quant black scholes

7/24/2019 Quant Black Scholes

1/26

Black-Scholes equationVariational formulation

LocalizationDiscretization

Non-smooth initial data

Computational Methods for Quant. Finance II

Finite difference and finite element methods

Lecture 4



2/26




Outline

Black-Scholes equation

Variational formulation

Localization

Discretization




3/26




From expectation to PDE

Goal: compute thevalue of European optionwith payoffg which isthe conditional expectation

V(t, x) = E

eRTt r(Xs)dsg(XT) | Xt=x

, (1)

whereX is the (unique) solution of the stochastic differentialequation (the dynamics of the underlying of the option)

dXt=b(Xt) dt + (Xt) dWt. (2)

It turns out: V(t, x) solves a PDE. We need the notion of theso-calledinfinitesimal generatorof the process X.



4/26




PropositionLetAdenote the differential operator which is, for functionsf C2(R) with bounded derivatives, given by

(Af)(x) := 1

22(x)xxf(x) + b(x)xf(x). (3)

Then, the processMt :=f(Xt) t0 (Af)(Xs)ds is a martingale

with respect to the filtration ofW.

The operator A is theinfinitesimal generatorof the process X,which solves the SDE

dXt=(Xt) dWt+ b(Xt) dt.


Bl k S h l i


5/26




We need a discounted version of the above Proposition.

Proposition

Letf C1,2(R R) with bounded derivatives in x, letAbe as in(3) and assume thatr C0(R) is bounded. Then the process

Mt:=eRt0r(Xs)dsf(t, Xt)

t0

eRs0 r(X)d(tf+Afrf)(s, Xs)ds

is a martingale with respect to the filtration ofW.

We now are able to link thestochastic representationof the optionprice

V(t, x) = E

eRT

t r(Xs)dsg(XT) | Xt=x

with aparabolic partial differential equation.


Bl k S h l ti


6/26




Theorem

LetV C1,2(J R) C0(J R) with bounded derivatives in xbe a solution of

tV + AV rV = 0 in J R,V(T, x) = g(x) in R,

(4)

withAas in (3). Then, V(t, x) can also be represented as

V(t, x) = E

eRT

t r(Xs)dsg(XT) | Xt=x

. (5)

RemarkThe converse of this Theorem is also true. AnyV(t, x) as in (5),which isC1,2(J R) C0(J R) with bounded derivatives in x,solves the PDE (4). This is known asFeynman-Kac Theorem.


Black Scholes equation


7/26




We apply the Feynman-Kac Theorem to theBlack-Scholes model.

In the Black-Scholes market with no dividends, the risky assetsspot-price is modelled by a geometric Brownian motion X, i.e., theSDE for this model is as in (2), with coefficients

b(x) =rx, (x) =x,

where >0 and r 0 denote the (constant)volatilitythe(constant)interest rate, respectively.

Therefore, the SDE is given by (use S instead ofX)

dSt=rStdt + StdWt,

with infinitesimal generator

A := 1

22s2ss+ rss. (6)


Black Scholes equation


8/26




The Black-Scholes PDE

We obtain: the discounted price of a European contract withpayoffg(s), i.e.,

V(t, s) = E[er(Tt)g(ST)|St =s],

is equal to a regular solution V(t, s) of the Black-Scholes equation

tV + 12

2s2ssV + rssV rV = 0 in [0, T) R+

V(T, s) = g(s) in R+.

The infinitesimal generator (and hence the Black-Scholes PDE)degeneratesat s= 0. Furthermore, the PDE isbackwardin time.




9/26




To obtain a nondegenerate equation, we switch to thelog-priceprocess Xt = log(St) which solves the SDE

dXt = r

1

22

dt + dWt.

The infinitesimal generator for this process has constantcoefficients:

ABS :=1

2

2xx+ r 1

2

2x. We furthermore change totime-to-maturity t T t, to

obtain a forward parabolic problem.




10/26

Black Scholes equationVariational formulation



The Black-Scholes PDE in log-price

Thus, by setting V(t, s) =:v(T t, log s), the BS equation (inreal price) satisfied by V(t, s) becomes theBS equationfor v(t, x)in log-price

tv ABSv+ rv = 0 in (0, T] R

V(0, x) = g(ex) in R, (7)

with

ABS :=1

2 2xx+

r 1

2 2

x.

We next study the variational formulation of (7).




11/26

c Sc o s qu t oVariational formulation



Outline



Localization

Discretization





12/26

qVariational formulation



Denote by (, ) the L2(R)-inner product, i.e., (, ) := R dx.Denote by aBS(, ) :H1(R) H1(R) R thebilinear formassociated to the operator ABS

aBS(, ) :=1

2

2(, ) + (2/2 r)(, ) + r(, ). (8)

The variational formulation of the Black-Scholes equation (7)reads:

Find u L2(J; H1(R)) H1(J; L2(R)) such that

(tu, v) + aBS(u, v) = 0, v H1(R), a.e. in J, (9)

u(0) =u0,

whereu0(x) :=g(ex).




13/26

Variational formulationLocalization

DiscretizationNon-smooth initial data

Ifu0 L2(R), then problem (9) admits a unqiue solution, since

a

BS

(, ) iscontinuousand satisfies aGarding inequalityonH

1

(R

).Proposition

There exist constantsCi =Ci(, r)> 0, i= 1, 2, 3, such thatthere holds for all, H1(R)

|aBS(, )| C1H1H1 , aBS(, ) C22H1C32L2 .

Proof.

|aBS(, )| 1

22L2

L2+ |2/2 r|L2

+rL2L2 C1(, r)H1H1 .

aBS(, ) = 1

222L2+ r

2L2 =

1

222H1+ (r

2/2)2L2

1

222H1 |r

2/2|2L2 .




14/26



Outline



Localization

Discretization



Black-Scholes equationV i i l f l i


15/26



u0(x) =g(ex) L2(R) implies an unrealistic growth

condition on the payoffg. For example, the payoff of both theput g(ex) = max{0, K ex} and the callg(ex) = max{0, ex K}are not in L2(R).

We can weaken this assumption by reformulating the problemon a bounded domain (which we have to do anyway for

discretizing the problem) The unbounded domain R of the log price x= log s is

truncated to a bounded domainG. In terms of financialmodeling, this corresponds to approximating the option priceby aknock-out barrier option.

LetG= (R, R), R >0, be an open subset and let

G = inf{t 0 | Xt Gc}

be thefirst hitting timeof the complement set Gc = R\G by

X.Computational Methods for Quant. Finance II

Black-Scholes equationV i ti l f l ti


16/26



The price of a knock-out barrier option in log-price with payoff

g(e

x

) is given byvR(t, x)= E

er(Tt)g(eXT)1{T0, q 1 such that the payoff functiong: R+ R satisfiesg(s) C(s + 1)

q for alls R+. Then, thereexistC(T, ), 1, 2>0, such that

|v(t, x) vR(t, x)| C(T, )e1R+2|x|.




17/26



We see from this Theorem that vR v exponentially for afixedx as R .

The artificial zero Dirichlet barrier type conditions at x= Rarenotdescribing correctly the asymptotic behavior of the

price v(t, x) for large |x|. Since the barrier option price vR is a good approximation to v

for |x| R, R should be selected substantially larger thanthe values ofx of interest.

The barrier option price vR can again be computed as the solutionof a PDE provided some smoothness assumptions.




18/26



Theorem

LetvR(t, x) C1,2(J R) C0(J R) be a solution of

tvR+ ABSvR rvR= 0, (11)

on [0, T) G where the terminal and boundary condition given by

vR(T, x) =g(ex), x G, vR(t, x) = 0, on (0, T) G

c.

Then,vR(t, x) can also be represented as in (10).

Now, we can restate the problem (9) on the bounded domain:

Find uR L2(J; H10 (G)) H

1(J; L2(G)) such that

(tuR, v) + aBS(uR, v) = 0, v H

10(G), a.e. inJ, (12)

uR(0) =u0|G .




19/26



Outline



Localization

Discretization





20/26



Finite difference discretization

We discretize the PDE (11) directly using finite differences onbounded domain with homogeneous Dirichlet boundary conditions.UsingxxvR(tm, xi) h

2(umi+1 2umi u

mi1),

xvR(tm, xi) (2h)1(um

i+1

um

i1

)we obtain the matrix problem:

Find um+1 RN such that for m= 0, . . . , M 1,I+ kGBS

um+1 =

I (1 )kGBS

um,

u0 =u0,

where GBS =2/2R +

2/2 rC+ rI, is given explicitly with

R :=h2tridiag

1, 2, 1

, C := (2h)1tridiag

1, 0, 1

.




21/26



Finite element discretization

We discretize (12) using the -scheme and the finite element spaceVN =S

1T H

10(G). Setting again u0(x) :=g(e

x) and assuminguniform mesh width h and constant time step k, we obtain thematrix problem:

Find um+1 RN such that for m= 0, . . . , M 1

(M+ kABS)um+1 = (M k(1 )ABS)um,

u0N =u0 ,

whereM

ij = (bj , bi)L2

(G) andABS

ij =aBS

(bj , bi). Using themethods we have developed in chapter 3, we findABS =2/2S +

2/2 r

B+ rM, explicitly with

S := h1tridiag

1, 2, 1

, B := 21tridiag

1, 0, 1

,

M

:= 6

1

htridiag

1, 4, 1

.Computational Methods for Quant. Finance II



22/26



Outline



Localization

Discretization





23/26



Graded mesh

Know: achievable convergence rate of linear continuous FEM+ -scheme is u uNL2(J;L2(G))= O(h

2 + kr) providedthe initial data u0(x) =g(e

x) (the payoff) satisfies

u0 H2(G). Here, r= 1 for [0, 1] \ {1/2} and r= 2 for= 1/2 (constant time step k is used).

However, for put and call contracts (u0 H3/2(G)) or for

digital options (u0 H1/2(G)), this regularity assumption

is not satisfied, and we expect areduction of the rate ofconvergencew.r. to time, i.e. we expect a lower r.

To recover the optimal convergence rate for u0 Hs(G),

0< s


24/26



SetT = 1. Let : [0, 1] [0, 1] be agrading functionwhich isstrictly increasing and satisfies

C0([0, 1]) C1((0, 1)), (0) = 0, (1) = 1.

We define for MN

the graded mesh by the time points,

tm=m

M

, m= 0, 1, . . . , M .

It can be shown that we obtain again the optimal convergence rate

if(t) = O(t) where depends on r and s, =(r, s).We give an example for a European call and digital (or binary)option.



L li i


25/26



Example

We set strike K= 100, volatility = 0.3, interest rate r= 0.01,maturity T = 1. For the discretization we use M= O(N),= 1/2, R= 3 and apply the L2-projection for u0.

We measure the discrete L2

(J; L2

(G))-error defined byMm=1 kih

m22 where

m22 :=N

i=1

|u(tm, xi) uN(tm, xi)|2.

both with constant time steps and with graded time steps. We usethe grading factor = 3 for the call option g(s) = max{0, s K}and = 25 for the digital option g(s) ={s>K}.



L li ti


26/26



101

102

103

104

104

103

102

101

100

s = 2.0

Call option

Digital option

N

L2-error

101

102

103

104

105

104

103

102

101

100

s = 2.0

Call option

Digital option

N

L2-err

or


quant black scholes

Documents