Extensions to the Black-Scholes bob/math472/extensions/main.pdf · Extensions to the Black-Scholes

Extensions to the Black-Scholes EquationMATH 472 Financial Mathematics

J. Robert Buchanan

2018

Objectives

In this lesson we will learn to:I model the value of a security paying dividends at a

continuous rate,I price European options on securities that pay dividends at

a continuous rate.

Dividends

I We have versions of the Put-Call Parity formula whichinclude the effects of dividends:

Pe + Se T = Ce + K er T (continuous)

Pe + S(0) n

i=1

S(ti )er ti = Ce + K er T (discrete)

I We do not have pricing formulas for the optionsthemselves. We explore modifications and extensions tothe Black-Scholes partial differential equation and itssolution in this lesson.

Basic Problem for the European Call

The non-dividend-paying stock is assumed to obey thestochastic process

dS = S dt + S dW (t)

and the European call solves the initial boundary valueproblem:

r F = Ft + r S FS +122S2FSS for (S, t) in [0,) [0,T ],

F (S,T ) = (S(T ) K )+ for S > 0,F (0, t) = 0 for 0 t < T ,F (S, t) = S K er(Tt) as S .

Stock Pays Continuous DividendsAssumption: the stock pays dividends at a continuous rateproportional to the value of the stock

I What is a suitable expression for the dividend yield(dividend paid per unit time)?

dividend per unit time = S

I How much dividend is paid in a short time interval dt?

dividend paid = S dt

I What stochastic differential equation would the value of thestock paying a continuous proportional dividend obey?

dS = ( )S dt + S dW (t)

Stock Pays Continuous DividendsAssumption: the stock pays dividends at a continuous rateproportional to the value of the stock

I What is a suitable expression for the dividend yield(dividend paid per unit time)?

dividend per unit time = S

I How much dividend is paid in a short time interval dt?

dividend paid = S dt

I What stochastic differential equation would the value of thestock paying a continuous proportional dividend obey?

dS = ( )S dt + S dW (t)

Suppose F (S, t) is the value of a European call option on thestock paying a continuous dividend. F obeys the followingstochastic differential equation:

dF =(

( )S FS +122S2FSS + Ft

)dt + S FS dW (t).

As before, we wish to eliminate the random part of this equationby creating a portfolio of a long position in the call option and ashort position in shares of the stock.

= F ()S

Suppose F (S, t) is the value of a European call option on thestock paying a continuous dividend. F obeys the followingstochastic differential equation:

dF =(

( )S FS +122S2FSS + Ft

)dt + S FS dW (t).

As before, we wish to eliminate the random part of this equationby creating a portfolio of a long position in the call option and ashort position in shares of the stock.

= F ()S

Change in Portfolio ValueOne share of stock pays S dt in dividends during a timeinterval of length dt , thus shares of stock pays

()S dt in dividends.

The portfolio changes in value

d = d(F ()S) ()S dt= dF ()dS ()S dt

=

(( )S FS +

122S2FSS + Ft

)dt + S FS dW (t)

() (( )S dt + S dW (t)) ()S dt

=

(( )S(FS ) +

122S2FSS + Ft ()S

)dt

+ S(FS ) dW (t).

Change in Portfolio ValueOne share of stock pays S dt in dividends during a timeinterval of length dt , thus shares of stock pays

()S dt in dividends.

The portfolio changes in value

d = d(F ()S) ()S dt= dF ()dS ()S dt

=

(( )S FS +

122S2FSS + Ft

)dt + S FS dW (t)

() (( )S dt + S dW (t)) ()S dt

=

(( )S(FS ) +

122S2FSS + Ft ()S

)dt

+ S(FS ) dW (t).

Eliminating RandomnessChoose = FS and the portfolio obeys the stochasticdifferential equation:

d =(

122S2FSS + Ft S FS

)dt .

In the absence of arbitrage the change in the value of theportfolio should be the same as the interest earned by aequivalent amount of cash.

d = r(F ()S) dt

Thus the Black-Scholes partial differential equation for thestock paying continuous dividends becomes

r F = Ft +122S2FSS + (r )S FS.

Eliminating RandomnessChoose = FS and the portfolio obeys the stochasticdifferential equation:

d =(

122S2FSS + Ft S FS

)dt .

In the absence of arbitrage the change in the value of theportfolio should be the same as the interest earned by aequivalent amount of cash.

d = r(F ()S) dt

Thus the Black-Scholes partial differential equation for thestock paying continuous dividends becomes

r F = Ft +122S2FSS + (r )S FS.

Similarities with Non-Dividend-Paying Stocks

I Payoff of the call option at expiry: F (S,T ) = (S(T ) K )+.I Boundary condition at S = 0 is F (0, t) = 0.I Boundary condition as S :

F (S, t) = Pe + S e(Tt) K er(Tt)

limS

F (S, t) = limS

(Pe + S e(Tt) K er(Tt)

)= S e(Tt) K er(Tt).

Change of Variables

Define the function G(S, t) = e(Tt)F (S, t), then

G(S,T ) = e(TT )F (S,T ) = (S(T ) K )+

G(0, t) = e(Tt)F (0, t) = 0

limS

G(S, t) = e(Tt)(

S e(Tt) K er(Tt))

= S K e(r)(Tt)

FS = e(Tt)GSFSS = e(Tt)GSS

Ft = e(Tt)(G + Gt ).

Substitute these expressions into the partial differentialequation, boundary conditions, and the final condition for theEuropean call option on the stock paying continuous dividends.

Initial Boundary Value Problem

(r )G = Gt +122S2GSS + (r )S GS

G(S,T ) = (S(T ) K )+

G(0, t) = 0lim

SG(S, t) = S K e(r)(Tt)

Remark: this is exactly the same initial boundary value problemwe have already solved except r has been replaced by r .

Black-Scholes Option Pricing Formulas

For a stock paying a continuous, proportional dividend at rate the value of a European options are given by the formulas

w =ln(S/K ) + (r + 2/2)(T t)

T tCe, = S e(Tt) (w) K er(Tt)

(w

T t

)Pe, = K er(Tt)

(

T t w) S e(Tt) (w)

Comparison

S(0) = 100, = 0.05, r = 0.0325, = 0.25, K = 100,T = 3/12, t = 0.

90 100 110 120S(T)

5

10

15

20

Ce

with dividendwithout dividend

Example: Call OptionSuppose the current price of a security is $62 per share. Thecontinuously compounded interest rate is 10% per year. Thevolatility of the price of the security is = 20% per year. Thestock pays dividends continuously at a rate of = 3% per year.Find the cost of a five-month European call option with a strikeprice of $60 per share.

T =512, t = 0, r = 0.10, = 0.20,

S = 62, K = 60, = 0.03

Using the formula for w and Ce, we have

w 0.544463Ce, $5.24

Without the dividend we calculate Ce $5.80.

T =512, t = 0, r = 0.10, = 0.20,

S = 62, K = 60, = 0.03

w 0.544463Ce, $5.24

T =512, t = 0, r = 0.10, = 0.20,

S = 62, K = 60, = 0.03

w 0.544463Ce, $5.24

Example: Put Option

Suppose the current price of a security is $97 per share. Thestock pays a continuous dividend at a yield of 6.5% per year.The continuously compounded interest rate is 8% per year. Thevolatility of the price of the security is = 45% per year. Findthe cost of a three-month European put option with a strikeprice of $95 per share.

T = 1/4, t = 0, r = 0.08, = 0.45, = 0.065, S = 97, K = 95.

Using the formulas for w and Pe, we obtain

w 0.221763Pe, $7.34

Example: Put Option

Suppose the current price of a security is $97 per share. Thestock pays a continuous dividend at a yield of 6.5% per year.The continuously compounded interest rate is 8% per year. Thevolatility of the price of the security is = 45% per year. Findthe cost of a three-month European put option with a strikeprice of $95 per share.

T = 1/4, t = 0, r = 0.08, = 0.45, = 0.065, S = 97, K = 95.

Using the formulas for w and Pe, we obtain

w 0.221763Pe, $7.34

Useful Result

Lemma

S e(Tt) (w) = K er(Tt)(

w

T t)

A New Greek

The rate of change in the price of a European call option on astock paying continuous dividends is

Ce, =Ce,

= S(T t)e(Tt) (w) .