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CHAPTER 7. MODIFICATION OF BLACK-SCHOLES MODEL 78

Chapter 7

Modification of Black-Scholes

model

7.1 Dividend

7.1.1 Continuous dividend yield model

Let q denote the constant continuous dividend yield which is known.1 In other

words, the holder receives a dividend, qSdt, within the time interval dt. The share

value is lowered after the dividend payout so that the expected rate of return µ

of a share becomes µ − q by the continuous dividend yield q. The geometric

Brownian motion model becomes

dS

S= (µ − q)dt + σdZ (7.1)

1The continuous yield model is extremely useful to options on foreign currencies where the

continuous dividend yield can be considered as the yield due to the interest earned by the foreign

currency at the foreign interest rate rf . In the pricing model for a foreign currency call option,

we can simply set q = rf . S is the domestic currency price of a unit of foreign currency and the

exchange rate is assumed to follow the lognormal diffusion process. The model in this section

applied to the currency option, assumes that both the domestic and foreign interest rates are

constant.


By having ∆ number of the share, a share holder gains a dividend, (qdt)∆S. The

value of a portfolio

−1 call option

+∆ underlying shares

is Π = −c + ∆S. The differential of the value of the portfolio is

dΠ = −dc + ∆dS + q∆Sdt

=

(

−∂c

∂t− σ2

2S2 ∂2c

∂S2+ q∆S

)

dt +

(

∆ − ∂c

∂S

)

dS (7.2)

where Eq.(5.1) has been used.

The portfolio becomes non-stochastic by choosing ∆ = ∂c∂S

. The portfolio is

then risk-free and should earn the risk-free interest rate, r. Thus

dΠ =

(

−∂c

∂t− σ2

2S2 ∂2c

∂S2+ qS

∂c

∂S

)

dt = r

(

−c + S∂c

∂S

)

dt (7.3)

By rearranging the equation above, we have the modified form of the Black-

Scholes equation:

∂c

∂t+

σ2

2S2 ∂2c

∂S2+ (r − q)S

∂c

∂S− rc = 0 (7.4)

Solving the equation, we find the call value

c = Se−qτN(d̂1) − Xe−rτN(d̂2), (7.5)

where τ = T − t is the remaining time to mature and

d̂1 =ln S

X+ (r − q + σ2

2 )τ

σ√

τ, d̂2 = d̂1 − σ

√τ (7.6)

Similarly, the value of a put option is

p = Xe−rτN(−d̂2) − Se−qτN(−d̂1). (7.7)

The put call parity can be derived using Eqs.(7.5) and (7.7),

p = c − Se−qτ + Xe−rτ , (7.8)

and the following put-call symmetry relation,

c(S, τ ;X, r, q) = p(X, τ ;S, q, r). (7.9)


7.1.2 Discrete dividend

Suppose the underlying asset pays N discrete dividends at known payments dates

t1, t2, · · ·, tN of amounts D1,D2, · · ·,DN , respectively. If the actual amounts of

dividends and ex-dividend dates are known then we can assume that the asset

price is composed of two components:

· risk-free component that will be used to pay the known dividends during

the life of the option.

· risky component which follows a stochastic process.

The risk-free component is taken to be the present value of all future dividends

discounted at the risk-free interest rate. The value of the risky component S̃t is

S̃t = St − D1e−rτ1 − D2e

−rτ2 − · · · − DNe−rτN for t < t1

S̃t = St − D2e−rτ2 − · · · − DNe−rτN for t1 < t < t2

·

·

·

S̃t = St for t > tN (7.10)

where τi = ti − t. The volatility of the risky component is customarily taken as

the volatility of the whole asset price multiplied by the factor St/(St −D), where

D is the present value of the future discrete dividends.

7.2 Time dependent parameters

The parameters, σ, r, q, are normally time-dependent. Suppose these parameters

are known functions of time, Eq.(7.4) becomes

∂c

∂τ=

σ2(τ)

2S2 ∂2c

∂S2+ [r(τ) − q(τ)]S

∂c

∂S− r(τ)c, τ = T − t. (7.11)


Introducing a new parameter, y = ln S, Eq.(7.11) is for w = exp[∫ τ0 r(τ ′)dτ ′]c,

∂w

∂τ=

σ2(τ)

2

∂2w

∂y2+

[

r(τ) − q(τ) − σ2(τ)

2

]

∂w

∂y. (7.12)

Its fundamental solution is

φ(y, τ) =1

√

2π∫ τ0 σ2(u)du

exp

−{y +∫ τ0 [r(u) − q(u) − σ2(u)

2 ]du}2

2∫ τ0 σ2(u)du

. (7.13)

The solution of Eq.(7.12)is

w(y, τ) =

∫ ∞

−∞w(ξ, 0)φ(y − ξ, τ)dξ (7.14)

which gives the call value

c = SN(d̃1) exp

(

−∫ τ

0q(u)du

)

− XN(d̃2) exp

(

−∫ τ

0r(u)du

)

(7.15)

where

d̃1 =1

√

∫ τ0 σ2(u)du

{

lnS

X+

∫ τ

0

[

r(u) − q(u) +σ2(u)

2

]

du

}

d̃2 = d̃1 −√

∫ τ

0σ2(u)du. (7.16)

7.3 Transaction costs

There have been continual efforts to construct hedging strategies that best repli-

cate the payoffs of derivative securities in the presence of transaction costs. The

transaction cost occurs in buying and selling. Suppose that the transaction cost

is proportional to the amount of money involved in the transaction and the rate

of proportion is k2 . For buying (+) or selling (-) of |α| shares at the price S, the

transaction cost is k2 |α|S.

Let us consider a portfolio of ∆ number of shares and B dollars in a risk-free

account whose value is

Π = ∆S + B (7.17)


Let δΠ, δS, δ∆ be the change in Π, S and ∆ in time δt, respectively. From

Eq.(7.17)

δΠ = ∆ δS + rδt B − k

2|δ∆|S, (7.18)

Let f(S, t) denote the value of the option which is replicated by the above port-

folio, i.e., f = Π. Using Ito’s lemma,

δf =∂f

∂SδS +

(

∂f

∂t+

σ2

2S2 ∂2f

∂S2

)

δt. (7.19)

Because f = Π, δf = δΠ. From Eqs.(7.18) and (7.19),

∆ =∂f

∂S(7.20)

rBδt − k

2|δ∆|S =

(

∂f

∂t+

σ2

2S2 ∂2f

∂S2

)

δt. (7.21)

Using Eq.(7.20) and the fact f = ∆S + B, we get

B = f − ∂f

∂SS. (7.22)

Leland2 found thatk

2S|δ∆| ≈ σ2

2Le S2

∣

∣

∣

∣

∣

∂2f

∂S2

∣

∣

∣

∣

∣

δt, (7.23)

where Le =√

2π

(

k

σ√

δt

)

is called the Leland number.

Using Eqs.(7.22) and (7.23), Eq.(7.21) is written as

∂f

∂t+

σ2

2S2 ∂2f

∂S2+

σ2

2Le S2

∣

∣

∣

∣

∣

∂2f

∂S2

∣

∣

∣

∣

∣

+ rS∂f

∂S− rf = 0 (7.24)

which can be rearranged as follows

∂f

∂t+

σ̃2

2S2 ∂2f

∂S2+ rS

∂f

∂S− rf = 0, (7.25)

with the modified volatility,

σ̃2 = σ2[1 + Le sign(Γ)] , Γ =∂2f

∂S2(7.26)

2H.E. Leland, ”Option pricing and replication with transaction costs,” Journal of Finance,

vol. 40 (1985) p.1283-p.1301


This form resembles the Black-Scholes equation.

Eq.(7.26) can be a problem because the modified volatility σ̃2 can become

negative when Γ < 0 and Le > 1. However, as we discussed earlier, Γ is positive

for a simple European call and put options. Assuming σ̃2 positive, we can solve

Eq.(7.26) as we solve the Black-Scholes equation with the modified volatility σ̃.

7.4 Futures options

Options of futures contracts are traded on many different exchanges, as some

examples are shown in Table 7.1. The futures options require the delivery of an

underlying futures contract when exercised.

· If a call futures option is exercised, the holder acquires

a long position in the underlying futures contract + cash

( = current futures price - strike price)

· If a put futures option is exercised, the holder acquires

a short position in the underlying futures contract + cash

( = strike price - current futures price)

Example

The strike price of an April futures call option on 1,000 tones of copper is

£1, 200 per ton. On the expiration date (say, March) of the option, the spot

copper futures price is £1, 300 per ton. The holder of the call option then re-

ceives £100, 000(= 1, 000 × (£1, 300 −£1, 200)) plus a long position in a futures

contract to buy 1,000 tonnes of copper on the April delivery date.

The maturity date of the options contract is generally on or a few days before

the earliest delivery date of the underlying futures contract.

Why are futures options popular?


Table 7.1: Some examples of futures options. (NYMEX: New York Mercantile Exchange, LME:

London Metal Exchange, CME: Chicago Mercantile Exchange.)

Futures options Exchanges

Treasury-bond futures CBOT

Crude oil futures NYMEX

Copper futures LME

Cattle live futures CME

· For most commodities, it is more convenient to deliver futures contract on

the asset rather than the asset itself. Example: crude oil.

· In most circumstances, the underlying futures contract is closed out prior

to delivery. Futures options are therefore settled in cash, which particu-

larly appeals to investors with limited capital because less amount of initial

capital is needed.

· Futures options facilitate hedging, arbitrage and speculation, which makes

the markets more efficient.

· Transaction costs for futures options are normally smaller than those for

spot options.

7.4.1 Black equation for futures option

The time-dependence of a futures price is the same as that of the underlying asset.

Thus we suppose the futures price F follows the geometric Brownian motion:

dF

F= µF dt + σF dZ (7.27)

where µF and σF are, respectively, the constant expected rate of return and the

constant volatility, and dZ is the standard Wiener process. By Ito’s lemma, the

volatility of the futures price is given by

σF =1

FσS

∂F

∂S=

σ

FSerτ = σ (7.28)


where σ is the volatility for the underlying asset. From Ito’s lemma,

df =

(

∂f

∂FµF F +

∂f

∂t+

1

2

∂2f

∂F 2σ2F 2

)

dt +∂f

∂FσFdZ (7.29)

where f is the value of the derivative. For the case of a call option f = c and for

the case of a put option f = p.

Consider a portfolio composed of

−1 derivative

+∂f

∂Fnumber of futures contracts

The value, Π, of the portfolio is

Π = −f, (7.30)

at the opening of the contract because it costs nothing to enter into a futures

contract. Let dΠ, df and dF be their changes of Π, f and F in time dt. In time

dt, the holder of the portfolio earns capital gains equal to −df from the derivative

and income of ∂f∂F

dF from the futures contract. The change, dΠ, of the wealth

of the portfolio in time dt is

dΠ =∂f

∂FdF − df (7.31)

Substituting Eqs.(7.27) and (7.29) into Eq.(7.31) we find the change in wealth

dΠ =

(

−∂f

∂t− 1

2

∂2f

∂F 2σ2F 2

)

dt (7.32)

which is non-stochastic and risk-free. Therefore the portfolio should give the

risk-free rate of return, r:

dΠ = rΠdt. (7.33)

We find the Black equation for futures derivatives by using Eqs.(7.30), (7.32) and

(7.33):∂f

∂t+

1

2

∂2f

∂F 2σ2F 2 = rf. (7.34)

This has the same form as Eq.(7.4) with q = r.


By solving the Black equation (7.34) for futures call and put options, we obtain

c = e−rτ [FN(d̆1) − XN(d̆2)]

p = e−rτ [XN(−d̆2) − FN(−d̆1)] (7.35)

where

d̆1 =1

σ√

τ

(

lnF

X+

σ2

2τ

)

, d̆2 = d̆1 − σ√

τ . (7.36)

We can easily derive the following put-call parity.

p + F e−rτ = c + Xe−rτ . (7.37)

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