derivatives theory and practice (2020) - chapter 8 options...

Chapter8 :Options Pricing (Financial Engineering : Derivatives And Risk Management) © K. Cuthbertson, D. Nitzsche

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CHAPTER 8

OPTIONS PRICING

LEARNING OBJECTIVES

To obtain a qualitative overview of what determines call and put premia.

To establish an arbitrage relationship between (European) put and call premia known as put-

call parity. For example, this enables us to easily calculate the put premium once we know

the call premium or to undertake riskless arbitrage with miss-priced calls or puts. It is an

example of financial engineering.

To demonstrate how the binomial option pricing model (BOPM) is used to determine option

premia by establishing a riskless portfolio consisting of a position in the underlying asset and

the option. This illustrates the principles of delta hedging and risk neutral valuation.

To illustrate how the Black-Scholes formula is used to provide an explicit equation to

determine call and put premia.

To show the linkages between the BOPM and the Black-Scholes approach.

In this chapter we first provide an intuitive feel for the determinants of call and put premia.

We then outline the put-call parity condition for European options. Next we present a detailed

account of the one and two-period BOPM which allows us to introduce the important idea of

constructing a riskless portfolio from two risky assets, namely a call and a stock. This is the

principle of delta hedging whereby changing proportions held in the stock and the option can

produce a riskless portfolio (over small intervals of time) and this enables us to price the option.

We generalise the BOPM to many periods. An alternative method of pricing the option is to

combine a stock and a specific amount of cash to replicate the payoff at maturity, of the option.

The option premium (at t=0) must then equal the known cost of this replication portfolio of the

“stock+cash”. Next we turn to the famous (and rather complex) Black-Scholes option pricing

formula and we try and extract as much intuition from it as we can. To help us get an intuitive feel

for the Black-Scholes approach we demonstrate its similarity to the BOPM when in the latter,

changes in the stock price are allowed to take place over a small interval of time.



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8.1. FACTORS INFLUENCING OPTION PREMIA

We have already discussed some of the factors which determine the payoffs from calls

and puts. Here we are concerned with intuitive arguments which help us to explain the

determination of option prices, that is how the price one pays (today) for a particular option varies

with the expiration date, strike price, etc. This intuitive approach will help us to understand the

mathematical formulae for option prices which we present later. The results of our intuitive

reasoning are summarised in table 8.1. Each factor is considered in turn, holding all the other

factors constant. In each case we assume the investor has a long position (ie. has purchased a

call or a put).

[Table 8.1 here - Word]

Table 8.1: Factors Affecting the Value of Options

Long

European

Call

Option

Long

European

Put

Option

Long

American

Call

Option

Long

American

Put

Option

Time to Expiration, T ? ? + + Current Stock Price, S + - + - Strike Price, K - + - +

Stock Return Volatility, + + + +

Risk Free Rate, r + - + -

Dividends - + - +

Notes:

“+” indicates a positive relationship between the option price and the variable chosen. That is, a rise(fall) in the variable is

accompanied by a rise (fall) in the option premium..

“-“indicates a negative relationship between the option price and the variable chosen. That is, a rise(fall) in the variable is

accompanied by a fall (rise) in the option premium.

“?” indicates the relationship could be “+” or “-“ depending on the precise circumstances.

TIME TO EXPIRATION (T)

Calls: An American call option with a long time to expiration T2 has all the exercise possibilities

of an equivalent option with a shorter time to maturity T1, but the former has the added

advantage that it can also be exercised between T1 and T2. Hence the American call

increases in value the longer the time to maturity. Also with a longer time to maturity the

option has more chance of ending up in-the-money. Since the buyer of the call has

limited downside risk, then an increase in T enhances the possibility of an upside profit,

so the American call premium increases.



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In general, a European call option will increase in value with time to maturity since the

option has more time to end up well in-the-money (and the downside is limited to the call

premium, should it end up out-of-the-money). However, a slightly pathological case

occurs if a very large dividend is expected in the near future as this will cause the stock

price to decline, so that a short life option could be worth less than a long life option.

Puts: An American put also benefits from a longer maturity for the same reasons as for the

American call. The put premium on a European option generally increases with time to

maturity but again pathological cases involving dividend payments are possible.

STRIKE PRICE (K) AND CURRENT STOCK PRICE (S)

Calls: European call options have a larger payoff the higher is the stock price ST at expiry,

relative to the strike price K. However, if stock price movements are random then the

higher the current stock price S relative to the strike price, the more likely it is that the

stock price at expiry will also be above the strike price. Hence the purchaser of a long

European call option will be willing to pay a higher price for the option, the higher is (S/K).

Similar arguments apply to American call premia which also increase with (S/K), since

they also have the advantage of early exercise. (Although it can be shown that for an

American call which does not pay dividends, early exercise is never optimal).

Puts: It should be obvious that the value (price) of a put option varies with S and K in the

opposite direction to that for a call. Hence the value of a long position in a European or

American put option depends negatively on (S/K).

VOLATILITY ()

Calls: The greater the volatility of the return on the share the greater the possible range of

share prices that might exist at the expiry date. But the owner of a European (or

American) long call option has limited downside risk. Hence, increased volatility

increases the chance of a high share price and hence high payoff for the call option (at

the expiry date) while downside risk is limited. The value of a long position in a call option

therefore increases with volatility.

Puts: The owner of a European (or American) long put benefits from price decreases but also

has limited downside risk from price increases. Hence the put premium increases with an

increase in volatility.



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At this point, it is useful to introduce a statistical result concerning the variance. The

variance of a sum of independent random variables is the sum of the variances of those variables.

Hence, if the variance of stock returns per day is 2

d and there are T-days to expiration, then the

variance over T- days is 22

dT T . Similarly if is the annual volatility (standard deviation) of

stock returns then the volatility over a shorter period (eg. T=1/4 of a year) is given by

T4/1 . The latter is often referred to as the “root-T rule”. Hence we might expect calls

and put premia (for an option which matures in 3 months) to depend on T4/1 .

RISK FREE INTEREST RATE

Calls: Firstly, an increase in interest rates reduces the present value of any future profits

from the option. This tends to directly reduce the value of a long call. Secondly, to the

extent that a general rise in interest rates increases the expected growth rate of stock

prices, this tends to increase the value of a long call. It can be shown that the second

effect dominates and the value of long position in a call increases with r.

Put: As a higher interest rate leads to a higher growth in the stock price this increases the

chance that the put will be out-of-the-money at expiry. A higher interest rate also reduces

the present value of any profits from the put. Hence a rise in interest rates reduces the

value of a put option.

It would be extremely useful if all of the above factors could be included in a single

equation to determine the call (C) or put premium (P). This ‘closed form’ solution is the Black-

Scholes equation for option premia and has the general form C (or P) = f(S, K, T, r, ). To work

out the exact functional form for the Black-Scholes equation is rather difficult. We therefore

postpone our heuristic presentation of Black-Scholes to later in this chapter after we have

explained put-call parity and the more intuitive binomial option pricing model, BOPM.

8.2. PUT-CALL PARITY: EUROPEAN OPTIONS

In this section we derive the put-call parity relationship. This enables us to calculate the

put premium if we know the call premium (for options with the same strike price and expiry date).

Hence if we know the formula for the price of a call (in terms of fundamental observable variables



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such as volatility, the underlying stock price, etc.) we can use the put-call parity relationship to

determine the price of a put (and vice-versa).

Simple Example

Basically put-call parity is an arbitrage relationship between (European) put and call

premia, the share price and holding a risk free asset (such as cash or a T-Bill which matures at

T).

Put-Call Parity (European Options) :

(long) share + (long) put = (long) call + cash (equal to Ke-rT

)

S + P = C + Ke-rT

We can nearly produce this result using our directional vectors to mimic the above

equation, namely {+1, +1} plus {-1, 0} equals {0, +1}, that is a long call - but we have ‘lost’ the

‘cash’. A more detailed analysis is given in figure 8.1. Consider the following portfolio:

Portfolio-A : Long Share + Long Put

[Figure 8.1 here - Powerpoint]

© K. Cuthbertson and D. Nitzsche

Figure 8.1 : Put-call parity

Profit (at expiry)

ST

$100

K = 100 110

Long share

Synthetic long call

plus K = $100 at T

Long put0

+1

+1

-1

At t = 0 : long share + long put = long call + cash of $Ke-rT

0

+1

Assume the current share price is S = $100 and (for expositional simplicity) the strike

price of the call and put are both K = $100 (However, note that put-call parity also holds when S

K). From figure 8.1 the payoff at expiry for portfolio-A is seen to be equivalent to holding a long

call plus $100 in cash (at t=T). To see more clearly consider the payoff for two alternative stock

prices at expiry, namely ST = 90 (<K) and ST = 110 (>K).

[Table 8.2 here – Word]



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Table 8.2: Value of Portfolios A and B at (t=T)

(Strike Price K = 100)

Stock Price Value of Stock +

Long Put at T

Value of long call B

+ cash 100e-rT

ST = 50 (< K) Vs + Vp = 50 + 50 = 100 $100 = K ST = 110 (> K) Vs + Vp = 110 + 0 = 110 $110 = ST

The payoff to the long put is Vp = max {K-ST, 0} and the value of the share is Vs = ST.

Portfolio-A therefore gives a dollar return of ST, for K < ST and of K(= (K-ST) + ST) for K > ST. We

will see that the “share plus put” has resulted in a payoff equivalent to that from portfolio-B:

Portfolio B: Long call + cash of $100e-rT

(at t=0 invested at r)

If you buy a ‘share+put’ this implies a payoff of either K = $100 (ie. exercise the put if ST <

K) or if ST > K the put is worthless (ie. don’t exercise the put) but you own one-share (worth ST). A

‘long call + cash of $100e-rT

also has a payoff of either one share worth ST (.ie. exercise the call if

ST > K and receive one share) or cash of $100 (ie. call expires worthless for ST < K) and ‘cash’

accrues to $100. Since the payoffs at T from portfolios A and B are the same, they must cost the

same today (at t=0) and this gives rise to the put-call parity relationship :

[8.1] S + P = C + Ke-rT

where S = stock price, P = put premium, C = call premium and Ke-rT

is held at t=0 in a risk free

asset. You must have cash of Ke-rT

at t=0 to ensure that you can pay the strike price K at t=T if

you choose to exercise the call. The cash amount Ke-rT

is often referred to as the ‘present value

of the strike price’ PV(K) and hence:

[8.2] PV(K) = Ke-rT

using continuously compounded rates

[8.3] PV(K) = K/(1+r)T using discrete compound rates

In analysing the above problem we have implicitly assumed that the options cannot be

exercised until t=T, that is, put-call parity only holds for European style options. Note that the

“signs” in [8.1] indicate whether you are long or short. They are all “+”, indicating long stocks, long

puts, long calls and long “cash” (or a T-Bill) of $Ke-rT

invested at the safe rate, r.

To demonstrate put-call parity more formally, consider the following two portfolios:



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Portfolio-A: One put option plus one share at t=0.

Portfolio-B: One call option plus an amount of cash equal to Ke-rT

at t=0.

The payoff for portfolio-A is given in figure 8.1 for K = 100. It can be seen that portfolio-A

has a payoff equivalent to holding one call and $100e-rT

in the risk-free asset at t=0. First,

consider the return from portfolio-B at the expiry date (see table 8.3).

First consider the value of portfolio-A at expiration, T (table 8.3). If the stock price at

expiry ST > K then the put option expires worthless but the share is worth ST. Alternatively, if ST <

K then put payoff is (K-ST) and the long share is worth ST, hence portfolio-A has a payoff = (K-

ST)+K = K.

[Table 8.3 here - Word]

Table 8.3 : Returns from Two Portfolios : Put-Call Parity

Case : ST > K Case : ST < K

Portfolio A(1)

ST K

Portfolio B(2)

(ST-K)+K=ST K

1. Portfolio A = One put option plus one share at t = 0 2. Portfolio B = One call option plus cash of Ke

-rT at t = 0

Now consider portfolio-B. If ST > K then the call option’s payoff is (ST-K), the cash amount

Ke-rT

has now accrued to $K and hence the payoff for portfolio-B is (ST-K + K) = ST. Alternatively,

if ST < K then the call option expires worthless but the “cash” will have accrued to $K, which is the

value of portfolio-B, at T.

These payoffs are shown in table 8.3 and it can be seen that both portfolios yield identical

outcomes at the expiry date, T. Since the options cannot be exercised prior to the expiry date, the

two portfolios must also have identical values today:

Long Call Option + Cash (or T-Bill) of $Ke-rT

= Long Put Option + Long Share

[8.4] C + Ke-rT

= P + S

At time t=0 we know K, r, and S and given an estimate of we can work out the value of

C using the Black-Scholes or binomial model: we then use equation [8.5] to deduce the value of

P. The put-call parity condition also demonstrates that calls can be "converted" into puts and vice-

versa by taking an appropriate position in the share and undertaking borrowing and lending. This



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is known as option conversion. Also we do not need calls, puts, shares and lending/borrowing:

given any three, we can always construct a portfolio which has an identical payoff to the "omitted

asset". This is an example of financial engineering.

Not only can we use [8.4] to “engineer” a synthetic put, we can use our synthetic put to

undertake profitable riskless arbitrage if we discover a put which is miss-priced. To see this first

note that from [8.4] we can use calls, stocks and cash (T-Bill) to create a synthetic put:

[8.5] P = -S + C + Ke-rT

From equation [8.5] it follows that we can “engineer” the payoff to a long put by short

selling one stock (-1), going long one call (+1) and holding either ‘cash’ (ie. a bank deposit) or a

T-Bill worth Ke-rT

today (ie. both of which will accrue to $K at time T). Here’s how we use this

synthetic put to take advantage of any miss-pricing between the actual put and the synthetic put.

Suppose at t=0, the quoted put premium is P and P -S + C + Ke-rT

. The put is overpriced. We

should therefore:

i) sell the actual put for P and short sell the stock for S, which gives a cash inflow at t=0 of

P+S, while simultaneously, we should:

ii) use these funds to buy a long call and a T-Bill (with face value of K, or simply place $Ke-rT

on deposit).

We will have a net cash inflow at t=0 since P + S > C + Ke-rT

by assumption. It is easy to

see (from table 8.3) that at expiry t=T the above two strategies have the same payoff at T, so the

net payoff is zero. Hence the overall strategy of (i)+(ii) has produced the equivalent of a net cash

inflow at t=0 which can be invested at the risk free rate to produce a positive amount at t=T. We

have therefore earned a positive risk free arbitrage return from the miss-pricing.

8.3. BINOMIAL OPTION PRICING MODEL

To get a handle on the use of delta hedging and risk-neutral valuation in pricing options

we first present a stylised example using an option which only has one period to expiration and

where we assume we know the possible outcomes for the stock price. The basic idea is to

construct a “synthetic “portfolio, which contains a call option and a share in such a way that this

portfolio is riskless (over each small interval of time). We can then equate the return from this

synthetic portfolio to the known risk free interest rate and solve for the 'unknown' price of the call

option. We assume the stock pays no dividends.



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ONE PERIOD BOPM (NON-DIVIDEND PAYING STOCK)

Consider a one period problem where there are only two possible outcomes for the share

price, namely 110 and 90. Let :

S = 100 = share price today

K = 100 = strike price of a call option

C = the unknown call premium (ie. price of the call)

r = 0.05 = risk free rate of interest


The payoffs to holding either one share or one long call are given in figure 8.2. The

difference between the two possible payoffs on the share is 20 (= 110 - 90). The possible payoffs

on the long call are Cu = 10 (= 110 - 100) and Cd = 0. The difference in the two payoffs for holding

a long call is +10 and hence the difference for a written one call is therefore -10. If we are willing

to be a little cavalier with notation we could write S = 20 and C = 10 and the ratio h = C/S =

½. In fact, we shall see below that h is the hedge ratio for the call. Consider a portfolio-A where

we are long 1/2-share and short 1-call. The cost of portfolio-A at t=0 is (1/2)S – C. It would appear

that the difference in the payoff on 1/2-share of 10 (=1/2x20) will just offset the difference in the

(negative) payoff of -10 on the written call:

Payoff to Portfolio-A (long 1/2 share, short 1-call) :

Payoff for price rise = (1/2) (110) - 10 = 45

Payoff for price fall = (1/2) (90) - 0 = 45

The payoff to portfolio-A at t=1 is known with certainty, no matter what the outcome for

the stock price (ie. either 110 or 90). We have created a risk free portfolio using a hedge ratio of



10

h = -1/2. Hence, the risk free payoff of 45 discounted back to 'today' at the risk free rate, must

equal the cost of portfolio-A :

[8.6] 05.1

45= (1/2) 100 - C

Hence C = 7.1428571

Equivalently, the payoff from the hedged portfolio is riskless and therefore must earn the

risk free rate:

[8.7] (1 + risk free rate) = 0

1

tatInvestmentInitial

tatPayoutCertain

1.05 = 45

1 2 100( / ) C and therefore C = 7.1428571

FORMAL DERIVATION

Again we assume there are only two outcomes for the stock price : Su = SU and Sd = SD

(where in our example U = 1.1 and D = 0.9 as shown in figure 8.2). The risk free rate must lie

between the rate of return if the stock goes up and its rate of return if the stock goes down so that

U > R > D (with D > 0). This ensures that no riskless arbitrage profits can be made. To see this,

reverse one of the inequalities, and assume R U > D. The riskless bond earns more than the

stock in either of the 2 states. Hence you could short $1 of the stock and invest the proceeds in

the bond, earning a return of R and a net return of either R-U or R-D both of which are positive. A

similar argument applies for U > D R.

Assume that if the stock price goes up the call will be in-the-money and if it goes down the

call will be out-of-the-money (as in figure 8.2). Let :

Cu = payoff to the long call if the stock price is Su = SU

Cd = payoff to the long call if the stock price is Sd = SD

Hence:

Cu = Su – K = SU - K

Cd = Sd – K = SD - K (for SD > K otherwise Cd = 0)

Portfolio-A (long h-shares, short 1-call) :

Payoff for price rise Vu = h Su - Cu



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Payoff for price fall Vd = h Sd - Cd

For the two payoffs to be equal at t=1:

[8.8] h Su - Cu = h Sd - Cd

[8.9] 2

1

)2.0(100

)010(

)(

)()(

DUS

CC

SS

CCh du

du

du

In the above analysis we are short one call and [8.9] indicates that the hedge will then

involve going long h-stocks. The hedge ratio h is also known as the option’s “delta” (see next

chapter) and hence this approach is also known as delta hedging. Because portfolio-A is risk

free at t=0 it is known as a delta neutral position. There is a lot of terminology in options theory!

(However, the importance and the latter terms will become clearer in chapter 9). Having

established the hedge ratio for our riskless portfolio we now determine the call premium C by

equating the PV of the known payoff at t=1, with the cost of portfolio-A at t=0:

PV of payoff for portfolio-A = Cost of portfolio-A at t=0

[8.10] )1( r

ChSU u

= hS – C

Substituting from equation [8.9] for h and rearranging, the call premium is:

[8.11] CR

qC q Cu d 1

1

where R = (1+r) and q = (R-D)/(U-D) = 0.75. The 'weight' q = 0.75 in [8.11] only depends on r, D

and U (and hence is constant).

The call premium C is a weighted average of the payoffs to the call (Cu, Cd) discounted using the

risk free rate. The weight q depends on the values of U, D (and r).

Note that the call premium is independent of the expected return on the stock and

depends only on the range of values for the change in S (ie. the values of U and D). In this simple

case (U-D) measures the volatility in the stock price and as we shall see volatility plays a key role



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in the Black-Scholes formula. Neither the actual probability of a fall of rise in the stock price nor

the risk preferences of investors, enter into the calculation of the price of the call. Hence all

investors, regardless of their differing degrees of risk aversion or their different guesses about the

probability of a rise or fall in future stock prices can agree on the 'fair' or 'correct' price for a call

option. The option is priced relative to the stock. While the probability of a rise or fall in the stock

price in the future affects the current price of the stock, it does not affect the price of the call. The

call is priced given the stock price.

The weight (q) is known as the risk neutral probability of a rise in the stock price but

this must not be confused with the actual probability of a rise in the stock price, which does not

affect the option premium. The risk neutral probability is simply a number which lies between 0

and 1 and is derived under the assumption that portfolio-A is riskless and therefore earns the risk

free rate. In some ways the term risk neutral probability could be misleading since it appears to

imply that we are assuming investors are risk neutral, which we are not ! An alternative is to call

q, an equivalent martingale probability and the latter is used frequently in the continuous time

literature. However, we will stick with the more common used term, “risk neutral probability”.

Using equation [8.11] we can price the option using the risk neutral probability q. This method of

pricing is known as risk neutral valuation (RNV) and plays a major role in the pricing of all types

of options.

It follows from the above analysis that if we have two otherwise identical call options but

the underlying stock for one has an expected growth rate of zero while the other has an expected

growth rate of 100% p.a. then the two options will have identical call premia. This is because in

each case we can create a riskless hedge portfolio, which by arbitrage arguments can only earn

the risk free rate. Put another way, the call premium is independent of the expected growth rate

of the underlying stock. Note however that we are not saying that the call premium is independent

of the volatility of the underlying stock (represented in the BOPM by ‘U-D’). Expected growth and

volatility are very different concepts. After all, we can have a stock with a zero expected growth

but a very high volatility or, vice versa. Mathematically the simple random walk ln(St) = + ln(St-1)

+ t has a mean growth rate and volatility given by the var(t) = . It is possible that =0 and is

very large, as well as the opposite case. The call premium depends on but is independent of .

(These rather subtle points are discussed in later chapters).

Of course, the above example has two key simplifying assumptions, namely that there are

only two possible outcomes for the stock price and that the option matures in one period.

However, by extending the number of branches in the binomial 'tree' we can obtain a large

number of possible outcomes for stock prices. If we consider each 'branch' as representing a



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short time period, then conceptually we can see how the BOPM for many periods 'approaches' a

continuous time formulation which forms the basis of the famous Black-Scholes approach.

ARBITRAGE

If the quoted option premium is $10 which exceeds the 'fair price' of C = 7.413 then a

riskless profit can be made. As usual, you sell high and buy low. At t=0, you sell (write) the call at

$10 and guarantee you have the payout at t=1 by also purchasing ½ a share. The guaranteed

payoff to this hedged ‘portfolio-A’ is $45. With a call premium of $10, portfolio-A costs $40 (=

(1/2)100-10) and hence the return is 12.5% (=45/40) which is in excess of the risk free rate of 5%.

Note that writing calls would tend to depress C, while borrowing to purchase shares would

increase S, thus tending to restore the equality given in equation [8.6] or [8.10].

If the call is underpriced at say $7 then we buy the call and hedge this position by short

selling ½ stock at a cost of $50 (= ½ of $100). The net cash inflow at t=0 is $43 (= 50-7). This

can be invested at the risk free rate to give receipts at t=1 of $45.15 (= 43 x 1.05). Your hedged

portfolio will be worth minus $45 at t=1, regardless of whether the stock prices rises or falls.

Hence you make an overall profit at t=1 of 0.15 (= 45.15 - 45).

PRICING A PUT OPTION

This follows a similar argument to that for calls. The risk-free hedge portfolio is obtained

by going long some shares and simultaneously going long one put. So, if S falls the loss on the

stock will be offset by the gain on the put, making the portfolio of “stock+put”, riskless. Let hp be

the number of shares to be purchased, (per put held), then:

Payoff to Portfolio-B (long hp-shares+long 1-put) :

Payoff for price rise hp(SU) + Pu

Payoff for price fall hp(SD) + Pd

If K = 100 and S at t=1 is either 110 or 90 (figure 8.3A) then for Su = 110 the put payoff is

Pu = 0 and for Sd = 90 we have Pd = 10. For a riskless portfolio we equate the two payoffs which

gives the hedge ratio :

[8.12] 2

1

)90110(

)100(

)(

)(

du

du

pSS

PPh

The hedge portfolio consists of buying 1/2 share for each long put held. Equating the

return on the risk-free hedge portfolio to the risk free rate gives :



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[8.13] 0

1

tatInvestmentInitial

tatPayoutCertain = (1 + risk free rate)

PSh

PSh

p

uup

= r1

Substituting for hp from [8.12] in [8.13] and using Su = SU, Sd = SD we obtain:

[8.14] PR

qP q Pu d

1

10 75 0 0 25 10

1052 381( )

( . ) ( . )

..

where again q = (R-D)/(U-D). Equation [8.14] has the same form as that for the call premium

except that the put payoffs Pu and Pd are used. The put premium P = 2.381 can be checked using

put-call parity.

[8.15] P = C - S + PV(K) = 7.143 - 100 + 100/(1.05) = 2.381

8.4 N-PERIOD BOPM AND DELTA HEDGING

Extending the BOPM to two periods where U = 1.1 and D = 0.9 gives the stock price

outcomes and the possible values of a long call at expiration as indicated in figure 8.2. Note that

we can price the option for any values of U and D (with D < (1+r) < U) and U and D need not be

symmetric changes (ie. U 1/D). For example in figure 8.3, U 1/D yet the lattice recombines

because to achieve the middle point at t=2 we either follow the path SUD or SDU and these are

both equal to 99, no matter what the values of U or D. Note that “99” is below the initial value of

S=100 but in the lattice we have (“cheated” and) drawn it symmetrically. We will see below and in

later chapters that there are a number ways of constructing a risk neutral lattice, one variant of

which does impose the symmetry property U = 1/D, but lets not worry about this here we have

enough to do!

[Figure 8.3 here – Powerpoint]



15


Figure 8.3 : Payoffs from the

two-period BOPM

A. Long one share

100

SU = 110

SD = 90

SU2 = 121

SUD = 99

SD2 = 81

B. Long one callK = $100

C = 10.714(h = 0.75)

Cu = 15(hu = 0.9545)

Cd = 0(hd = 0)

Cuu = 21

Cud = 0

Cdd = 0

Because we know that we can create a riskless hedge portfolio at any node we can

calculate C using 'backward induction' from the known values of Cuu, Cud and Cdd in our binomial

formula [8.11]. Here we are implicitly using “RNV”. For example, considering the two upper

branches in figure 8.3 have :

[8.16] CR

qC q Cu uu ud

1

10 75 21 0 25 0

10515( )

. ( ) . ( )

.

From the two lower branches we obtain :

[8.17] CR

qC q Cd ud dd 1

1 0( )

We can now solve for C the call premium for this two period option problem :

[8.18] 71423.1005.1

0)15(75.0)1(

1

du CqqC

RC

Note that the call premium for the option with two periods to maturity has a higher value

than our ‘identical’ option with one period to maturity, where we found C = 7.143. The hedge

ratios at each node are easily calculated from equations [8.19] (see figure 8.3 ).



16

[8.19a] 954545.0)2.0(110

21

)(

DUS

CCh uduu

u

[8.19b] hC C

S U Dd

ud dd

( )0

[8.19c] 75.0)2.0(100

015

)(

DUS

CCh du

The hedge ratio is initially 0.75 and then if the upper branch ensues, it rises to 0.9545

whereas on the lower branch it is zero. Let us follow through the actions required at the various

nodes to maintain our delta neutral position and hence earn the risk free rate.

At t=0 (h = 0.75, C = $10.714, S = $100)

Write 1,000 calls and buy 750 shares

Buy 750 shares @ $100 = $75,000

Write 1,000 calls at $10.714 = $10,714

Net investment = $64,285

Cu node (hu = 0.9545, Cu = $15, S = $110)

Value of portfolio Vu = 750 ($110) – 1,000($15) = $67,500

Return over period-1 = $67,450 / $64,285 = 1.05 (5%)

New hedge ratio hu = 0.9545 = Ns/Nc :

Either write 1,000 calls and hold 954.5 shares (case-A)

or write 785.7 calls (= 750/0.9545) and hold the ‘original’ 750 shares (case-B)

The first possibility involves increasing the number of shares held from 750 to 954.5 (at a

price of $110 per share) while the second appears to be the cheaper alternative since it involves

buying back 214.3 calls (= 1,000 – 785.7) at Cu = 15 per contract. For the moment lets assume

we take the second course of action (Case B). We do not increase our “own funds” in the hedge

so we borrow the funds required at the risk free rate (r) of 5%.

Case B : Buy back 214.3 calls @ $15 = $3,214 (= borrowed funds)

Cuu node (Cu = $21, S = $121, r = 0.05)

Value of shares = 750 ($121) = $90,750



17

Less call payoff = 785.7 ($21) = $16,500

Less loan outstanding 3,214 (1.05) = $3,375

Value of portfolio Vuu = $70,875

Return over period-2 = $70,875 / $67,500 = 1.05 (or 5%)

If we now examine case A, we find that the outcome is Vuu = $70,875 and again the return is 5%.

Case A : Continue to hold 1,000 written options and increase shares to 954.5

Buy additional (954.5-750) stocks @ $110 = $22,495 (= borrowed funds)

Cuu node (Cu = $21, S = $121, r = 0.05)

Value of shares = 954.5 ($121) = $115,494.50

Less call payoff = 1,000 ($21) = $21,000

Less loan outstanding $22,495 (1.05) = $23,619.75

Value of portfolio Vuu = $70,874.75 (= $70,875)


Vuu is the same as for case A and B. In case-A we hold more shares and we have also

written more calls. Since S increases, the higher number of shares therefore increases the value

of the portfolio while the increased number of written calls reduces its value, as do the higher

interest payments. The net effect is equivalent to case B. However, in practice, transactions

costs (eg. bid-ask spreads for stocks and calls) would imply different final values from the two

alternative hedging schemes and a computer program would be written to calculate the cheaper

of the two hedging scenarios. In our exposition we have assumed transactions costs are zero.

Let us now see what happens if, starting at the Cu-node with hu = 0.9545 (case A) we had moved

to the Cud – node.

Cud node (Cud = 0, S = 99, r = 0.05)

Value of shares = 750 ($99) = $74,250

Less call payoff = 785.7 (0) = 0

Less loan outstanding 3,214 (1.05) = $3,375

Value of portfolio Vud = $70,875

Not surprisingly the payoff Vud of $70,875 is the same as Vuu since we have been delta

hedging to achieve a risk-free outcome. Note however that the composition of the payoff is

different in these two cases. We can now return to t=0 and fill in the remainder of the tree.

Cd node (hd = 0, Cd = 0, S = $90)



18

At t=0 we wrote 1,000 calls and bought 750 shares, hence now :

Value of portfolio Vd = 750 ($90) – 1,000(0) = $67,500


This of course is the same value as Vu. The new hedge ratio hd = 0 so we now need to

write zero calls and hold zero shares. Hence we need to sell 750 shares and nominally we buy

back 1,000 calls at a cost of Cd = 0. The total receipts from the sale of shares of $67,500 we

invest at the risk free rate and naturally the outcome in the second period is independent of the

share price of either $99 or $81 (figures 8.3.). So if a call moves deeply out-of-the-money the

hedge ratio becomes zero and if the call is well-in-the-money the hedge ratio becomes unity. The

above analysis highlights the need to continually 'rebalance' the portfolio through time, if it is to

remain fully hedged and in the table 8.5 this is done for n = 10 nodes and the above analysis gives

a call premium of C10 = 30.086.

Making arbitrage profits from a mispriced call with two periods to maturity is similar to that

for the one-period case except the ‘excess profit’ (ie. in excess of the risk free rate) may accrue in

either or both of the two periods depending on when the mispricing is corrected. Clearly if non of

the mispricing is corrected in the first period the hedged position may earn less than the risk free

rate in this period. But in the second period the mispricing must be corrected since the option

reaches maturity. Then a return in excess of the risk free rate will be earned between t=1 and t=2

and hence over the two periods, you will earn more than the risk free rate. So if R01 and R12 are

the returns earned over the two separate periods then (1+R01) (1+R12) will exceed (1+r)2 if the

option was initially mispriced. For example suppose the call is initially overpriced. You therefore

at t=0 sell 1-call and buy h-shares. If the call does not fall sufficiently in period one then you can

earn less than the risk free rate over the 1st period. If the call becomes correctly priced at the end

of the 1st period you earn more than the risk free rate over period-1 (and earn the risk free rate in

period-2). Finally, if the call at the end of the 1st period falls below its ‘fair value’ and becomes

underpriced you earn (much) more than the risk free rate in the 1st period. But you now have to

close out the hedge and take a long position in the underpriced call and short sell the stock,

investing any excess funds at the risk free rate. Hence in the 2nd

period you will also earn more

than the risk free rate.

MANY PERIODS

Equations [8.16] and [8.17] give the values of Cu and Cd in terms of the final payoffs Cuu,

Cud and Cdd and if we substitute in [8.18] we obtain :



19

[8.20] C = 1

1 2 1 1 12

2 2

Rq C q q C q Cuu ud dd( ) ( ) ( )( )

The ‘2’ in the above formula represents the 2 possible paths to achieve the stock price

SUD (that is paths UD or DU) and the ‘1’s’ represent the single path to achieve either SU2 or SD

2.

We can interpret q as the probability of an ‘up move’ for S and hence (1-q) the probability of a

‘down move’, in a risk neutral world. Then probability of out-turns at nodes UU, UD and DD are

q2, 2q(1-q) and (1-q)

2 respectively. The BOPM formula (equation [8.20]) implies that the call

premium is the (risk neutral) probability weighted average of the final option payoffs at t=2. Hence

C is the expected value of the option payoffs (using risk neutral probabilities), discounted at the

risk-free rate. In general the number of possible paths to any final stock price are given by the

binomial coefficients.

[8.21] n

k

n

n k k

!

( )! !

where n! = n(n-1)(n-2) …. 1 and 0! = 1

n is the number of periods in the binomial tree

k is the number of upward price movements

Lets try it out for n = 2 :

Number of paths with k = 2 “ups” = (2!)/(0! 2!) = 1 (ie. UU)

Number of paths with k = 1 “ups” = (2!)/(1! 1!) = 2 (ie. UD and DU)

Number of paths with k = 0 “ups” = (2!)/(2! 0!) = 1 (ie. DD)

The formula therefore works for n = 2. The reader might like to draw a tree over n = 3

periods and verify that the number of possible paths to achieve k=1 “up” move is

knk qqk

n

)1( = (3!) / (2! 1!) = 3 and these are UUD, DUD and DDU (see table 8.5). In

general therefore over n-periods the BOPM formula becomes :

[8.22] CR

n

kq q MAX SU D K

n

k n k k n k

k

n

1

1 00

( ) [ , ]



20

The probability of the stock price reaching the value SUkD

n-k after n-time periods is

knk qqk

n

)1( . Note that the term in square brackets in [8.22] is just another way of writing

the payoff at the nodes. For example for n = 2, these are :

[8.23] Cuu = max[0, SU2 - K] Cud = max[0, SUD - K] Cdd = max[0, SD

2 - K]

The binomial formula for the option price is equal to the expected value (using risk neutral

probabilities) of the option payoffs at expiry, discounted at the risk-free rate of interest.

The hedge ratios in the BOPM can be calculated at each node and this is done for n=10

time periods in the Excel file provided. The BOPM assumes:

the 'up' and 'down' movements (U, D) are known and although not necessarily symmetric (ie.

U 1/D) nevertheless, they are assumed to be constant over time (ie. multiplicative binomial).

U and D depend on the actual (real world) volatility of the stock but not on the growth rate of

the stock price.

no transactions costs arise when implementing the hedge portfolio (even though it is

continually rebalanced).

no early exercise and no dividend payments on the stock. (But these assumptions can easily

be relaxed – see below).

The price of a put option is also given by equation [8.22] but with the term max[...]

replaced by the sequence of put-option payoffs, namely max[0, K-SUkD

n-k]. Equation [8.22]

indicates that the price of an option depends on the strike price K, the underlying asset price S,

the maturity n, the riskless rate r and the assets volatility (U, D), but it does not depend on the risk

preferences of individuals or the actual (“real world”) probability of a price increase (or decrease).

WHERE DO “U” AND “D” COME FROM ?

At t=0, r, K and S are known. Above we have shown that if we know U and D, then we

can price an option by invoking RNV which is embodied in the BOPM formula. However, since

different stocks have different volatilities (heuristically represented by U and D) then different

stocks will have different values of U and D and hence different option premia. The size of U and

D must be linked to the stock’s actual measured volatility. In fact it can be shown (see chapter

17) that one method of achieving this is to set :

[8.24a] U = te and [8.24b] D =

te



21

where = the observed annual standard deviation of the (continuously compounded) stock return,

t = T/n, T is the time to expiration in years (or fraction of a year) and n is the number of steps

chosen for the binomial tree.

We divide the T years (or fraction of a year) into n steps, each of which represents a small

time period t (= T/n). Note that the “spread” of the lattice at any two adjacent points is ln(SU) –

ln(SD) = 2 t , so the proportionate gap between SU and SD does depend directly on the

observed value of for the stock. This particular choice for U and D imposes symmetry that is U

= 1/D but it can be shown that this is not restrictive if our aim is to construct a “risk neutral” lattice.

Note also that when U = 1/D then for example the nodes SUD or SDU both have a value of S and

hence this particular choice of risk neutral lattice not only recombines but is geometrically

symmetric. Hence if S (at t=0) is 100 it will also be 100 in the middle node at t=2. We will meet

this “symmetric risk neutral lattice” in other parts of the book. Finally, note that U and D do not

depend on the actual mean growth rate of the stock and hence neither does the option premium.

This is RNV again.

REPLICATION PORTFOLIOS

We noted above that if you short h-shares and write one call you obtain a known amount

at the end of one period. This known amount at t=1 could also be obtained by investing in a risk

free bond for one period. Hence our above analysis involved a replication portfolio which

schematically is :

[8.25] Value of “long” N-stocks + Short Call = Risk Free Borrowing

N. S + (-C) = B

where N = number of stocks held (i.e. the hedge ratio for one call). Equation [8.25] allowed us to

calculate the (one period) call premium as :

[8.26] C = 1

1R

qC q Cu d

where R = (1+r) and q = (R - D)/(U-D) is the risk neutral probability.



22

REPLICATING A LONG CALL

In our original example we priced a call by establishing a risk free portfolio and we had to

readjust the hedge ratio at each node. At this point it is useful to obtain equation [8.26] by using a

replication portfolio for the call. This is financial engineering again.

Consider replicating the pay-off from one long call with a portfolio of stocks and bonds (ie. a risk

free asset).

To see how to replicate a long call we re-arrange [8.25] to give C=N.S+B. Here we will determine

the call premium by equating C with the cost of the replication portfolio of stocks and bonds which

is designed to have the same payoff structure as the long call at t=1.

Consider purchasing N-stocks at a price S and $B of risk free bonds with a return R =

(1+r) – see figure 8.4. When B > 0 this implies a bond purchase and B < 0 implies issuing bonds,

equivalently you can think of B > 0 as lending and B < 0 as borrowing (from a bank say). At t=0

the cost of this portfolio is:



Figure 8.4 : Stock + bond replicates

long call

Value of call at t = 0 equals the value of the replication portfolio at t = 0

C = NS + B = [qCu + (1 - q)Cd] / R

(NS + B)

NSu + BR = Cu

NSd + BR = Cd

Hedge ratio and bonds held at t = 0

N = (Cu - Cd) / (Su - Sd)

B = (Cu - NS) / R

At t = 1, values of

replication portfolio

equal

to outcomes for the

long call

Value of

replication portfolio

at t = 0

Replication Portfolio-A: Stocks plus Bonds

[8.27] Cost of Replication Portfolio = $ (NS + B)

At t=1, portfolio-A pays out either (NSu + BR) or (NSd + BR). Now we replicate the payoff

from a long call by equating the payoff from portfolio-A with Cu and Cd.



23

[8.28a] (NSu + BR) = Cu [8.28b] (NSd + BR) = Cd

Subtracting equation [8.28b] from [8.28a] gives :

[8.29] N =

du

du

SS

CC

From [8.28a] and [8.28b].

[8.30a] B =

R

SNC uu ).(

R

SNC dd ).(

Substituting for N from [8.29] and using Su = SU, Sd = SD:

[8.30b] B = )( du

uddu

SSR

CSCS

)( DUR

CDCU ud

It is easy to see that the two expressions for B in [8.30a] are equal by noting that the first

two equalities imply Cu – Cd = N(Su – Sd) and given the definition of N in [8.29] these two

expressions must be equal. Our chosen ‘stocks+bond’ replication portfolio reproduces the payoff

of the actual call option at t=1 and therefore the call premium C (at time t=0) must equal the cost

of the replication portfolio at t=0. Hence substituting [8.30b] in [8.27], after some manipulation we

obtain:

[8.31] C = (NS + B) = 1

R[qCu + (1-q)Cd]

where q = (R-D)/(U-D), as expected. The number of shares N is equivalent to the hedge ratio h in

our earlier derivation. The term R = (1+r) is the one period risk free rate. If we use continuously

compounded annual rates rc, then R = tcre

, where t = T/n.

Now let us replicate the value of the call over t=2 periods. Consider what is happening at

t=0 (figure 8.3b). From [8.29] and [8.30] we have:



24

[8.32] N0 = 75.0)2.0(100

015

du

du

SS

CC (as before)

[8.33] B0 = R

SNC uu ).( 0286.64

05.1

)110(75.015

Thus at t=0 the replication portfolio consists of borrowing $64.286 and purchasing $75 in

shares (ie. NoSo = 0.75 shares x $100). This is a net investment of $10.716 which, not

surprisingly, we have earlier found is the value of the option premium C at t=0 for a two-period

call. Note that here we are replicating the payoff of the long call with a long position in 0.75 stock

and a short position in the bond. Note that, if we were trying to hedge the long call this would

require exactly the opposite position to this replication portfolio, namely short 0.75 of the stock and

go long the bond (ie. lending). The outcome in the ‘up’ and ‘down’ nodes for our ‘stock+bond’

replication portfolio is :

Node-U: NoSu+BoR = 0.75(110) – 64.286(1.05) = 75 – 67.5 = 15

Node-D: NoSd+BuR = 0.75(90) – 64.286(1.05) = 67.5 – 67.5 = 0

which, of course, are the outcomes for the value of the call, Cu =15, and Cd = 0 at the first two

nodes. We now rebalance our replication portfolio so at the U-node :

[8.34] Nu =

uduu

uduu

SS

CC

= 0.9545

[8.35] Bu = R

SNC uuuuu ).(

05.1

)121(9545.021-90

The reason you borrow 90 is that you must increase your share holdings by (0.9545-0.75)

= 0.2045 at a cost of $110 per share. The total cost is $22.5 which when added to you existing

outstanding debt of 67.5 brings your debt level up to $90. The outcome is moving from the U-

node to the nodes UU and UD are :

Node-UU: NuSuu+ BuR = 0.9545(121) – 90(1.05) = 75 – 67.5 = 21

Node-UD: NuSud+ BuR = 0.9545(99) – 90(1.05) = 94.5 – 94.5 = 0



25

Again we have replicated the value of the call at these two nodes (see figure 8.3b).

Finally consider the D-node. Here Nd = (Cud-Cdd)/(Sud - Sdd) = 0 and the replication portfolio

consists entirely of bonds $Bd = Cud/R but because Cud = 0 (figure 8.38) we hold zero bonds. The

replication portfolio at node-D is therefore worth zero and hence at t=2 is also worth zero, but this

exactly replicates the value of the call at the nodes UD and DD (figure 8.3B)

In practice we would not perfectly replicate the payoffs to the call because the hedge ratio

would have to be calculated at t=0 before knowing Cu and Cd (and Su and Sd). There will be some

hedging error because the actual stock price and hence the actual out-turn values may differ

slightly from Cu and Cd although this error becomes smaller as t 0. Of course if we extend the

above analysis to n > 2 periods then the hedge ratio N will alter at each future node in the lattice

(and therefore so will the amount borrowed or lent, B). This is just another example of dynamic

hedging where in this case we are trying to replicate the payoff from one long call.

Naturally, we obtain the same BOPM formula for the price of the call using either the

“replication portfolios” of stocks plus bonds or by using our earlier “delta hedge” riskless portfolio.

Note also that even if you were a legislator and succeeded in banning the trading of (call) options

because they were “dangerous”, they could be replicated with stocks plus bonds and hence you

would have accomplished nothing – although note that this argument assumes zero transactions

costs.

Up to now we have used the notation Si where i = U, D, UU, DD, UD, etc. mainly because

we are used to it. However, this can obviously become cumbersome and therefore it is more

usual to designate i as the actual number of ‘up’ moves in the lattice (eg. UU = 2). A ‘down’ move

is designated the number zero, which geometrically is represented as a horizontal move. We also

need to note the time period for each node, so the full notation would be St,k. In general the

notation is “St,k” where t is the time period in the lattice and k is the node corresponding to the

number of ‘up’ moves (at time t). The use of this notation allows the lattice to be derived using a

computer programme. We can now expression our risk neutral recursion as (see figure 8.5) :

[8.36] Ct, k = [qCt+1, k+1 + (1-q)Ct+1, k] / R




26


Figure 8.5 : Notation for nodes

t t + 1

(t, k) (t+1, k)

(t+1, k+1)

Value of call at t is given by the recursion

C(t, k) = [qC(t + 1, k) + (1 - q) C(t + 1, k + 1)] / R

Also, the recursion for the ‘stocks+bonds’ replication portfolio which reproduces the payoff

from holding Nc calls (where in our example above N

c = 1) is:

[8.37] Value of this replication portfolio at node (t,j) : jt

c

jtjt

s

jt CNBSN ,,,,

Hedge parameters:

[8.38] jth ,

jtjt

jtjt

c

s

jt

SS

CC

N

N

,11,1

,11,1,

and

[8.39] R

SNCNB

jt

s

jtjt

c

jt

,1,1,1

,

8.5 EARLY EXERCISE, AMERICAN OPTIONS AND

DIVIDEND PAYMENTS

So far we have been analysing European options which can only be exercised at

expiration. American options can be exercised at any time. The question then arises as to when

it is optimal to exercise an American option prior to its expiration date and how this affects the

valuation of the American option. This is not always easy to demonstrate but we shall outline the



27

key issues using the binomial formulation. The reader might like to bear in mind the following

results for American options:

For a call option on a non-dividend paying stock, early exercise is never optimal.

For a call option on a dividend paying stock, early exercise is sometimes optimal.

For a put option on a stock (with or without dividends), early exercise is sometimes optimal.

AMERICAN OPTIONS (NO DIVIDENDS)

To value an American put option using the BOPM, we first note that at maturity, the option

is European and for a two period put option the payoffs would be Puu, Pud and Pdd. We now use

the recursive formula to calculate the value of the put at the previous nodes, namely Pu and Pd.

The possibility of early exercise now enters the picture. For example, at node-U, if the put were

exercised it would be worth max{K-Su, 0}. So if K-Su > Pu, we replace Pu in the binomial tree by

the value K-Su, and if K-Su < Pu we leave the value Pu at the U-node. We repeat this procedure

for node-D. We now have two values at these nodes which may be different from those for the

European option if Pu or Pd have been replaced in the lattice. We again apply our recursive

formula to the values at the U and D nodes to get a ‘trial value’ for the put premium P* at t=0.

Finally, we compare P* with max(K-S0, 0} and take the larger of these two values as the value of

the put at t=0. Clearly if K-S0 exceeds P* then the option will be immediately exercised, otherwise

early exercise is not optimal and the put premium will be P*. As we see below, this general

principle of working backwards through the tree and seeing if early exercise is worthwhile applies

in more complex cases.

DIVIDENDS AND THE BOPM

The BOPM tree can get quite complex when dividends are paid and usually we make

some simplifying assumptions to make the problem tractable. For example, it can be shown that

when dividends are paid continuously at a rate per period then a very similar recursive formula

for the price of the option applies, namely for two-period European call on a dividend paying

stock :

uduuu CqqCR

C )1(1

and ddudd CqqCR

C )1(1

where DU

Drq

)1(



28

The only change is in the definition of q where we use r- in place of r. This has to do with

risk neutrality again, since in a risk neutral world if the stock is to earn a return equal to the risk

free rate r but you receive dividends at a rate , then the expected growth of the stock must be r-.

Of course, the values Cu and Cd must be compared with (and possibly replaced by) their intrinsic

values (ie. early exercise may be optimal), before moving backwards through the tree.

KNOWN DIVIDEND YIELD

This is a slightly more complicated case. Assume the tree has n-periods and the nodes

are t apart, so that n = T/t. For example if T = 30/360 and if we choose t = 1/360 then n = 30

and each node represents 1-day of “real time”. Now let us apply the BOPM to a call option where

the underlying share pays out one dividend at time t = i(t), where i = 0, 1, 2, …, n. The nodes

prior to time t = i(t), would have values jij DSU (for j = 0, 1, 2, …, i) as usual, but after the

stock goes ex-dividend the nodes would have values jij DUS )1( , where is the known

dividend yield. If there are several known dividend yields i over the life of the option, then the

nodes after the ex-dividend dates would be jij

i DUS )1( . For example, given a single

dividend payment prior to the 2nd node, the binomial tree is shown in figure 8.6. In practice has

to be estimated and clearly while the assumption of a constant dividend rate is not unreasonable

for a stock index, it is less plausible for individual stocks where dividend payments tend to be

bunched in certain months of the year.

To incorporate the possibility of early exercise and price an American call on a dividend

paying stock we proceed as follows.



Figure 8.6 : Binomial tree :

dividend payout

S

SU

SD

SU2(1 - )

SU3(1 - )

SU2D(1 - )

SUD(1 - )

SUD2(1 - )

SD2(1 - )

SD3(1 - )

Ex-dividend date just before 2nd node

Single dividend payable with dividend yield known, then the tree recombines



29

At maturity T we know the payoffs from the call. We now work backwards through the

binomial tree taking each node at T-1 and calculate the “recursive values” for the call CT-1,u ,

CT-1,d using the above formula for a European call.

For each node at T-1 (calculated as above) compare the recursive value for C with the

intrinsic value IV = max[price of underlying – K, 0]. We take the maximum of these two

values and use these for further calculations in the binomial tree, always working backwards.

For example, if Cuu = 4.5 but SU2 (1-) = 105 and K = 100, then early exercise yields a value

of 5 (> 4.5) and hence '5' would replace 4.5 in the tree. Hence, Cu would be calculated as :

[8.40] )(7.3)1()5(1

sayCqqR

C udu

which would then be compared with the intrinsic value at node-U.

We work backwards through the tree until finally we have to check that early exercise is not

possible at t=0 by comparing the “recursive value” C0 with its intrinsic value max[0, S0 -K].

The option premium is the larger of these two values.

The call premium on an American option will always be at least as high as that on an

equivalent European option because of the possibility of early exercise.

KNOWN DOLLAR DIVIDEND

When dividends are paid in discrete amounts the problem is a little more complex. First

note that when a dividend is paid the stock price falls by the amount of the dividend payment D.

(We ignore any tax issues here). This occurs because if the stock price were to fall by less than

D (eg. Z < D) then you could buy the share for S immediately prior to the ex-dividend date, capture

the dividend D and immediately sell the share for S-Z. Your profit = (D+S-Z)-S = D-Z > 0 (ignoring

any problems due to discounting if dividends are paid with a lag).

If we let U and D apply to the stock price S, then unfortunately the binomial tree does not

recombine and there are a very large number of nodes to evaluate. To avoid this problem and

obtain a recombining tree we proceed as follows. We let U and D apply to S*, the stock price

minus the present value of all known future dividends. Suppose the ex-dividend date is . Then

Before ex-div date: S* = S - D)( tire

for it



30

After ex-div date: S* = S for it>

Also, the value of for U and D is replaced by * the volatility of S* , so that U=

te * ,

D =1/U and q =(R - D)/(U-D) is the (usual) risk neutral probability. This gives us the recombining

tree for S* and to obtain the tree for S we add back the PV of future dividends. If *

0S is the value

of S* at t=0, then the tree for the stock price S is:

Before ex-div date: S = *

0S jij DU + D

)( tire for it

After ex-div date: S = *

0S jij DU for it>

Suppose there is one dividend payout of $10 at the end of the 5th period then

PV(Dividends) = 10/(1+r)5

= $2 say. If S0 = 100 then *

0S = 98. We then calculate the lattice for S*

using the above formulae and then derive the lattice for S from that for S*. Working recursively

backwards through the lattice for S, from the maturity date of the option, then given the European

call (or put) premium.

To value an American call on the dividend paying stock we calculate the intrinsic value at

each node and of course the early exercise decision is based on S (not S*). For example, if the

stock has just gone ex-dividend and at the next “upper” node S* = 110 and the dividend at is D =

$3 then S = 113 (since the present value of the dividend at is the dividend itself of $3). For an

instant S = 113 and then “immediately” falls to 110 as it goes ex-dividend. But just before the

stock goes ex-dividend the call has an intrinsic value of 3 (= 113-110). If the recursive value C is

less than the intrinsic value of $3 then we replace value C with the intrinsic value:

[8.41] dCqqR

C ,11 )1(201

A similar calculation is required at each node to see if the intrinsic value exceeds the

recursive value given by working backwards through the tree. This allows us to price the

American option on the dividend paying stock in the usual way.

SUMMARY: BOPM

The option premium can be determined by creating a riskless portfolio consisting of a long

position in stocks and a short (written) call.



31

The option premium is determined by U, D, r, q and the time to maturity. U and D depend on

the actual ‘real world’ measured volatility of the stock (but not on the mean growth rate of

the stock).

The parameter q can be interpreted as a risk neutral probability. Hence it is valid to assume

that the world is risk neutral, when valuing options. But (surprisingly) the option premium

which results, is valid in the real world! The BOPM formula is a recursion which works

backwards, starting with the option payoffs at expiry. These conceptual issues are taken up

in greater detail in chapter 17. Both American (and European) options can be valued using the BOPM and therefore the

method is very flexible (and intuitive).

8.6 BLACK-SCHOLES MODEL

In the 1960s, options were being traded in the US over-the-counter but rather bizarrely

no-one knew how to correctly price them. It was easy to work out that for a call option say, the

premium should be higher: the lower the strike price, the longer the time to maturity, the higher the

interest cost and the greater the volatility of the underlying stock. But how could all of these be

combined to give an explicit equation that could be used to quickly give the correct or fair price for

the option? A brief history of the route to the Black-Scholes equation is given in box 8.1.

[Box 8.1 here - WORD]

BOX 8.1 : NOBLE PURSUITS

It was the combined work of Fischer Black, Myron Scholes and Robert Merton at MIT in Boston that finally solved the option pricing problem in the early 1970s. Black (after his degree in Physics) initially worked on the pricing of warrants (ie. options on a company’s stocks, issued by the company itself) since these (unlike options) were at this time traded in liquid markets on US exchanges. Black initially used the CAPM to evaluate the performance of a portfolio consisting of the underlying stock and a warrant on the stock. A crucial result the calculations showed was that the warrant’s value did not depend on the return on the stock.

Scholes was also working on options pricing in the late 1960s at the Sloan School of Management (MIT), when he met Black and they began working together. Meanwhile, a young applied mathematician, Robert Merton, joined MIT as a research assistant to Paul Samuelson in the economics faculty. Samuelson, based on his own earlier work encouraged Merton to explore the theory of warrant pricing. At this time, Merton also developed an intertemporal CAPM using continuous time finance, which utilised Ito’s lemma. These mathematical ideas were to provide the basis for the Black-Scholes formula.

Merton, Black and Scholes exchanged ideas over several years at MIT. Their path to success might be described as a ‘random walk with positive drift’. There were many dead-ends encountered but ultimately the problem was solved.



32

In 1970, Black and Scholes completed their options pricing paper. They acknowledged Merton’s suggestion of combining the option and the underlying asset, to yield a risk free portfolio. Black-Scholes’ paper was initially rejected by the Journal of Political Economy (JPE) (a publication of the economics faculty of the University of Chicago) because it was too specialised. It was then rejected by Harvard’s Review of Economics and Statistics but finally, with the support of Eugene Fama and Merton Miller, it was accepted by the JPE and published in May/June 1973 under the title “The Pricing of Options and Corporate Liabilities”. Merton, who had collaborated with Black and Scholes, also produced a paper on options pricing in the Bell Journal of spring 1973.

Scholes and Merton both received the Nobel Prize for their work but unfortunately Fischer Black died before the prize was awarded (it cannot be awarded posthumously). Some of the above academics have also “put their money where their mind is” and used continuous time mathematics to try and beat the market, using options to leverage their bets. This was often successful. But not always, as was the case with Long Term Capital Management (LTCM), a hedge fund in which both Scholes and Merton were involved, but which effectively went bust in 1998.

Coincidentally, the Chicago Board Options Exchange (CBOE) began trading options (initially in the large smoking room of the Chicago Board of Trade) in April 1973 and the “new” Black-Scholes formula was soon in use by traders (for more details of this ‘story’ see the excellent book by Berstein 1992). The Ivory Towers of academia produced something of real practical value (as well as aesthetically pleasing). Whether it be ‘Black Holes’ or ‘Black-Scholes’, the power of mathematics to solve real world problems is impressive, not least in modern finance. This will become apparent in later chapters, where we use some continuous time mathematics to price options.

The Black-Scholes (1973) option price formula applies to European options. It is derived

(using stochastic calculus) by assuming a continuous stochastic process for the asset price at

each point in time (with a given probability distribution). As with the BOPM a riskless hedge

portfolio consisting of a long position in stocks and a short position in a call is constructed and

continuously rebalanced. In fact the Black-Scholes formula is a limiting case of the BOPM, when

the time intervals t = T/n for the change in stock prices in the BOPM become infinitesimally

small. More explicitly the assumptions of the model are:

all riskless arbitrage opportunities are eliminated.

no transactions costs or taxes.

trading is continuous, stocks are perfectly divisible and no dividend is paid on the stock.

Investors can borrow and lend unlimited amounts at the risk free interest rate which is

constant over the life of the option.

stock prices follow a particular stochastic process known as a Geometric Brownian Motion.

This gives a lognormal distribution for stock prices with a constant expected return () and

variance . (An extended model can accommodate cases where r and S are specific

functions of time).

CALL OPTION

By assuming a lognormal process for the stock price and creating a riskless hedge

portfolio (consisting of the option and the underlying asset), Black and Scholes were able to obtain

an exact solution for the premium on a European call option (and put-call parity can then be used



33

to price the put). The Black-Scholes formula for the price of a European call option looks rather

formidable:

[8.42] C = S N(d1) – N(d2) PV = S N(d1) – N(d2) Ke-rT

where dS PV

T

T1

2

ln( / )

T

TrKS

)2/()/ln( 2

Tdd 12T

TrKS

)2/()/ln( 2

C is the price of call option (call premium)

r is the safe rate of interest for horizon T (continuously compounded).

S is the current share price

T is the time to expiry (as proportion of a year)

PV is the present value of the strike price (= K e-rT

)

is the annual standard deviation of the (continuously compounded) return on the stock.

The term N(d1) is the cumulative probability distribution function for a standard normal

variable. It gives the probability that a variable with a standard normal distribution ~ N(0,1) will

have a value less than d1 (figure 8.7). In fact when d1 = -1.96 then N(d1) = 0.025 which may be

familiar as the so called 5% ‘critical value’ used in statistical tests (remember that 5% = 2 x 2.5%

corresponding to the two ‘tails’ in the normal distribution). Also, for example, N(- ) = 0, N(0) =

0.5, N(+ ) = 1 and as N(-1.96) = 0.025, and N(1.96) = 0.975 then we see that N(-x)+N(x) = 1.

In table 8.6 we price an option where S=45, K=43, r=0.1, =0.20, T=0.5 (years). We find

d1 = 0.7457, and hence N(d1) = 0.7721. This implies that the area under the standard normal

curve between - and d1 = 0.7457 is 0.7721. Given d1 the value of N(d1) can be found from

tables of the standard normal distribution (by interpolation) or by using a power series

approximation (see table 8.7) or directly using the appropriate statistical function in spreadsheets

such as Excel (ie. “NORMSDIST(d1)”) .


Table 8.6 : Call Premium Using Black-Scholes

Data : S = $45 K = $43

r = 0.1 (= 10% p.a.) = 0.2 (20% p.a.) T = 0.5 (6 months to maturity)

Question : Calculate the Call Premium, C.



34

Answer: Present Value of Excise Price

PV = K Tre .

= 43 )5.0).(1.0(e = 40.9029

d1 = 7457.00707.06750.02

5.02.0

5.02.0

)9029.40/45ln(

d2 = 0.7457 - 0.2(0.5)1/2

= 0.6043

From tables of the cumulative normal distribution (interpolated) or using a power function approximation: N(d1 = 0.7457) = 0.7721 N(d2 = 0.6042) = 0.7271

Hence C = S N(d1) - N(d2) PV = 45 (0.7721) - 0.7271 (40.9029) = $5.00


Table 8.7 : Power Series Approximation for N(d1) and N(d2)

A power series provides an approximation to any (continuous) function: the more terms in

the power function the greater the accuracy. One useful approximation (which can be

programmed into a computer and in a spreadsheet like Excel) is:

N(x) = 1 - (a1 k + a2 k2 + a3 k

3) Z(x)

k = 1/(1 + x)

0.33267

a1 = 0.4361836

a2 = -0.1201676 a3 = 0.9372980

Z(x) = 1 2/ exp[-(x2/2)]

For example for d1 = 0.7547 we have :

k = 0.79932

Z(0.7547) = 0.3001

N(0.7547) = 0.77478

The answer of 0.7748 is close to that obtained (by interpolation) using the cumulative standard

normal table.

Suppose x < 0

The standard normal distribution is symmetric around its mean, hence for x = -0.7547 we

calculate N(x) for x = 0.7547 and N(-x) is given by :

N(-x) = 1 - N(x)

N(-0.7547) = 1 - N(0.7547) = 0.225219



35


It should be intuitively obvious from figure 8.7 that :

As d1 then N(d1) 1

As d1 - then N(d1) 0

It can be shown that N(d1) [= 0.7721] is the probability that the call option will be in-the-

money at expiry. Here, this probability is in excess of 77% because the option is currently well in-

the-money since S = 45 > K = 43 and 2 is relatively small at 20% p.a. so S is unlikely to fall below

K = 43 over the next 6-months. We now have all the information required for the Black-Scholes

formula (8.42), to determine the price of the call option which is C = $5 (see table 8.6).

PUT OPTION

To find the price of a European put (P) with the same expiration date and strike price (as

the call) we can use the put-call parity condition and this yields :

[8.43] P = C - S + K e-rT

P = 5 - 45 + 43 exp[(-0.1) (0.5)] = 0.9029

Alternatively we can derive P using the Black-Scholes model directly and this gives:

[8.44] P = PV N(-d2) – S N(-d1)



36

where d1 = 0.7457

d2 = 0.6042

N(-d1) = 1-N(d1) = 1 - 0.7721 = 0.2279

N(-d2) = 1-N(d2) = 1 - 0.7271 = 0.2729

Hence: P = 40.9029 (0.2729) - 45(0.2279) = 0.9069

In practice of course, traders have all of these calculations programmed using 'in house'

software and only the results appear on trading screens. The calculations can be done in EXCEL

or directly inputting the formulae in a procedure (subroutine) of standard statistical or

mathematical software (eg. GAUSS, C++, Visual Basic, MATHEMATICA). The formulae can then

be used to calculate the call or put premia for different values of the spot price S, the exercise

price K, time to maturity T, the safe rate r and the volatility . For example, an element of code is

provided below in GAUSS to calculate call and put premia for alternative values of S. (Other

programs would have a similar structure):

GAUSS CODE

@ Calculating Black-Scholes for call and put (no dividends) @

tau = 180/365; @ time to maturity @

r = 0.05 ; @ continuously compounded @

k = 100 ; @ strike price @

sigma = 0.10 ; @ annual volatility @

s ={98,99,100,101,102} ; @ inputs for underlying asset @

@ === Use Gauss procedures for Black-Scholes call and put premia == @

{ call_bs } = bscall(k, s, sigma, tau, r) ;

{ put_bs } = bsput( k, s, sigma, tau, r) ;

@ ======= Black-Scholes Procedures/Subroutines Follow ========= @

proc bscall(k,s,sigma,tau,r);

local d1,d2,c ;

d1=(ln(s./k) + ( r + sigma^2/2).*tau )./(sigma.*sqrt(tau)) ;

d2=d1 - sigma.*sqrt(tau);

c=s.*cdfn(d1)-k.*exp(-r.*tau).*cdfn(d2);

retp(c);

endp;

proc bsput(k,s,sigma,tau,r);

local d1,d2,p ;

d1=(ln(s./k) + ( r + sigma^2/2).*tau )./(sigma.*sqrt(tau)) ;

d2=d1 - sigma.*sqrt(tau);

p= k.*exp(-r.*tau).*cdfn(-d2) - s.*cdfn(-d1);

retp(p);

endp;

end;



37

After setting the (scalar) inputs K, r, tau (time to maturity), (= volatility) and the 5x1

vector of alternative values for S, the program uses the procedures ‘bscall’ and ‘bsput’. These

calculate 5 values for each of the vectors call_bs and put_bs, the call and put premia

corresponding to S={98,99,100,101,102}. The call premium “call_bs” can then by plotted against

S giving the curve AB in figure 8.8. (The ‘procedures’ (subroutines) for bscall and bsput are listed

at the bottom of the program and are just the textbook equations given above).

As the option approaches its expiration date, the curve AB moves towards the familiar

‘kinked’ dashed line representing the payoff at expiration. At any given stock price such as S

the distance CD is the intrinsic value and BC is the time value of the option. One can see from

figure 8.8 that the loss in time value (if the stock price remained unchanged) is greatest for a call

which is purchased at-the-money. However, the actual payoff depends on the holding period and

the movement in the stock price. For example, if you purchased an existing long call (form

another investor) at point-X and the stock price subsequently increased to S then at maturity you

would have made a profit of Cc – CX (where Cc = KS ).


Also, the above values of S could be altered to represent the impact of an “extreme

event” on the call premium (eg. the market crash of October 1987). This is a form of “scenario

analysis” (or stress testing). It is extremely useful in calculating the change in value of a



38

portfolio containing calls and puts, to changes in the price of the underlying asset and is discussed

in later chapters on risk management.

ANALYSIS OF THE BLACK-SCHOLES FORMULA

The B-S formula (equation [8.42]) is somewhat complex. Let us see if the B-S equation

fits in with our intuitive ideas about the factors affecting the price of a call option, given earlier.

There we found that the call premium is positively related to S and and negatively related to K.

STOCK PRICE AND STRIKE PRICE

Prior to maturity the relationship between the call premium and the stock price, given by

B-S is non-linear or convex, (see figure 8.8). If S falls to zero then so does C (see figure 8.8, point

A) since the option is not likely to be exercised. Does B-S give this result? If S = 0 then from

equation [8.42], d2 = -, N(d2) = 0 and hence C = S N(d1) - N(d2) Ke-rT

= 0.

Alternatively, if S/K is large then C is slightly larger than S-K as at point B. If S is very

large the call option is almost certain to be exercised and hence it becomes like a forward contract

with delivery price K, which has a value C = S – Ke-rT

. Does B-S give this result? As S we

see that d1 and d2 and N(d1) and N(d2) 1 and therefore equation [8.42] gives C = S – Ke-rT

,

as expected.

VOLATILITY

It is not obvious from the B-S formula that the call premium (C) depends positively on 2.

However, we can show that the B-S formula produces ‘sensible’ results as 2 0. In this case,

the stock is virtually riskless and hence the stock price at expiration will be SerT

and the payoff

from the call option is :

[8.45a] max[SerT

-K, 0]

Discounting, the value of the call at t=0 is

[8.45b] C = e-rT

max (SerT

-K,0) = max [S –Ke-rT

, 0]

Let us see if [8.45b] is consistent with the B-S formula when 0. With no uncertainty

we would know if the call would expire in-the-money or out-of-the-money. Consider these two

cases.



39

Case A : Expiry in-the-money (SerT

>K)

At t=0 a riskless profit could be made by ‘buying low’ and ‘selling high’

Borrow $Ke-rT

and purchase the call for $C

At expiry, pay $K, take delivery and sell the share for $S

The profit at T is = S - (Ke-rT

+ C). Arbitrage would ensure that such riskless profits are

zero, hence setting = 0, we have :

[8.46] C = S - Ke-rT

The B-S equation [8.42] is consistent with this result since for = 0, we have d1 = d2 = +

and hence N(d1) = N(d2) = 1 so that C = S - Ke-rT

. When the call is well in-the-money, the call

(which delivers one share at expiration) is equivalent to a long forward contract. Hence it is not

surprising that the RHS of equation [8.46] is the value of a forward contract (on a non-dividend

paying stock) with a delivery price of K.

Case B : Expiry out-of-the-money (SerT

<K)

Clearly the value of the call is zero. This is consistent with the B-S equation [8.42] because:

[8.47] S < Ke-rT

implies ln(S/KerT

) < 0

Thus as 0, d1 -, so that N(d1) and N(d2) 0. It follows from the B-S formula

[8.42] that C = 0, which is consistent with our assumption that the option ends up out-of-the-

money.

BLACK SCHOLES WITH DIVIDENDS

To incorporate dividends in the B-S model for European options (B-S cannot handle

American style options and early exercise) two procedures are possible and these closely

resemble the procedures we used for the BOPM. First assume the known dividend payments D i

are at discrete intervals, where ti (measured as fraction of a year) are the ex-dividend dates over

the remaining life of the option. To price this option we merely substitute S’ = S – DPV(dividends)

= S - iitr

i

ieD

in place of “S” in the B-S formula above.

An alternative to the above is to assume that dividends are paid continuously and the

(annual continuously compounded) dividend rate (or dividend yield) is constant. This is a

reasonable assumption for stock index options. The B-S formula can then be used with S* = Se-T



40

in place of ‘S’ in the usual B-S formula. This expression for S* arises as follows. If S remains

constant and the dividends paid are reinvested at the rate then the stocks at T will be worth SeT

so that the value at T, of the dividends paid out is (SeT - S). If, as before, we let S* represent S

minus the PV of the value of the dividends then S* = S – e-T

(SeT – S) = Se

-T as required.

In both of the above cases the call premium on a European option on a dividend paying

stock will be less than that for an identical European option which does not pay dividends. This is

because when a stock goes ex-dividend, the stock price and hence the call premium on a

dividend paying stock will fall.

8.7 FROM THE BOPM TO BLACK SCHOLES

We can give some intuitive feel to the Black-Scholes option pricing formula based on

continuous time mathematics, by considering the BOPM approach where we increase the number

of steps ‘n’ in the binomial tree. Assume returns are continuously compounded Rt = ln(St/St-1). A

well known result in statistics is that if continuously compounded stock returns are normally

distributed, then stock prices will be log-normally distributed. The latter is the assumption used by

Black-Scholes in their continuous time approach.

In the BOPM, suppose we start with the share price S. If the down multiplier D is set

equal to the inverse of the ‘up’ multiplier, D = 1/U then returns will be symmetric. (This is a

multiplicative geometric process). To see this note that Ru = ln(SU/S) = ln(U) and Rd = ln(SD/S) =

ln(D) = ln(1/U) = -ln(U). In addition, after n successive falls, the stock price will be SDn = S/U

n and

since U > 1, stock price in this geometric process never falls below zero. Hence the left tail of the

distribution is truncated at zero. Also, after n-successive rises the stock price will be SUn which

continually increases as n increases (ie. the right tail of the distribution is elongated). Therefore

as we increase the number of steps n in the binomial tree, the price distribution at the expiration

date approaches the lognormal – as in the Black-Scholes model. Suppose we have:

S = 99 K = 100 rc = 0.10 = 0.20 T = 0.3 (years)

To translate these inputs into the BOPM we divide the time to expiration into shorter time

periods t = T/n where n is the number of steps chosen for the binomial tree and calculate U and

D as follows :

[8.48a] U = te and [8.48b] D =

te



41

We also convert the continuously compounded rate rc = 0.10 into a discrete periodic rate

rp to match the time period t = T/n of our analysis:

[8.49] R = (1 + rp) = tcre

.

For n = 1 :

t = T/n = 0.3/1 = 0.3 R = (1 + rp) = tcre

. = 1.0304

U = 3.020.0e = 1.0618

D =

03.020.0e = 0.9418

The call price for n = 1 is then :

[8.50] C(1)

=

R

CqqC du )1(6.21

where q = (R-D)/(U-D), Cu = max[SU-K, 0], Cd = max[SD-K,0].

Thus for n=1 the BOPM gives C(1)

= 6.21 which is not particularly close to the B-S value

of CBS

= 5.33. However as we increase n from 1 to 2, 3, … etc. we see that the BOPM call price

for about n=30 is C(30)

= 5.345 and approaches the B-S value of CBS

= 5.33 (and then ‘bounces

around’ this value) – see figure 8.9. In general, for plain vanilla options (but not necessarily for

complex exotic options) choosing n = 30 in the BOPM gives reasonably accurate results. (The

GAUSS code to reproduce the above can be found on the Web site. Of course, one of the

obvious problems with a numerical method like the BOPM is that it may not converge (at all), it

may ‘bounce around’ the true value for the option premium. This is the ‘price’ you pay for the

flexibility of the binomial ‘tree’ approach. Another intuitive yet slightly more mathematical way of

illustrating the common basic ideas in the BOPM approach and the Black Scholes equation is

given in appendix 8.1.

[Figure 8.9 here – Powerpoint taken from Gauss+Excel]



42

Finally, it is worth enquiring about the link between the above binomial process for the

stock price and the continuous time stochastic process assumed by Black-Scholes. The latter

assumes that the stock price follows a process known as a Geometric Brownian Motion (GBM),

which gives a lognormal distribution (at expiration of the option). Here, all we can do is to illustrate

what this process looks like. In fact it is relatively easy to simulate a random path for the stock

price which follows a GBM and which is (approximately) log-normally distributed by using:

[8.51] St = St-1 [1 + t + t t

where is the annual growth rate of stock prices, is the annual standard deviation of stock

returns and t is a small interval of time. For example if the total simulation period is T=1 year

and the number of periods in the simulation N = 100, then t = T/N = 0.01 (approximately 3.5

days). In equation [8.51] the stock price (in a small interval of time t) grows at the rate t and

has a variance equal to .t.

We generate a random series for S using = 20% p.a., = 15% p.a. We set So = 100

and draw successive values for from N(0,1) distribution. (In Excel RAND(.) draws a

number lying between 0 and 1 with equal probability.) It can be shown that =

12

1

6(.)i

RAND

generates an niid(0,1) variable in Excel. Our simulation gives the series shown in figure 8.10.

(Note that the generated returns series R = ln(St/St-1) will be a normally distributed random

variable.)



43

[Figure 8.10 here – Excel]

Generating a random series for the stock price is the first step in valuing an option under

risk neutral valuation (RNV), using Monte Carlo simulation and this is further discussed in chapter

17. However, note that when valuing an option under RNV we would assume that the growth rate

of the stock equals the risk free rate r. Hence when pricing the option using MCS we would set

= r in equation [8.51].

The reason the option premium in the BOPM approaches that given by the Black-Scholes

formula is that as n (ie. t = T/n 0), then the “up-down” lattice of the BOPM mimics the

behaviour of S in the GBM of equation [8.51] and the latter is used in the Black-Scholes approach.

8.8 SUMMARY

The main conclusions of this chapter are:

By constructing a risk free portfolio (from the option and the underlying asset) either the

BOPM or the Black-Scholes (continuous time) framework can be used to determine call and

put premia.

In the BOPM and the Black-Scholes approach we can value the option under the assumption

that the growth in the stock price equals the risk free rate. This is the principle of risk neutral

valuation and is reflected in the use of the risk neutral probability q, in the BOPM.

Call and put premia depend on the current stock price relative to the strike price (S/K), the risk

free rate r the volatility of the underlying asset and the time to expiry T.

Put-call parity is an arbitrage condition that can be used to derive the fair price of a put, given

the fair price of a call. This represents a form of ‘option conversion’ or financial engineering.



44

It can be shown that as the number of steps n in the BOPM increases then the BOPM formula

gives a value for the option premium, which approaches that given by the Black-Scholes

formula. This is because the binomial distribution for the stock price at expiry approaches the

continuous (lognormal) distribution as t = T/n 0, and the latter is the assumption made in

the Black-Scholes (continuous time) approach.

The BOPM is more flexible than the Black-Scholes approach. For example, the BOPM can

deal with early exercise and hence the pricing of American options as well as many other

types of ‘exotic options’ discussed later in the book. This flexibility comes at a ‘price’.

Although the option premium in the BOPM approaches that given by the Black-Scholes

formula for a European option (as n ), the BOPM being a numerical technique may

suffer from convergence problems and may only give an approximation to the true price.

APPENDIX 8.1

COMPARING THE BOPM AND THE BLACK-SCHOLES

EQUATION

The BOPM formula for the call premium can be re-arranged and interpreted in a similar

way to that for the Black Scholes formula in the text (see Fortune 1995). In the BOPM the time to

maturity T is divided up into n-periods. The payoff to a call after k “up” moves is :

[A8.1] Payoff after k “ups” = max[SUk D

n-k – K, 0]

The number of “up” moves to ensure the option ends up in-the-money is that value of k for which:

[A8.2] SUk D

n-k – K = 0

Let this specific number of up moves be denoted k* where

]/ln[/]/ln[* DUSDKk kn . We can rewrite the BOPM formula [8.22]

[A8.3] CR

n

kq q MAX SU D K

n

k n k k n k

k

n

1

1 00

( ) [ , ]

as:



45

[A8.4] 21 NRKNSC n

where

n

k

knkknk

nKDSUqq

RN

01 ][)1(

1

n

k

knk qqN0

2 )1( for i = max(k*, 0), …, n

Note the similarity of the above formulation of the BOPM to the Black-Scholes formula:

[A8.5] C = SN(d1) – N(d2)Ke-rT

We can interpret SN1 in the BOPM as the expected present value of the stock, conditional

on the option ending in-the-money. The expression for N2 can be interpreted as the probability

that the option will have enough “up” moves (ie. k* or more) to end up in-the-money and hence be

exercised at expiry. It follows that KR-n

N2 is the present value of the strike price. Hence the

BOPM formula implies that the call premium is the expected present value of the net payoff at

expiration. The latter is the interpretation we gave to the Black Scholes formula.

derivatives theory and practice (2020) - chapter 8 options...

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