Free boundary value problems in mathematical finance

presented by

Yue Kuen Kwok

Department of Mathematics

December 6, 2002

* Joint work with Min Dai and Lixin Wu

Rocket science on Wall Street

• The volatility of stock price returns is related to the viscosity effect in molecular diffusion. • The expected rate of returns is related to convective velocity. The option pricing equation is closely related to the convective diffusion equation in engineering.

Brownian motion was first studies in theoretical aspects by Bachelier (1900) in the modeling of stock price movements, a few years earlier than Einstein (1905).

OPTIMAL SHOUTING POLICIES OF OPTIONS

WITH RESET RIGHTS

presented by

Yue Kuen Kwok

Hong Kong University of Science and Technologyat

International Conference on Applied Statistics,

Actuarial Science and Financial Mathematics

December 17-19, 2002

* Joint work with Min Dai and Lixin Wu

The holders are allowed to reset certain terms of the derivativecontract according to some pre-specified rules.

Terms that are resettable• strike prices• maturities

Voluntary (shouting) or automatic resets; other constraints onresets may apply.

Examples

1. S & P 500 index bear market put warrants with a 3-month reset(started to trade in 1996 in CBOE and NYSE)- original exercise price of the warrant, X

= closing index level on issue date- exercise price is reset at the closing index level St on the reset

date if St > X (automatic reset)

Reset-strike warrants are available in Hong Kong and Taiwan markets.

2. Reset feature in Japanese convertible bonds

• reset downward on the conversion price

Sumitomo Banks 0.75% due 2001

Issue date May 96

First reset date 31 May 1997

Annual reset date thereafter 31 May

Reset calculation period 20 business day period, excluding holidays in Japan, ending 15 trading days before the reset day

Calculation type Simple average over calculation period

3. Executive stock options

• resetting the strike price and maturity

4. Corporate debts

• strong incentive for debtholders to extend the maturity of adefaulting debt if there are liquidation costs

5. Canadian segregated fund

- Guarantee on the return of the fund (protective floor);guarantee level is simply the strike price of the embedded put.

- Two opportunities to reset per year (at any time in the year) for10 years. Multiple resets may involve sequentially reduced guarantee levels.

- Resets may require certain fees.

Objectives of our work

Examine the optimal shouting policies of options with voluntary reset rights.

Free boundary value problems

• Critical asset price level to shout;• Characterization of the optimal shouting boundary for one-shout

and multi-shout models (analytic formulas, numerical calculations and theoretical analyses)

Resettable put optionThe strike price is reset to be the prevailing asset price at the shouting moment chosen by the holder.

tS(S

,S(X

Tt

T

at time occurs shouting if)0,max

occursshout no if)0max payoff Terminal

Shout call option

occursshout no if)0,max

, at time occurs shouting if)max payoff Terminal

X(S

XStXX,S(S

T

ttT

Shout floor (protective floor is not set at inception)Shout to install a protective floor on the return of the asset.

occursshout no if0

at time occurs shouting if)0max payoff Terminal

t,S(S Tt

Relation between the resettable put option and the shout call option

,)0,max(

)0,max( )(

)0,max(

),max(

Tt

TT

T

tT

SS

SXXS

XS

XSXSSince

so

price of one-shout shout-call option= price of one-shout resettable put option + forward contract.

• Both options share the same optimal shouting policy.

• Same conclusion applied to multi-shout options.

Formulation as free boundary value problems• Both one-shout resettable put option and one-shout shout floor

become an at-the-money put option upon shouting.• Price function of an at-the-money put option is SP1(), where

)()()( 121 dNedNeP qr

where . and 122

1

2

ddqr

d

Linear complementarity formulation of the pricing function V(S, )

),(),( ,0)(2 12

22

2

SPSVrVV

SqrS

VS

V

.floor shout for 0

put resettablefor )0,max()0,(

0)()(2 12

22

2

SXSV

SPVrVS

VSqr

S

VS

V

Properties of P1()

(i) If r q, then

).,0( for 0)(1

Ped

d q

(ii) If r > q, then there exists a unique critical value such that),0(1

,0)(1

|1

Pe

d

d q

and

),(for 0 )(

),0(for 0 )(

11

11

Ped

d

Ped

d

q

q

eqP1()eqP1()

r qr > q

*

1

Price of the one-shout shout floor, R1(S, )

)( ),(1 gSSR

Substituting into the linear complementarity formulation:

.0)0(

0)()( )(

)()( ,0 )(

1

1

g

Pgged

d

Pgqed

d

q

q

(i) r q

;0)0( and 0for 0 )( 11 PPed

d q

so).,0( ),()( 1 Pg

(ii) r > q

].,0(for )()( 11 Pg

When > , we cannot have g() = P1() since this leads to

a contradiction. Hence,

Solving

*

1

,0)( )( 1

Ped

dge

d

d qq

).,(for 0)( 1

ged

d q

).,(for )()( 111)( 1

Peg q

*

1

g(

P1()

P1( )*

1eq(

1

Optimal shouting policy of the shout floor

• does not depend on the asset price level (due to linear homogeneityin S)

• when ,

inferring that the holder should shout at once.

)(),(,0)( 111

SPSRPed

d q

Summary(i) r q, (0, ) or (ii) r > q, *

holder should shout at once

(ii) r > q, *

holder should not shout at any asset price level.

1

1

Optimal shouting boundary for the reset put option

Asymptotic behavior of S*() as 0+

1S*(0+) = X, independent of the ratio of r and q.

1Proof

).0,max()0,( ,0)),((

0)),(( and ),0(

0 ),(0

)()(')(2

)(),(),(

111

111

1

1111

21

22

21

111

SXSDSS

D

SDXeD

SS

qPPSrDS

DSqr

S

DS

D

SPSVSD

r

1

Note that D1(S,) 0 for all ; S[P() + qP1()] as 0+. Assuming S*(0+) > X, then for S (X, S*(0+)), we have

1 1

,0)0()(')0,( 111

qPPSSD

a contradiction.

Financial intuition dictates that S*(0+) X.1

Asymptotic behaviour of S*() as 1

S*

1, exists when r < q; this is linked with the existence of the limit:1)(lim 1

Pee

for r < q.

Write

.1)(

,)( ,)0(

0 ,0)(2

).,(lim)( );,(),(

,11

,1,111

,11

21

22

2

111

SdS

dW

SSWXW

SSdS

dWSqr

dS

WdS

SWSWSVeSW r

,0 ,)1(

)( ,11

11

SSSXXSW

XS

1

1.1where and ( becomes zero when r = q). 2

)(2

rq

When r > q, V1(S,) R1(S, ) > SP1() for > *.It is never optimal to shout at > *.

Lemma For r > q and 0 < *, there exists a critical asset price S*(0)such that V1(S, 0) = SP1(0) for S S*(0).

1

1

1

1

1

Integral equation for S*()1

0 1.11111 )]([)()()),(()()( duuPedu

ddNeSSPPS qu

uq

E

where

.

)(ln 2)(

)(

,1

2

1

1

u

uqrd uS

S

u

Pricing formulation of the n-shout resettable put option

Terminal payoff where t is the last shouting instant, 0 n.

),max( 0,Tt SS

Define then

).0,max()0,(

0)( )(2

),(),( ,0)(2

),1;,1()(

2

22

2

2

22

2

1

SXSV

SPVrVS

VSqr

S

VS

V

SPSVrVS

VSqr

S

VS

V

XVP

n

nnnnn

nnnnnn

nn

* The analytic form for Pn(), n > 1.

Properties of the price function and optimal shouting boundaries

(i) r < q, S*() is defined for (0, )n

XS

nSS

n

nn

)0(

2,1 ),()(1

nS*() is an increasing function of with a finite asymptotic valueas .

(ii) r < q, S*() is defined only for (0, *), where * is given bythe solution ton n n

. 11

1 and

1

1

111

)1()1(

1

,3

)1(

1

,2

SS

.0)(

n

q Ped

d

• The behaviors of the optimal shouting boundaries of the resettableput options depends on r > q, r = q or r < q.

• Monotonic properties(i) an option with more shouting rights outstanding should have

higher value;(ii) the holder shouts at a lower critical asset price with more

shouting rights outstanding;(iii) the holder chooses to shout at a lower critical asset price for

a shorter-lived option;(iv) the critical time earlier than which it is never optimal to shout

increases with more shouting rights outstanding.• Analytic price formula of the one-shout shout floor and integral

representation of the shouting premium of the one-shout resettableput are obtained.

Summaries and conclusions