The Black-Scholes market
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
The Black-Scholes market
STOCK
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
The Black-Scholes market
CASH
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}
The Black-Scholes market restrictions
r≥,0
S_0 ≥ 0 (stock price)
μ∈R
σ strictly bigger than 0 (o/w stock price isn’t random, difficulties with arbitrage)
in continuous time:
*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
filtered space generated by the Brownian motion Bt
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
DEF 14.1.2 portfolio for Black Scholes model
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
DEF 14.1.3
Black scholes process
portfolio strategy
*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)
amounts of assets in BS
The amounts of cash xt and stock yt that we hold at a given time are stochastic processes. (Just like in discrete time.)
DEF 14.1.4 self-financing
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
DEF 14.1.5 arbitrage possibility in BS
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
DEF 14.1.6 Contingent claim BS
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)
DEF 14.2.2
The risk-netural world in continuous time
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)
we’ll tend to write X (instead of our usual Φ(ST)) for contingent claims.
we’ll tend to write X (instead of our usual Φ(ST)) for contingent claims.
LEMMA 14.2.4
specific case
when there exists a replicating portfolio
(X= Φ(S_T)
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
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,
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
THM 14.2.5
contingent claim in BS market and replicating portfolio
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.
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
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
NOTES THM 14.2.5:
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.
The Black-Scholes equation
- 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
Theorem 14.3.1
risk neutral valuation formula
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
NOTES ABOUT:
Theorem 14.3.1
risk neutral valuation formula
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
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)=
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.
Corollary 14.3.3
replicating portfolio given in Theorem 14.3.1
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.
Remark:he Feynman-Kac
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.
Lemma 14.4.1
martingale in risk neutral world
The stochastic process
X_t= S_t/C_t
= e^{-rt}S_t
is a martingale in the risk-neutral world Q.
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.
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)
PROPOSITION 14.4.2
value of contract and stock
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
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
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)
Risk neutral valuations summary
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
EXAMPLE:
We look to find the price, at time t ∈ [0, T], of the contingent claim
Φ(S_T ) =
2S_T + 1
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
Note compute explicit pricing formula
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
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).
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)}
