4.5.2 The Vasicek model33

In this model, the state variable rt develops according to an Ornstein“Uhlenbeck process

drt = δ(θ ’ rt ) · dt + σ · dwt in which the parameters δ, θ and σ are strictly positive

constants and the rate risk unit premium is also a strictly positive constant »t (rt ) = » > 0.

The essential property of the Ornstein“Uhlenbeck process is that the variable rt is

˜recalled™ back towards θ if it moves too far away and that δ represents the ˜force

of recall™.

Example

Figure 4.6 shows a simulated trajectory (evolution over time) for such a process over

1000 very short time periods with the values δ = 100, θ = 0.1 and σ = 0.8 with a start

value for r0 of 10 %.

33

Vasicek O., An equilibrium characterisation of the term structure, Journal of Financial Economics, Vol. 5, No. 2, 1977,

pp. 177“88.

Bonds 143

0.3

0.25

0.2

0.15

0.1

0.05

0

1 101 201 301 401 501 601 701 801 901 1001

“0.05

“0.1

“0.15

“0.2

Figure 4.6 Ornstein“Uhlenbeck process

The partial derivatives equation for the price is shown here:

σ2

Pt + (δ(θ ’ r) + »σ )Pr + P ’ rP = 0

2 rr

The solution to this equation and its initial condition is given by

k ’ rt σ2

’δ(s’t)

) ’ 3 (1 ’ e’δ(s’t) )2

Pt (s, rt ) = exp ’k(s ’ t) + (1 ’ e

δ 4δ

where we have:

»σ σ2

k=θ+ ’2

δ 2δ

The average instant return rate is given by:

σ2

Pt + δ(θ ’ rt )Pr + P

2 rr

µt (s, rt ) =

P

rt · P ’ »σ · Pr

=

P

»σ

(1 ’ e’δ(s’t) )

= rt +

δ

This average return increases depending on the maturity date, and presents a horizontal

asymptote in the long term (Figure 4.7).

The spot rate is given by:

1

Rt (s, rt ) = ’ ln Pt (s, rt )

s’t

k ’ rt σ2

’δ(s’t)

(1 ’ e’δ(s’t) )2

=k’ (1 ’ e )+ 3

δ(s ’ t) 4δ (s ’ t)

144 Asset and Risk Management

µt(s, rt)

rt + »σ/δ

rt

t s

Figure 4.7 The Vasicek model: average instant return

On one hand, this expression shows that the spot rate is stabilised for distant maturity

Rt (s, rt ) = k.

dates and regardless of the initial value of the spot rate: lim

(s’t)’’+∞

On the other hand, depending on the current value of the spot rate in relation to the

parameters, we can use this model to represent various movements of the yield curve.

Depending on whether rt belongs to the

σ2 σ2 σ2 σ2

0; k ’ 2 , k ’ 2;k + 2 , k + 2 ; +∞

4δ 4δ 2δ 2δ

we will obtain a rate curve that is increasing, humped or decreasing.

Example

Figure 4.8 shows spot-rate curves produced using the Vasicek model for the following

parameter values: δ = 0.2, θ = 0.08, σ = 0.05 and » = 0.02. The three curves correspond,

from bottom to top, to r0 = 2 %, r0 = 6 % and r0 = 10 %.

The Vasicek model, however, has two major inconvenients. On one hand, the Orn-

stein“Uhlenbeck process drt = δ(θ ’ rt ) · dt + σ · dwt , on the basis of which is con-

structed, sometimes, because of the second term, allows the instant term rate to assume

negative values. On the other hand, the function of the spot rate that it generated, Rt (s),

may in some case also assume negative values.

0.12

0.1

0.08

0.06

0.04

0.02

0

1 11 21 31 41 51

Figure 4.8 The Vasicek model: yield curves

Bonds 145

4.5.3 The Cox, Ingersoll and Ross model34

This model is part of a group also known as the equilibrium models as they are based on

a macroeconomic type of reasoning, based in turn on the hypothesis that the consumer

will show behaviour consistently aimed at maximising expected utility.

These considerations, which we will not detail, lead (as do the other arbitrage models)

to a speci¬c de¬nition of the stochastic process that governs the evolution of the instant

term rate rt as well as the market price of risk »t (rt ).

If within the Ornstein“Uhlenbeck process the second term is modi¬ed to produce

drt = δ(θ ’ rt ) · dt + σ r ± t · dwt , with ± > 0, we will avoid the inconvenience mentioned

earlier: the instant term rate can no longer become negative. In fact, as soon as it reaches

zero, only the ¬rst term will subsist and the variation in rates must therefore necessarily

be upwards; the horizontal axis then operates as a ˜repulsing barrier™.

Using the macroeconomic reasoning on which the Cox, Ingersoll and Ross model is

based, we have a situation where ± = 1/2 and the stochastic process is known as the

square root process:

√

drt = δ(θ ’ rt ) · dt + σ rt · dwt

γ√

The same reasoning leads to a rate risk unit premium given by »t (rt ) = rt , where

σ

γ is a strictly positive constant; the market price of risk therefore increases together with

the instant term rate in this case.

Example

Figure 4.9 represents a square root process with the parameters δ = 100, θ = 0.1 and

σ = 0.8.

The partial derivative equation for the price is given by

σ2

Pt + (δ(θ ’ r) + γ r)Pr + rP ’ rP = 0

2 rr

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

1 101 201 301 401 501 601 701 801 901 1001

Figure 4.9 Square root process

34

Cox J., Ingersoll J. and Ross J., A theory of the term structure of interest rates, Econometrica, Vol. 53, No. 2, 1985, pp.

385“406.

146 Asset and Risk Management

The solution to this equation and its initial condition is given by

Pt (s, rt ) = xt (s) · e’yt (s)r

where we have: ± « 2δθ

σ2

1

2ke 2 (δ’γ +k)(s’t)

x (s) =

t

zt (s)

2(ek(s’t) ’ 1)

yt (s) =

zt (s)

z (s) = 2k + (δ ’ γ + k)(ek(s’t) ’ 1)

t

k = (δ ’ γ )2 + 2σ 2

The average instant rate return is given by:

σ2

Pt + δ(θ ’ rt )Pr + rP

2 rr

µt (s, rt ) =

P

rt · P ’ γ rt · Pr

=

P

= rt (1 + γ yt (s) )

In this case, the average rate of return is proportional to the instant term rate.

Finally, the spot rate is given by:

1

Rt (s, rt ) = ’ ln Pt (s, rt )

s’t

1

=’ (ln xt (s) ’ rt yt (s))

s’t

Example

Figure 4.10 shows the spot-rate curves produced using the Cox, Ingersoll and Ross model

for the following parameter values: δ = 0.2, θ = 0.08, σ = 0.05 and γ = 0.02. The three

curves, from bottom to top, correspond to r0 = 2 %, r0 = 6 % and r0 = 10 %.

0.12

0.1

0.08

0.06

0.04

0.02

0

0 10 20 30 40 50 60

Figure 4.10 The Cox, Ingersoll and Ross model: rate curves

Bonds 147

Finally, we should point out that in contrast to the Vasicek model, the Cox, Ingersoll

and Ross model can never produce a negative spot rate. In addition, as with the Vasicek

model, the spot rate stabilises for distant maturity dates regardless of the initial value of

the spot rate:

2δθ

lim Rt (s, rt ) =

δ’γ +k

(s’t)’+∞

4.5.4 Stochastic duration

Finally, to end this section dedicated to random models, we turn to a generalisation of

the concept of duration.

Duration and convexity of rate products are ongoing techniques used to assess the

sensitivity and alteration of the price of an asset following an alteration to its rate. Duration

allows the variation in value to be estimated for more signi¬cant variations. These concepts

are used not only in bond portfolio management but also in asset and liability management

in the context of immunisation of interest margins. What happens is that part of the

balance-sheet margin is produced by a spread between the interest paid on assets (long-

term deposits) and interest received on assets (the bank™s own portfolio with a ¬xed

income). This margin is immunised against variations in rate if convexity and duration

are identical on both sides of the balance sheet. This identity of duration and convexity,

also known as mutual support, does not necessarily mean that cash ¬‚ows are identical in

assets and liabilities. In this case, distortion of the rate curve could lead to non-identical

alterations in asset and liability values.

The transition from a deterministic rate curve to a stochastic rate model provides the

solution to this problem. The random evolution of rates allows the stochastic duration to

be calculated. There are several stochastic rate models (see Sections 4.5.1“4.5.3), but the

type most frequently used in ¬nancial literature is the classical Vasicek model.

4.5.4.1 Random evolution of rates

The classical Vasicek model is based on changes in the instant term rate governed by an

Ornstein“Uhlenbeck process: drt = δ(θ ’ rt ) · dt + σ · dwt .

The forward long rate r+∞f w is of course a function of the parameters δ, θ and σ of the

model. Variations in a bond™s price depend on the values taken by the random variable rt

and the alterations to the model™s parameters. The natural way of approaching stochastic

duration is to adjust the parameters econometrically on rate curves observed.

4.5.4.2 Principle of mutual support

The total variation in value at the initial moment t is obtained by developing Taylor in

the ¬rst order:

dVt (s, rt , r+∞f w , σ ) = Vrt drt + Vr+∞f w dr+∞f w + Vσ dσ

The principle of mutual support between assets and liabilities requires two restrictions to

be respected: ¬rst, equality of values between assets and liabilities:

VA,t (s, rt , r+∞f w , σ ) = VL,t (s, rt , r+∞f w , σ )

148 Asset and Risk Management

and second, equality of total variations: regardless of what the increases in drt , dr+∞f w

and dσ may be, we have

dVA,t (s, rt , r+∞f w , σ ) = dVL,t (s, rt , r+∞f w , σ )

This second condition therefore requires that:

VA,rt (s, rt , r+∞f w , σ ) = VL,rt (s, rt , r+∞f w , σ )

VA,r+∞f w (s, rt , r+∞f w , σ ) = VL,r+∞f w (s, rt , r+∞f w , σ )

VA,σ (s, rt , r+∞f w , σ ) = VL,σ (s, rt , r+∞f w , σ )

4.5.4.3 Extension of the concept of duration

Generally speaking, it is possible to de¬ne the duration D that is a function of the variation

in long and short rates:

1

Dt (s, rt , r+∞f w ) = (V + Vr+∞f w )

2V rt

This expression allows us to ¬nd the standard duration when the rate curve is deterministic

and tends towards a constant curve with σ = 0 and θ = rt , with the initial instant term

rate for the period t:

1

Dt (s, rt , r+∞f w ) = · Vrt

V

Generally, the duration is sensitive to the spread S between the short rate and the long rate.

Sensitivity to spread allows the variation in value to be calculated for a spread variation:

1 1

St (s, rt , r+∞f w ) = (’Vrt + Vr+∞f w ) = · Vrt

V

2V

In this case, if s is stable and considered to be a constant, the mutual support will

correspond to the equality of stochastic distribution and of sensitivity of spread for assets

and liabilities.

The equality dVA,t (s, rt , r+∞f w ) = dVL,t (s, rt , r+∞f w ), valid whatever the increases in

drt and dr+∞f w is equivalent to:

DA = DL

SA = SL

5

Options

5.1 DEFINITIONS

5.1.1 Characteristics

An option 1 is a contract that confers on its purchaser, in return for a premium, the right

to purchase or sell an asset (the underlying asset) on a future date at a price determined

in advance (the exercise price of the option). Options for purchasing and options for

selling are known respectively as call and put options. The range of assets to which

options contracts can be applied is very wide: ordinary equities, bonds, exchange rates,

commodities and even some derivative products such as FRAs, futures, swaps or options.

An option always represents a right for the holder and an obligation to buy or sell

for the issuer. This option right may be exercised when the contract expires (a European

option) or on any date up to and including the expiry date (an American option). The

holder of a call option will therefore exercise his option right if the price of the underlying

equity exceeds the exercise price of the option or strike; conversely, a put option will be

exercised in the opposite case.

The assets studied in the two preceding chapters clearly show a degree of random

behaviour (mean-variance theory for equities, interest-rate models for bonds). They do,

however, also allow deterministic approaches (Gordon-Shapiro formula, duration and con-

vexity). With options, the random aspect is much more intrinsic as everything depends

on a decision linked to a future event.

This type of contract can be a source of pro¬t (with risks linked to speculation) and a

means of hedging. In this context, we will limit our discussion to European call options.

Purchasing this type of option may lead to an attractive return, as when the price of

the underlying equity on maturity is lower than the exercise price, the option will not be

exercised and the loss will be limited to the price of the option (the premium). When the

price of the underlying equity on maturity is higher than the exercise price, the underlying

equity is received for a price lower than its value.

The sale (issue) of an equity option, on the other hand, is a much more speculative

operation. The pro¬t will be limited to the premium if the price of the underlying equity

remains lower than the exercise price, while considerable losses may arise if the rise is

higher than the price of the underlying equity. This operation should therefore only be

envisaged if the issuer has absolute con¬dence in a fall (or at worst a reduced rise) in the

price of the underlying equity.

Example

Let us consider a call option on an equity with a current price of 100, a premium of

3 and an exercise price of 105. We will calculate the pro¬t made (or the loss suffered)

1

Colmant B. and Kleynen G., Gestion du risque de taux d™int´ rˆ t et instruments ¬nanciers d´ riv´ s, Kluwer, 1995. Hull

ee ee

J. C., Options, Futures and Other Derivatives, Prentice Hall, 1997. Hicks A., Foreign Exchange Options, Woodhead, 1993.

150 Asset and Risk Management

Table 5.1 Pro¬t on option according to price

of underlying equity

Price of underlying equity Gain

Purchaser Issuer

’3

90 3

’3

95 3

’3

100 3

’3

105 3

’2

106 2

’1

107 1

108 0 0

’1

109 1

’2

110 2

’7

115 7

’12

120 12

by the purchaser and by the issuer of the contract according to the price reached by the

underlying equity on maturity. See Table 5.1.

Of course, issuers who notice the price of the underlying equity rising during the

contractual period can partly protect themselves by purchasing the same option and thus

closing their position. Nevertheless, because of the higher underlying equity price, the

premium for the option purchased may be considerably higher than that of the option that

was issued.

The price (premium) of an option depends on several different factors:

• price of the underlying equity St at the moment t (known as the spot);

the

• exercise price of the option K (known as the strike);

the

• duration T ’ t remaining until the option matures;2

the

• volatility σR of the return on the underlying equity;

the

• risk-free rate RF .3

the

The various ways of specifying the function f (which will be termed C or P depending

on whether a call or a put is involved) give rise to what are termed models of valuation.

These are dealt with in Section 5.3.

5.1.2 Use

The example shown in the preceding paragraph corresponds to the situation in which the

purchaser can hope for an attractive gain. The pro¬t realised is shown in graphic form in

Figure 5.1.

This residual duration T ’ t is frequently referred to simply as „

2

3

This description corresponds, for example, to an option on an equity (with which we will mostly be dealing in this

chapter). For an exchange option, the rate of interest will be divided in two, into domestic currency and foreign currency. In

addition, this rate, which will be used in making updates, may be considered either discretely (discounting factor (1 + RF )’t )

or continuously (the notation r: e’rt will then be used).

Options 151

Profit

K

ST

“p

Figure 5.1 Acquisition of a call option

Alongside this speculative aspect, the issue of a call option can become attractive if it

is held along with the underlying equity. In fact, if the underlying equity price falls (or

rises little), the loss suffered on that equity will be partly offset by receipt of the premium,

whereas if the price rises greatly, the pro¬t that would have been realised will be limited

to the price of the option plus the differential between the exercise price and the price of

the underlying equity at the start of the contract.

Example 1

Following the example shown above, we calculated the pro¬t realised (or the loss suffered)

when the underlying equity alone is held and when it is covered by the call option

(Table 5.2).

Example 2

Let us now look at a more realistic example. A European company X often has invoices

expressed in US dollars payable on delivery. The prices are of course ¬xed at the moment

of purchase (long before the delivery). If the rate for the dollar rises between the moment

of purchase and the moment of delivery, the company X will suffer a loss if it purchases

its dollars at the moment of payment.

Table 5.2 Pro¬t/loss on equity covered by call

option

Price of underlying equity Pro¬t/loss

Purchaser Issuer

’3

90 3

’3

95 3

’3

100 3

’3

105 3

’2

106 2

’1

107 1

108 0 0

’1

109 1

’2

110 2

’7

115 7

’12

120 12

152 Asset and Risk Management

Let us assume, more speci¬cally, that the rate for the dollar at the moment t is

St (US$1 = ¤St ) and that X purchases goods on this day (t = 0) valued at US$1000,

the rate being S0 = x (US$1 = ¤x), for delivery in t = T . The company X, on t = 0,

acquires 1000 European US$/¤ calls maturing on T , the exercise price being K = ¤x

for US$1.

If ST > x, the option will be exercised and X will purchase its dollars at rate x

(the rate in which the invoice is expressed) and the company will lose only the total

of the premium. If ST ¤ X, the option will not be exercised and X will purchase its

dollars at the rate ST and the business will realise a pro¬t of 1000 · (x ’ ST ) less

the premium.

The purchase of the option acts as insurance cover against changes in rates. Of course

it cannot be free of charge (consider the point of view of the option issuer); its price is

the option premium.

The case envisaged above corresponds to the acquisition of a call option. The same kind

of reasoning can be applied to four situations, corresponding to the purchase or issue

of a call option on one hand or of a put option on the other hand. Hence we have

Figures 5.2 and 5.3.

In addition to the simple cover strategy set out above, it is possible to create more

complex combinations of subsequent equity, call options and put options. These more

involved strategies are covered in Section 5.4.

Profit

p

ST

K

Figure 5.2 Issue of a call option

Profit

Profit

p

K

ST ST

K

“p

Figure 5.3 Acquisition and issue of a put option

Options 153

5.2 VALUE OF AN OPTION

5.2.1 Intrinsic value and time value

An option premium can be spilt into two terms: its intrinsic value and its time value.

The intrinsic value of an option at a moment t is simply the pro¬t realised by the

purchaser (without taking account of the premium) if the option was exercised at t. More

speci¬cally, for a call option it is the difference, if that difference is positive,4 between the

price of the underlying equity St at that moment and the exercise price5 K of the option.

If the difference is negative, the intrinsic value is by de¬nition 0. For a put option, the

intrinsic value will be the difference between the exercise price and the underlying equity

price.6 Therefore, if the intrinsic value of the option is termed VI, we will have VI t =

max (0, St ’ K) = (St ’ K)+ for a call option and VI t = max (0, K ’ St ) = (K ’ St )+

for a put option, with the graphs shown in Figure 5.4.

The price of the option is of course at least equal to its intrinsic value. The part of

the premium over and above the intrinsic value is termed time value and shown as VT,

hence: V Tt = pt ’ VI t .

This time value, which is added to the intrinsic value to give the premium, represents

payment in anticipation of an additional pro¬t for the purchaser. From the point of view

of the issuer, it therefore represents a kind of risk premium.

The time value will of course decrease as the time left to run decreases, and ends by

being cancelled out at the maturity date (see Figure 5.5).

VIt VIt

(call) (put)

K St K St

Figure 5.4 Intrinsic value of a call option and put option

VTt

T t

Figure 5.5 Time value according to time

4

The option is then said to be ˜in the money™. If the difference is negative, the option is said to be ˜out of the money™. If

the subjacent share price is equal or close to the exercise price, it is said to be ˜at the money™. These de¬nitions are inverted

for put options.

5

The option cannot in fact be exercised immediately unless it is of the American type. For a European option, the exercise

price should normally be discounted for the period remaining until the maturity date.

This de¬nition is given for an American option. For a European option, it is suf¬cient, within the interpretation of St , to

6

replace the price at the moment t by the maturity date price.

154 Asset and Risk Management

VT

K S

Figure 5.6 Time value according to underlying equity price

p

OTM ATM ITM

VT

VI

K S

Figure 5.7 Splitting of call option premium

It is easy to see, as the other parameters are constant, that the time value will be greater

as the underlying equity price comes near to the exercise price, as shown in Figure 5.6.

To understand this property, let us view things from the call issuer™s point of view to

lay down the ideas. If the option is out of the money, it will probably not be exercised and

the issuer may dispense with acquiring the underlying equity; his risk (steep rise in the

underlying equity price) will therefore be low and he will receive very little reward. In the

same way, an in-the-money option will probably be exercised, and the issuer will therefore

have an interest in acquiring the underlying equity; a sharp drop in the underlying equity

price represents a highly improbable risk and the time value will also be low. Conversely,

for an at-the-money option the issuer will have no degree of certainty with regard to

whether or not the option should be exercised, or how the underlying equity price will

develop; the risk of the underlying equity price falling after he acquires the equity (or of

a price surge without the underlying equity being acquired) is therefore high and a risk

premium will be requested in consequence. This phenomenon is shown in Figure 5.7.

In addition, it is evident that the longer the period remaining until the option contract

matures, the higher the risk and the greater the time value (see Figure 5.8).

Of course, the value of an option at maturity is identical to its intrinsic value:

CT = (ST ’ K)+

PT = (K ’ ST )+

5.2.2 Volatility

Of the parameters that de¬ne the price of an option, let us now look more speci¬cally

at the volatility σR of the return of the underlying equity. The volatility of an option

Options 155

p

(a)

(b)

K S

Figure 5.8 Call premium and high (a) and brief (b) maturity

is de¬ned as a measurement of the dispersion of the return of the underlying equity.

In practice, it is generally taken for a reference period of one year and expressed as a

percentage. This concept of volatility can be seen from two points of view: historical

volatility and implied volatility.

Historical volatility is simply the annualised standard deviation on the underlying equity

return, obtained from daily observations of the return in the past:

n

1

σR = J· (Rt ’ R)2

n t=1

Here, the factor J represents the number of working days in the year; n is the number

of observations and Rt is the return on the underlying equity. It is easy to calculate, but

the major problem is that it is always ˜turned towards the past™ when it really needs to

help analyse future developments in the option price.

For this reason, the concept of implied volatility has been introduced. This involves

using a valuation model to estimate the dispersion of the return of the underlying equity

for the period remaining until the contract matures. The value of the option premium is

determined in practice by the law of supply and demand. In addition, this law is linked to

various factors through a binomial model of valuation: pt = f (St , K, T ’ t, σR , RF ) or

through Black and Scholes (see Section 5.3). The resolution of this relation with respect

to σR de¬nes the implied volatility. Although the access is more complicated, this concept

is preferable and it is this one that will often be used in practice.

5.2.3 Sensitivity parameters

5.2.3.1 ˜Greeks™

The premium is likely to vary when each of the parameters that determine the price of the

option (spot price, exercise price, maturity etc.) change. The aim of this paragraph is to

study the indices,7 known as ˜Greeks™, which measure the sensitivity of the premium to

¬‚uctuations in some of these characteristics through the relation pt = f (St , K, „, σR , RF ).

7

In the same way as duration and convexity, which measure the sensitivity of the value of a bond following changes in

interest rates (see Chapter 4).

156 Asset and Risk Management

Here, we will restrict ourselves to examining the most commonly used sensitivity

coef¬cients: those that bring the option price and namely the underlying equity price

time, volatility and risk-free rate into relation. In addition, the sign indications given are

valid for a non-dividend-paying equity option.

The coef¬cient (delta) represents the sensitivity of the option price with respect to

the underlying equity price. It is measured by dividing the variations in these two prices

for a small increase δSt in the underlying equity price:

f (St + δSt , K, „, σR , RF ) ’ f (St , K, „, σR , RF )

=

δSt

Or, more speci¬cally:

f (St + δSt , K, „, σR , RF ) ’ f (St , K, „, σR , RF )

= lim

δSt

δSt ’0

= fS (St , K, „, σR , RF )

Thus, for a call, if the underlying equity price increases by ¤1, the price of the option

will increase by ¤ . It will be between 0 and 1 for a call and between ’1 and 0 for

a put.

Another coef¬cient expresses the sensitivity of the option price with respect to the

underlying equity price, but this time in the second order. This is the coef¬cient

(gamma), which is expressed by the ratio of variations in on one hand and the price

St on the other hand.

= fSS (St , K, „, σR , RF )

If one wishes to compare the dependency of the option premium vis-` -vis the underlying

a

equity price and the price of a bond according to the actuarial rate, it can be said that

is to the duration what is to convexity. This coef¬cient , which is always positive, is

the same for a call option and for a put option.

The following coef¬cient, termed (theta), measures the dependence of the option

price according to time:

= ft (St , K, T ’ t, σR , RF )

or, by introducing the residual life span „ = T ’ t of the contract,

= ’f„ (St , K, „, σR , RF )

When the maturity date for the option contract is approaching, the value of the contract

will diminish, implying that is generally negative.

The coef¬cient V (vega)8 measures the sensitivity of the option premium with respect

to volatility:

V = fσ (St , K, „, σR , RF )

It is always positive and has the same value for a call and for a put. It is of course

interpreted as follows: if the volatility increases by 1 %, the option price increases by V .

Also termed κ (kappa) on occasions “ possibly because vega is not a Greek letter!

8

Options 157

Finally, the coef¬cient ρ (rho) expresses the manner in which the option price depends

on the risk-free rate RF :

ρ = fRF (St , K, „, σR , RF )

This coef¬cient will be positive or negative depending on whether we are dealing with a

call or a put.

5.2.3.2 ˜Delta hedging™

As these coef¬cients have now been de¬ned, we can move onto an interesting interpre-

tation of the delta. This element plays its part in hedging a short-term position (issue)

of a call option (referred to as ˜delta hedging™). The question is: how many units of the

underlying equity must the issuer of a call acquire in order to hedge his position? This

quantity is referred to as X. Although the current value of the underlying equity is X, the

value of its portfolio, consisting of the purchase of X units of the underlying equity and

the issue of one call on that equity, is:

V (S) = X · S ’ C(S)

If the price of the underlying equity changes from S to S + δS, the value of the portfolio

changes to:

V (S + δS) = X · (S + δS) ’ C(S + δS)

C(S + δS) ’ C(S)

≈

As , the new value of the portfolio is:

δS

V (S + δS) = X · (S + δS) ’ [C(S) + · δS]

= X · S ’ C(S) + (X ’ ) · δS

= V (S) + (X ’ ) · δS

The position will therefore be hedged against a movement (up or down) of the underlying

equity price if the second term is zero (X = ), that is, if the issuer of the call holds

units in the underlying equity.

5.2.4 General properties

5.2.4.1 Call“put parity relation for European options

We will now draw up the relation that links a European call premium and a European put

premium, both relating to the same underlying equity and both with the same exercise

price and maturity date: this is termed the ˜call“put parity relation™.

We will establish this relation for a European equity option that does not distribute a

dividend during the option contract period.

Let us consider a portfolio put together at moment t with:

• the purchase of the underlying equity, whose value is St ;

• the purchase of a put on this underlying equity, with exercise price K and maturity T ;

its value is therefore Pt (St , K, „, σR , RF );

158 Asset and Risk Management

• the sale of a call on the same underlying equity, with exercise price K and maturity

T ; its value is therefore Ct (St , K, „, σR , RF );

• the borrowing (at risk-free rate RF ) of a total worth K at time T ; the amount is therefore

K · (1 + RF )’„ .

The value of the portfolio at maturity T will be ST + PT ’ CT ’ K. As we have shown

previously that CT = (ST ’ K)+ and that PT (K ’ ST )+ , this value at maturity will equal:

if ST > K, ST + 0 ’ (ST ’ K) ’ K = 0

if ST ¤ K, ST + (K ’ ST ) ’ 0 ’ K = 0

This portfolio, regardless of changes to the value of the underlying equity between t and

T and for constant K and RF , has a zero value at moment T . Because of the hypothesis

of absence of arbitrage opportunity,9 the portfolio can only have a zero value at moment t.

The zero value of this portfolio at moment t is expressed by: St + Pt ’ Ct ’ K · (1 +

RF )’„ = 0.

Or, in a more classic way, by:

Ct + K · (1 + RF )’„ = Pt + St

This is the relation of parity declared.

Note

The ˜call“put™ parity relation is not valid for an exchange option because of the interest

rate spread between the two currencies. If the risk-free interest rate for the domestic

currency and that of the foreign currency are referred to as RF and RF ) (they are

(D) (F

assumed to be constant and valid for any maturity date), it is easy to see that the parity

relation will take the form Ct + K · (1 + RF )’„ = Pt + St · (1 + RF ) )’„ .

(D) (F

5.2.4.2 Relation between European call and American call

Let us now establish the relation that links a European call to an American call, both for

the same underlying equity and with the same exercise price and maturity date. As with

the parity relation, we will deal only with equity options that do not distribute a dividend

during the option contract period.

As the American option can be exercised at any moment prior to maturity, its value will

always be at least equal to the value of the European option with the same characteristics:

Ct(a) (St , K, T ’ t, σR , RF ) ≥ Ct(e) (St , K, T ’ t, σR , RF )

The parity relation allows the following to be written in succession:

Ct(e) + K · (1 + RF )’„ = Pt(e) + St

Ct(e) ≥ St ’ K · (1 + RF )’„ > St ’ K

Ct(a) ≥ Ct(e) > (St ’ K)+

Remember that no ¬nancial movement has occurred between t and T as we have excluded the payment of dividends.

9

Options 159

As (St ’ K)+ represents what the American call would return if exercised at moment t,

its holder will be best advised to retain it until moment T . At all times, therefore, this

option will have the same value as the corresponding European option:

Ct(a) = Ct(e) ∀t ∈ [0; T ]

We would point out that the identity between the American and European calls does not

apply to puts or to other kinds of option (such as exchange options).

5.2.4.3 Inequalities on price

The values of calls and puts obey the following inequalities:

[St ’ K(1 + RF )’„ ]+ ¤ Ct ¤ St

[K(1 + RF )’„ ’ St ]+ ¤ Pt(e) ¤ K(1 + RF )’„

[K ’ St ]+ ¤ Pt(a) ¤ K

These inequalities limit the area in which the graph for the option according to the under-

lying equity price can be located. This leads to Figure 5.9 for a European or American

call and Figure 5.10 for puts.

The right-hand inequalities are obvious: they state simply that an option cannot be worth

more than the gain it allows. A call cannot therefore be worth more than the underlying

equity whose acquisition it allows. In the same way, a put cannot be worth more than the

Ct

K(1 + RF)“„ St

Figure 5.9 Inequalities for a call value

P(e) P(a)

t t

K

K(1 + RF)“„

K(1 + RF)“„ K

St St

Figure 5.10 Inequalities for the value of a European put and an American put

160 Asset and Risk Management

exercise price K at which it allows the underlying equity to be sold; and for a European

put, it cannot exceed the discounted value of the exercise price in question (the exercise

can only occur on the maturity date).

Let us now justify the left-hand inequality for a call. To do this, we set up at moment

t a portfolio consisting of:

• the purchase of one call;

• a risk-free ¬nancial investment worth K at maturity: K(1 + RF )’„ ;

• the sale of one unit of the underlying equity.

Its value at moment t will of course be: Vt = Ct + K(1 + RF )’„ ’ St .

Its value on maturity will depend on the evolution of the underlying equity:

if ST > K, (ST ’ K) + K ’ ST = 0

VT =

if ST ¤ K, 0 + K ’ ST

In other words, VT = (K ’ ST )+ , which is not negative for any of the possible evo-

lution scenarios. In the absence of arbitrage opportunity, we also have Vt ≥ 0, that is,

Ct ≥ St ’ K(1 + RF )’„ .

As the price of the option cannot be negative, we have the inequality declared.

The left-hand inequality for a European put is obtained in the same way, by arbitrage-

based logic using the portfolio consisting of:

• the purchase of one put;

• the purchase of one underlying equity unit;

• the purchase of an amount worth K at maturity: K(1 + RF )’„ .

The left-hand inequality for an American put arises from the inequality for a European

put. It should be noted that there is no need to discount the exercise price as the moment

at which the option right will be exercised is unknown.

5.3 VALUATION MODELS

Before touching on the developed methods for determining the value of an option, we

will show the basic principles for establishing option pricing using an example that has

been deliberately simpli¬ed as much as possible.

Example

Consider a European call option on the US$/¤ exchange rate for which the exercise

price is K = 1. Then suppose that at the present time (t = 0) the rate is S0 = 0.95

(US$1 = ¤0.95). We will be working with a zero risk-free rate (RF = 0) in order to

simplify the developments.

Let us suppose also that the random changes in the underlying equity between moments

t = 0 and t = T can correspond to two scenarios s1 and s2 for which ST is ¤1.1 and ¤0.9

respectively, and that the scenarios occur with the respective probabilities of 0.6 and 0.4.

1.1 Pr(s1 ) = 0.6

ST =

0.9 Pr(s2 ) = 0.4

Options 161

The changes in the exchange option, which is also random, can therefore be de-

scribed as:

0.1 Pr(s1 ) = 0.6

CT =

0.0 Pr(s2 ) = 0.4

Let us consider that at moment t = 0, we have a portfolio consisting of:

• the issue of a US$/¤ call (at the initial price of C0 );

• a loan of ¤X;

• the purchase of US$Y .

so that:

• The initial value V0 of the portfolio is zero: the purchase of the US$Y is made exactly

with what is generated by the issue of the call and the loan.

• The portfolio is risk-free, and will undergo the same evolution whatever the scenario

(in fact, its value will not change as we have assumed RF to be zero).

The initial value of the portfolio in ¤ is therefore V0 = ’C0 ’ X + 0.95Y = 0.

Depending on the scenario, the ¬nal value will be given by:

VT (s1 ) = ’0.1 ’ X + 1.1 · Y

VT (s2 ) = ’X + 0.9 · Y

The hypothesis of absence of opportunity for arbitrage allows con¬rmation that

VT (s1 ) = VT (s2 ) = 0 and the consequent deduction of the following values: X = 0.45

and Y = 0.5. On the basis of the initial value of the portfolio, the initial value of the

option is therefore deduced:

C0 = ’X + 0.95Y = 0.025

It is important to note that this value is totally independent of the probabilities 0.6 and 0.4

associated with the two development scenarios for the underlying equity price, otherwise

we would have C0 = 0.1 — 0.6 + 0 — 0.4 = 0.06.

If now we determine another law of probability

Pr(s1 ) = q Pr(s2 ) = 1 ’ q

for which C0 = Eq (CT ), we have 0.025 = 0.01 · q + 0 · (1 ’ q), that is: q = 0.25.

We are in fact looking at the law of probability for which S0 = Eq (ST ):

Eq (ST ) = 1.1 · 0.25 + 0.9 · 0.75 = 0.95 = S0

162 Asset and Risk Management

We have therefore seen, in a very speci¬c case where there is a need for generalisation,

that the current value of the option is equal to the mathematical expectation of its future

value, with respect to the law of probability for which the current value of the underlying

equity is equal to the expectation of its future value.10 This law of probability is known

as the risk-neutral probability.

5.3.1 Binomial model for equity options

This model was produced by Cox, Ross and Rubinstein.11 In this discrete model we look

simply at a list of times 0, 1, 2, . . . , T separated by a unit of time (the period), which is

usually quite short.

Placing ourselves in a perfect market, we envisage a European equity option that does

not distribute any dividends during the contract period and with a constant volatility

during the period in question.

In addition, we assume that the risk-free interest does not change during this period,

that it is valid for any maturity (¬‚at, constant yield curve), and that it is the same for a

loan and for an investment. This interest rate, termed RF , will be expressed according

to a duration equal to a period; and the same will apply for other parameters (return,

volatility etc.).

Remember (Section 3.4.2) that the change in the underlying equity value from one time

to another is dichotomous in nature: equity has at moment t the value St , but at the next

moment t + 1 will have one of the two values St · u (greater than St ) or St · d (less than

St ) with respective probabilities of ± and (1 ’ ±). We have d ¤ 1 < 1 + RF ¤ u and the

parameters u, d and ±, which are assumed to be constant over time, should be estimated

on the basis of observations.

We therefore have the following graphic representation of the development in equity

prices for a period:

S = St · u (±)

’ ’ t+1

St ’ ’

’

’’

’ S = S · d (1 ’ ±)

t+1 t

and therefore, more generally speaking, as shown in Figure 5.11.

Now let us address the issue of evaluating options at the initial moment. Our reasoning

will be applied to a call option.

It is known that the value of the option at the end of the contract will be expressed

according to the value of the equity by CT = (ST ’ K)+ .

S0 • u3 ¦

S0 • u2

S0 • u2d ¦

S0 • u

S0 S0 • ud

S0 • ud2 ¦

S0 • d

S0 • d2

S0 • d2 ¦

Figure 5.11 Binomial tree for underlying equity

10

When the risk-free rate is zero, remember.

11

Cox J., Ross S. and Rubinstein M., Option pricing: a simpli¬ed approach, Journal of Financial Economics, No. 7, 1979,

pp. 229“63

Options 163

After constructing the tree diagram for the equity from moment 0 to moment T , we

will now construct the tree from T to 0 for the option, from each of the ends in the

equity tree diagram, to reconstruct the value C0 of the option at 0. This reasoning will be

applied in stages.

5.3.1.1 One period

Assume that T = 1. From the equity tree diagram it can be clearly seen that the call C0

(unknown) can evolve into two values with the respective probabilities of ± and (1 ’ ±):

+

’ C1 = C(u) = (S0 · u ’ K)

’

’

’

C0’

’’’ C = C(d) = (S · d ’ K)+

1 0

As the value of C1 (that is, the value of C(u) and C(d)) is known, we will now determine

the value of C0 . To do this, we will construct a portfolio put together at t = 0 by:

• the purchase of X underlying equities with a value of S0 ;

• the sale of one call on this underlying equity, with a value of C0 .

The value V0 of this portfolio, and its evolution V1 in the context described, are given by:

V = X · S0 · u ’ C(u)

’’ 1

V0 = X · S0 ’ C0’ ’

’

’’

’V = X · S0 · d ’ C(d)

1

We then choose X so that the portfolio is risk-free (the two values of V1 will then be

identical). The hypothesis of absence of arbitrage opportunity shows that in this case, the

return on this portfolio must be given by the risk-free rate RF .

We therefore obtain:

V1 = X · S0 · u ’ C(u) = X · S0 · d ’ C(d)

V1 = (X · S0 ’ C0 )(1 + RF )

The ¬rst equation readily provides:

C(u) ’ C(d)

X · S0 =

u’d

and therefore:

d · C(u) ’ u · C(d)

V1 =

u’d

The second equation then provides:

d · C(u) ’ u · C(d) C(u) ’ C(d)

= ’ C0 (1 + RF )

u’d u’d

164 Asset and Risk Management

This easily resolves with respect to C0 :

(1 + RF ) ’ d u ’ (1 + RF )

C0 = (1 + RF )’1 C(u) + C(d)

u’d u’d

The coef¬cients for C(u) and C(d) are clearly between 0 and 1 and total 1. We therefore

introduce:

(1 + RF ) ’ d u ’ (1 + RF )

q= 1’q =

u’d u’d

They constitute the neutral risk law of probability.

We therefore have the value of the original call:

C0 = (1 + RF )’1 [q · C(u) + (1 ’ q) · C(d)]

Note 1

As was noted in the introductory example, the probability of growth ± is not featured

in the above relation. The only law of probability involved is the one relating to the

risk-neutral probability q, with respect to which C0 appears as the discounted value of

the average value of the call at maturity (t = 1).

The term ˜risk-neutral probability™ is based on the logic that the expected value of the

underlying equity at maturity (t = 1) with respect to this law of probability is given by:

Eq (S1 ) = q · S0 · u + (1 ’ q) · S0 · d

(1 + RF ) ’ d u ’ (1 + RF )

= S0 u+ d

u’d u’d

= S0 (1 + RF )

The change in the risk-free security is the same as the expected change in the risked

security (for this law of probability).

Note 2

When using the binomial model practically, it is simpler to apply the reasoning with

respect to one single period for each node on the tree diagram, progressing from T to 0.

We will, however, push this analysis further in order to obtain a general result.

5.3.1.2 Two periods

Let us now suppose that T = 2. The binomial tree diagram for the option will now be

written as:

+

’ C2 = C(u, u) = (S0 · u ’ K)

2

’

C = C(u) ’’’

’’

’

’’

’1 C2 = C(u, d) = C(d, u) = (S0 · ud ’ K)+

’

C0’

’’’ C = C(d) ’ ’

’’

’’

’’

1

C2 = C(d, d) = (S0 · d 2 ’ K)+

Options 165

The previous reasoning will allow transition from time 2 to time 1:

C(u) = (1 + RF )’1 [q · C(u, u) + (1 ’ q) · C(u, d)]

C(d) = (1 + RF )’1 [q · C(d, u) + (1 ’ q) · C(d, d)]

And from time 1 to time 0:

C0 = (1 + RF )’1 q · C(u) + (1 ’ q) · C(d)

= (1 + RF )’2 q 2 · C(u, u) + 2q(1 ’ q) · C(u, d) + (1 ’ q)2 · C(d, d)

Consideration of the coef¬cients for C(u,u), C(u,d) and C(d, d) allows the above note

to be speci¬ed: C0 is the discounted value for the expected value of the call on maturity

(t = 2) with respect to a binomial law of probability12 for parameters (2; q).

5.3.1.3 T periods

To generalise what has already been said, it is seen that C0 is the discounted value of

the expected value of the call on maturity (t = T ) with respect to a binomial law of

probability for parameters (T ; q). We can therefore write:

T

T

’T

q j (1 ’ q)T ’j C(u, . . . , u, d, . . . , d )

C0 = (1 + RF )

j

j =0 j T ’j

T

T

’T

q j (1 ’ q)T ’j (S0 uj d T ’j ’ K)+

= (1 + RF )

j

j =0

As uj d T ’j is an increasing function of j , if one introduces j = min {j :S0 uj d T ’j ’ K > 0},

ln K ’ ln(S0 d T )

that is, the smallest value of j that is strictly higher than , the evaluation of

ln u ’ ln d

the call takes the form:

T

T

’T

q j (1 ’ q)T ’j (S0 uj d T ’j ’ K)

C0 = (1 + RF )

j

j =J

T T

j T ’j

uq d(1 ’ q)

T T

’T

q j (1 ’ q)T ’j

= S0 ’ K(1 + RF )

j j

1 + RF 1 + RF

j =J j =J

Because:

uq d(1 ’ q) u[(1 + RF ) ’ d] + d[u ’ (1 + RF )]

+ = =1

1 + RF 1 + RF (1 + RF )(u ’ d)

we introduce:

uq d(1 ’ q)

q= 1’q =

1 + RF 1 + RF

12

See Appendix 2.

166 Asset and Risk Management

By introducing the notation B(n; p) for a binomial random variable with parameters

(n, p), we can therefore write:

C0 = S0 · Pr B(T ; q ) ≥ J ’ K(1 + RF )’T · Pr B(T ; q) ≥ J

The ˜call“put™ parity relation C0 + K(1 + RF )’T = P0 + S0 allows the evaluation for-

mula to be obtained immediately for the put with the same characteristics:

P0 = ’S0 · Pr[B(T ; q ) < J ] + K(1 + RF )’T · Pr[B(T ; q) < J ]

Note

The parameters u and d are determined, for example, on the basis of the volatility σR

of the return of the underlying equity. In fact, as the return relative to a period takes the

values (u ’ 1) or (d ’ 1) with the respective probabilities ± and (1 ’ ±), we have:

ER = ±(u ’ 1) + (1 ’ ±)(d ’ 1)

σR = ±(u ’ 1)2 + (1 ’ ±)(d ’ 1)2 ’ [±(u ’ 1) + (1 ’ ±)(d ’ 1)]2

2

= ±(1 ’ ±)(u ’ d)2

By choosing ± = 1/2, we arrive at u ’ d = 2σR . Cox, Ross and Rubinstein suggest

taking d = 1/u, which leads to an easily solved second-degree equation or, with a Talor

approximation, u = eσR and d = e’σR .

Example

Let us consider a call option of seven months™ duration, relating to an equity with a

current value of ¤100 and an exercise price of ¤110. It is assumed that its volatility is

σR = 0.25, calculated on an annual basis, and that the risk-free rate is 4 % per annum.

We will assess the value of this call at t = 0 by constructing a binomial tree diagram

with the month as the basic period. The equivalent volatility and risk-free rate as given

by:

1

σR = · 0.25 = 0.07219

12

√

RF = 1.04 ’ 1 = 0.003274

12

We therefore have u ’ 1/u = 0.1443, for which the only positive root is13 u = 1.07477

(and therefore d = 0.93043). The risk-neutral probability is:

1.003274 ’ 0.93043

q= = 0.5047

1.07477 ’ 0.93043

If we had chosen ± = 1/3 instead of 1/2, we would have found that u = 1.0795, that is, a relatively small difference;

13

the estimation of u therefore only depends relatively little on ±.

Options 167

Let us ¬rst show the practical method of working: the construction of two binomial tree

diagrams (forward for the equity and backward for the bond). For example, we have for

the two values of S1 :

S0 · u = 100 · 1.07477 = 107.477

S0 · d = 100 · 0.93043 = 93.043

The binomial tree for the underlying equity is shown in Table 5.3.

The binomial tree diagram for the option is constructed backwards. The last column is

therefore constructed on the basis of the relation CT = (ST ’ K)+ .

The ¬rst component of this column is max (165.656 ’ 110; 0) = 55.656, and the ele-

ments in the preceding columns can be deduced from it, for example:

1

[0.5047 · 55.656 + 0.4953 · 33.409] = 44.491

1.003274

This gives us Table 5.4.

The initial value of the call is therefore C0 = ¤4.657.