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

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  • 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.

  • 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.

  • 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.

  • 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) .

    For a European put option

    Pe, =Pe,

    = S(T t)e(Tt) (1 (w)) .

  • Dividends Influence on Other Greeks

    The presence of the continuous dividend rate , in the Call andPut formulas alters the previously discussed Greeks.

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

    T tCe, = S e(Tt) (w) Ker(Tt)

    (w

    T t

    )Pe, = Ker(Tt)

    (w

    T t

    ) S e(Tt) (w)

    Find Delta, Gamma, Rho, Theta, and Vega.