LECTURE 8.I

Options on dividend-paying assets

Adriana Breccia

February 23, 2009

1

1 Options on Dividend-paying Stocks

The market includes a stock St and a riskless asset Bt with dynamics:

dSt = µStdt+ σStdW (1)

dB = rBdt (2)

Assume that the stock pays a dividend at some fixed rate k and define a self-financing portfolio.

For the portfolio to be self-financing we need to fully reinvest the dividend payment. Therefore

Vt = φSt + ψBt (3)

dV = φ(dS + kSdt) + ψBt (4)

dV = φdS + (φkS + ψrB)dt (5)

As usual, the option payoff is a function of S and t, and by Ito’s lemma (for sake of brevity I

might omit the time subscript)

df(S, t) =∂f

∂SdS +

∂f

∂tdt+

∂2f

∂S2

σ2

2S2dt (6)

Matching coefficients of dS and dt in 5 and 6 gives:

φ =∂f

∂S(7)

φkS + ψrB =∂f

∂t+∂2f

∂S2

σ2

2S2 (8)

and therefore

ψ =1

rB

(∂f

∂t+∂2f

∂S2

σ2

2S2 − ∂f

∂SkS

).

Having found the self-financing strategy we can easily derive the Black and Scholes PDE for

the dividend case:

V = φS + ψB (9)

=∂f

∂SS +

1

rB

(∂f

∂t+∂2f

∂S2

σ2

2S2 − ∂f

∂SkS

)B (10)

rV = (r − k)∂f

∂SS +

∂f

∂t+∂2f

∂S2

σ2

2S2 (11)

2

1.1 Risk neutral valuation

We can now turn to the risk neutral valuation and see how the presence of a dividend yield

changes the martingale pricing (with respect to the result in lecture 1). First, note that the

total gain from holding the stock, say G is

Gt = St + cumulative dividend (12)

= St +∫ t

0kSτdτ (13)

that is,

dGt = dSt + kStdt

Second, define an auxiliary process, S∗ = ektS, and notice that by applying Ito’s lemma to S∗

gives

dS∗ = (µ+ k)S∗ + σS∗dW.

Thus we can write dG as function of the auxiliary process

dG = (µ+ k)Sdt+ σSdW = e−ktdS∗.

Now, back to the self-financing portfolio, we can use the notation G in the self-financing

portfolio and derive the dynamics of the discounted portfolio1

dV = φdG+ ψdB (14)

= φe−ktdS∗ + ψdB (15)

dV = φe−ktdS∗ (16)

where a proof of the last equation is in Appendix. It is easy to show (by Girsanov theorem)

that there exist an equivalent probability measure Q such that the process S∗ is a martingale

1We use the usual˜notation to denote discounting, e.g dVt = d(Vt/Bt), dS∗t = d(S∗

t /Bt).

3

(that is, with driftless dynamics) under Q. As under Q dWQ = dW P + γdt is a standard

Brownian motion, then

dS∗ = (µ+ k − r)S∗dt+ σS∗(dWQ − γdt) (17)

dS∗ = (µ+ k − r − σγ)S∗dt+ σS∗dWQ (18)

and with γ = µ+k−rσ

the drift is zero. Therefore we can rewrite equ. 16 as

dV = φe−ktσS∗dWQ (19)

= φe−ktσS∗

Bt

dWQ (20)

= φe−ktσektS

Bt

dWQ (21)

= φσS

Bt

dWQ (22)

= φσSdWQ (23)

Therefore dV is also a martingale under Q (however notice that S is not a martingale under

Q). This tells us that we can still price any claim with payoff X = VT by martingale pricing,

that is

EQ(VT | Ft) = Vt (24)

EQ(VTBT

Bt | Ft) = EQ(VT e−r(T−t) | Ft) = Vt (25)

however, because we have found γ = µ+k−rσ

, then the dynamic of St under Q has drift r − k(instead of r as in the non-dividend case). This is all we need in practice in order to price

under Q.

4

Remark

Why St is not a martingale under Q? The economic intuition goes through the dynamic of G

under Q. In particular, define the discounted gain G as

Gt =StBt

+∫ t

0kSτBτ

dτ (26)

= St +∫ t

0kSτdτ (27)

hence

dG = dS + kSdt (28)

= (µ+ k − r)Sdt+ σSdW (29)

= e−ktdS∗ (30)

therefore we could have also written dV = φdG. It is interesting to notice that dG under Q

has zero drift, therefore G is a martingale under Q. We can then write the following

Gt = EQ(GT | Ft) (31)

e−rtSt +∫ t

0e−rτkSτdτ = EQ(e−rTST +

∫ T

0e−rτkSτdτ | Ft) (32)

St = EQ(e−r(T−t)ST +∫ T

te−r(τ−t)kSτdτ | Ft) (33)

where the last line tell us that the risk neutral valuation principle still applies, that is, under

Q the current price of an asset is equal to ALL future expected payoffs (discounted by r)

generated by the asset. It should be clear that of course this is why St cannot be a martingale.

5

Appendix

dV =dV

B− V

B2dB (34)

=φe−ktdS∗ + ψdB

B− V

B2dB (35)

= φe−kt(dS∗ +

S∗

B2

)+ ψ

dB

B− V

B2dB (36)

= φe−ktdS∗ + (φe−ktS∗ + ψB − V )︸ ︷︷ ︸ dBB2(37)

= φe−ktdS∗ + 0 (38)

6

2 Exercise

Ex.1. Options on an exchange rate. Suppose the US dollar risk-free rate is r; the British

pound risk-free rate is q, and the dollar value of one pound is lognormal, i.e. the exchange

rate Ct with units dollars/pound satisfies

dC = µCdt+ σCdw

.

1.1 Price an option to buy a British pound at time T for K dollars.

(Hint: To a dollar investor the pound looks like a “stock with continuous dividend yield q”.)

1.2 What is the relation between the risk neutral probability of a dollar investor (say Q) and

that of a pound investor (Q)?

1.3 Consider again the option to buy a pound for K dollars, denote its payoff at T as f. The

option payoff in pounds is C−1T f . To the dollar investor its value at time 0 is e−rTEQ(f)

dollars. To the pound investor its value at time 0 is e−qTEQ(C−1T f ] pounds. Show the prices

are consistent. (Hint: you should verify that C0e−qTEQ(C−1

T f) = e−rTEQ(f).

Ex.2 Prove that dV = φdG without using the auxiliary process S∗ but only the definition in

equ. 26

7