Free boundary value problems in finance

7/28/2019 Free boundary value problems in finance

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Free boundary value problems in finance

presented by

Yue Kuen Kwok

Department of Mathematics

Hong Kong University of Science & Technology

Hong Kong, China


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Examples of free boundary value problems

In physics

burning of a cigarette

melting of an iceberg

solidification of molten metal

The boundary of the domain of the problem is not fixed, and it has to be

determined as part of the solution procedure.

In finance

decision to terminate a contract

decision to exercise certain right, like early exercising an American option,resetting terms in an option contract, or conversion into shares in a con-

vertible bond.


3/30

Reset features in options

Strike reset

new strike = S, S is the stock price at shout moment

Maturity reset

1. The new maturity date is e time periods beyond the original maturity date

T = T + e.

2. The maturity date is reset to be r time periods beyond the reset moment

, but bounded above by Tmax (Tmax > T).

T = min( + r, T + m).


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In his Nobel prize winning paper, Merton (1973) wrote:

... because options are specialized and relatively unimportant financial secu-

rities, the amount of time and space devoted to the development of a pricingtheory might be questioned

Ross (1987) wrote in New Palgrave Dictionary of Economics

This does not mean, however, that there are no important gaps in the (option

pricing) theory. Perhaps of most important, beyond numerical results ... very

little is known about most American options which expire in finite time... Despite such gaps, when judged by its ability to explain the empirical data,

option pricing theory is the most successful theory not only in finance, but in

all of economics.


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Continuation region and stopping region of American options

Consider an American call option, upon early exercise, C(S, t) = S X. Sub-stituting into the Black-Scholes equation

Ct

+2

2S2

2CS2

+ rSCS

rC = qS rX.When qS rX > 0, this would imply

d < r dt

where = V S = V VS

S.

Financially, this indicates that the rate of return from the portfolio is less thanthe riskfree rate.

Vt +

2

2 S22VS2

+ rSVS rV = 0 in the continuation region

Vt +

22 S

22VS2

+ rSVS rV < 0 in the stopping region.


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Smooth paste condition

At the critical asset price S(t),

C(S(t), t) = S(t) X

C

S(S

(t), t) = 1.

One boundary condition is used to determine the option value on the freeboundary, while the other boundary condition is used to determine the location

of the free boundary.


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Proof of the smooth paste condition

C(S, t) = maxb(t)

f(S, ; b(t)); write F(S, b) = f(S, ; b(t)).

The total derivative of F with respect to b along the boundary S = b is

dF

db=

F

S(S, b)|S=b+

F

b(S, b)|S=b.

Let b be the value of b that maximizes F. When b = b, we have

Fb

(S, b) = 0 (first order condition).

For a call, we write h(b) = F(b, b) = b X and

dhdbb=b= ddb

(b x)b=b= 1so that

F

S

(b, b) = 1

C

S

(S(t), t) = 1.


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Obstacle problem

The string is held fixed at A and B, and it must pass smoothly over the obstacle.

u(u

f) = 0, f(x) is the obstacle function;

u

0 and u

f.

string must be above or on the obstacle

string must have negative or zero curvature

string and its slope must be continuous at contact points

Free boundary value problem: We do not know a priori the contact points ofthe string on the obstacle.


9/30

Linear complimentarily formulation

Write L = t

2

2S2

2

S2 rS

S+ r

and V0(S) as the option payoff upon exercising.min ((LV(S, t), V(S, t) V0(S)) = 0LV(S, t) 0 and V(S, t) V0(S) .


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American put price function

P(S, t) = erX

0(X ST)(ST; S) dST

+ 0

er S()0

(rX qS)(S; S) dSdwhere = T t and is the time lapsed after the current time t.

P(S, t) = Xer

N(d2) Seq

N(d1)+

0rX erN(d,2) qSe1N(d,1) d

where

d1 =ln SX + r q + 22

, d2 = d1

d,1 =

ln SS()

+ r q +2

2 , d,2 = d,1 .


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Delayed exercise compensation

The holder of the American put should be compensated by a continuous cash

flow when the put should be exercised optimally. The discounted expectation

for the above continuous cash flow is

erS()

0(rX qS)(S; S) dS.

This is because the holder would earn interest of amount rXd from the cash

received and would lose dividends of amount of qS d from the short position

of the asset.


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Proof of the integral representation of the early exercise premium

Let G(S, t; , T) =er(Tt)

2(T t)

exp 1

22(T t)

lnS

+

r q 2

2

(T t)

2 .Write V(, u) = G(S, t; , u), which is considered as a function of and u; then

V(, u) satisfies the adjoint equation:

LV = Vu

+2

2

2

2(2V) (r q)

(V) rV = 0

V(, t) = (

S).

We have

LP(S, t) =

0 (S, t) continuation regionrX

qS (S, t)

stopping region

.


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Consider Tt+

duS(u)

0(rX q)G(, u; S, t) d

= Tt+

du 0

G(, u; S, t)LP d

= Tt+

du

0[G(, u; S, t)LP PLG(, u; S, t)] d

= Tt+

du 0

u

(GP) + 2

2

2GP

2

2

P

(2G)

+ (r q)

(P G)

d.

Note that when 0 and , we have

2GP

0, P

(2G) 0 and P G 0.


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Hence, we obtain

0 G(, t + ; S, t)P(, t + ) d =

0 G

(, T; S, t)P(, T) d

+Tt+

duS(u)

0(rX q)G(, u; S, t) d.

Lastly, we let

0 so that

P(S, t) =

0G(S, t; , T)(X ) d

+

T

tdu

S(u)

0(rX q)G(S, t; , u) d.


15/30

Optimal stopping time

P(S, t) = suptT

E{er(t)(X S)]

where

= inf

{t T : P(S, ) = X S},

that is, is the first time that the price of the derivative drops down to its

exercise payoff.


16/30

The price of an American put satisfies the following linear complimentary prob-lem:

P

t +2

2 S22P

S2 + rSP

S rP 0 (i)

P X S (ii)

Pt

+ 2

2S2

2

PS2

+ rSPS

rP [P (X S)] = 0 (iii)to be solved in {(S, t) : S > 0, 0 < t < T} and P(S, T) = (X S)+.

Sketch of the proof : For any stopping time , t < < T, by Itos calculus,

erP(S, ) = ertP(St, t) +

t

eru

t+

2

2S2

2

S2+ rS

S r

P(Su, u) du

+ t eruSuPS(Su, u) dWsE{er(t)P(S, )} P(St, t) by (i) & Doobs optional stopping theorem

E

{er(t)(X

S)

} P(St, t) by (ii)

If = , is the optimal stopping time, then we have equality.


17/30

American Asian Options

Terminal payoff:

C(ST, GT, T) = max(ST

GT, 0)

where ST is the terminal asset price and

Gt = exp

1

t

t0

ln Su du

, 0 < t T.

Suppose St follows the following risk neutral lognormal process

dSt

St= (r q) dt + dZ(t)

then

ln GT =t

Tln Gt +

1

T

(T t) ln St +

r q

2

2

(T t)2

2

+

T T

t [Z(u) Z(t)] du.


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Ct

+ Gt

ln SG C

G+

2

2S2

2

CG2

+ (r q)SCG

rC

=

0 if S S(G, t)qS dGdt + rG if S > S(G, t)

.

Use the asset price S as numeraire, and define

y = t lnG

Sand V(y, t) =

C(S,G,t)

S

then

V

t+

2t2

2

2V

y2

r q + 2

2

t

V

y qV

= 0 if y y(t)q + rey/t + yt2ey/t if y < y(t)

.

and

V(y, T) = 1 ey/t, y y(t).In the stopping region, V(y, t) = 1 ey/t, y y(t).


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Stochastic movement of the primary fund S2(t)and upgraded fund S1(t)

A primary fund with value process St2, which is protected with reference toanother (guaranteed) fund with value process St1.

The holder has the right to reset the value of the fund to that of theguaranteed fund upon exercising the reset rights.


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n(t) = number of units at time t

F(t) = value of the profected fund


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Pricing model of the perpetual protection fund

Let Vn(S1, S2) denote the value of the perpetual protection fund with nreset rights and withdrawal right.

We take advantage of the linear homogeneity property of Vn(S1, S2); andaccordingly, we define

Wn(x) =Vn(S1, S2)

S2, x =

S1

S2.

This corresponds to the choice of S2 as the numeraire.

In the continuation region, the governing equation for Wn(x) takes the form2

2x2

d2Wn

dx2+ (q1 q2)x

dWn

dx q2Wn = 0, xwn < x < xrn,

where 2

= 2

1 212 + 2

2, xw

m and xr

n are the threshold values for x atwhich the holder should optimally withdraw and reset, respectively.


22/30

Boundary conditions

Wn(xwn ) = 1 and W

n(x

wn ) = 0,

Wn(xrn) = x

rnWn1(1) and W

n(x

rn) = Wn1(1).

Upon withdrawal, Vn(S1, S2) becomes S2 and so we have Wn(xwn ) = 1.

When the holder resets at x = xrn

, the option writer has to supply enough funding toincrease the number of units of the primary fund such that the new fund value equals S1.

The corresponding number of units equals xrn

, which is the ratio of the fund values atthe reset threshold xr

n. Subsequently, the protection fund has one reset right less and x

becomes 1 since the values of the new upgraded fund and guaranteed fund are identical

upon reset.

Hence, the value of the protection fund at reset threshold becomes xrnWn1(1).


23/30

Limiting case: infinite number of reloads

Consider the limiting case n

, we have W(xr) = x

rW(1).

The equation is seen to be satisfied by xr = 1.

This represents immediate reset whenever St2 falls to St1, given that theholder has infinite number of reset rights.

The corresponding derivative condition becomes W(1) = W(1). Thisis because the value of V(S1, S2) is unaffected by marginal changes in S2when S2 is close to S1.


24/30

Monotonicity properties on price functions and threshold values

Price functions

W1(x) < W2(x) < < W(x)

Threshold values

xw1 > xw2 > x2

xr1 > xr2 >

> xr = 1.


25/30

Price function: W1

(x)

W1(x) = g x

xw1 , where g(z) =2z

1

1z

2

2 1 , z > 0, 1 0 and 2 1.

Here, 1 and 2 are the roots of2

2( 1) + (q2 q1) q2 = 0.

Note that W1(xw1 ) = 1 and W

1(x

w1 ) = 0 are automatically satisfied.

The other two boundary conditions, W1(xr1) = x

r1 and W

1(x

r1) = 1, lead to

xw1 = 1

1 1(11)/(21) 2

2 1(21)/(21)

,

xr

1=

11 1

1/(21)

2

2 12/(21)

.


26/30

Price function: Wn(x)

By setting

xr1 = Wn1(1)xrn and x

w1 = Wn1(1)x

wn

and considering the function g x

xwn , we observe thatg

xrnxwn

= xrnWn1(1) and g

xrnxwn

= Wn1(1).

Hence, we deduce that

Wn(x) = g

x

xwn

, xwn < x < x

rn.

Once we know xw1 , xr1 and W1(x), we can compute W1(1) = g 1xw1 ; then weapply the recursive relations to obtain xw2 and x

r2, also W2(x) = g

x

xw2

and

W2(1) = g 1xw2

, and so forth.


27/30

Price function: W(x)

2

2 x2dW2

dx2 + (q1 q2)x

dW

dx q2W = 0, xw < x < 1,subject to the auxiliary conditions:

W(xw) = 1 and W

(x

w) = 0,

W(1) = W(1).

The solution to W(x) is easily seen to be

W(x) =

h(x)

h(xw), xw < x < 1,

where

h(x) = (2 1)x1 (1 1)x2, x > 0.


28/30

The boundary condition W(xw) = 1 is satisfied by the inclusion of the

multiplicative factor 1/h(xw) in W(x).

The optimality condition, W

(xw

) = 0, gives the following algebraic equa-tion for xw:

h(xw) = 1(2 1)(xw)1 2(1 1)(xw)2 = 0.

Alternatively, from the recursive relationsxw1xr1

=xwxr

= xw.

In summary,

W(x) = g

x

xw

=

h(x)

h(xw), xw < x < 1.


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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.81

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Wn

(x)

x

n=1n= n=2n=3

The figure shows the plots of the price functions of the protection fund Wn(x)

against x, with varying number of reset rights n.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1/n

Threshold

values

xn

r

xn

w

We plot the threshold values, xwn and xrn, at which the holder of the protection

fund should optimally withdraw and reset, respectively, against the reciprocalof the number of reset rights, 1/n.

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Free boundary value problems in finance