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4 Hedging Options: Volatility of Spot and Forward Processes

4 Hedging Options: Volatility of Spot and Forward Processes

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3.4 HEDGING OPTIONS: VOLATILITY OF SPOT AND FORWARD PROCESSES



81



options prices, Opt1 , Opt2 , . . . , Optn , on the same underlying for different strikes and

maturities. In a Black-and-Scholes framework, the stochastic evolution of the underlying

spot process in the real world is given by a stochastic differential equation of the form

dS(t)

= µ(S, t) dt + σ (t) dz(t)

S(t)



(3.12)



where S(t) denotes the stock price at time t, µ its real-world drift, σ the instantaneous percentage volatility of the spot process, and dz(t) is the increment of a standard

Brownian motion. In moving from the real-world to the pricing (risk-neutral) measure,

Equation (3.12) becomes

dS(t)

= r(t) dt + σ (t) dz(t)

S(t)



(3.13)



where all the symbols have the same meaning as in Equation (3.12), and r(t) is the short

(riskless) rate at time t. The solution to this stochastic differential equation is given by



(3.14)

S(T , ) = S(0) exp[rT − 12 σ 2 T + σ T ]

with

T



σ 2T =



σ (u)2 du



(3.15)



r(u) du



(3.16)



0

T



rT =

0



and ∈ N (0, 1). The quantity σ , which is the root-mean-square of the instantaneous

volatility, σ (t), will be shown to play a central role in option pricing when the underlying

process follows a diffusion.

Since we are for simplicity working with deterministic interest rates, we can write

T



exp −



r(u) du = exp [−rT ] = exp −r0,T T ≡ P (0, T )



(3.17)



0



Using the definitions introduced above, Equation (3.14) can therefore be re-written as



(3.18)

S(T , ) = F S(0, T ) exp[− 12 σ 2 T + σ T ]

and, since at expiry F (T , T ) = S(T ), we can write



F S(T , T , ) = F S(0, T ) exp[− 12 σ 2 T + σ T ]



(3.19)



Comparing Equation (3.19) with Equation (3.14) we can recognize that it implies that the

stochastic differential equation for the forward price displays no drift, i.e. that it can be

written as

dF S

= σF S (t) dz(t)

FS



(3.20)



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CHAPTER 3 THE BUILDING BLOCKS



where σF S (t) is now the volatility of the forward price. This observation will become

important in what follows when we discuss under what circumstances it is possible to

move freely from variance to volatility and vice versa. In the meantime one should notice

that, given the assumption of deterministic discounting that we are enforcing so far, the

percentage volatility of the forward rate is the same as the volatility of the spot process.

(This can be immediately seen by applying Ito’s lemma to F S = f (S) = S/P , and

treating P as a deterministic function of time.) Therefore, if we denote by σF S (t) the

volatility at time t of the forward price, in a deterministic-discounting setting we can put

σF S (t) = σ (t).

We can draw on these observations to say that the present value of the ith options in

our basket can equivalently be written in either of the following equivalent ways:

P V (Opti ) = E[S(T ) − Xi ]+ P (t0 , Ti )

=E





S(0) exp[rT − 12 σ 2 T + σ T ] − Xi



+



P (t0 , Ti )



(3.21)



or

P V (Opti ) = E[(ST − Xi )+ ]P (t0 , Ti )

= E (F S(T , T ) − Xi )+ P (t0 , Ti )

=E





F S(0, T ) exp[− 12 σF2 S T + σF S T ] − Xi



= F S(0, T )E





exp[− 12 σ T + σ T ] −



Xi

F S(0,T )



+



+



P (t0 , Ti )

P (t0 , Ti )



(3.22)



where the symbol E denotes the expectation operator, the notation [a−b]+ means max[a−

b, 0], the root-mean-squared quantity σF S is defined analogously to the quantity σ in

Equation (3.15) and the definitions introduced at the beginning of this section have been

made use of.

Looking at the expressions (3.21) and (3.4) above one can see that the present value

of the option can be expressed as an expectation over the terminal values either of the

forward or of the spot price. By the way it has been presented, a treatment in terms of

the spot or in terms of the forward process appears to be perfectly equivalent. But is it

also equivalent from a financial point of view? Is there a ‘preferred set of co-ordinates’?

We have only been able to arrive at the volatility of the forward price, σF S , using no

other information than our assumed knowledge of the volatility of the spot process for the

stock, σ , because we made the assumption that the discount factor was deterministic (in

which case, as pointed out, the volatility of the forward price is identical to the volatility

of the spot price: σF S = σ ). This way of looking at the pricing of options, although

common and ‘pedagogically’ more straightforward, can however be somewhat lopsided,

in that it assumes that we start from the knowledge of the volatility of the spot process for

the stock and we derive from this the volatility of the forward price. In reality, if interest

rates were not deterministic, the volatility of the forward price would include a component

arising from the volatility of the discount bond that connects the spot and the forward

price: F S = S/P . If we assume that the dynamics of the (strictly positive) bond price is



3.4 HEDGING OPTIONS: VOLATILITY OF SPOT AND FORWARD PROCESSES



83



also of a diffusive nature, its stochastic differential equation can be written in the form

dP

= rdt + v(t, T ) dw

P



(3.23)



(with v(t, T ) denoting the percentage volatility at time t of a T -maturity bond, and

E[dzt dwt ] = ρ). A straightforward application of the two-dimensional Ito’s lemma

applied to the function F S(t, T ) = f (S(t), P (t, T )) gives for the volatility of the forward

price, σF S ,

σF S =



σ 2 + v 2 − 2ρσ v



(3.24)



where ρ is the correlation between the discount bond and the spot process.4 From Equation

(3.24) we can see that in the case of stochastic interest rates the volatility of the forward

price no longer coincides with the volatility of the spot process. So, we should prefer

the spot (Black-and-Scholes) formulation if we derive our information about the volatility from a price history of the spot underlying and of the bond (in which case the spot

co-ordinates are most appropriate); we should prefer the forward (Black) formulation if

we impute the volatility from the traded prices of plain-vanilla options (in which case we

should work directly in the forward co-ordinates). Since, in this book, I am mainly looking at volatility and correlation from the perspective of a trader of complex derivatives

(as opposed to a plain-vanilla option trader) I will in general favour the forward-price

perspective.

Summarizing: from a quoted option price one can extract, by inverting the implied

volatility, a single number, i.e. the combination of the spot price volatility, of the discount

bond volatility and of the correlation between the two processes that gives Expression

(3.24). In other words, the practice of ‘implying’ a volatility from an option price gives

access to the volatility of the forward price (that enters the Black formula), but not of the

spot price (that enters the Black-and-Scholes formula).

Exercise 1 Estimate the volatility of the bond price and gauge the relative importance of

the discount bond volatility to the volatility of a 3-month, 1-year, and 10-year forward

price.

Because of this, Black’s formula has a much wider applicability than the deterministicdiscounting case, and still applies even if the discount bond is stochastic. (See, in this

respect, Merton’s fundamental paper (1973) for a ‘traditional’ PDE treatment, or, for

instance, Baxter and Rennie (1996) for a ‘modern’ numeraire approach.)

Traders in the market are obviously aware of the stochastic nature of discounting and

embed this knowledge in their quotes of the vanilla option prices (from which the implied

volatility is obtained). If, however, a naăve trader erroneously believed that these quotes

had been made assuming deterministic discounting, she would be implying from each

option price a volatility that she would identify with the volatility of the spot process.

If, therefore, this trader delta-hedged herself using a spot position in the underlying, she

would be buying or selling the wrong delta amount of the spot underlying, since she would

4 Note that if ρ denotes the correlation between the underlying and the bond price, the correlation between

the underlying and forward rates will be −ρ.



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CHAPTER 3 THE BUILDING BLOCKS



be ‘misinterpreting’ the component of the volatility arising from the discount factor. If

the same trader, on the other hand, recognized that the option quote is made in the market

taking the stochastic nature of discounting into account, she would correctly interpret the

implied volatility as the volatility of the forward price. She would therefore put in place a

delta hedge in a forward contract with the same strike and expiry date. This strategy would

automatically take the stochastic nature of the discounting bond into account, and would

therefore provide not only a more practical, but also a theoretically more satisfactory hedging strategy. Alternatively, if she wanted to, she could still carry out spot hedging transactions, but she would have to do so both in the underlying and in the discount bond maturing

at the option expiry. In either case the correct hedge ratios would have to be determined

using the difficult-to-estimate correlation between the underlying and the discount bond.

See Equation (3.24). Needless to say, for short maturities the two volatilities virtually

coincide, but the difference becomes far from trivial for long-dated equity and FX options.

For this reason, if one follows an ‘implied’ rather than a statistical approach to estimating volatility, it is intellectually more satisfactory and more effective in practice to regard

all plain-vanilla options as calls or puts on a forward price. A quoted market price gives

direct information on the root-mean-squared volatility of the associated forward price, not

on the root-mean-squared volatility of the spot process.



3.5



The Link Between Root-Mean-Squared Volatilities

and the Time-Dependence of Volatility



We go back to the case of purely deterministic discounting to explore the link between

the quoted prices of plain-vanilla options and the time dependence of the instantaneous

volatility of the underlying. We know that, for equities, all the forwards are a function of

the same underlying. Therefore, if discounting is deterministic, a series of option prices

can give direct information about the future volatility of the (same!) spot process. This can

be seen more precisely as follows. From Equation (3.20) and from the definition of rootmean-squared volatility we can see that each option in our portfolio can be written as some

function, f , of the time integral of the square of the (a priori unknown) time-dependent

volatility of the spot price:

t1



Opt1 = f

0



t2



Opt2 = f

0



2

σ 2 du = f σ0→1



(3.25)



2

σ 2 du = f σ0→2



(3.26)



···

tn



Opt2 = f

0



2

σ 2 du = f σ0→n



(3.27)



(The quantity σ0→k is proportional to the root-mean-squared volatility from time 0 to

time tk .) Since we know that the solution to the Black pricing formula is a function of the

root-mean-squared volatility of the forward rate, we can impute (imply) from the market

price of the option the implied Black volatility of the forward rate. By so doing we can



3.6 ADMISSIBILITY OF A SERIES OF ROOT-MEAN-SQUARED VOLATILITIES



85



therefore construct

2

2

2

σ1→2

= σ0→2

− σ0→1



=



t2

t1



t2



σ 2 du =



t1



σ 2 du −



0



σ 2 du



(3.28)



0



In the limit as time t2 becomes closer and closer to time t1 (t2 = t1 + ) we can write

t1 +



σ (t1 ) = lim

2



0



t1



σ 2 du −



σ 2 du



0



→0



(3.29)



But, if we have access to a continuum of option prices,5 the RHS can be written as a

function of market-related quantities:

σ 2 (t1 ) = lim



2

2

σ0→1+

− σ0→1



→0



(3.30)



Under our working assumptions (a Black world without smiles and deterministic discounting) this equation links the not-directly-observable future instantaneous volatility of

the spot process from time t1 to time t2 = t1 + with the market-observable option prices

for the corresponding expiries. The relationship in Equation (3.30) expresses an important relationship between a set of current quantities (the root-mean-squared volatilities

for different expiries) and the future instantaneous volatility. This relationship is often

referred to as the ‘balance-of-variance condition’.



3.6



Admissibility of a Series of Root-Mean-Squared

Volatilities



3.6.1 The Equity/FX Case

From the results above we can immediately deduce that, if we live in a deterministic2

T is not a strictly increasing function of

discounting Black world and the quantity σ0→T

the equity or FX option expiry, T , there is an arbitrage. (To simplify notation, in what

follows I will write σ0→T = σT .) Finding the arbitrage is very easy, and only requires

recalling that the Black option price is a monotonically increasing function of the rootmean-squared volatility: if the quantity σT22 T2 is smaller than σT21 T1 , it would mean that

the uncertainty about the underlying out to time T2 is less than the uncertainty out to time

T1 (T1 < T2 ). It is not difficult to set up a model-independent arbitrage to exploit this

state of affairs. Therefore, in a Black world, a series of option prices whose associated

5 This assumption might appear unrealistic, but, in reality, an interpolation procedure between actually quoted

prices can often be employed (see Chapter 9), and a continuum of ‘synthetic’ market prices can therefore be

assumed to be available without making too drastic an assumption.



86



CHAPTER 3 THE BUILDING BLOCKS



root-mean-squared volatilities do not satisfy the relationship

∂[σT2 T ]

>0

∂T



(3.31)



lend themselves to model-independent arbitrage. Such a collection of option prices, and

the associated implied volatilities, therefore constitutes a first example of non-admissible

prices (volatilities). The concept of admissibility will be revisited at length in Chapter 17.

In closing this section, I should point out that the reasoning just presented does not

apply to the case of interest rates: caplet-implied volatilities, for instance, need not be

such that σT22 T2 > σT21 T1 . I discuss this point immediately below.



3.6.2 The Interest-Rate Case

Let us now compare the situation just described with the case of interest rates. We shall

do so by looking in some detail at options on a closely-related set of futures contracts.

Because of margin calls, futures contracts pay out in a different way than forward-rate

agreements (FRAs). These differences give rise to an adjustment to the equilibrium forward rate. For the purpose of the following discussion, however, the distinction between

a futures rate and an equilibrium forward rate is immaterial, and I shall ‘pretend’ that

futures contracts are simply standardized FRAs.

Let us begin by considering the following series of futures contracts:

MAR2008

JUN2008

SEP2008

(At the time of writing, the dates above are well into the future. If this book is still in

print – and read – close to the expiry dates of these contracts, the reader should mentally

substitute some expiry dates a few years in the future.) If we observe the behaviour

of the forward rates implied by these futures prices over time, we expect them to move

roughly ‘in step’, i.e. to display a high degree of correlation and a similar volatility. Let us

suppose, however, that we observe today that the market volatility of, say, the JUN2008

caplet trades significantly below the volatility of the MAR2008 and SEP2008 caplets.

In particular, if T1 , T2 and T3 denote expiry times MAR2008, JUN2008 and SEP2008,

respectively, the market-implied volatilities are such that

σT22 T2



σT21 T1



(3.32)



σT22 T2



σT23 T3



(3.33)



A trader therefore would be very tempted to sell options on the MAR2008 and SEP2008

contracts (trading at a ‘high’ implied volatility) and buy options on the JUN2008 contract (trading at a relatively low implied volatility). However, much as the trade might

seem common-sensically attractive, the market-observed relationships (3.32) and (3.33)

do not imply any violation of arbitrage. Since the different futures contracts refer to different assets, there is no logically compelling reason why the JUN2008 contract should

not ‘vibrate’ throughout its life by much less than its MAR2008 and SEP2008 cousins.

Therefore it might well be reasonable to expect that we shall make money from the



3.7 SUMMARY OF THE DEFINITIONS SO FAR



87



sell–buy–sell strategy, and the more so the greater the discrepancy between the JUN

volatility on the one hand and the MAR and SEP volatilities on the other. However, in

the case of interest rates the violation of the condition

σT21 T1 < σT22 T2 ,



T1 < T2



(3.34)



no longer gives rise to an arbitrage, even in a purely Black world. The implications for

the joint evolution of interest rates reflected by these market prices might be deemed to

be extremely unlikely, but it cannot be ruled out as an impossibility in an arbitrage-free

world in the same way as the situation discussed in the previous section could. This

conclusion should be contrasted with the arbitrage strategy presented above in the case

of equities or FX, where, even in the most far-fetched scenarios, we would make money

if the total variance from today to option expiry were not a strictly increasing function

of expiry. We shall explore later (see Chapter 19) what the violation of condition (3.34)

implies about the evolution of the term structure of volatilities.

From this discussion one can reinforce the conclusion already drawn above: N forward

contracts truly represent positions in 2N different underlying spot assets. Therefore one

must specify the volatility of each of these assets. In addition, each of these assets can

have a time-dependent volatility. The problem is no longer how to specify a one-variable

function (the volatility of the underlying spot process as a function of calendar time),

but how to specify a surface (the volatilities of different forward rates – which cannot be

obtained from each other – at different points in time). In addition one also has to specify

the (possibly time-dependent) correlation amongst the forward rates. In moving from the

equity/FX to the interest-rate world the problem therefore explodes in complexity: the

user effectively has to supply a full time-dependent covariance matrix. We shall see in

Chapter 5 that for a discrete-look option problem, where the payoff is determined by

N realizations of different forward rates (i.e. of different assets), O(N 3 ) covariances

must be supplied: O(N 2 ) covariances for each of the N price-sensitive times. This is,

ultimately, the reason why low-dimensionality (where ‘low’ sometimes means ‘one’ or

‘two’) term-structure models have often been employed in trading practice.



3.7



Summary of the Definitions So Far



For future reference, it is useful to collect the definitions introduced so far, with some

further comments where appropriate.

1. When, at time t, we speak about a price process (typically, for equities or FX)

described by a diffusive equation of the form

dS(t)

= µ(S, t) dt + σ (S, t) dz(t)

S(t)



(3.35)



the volatility that appears in the equation above is called the present (if t = t0 ) or

future (if t > t0 ) volatility of the spot process. Therefore:

Definition 4 The future or present volatility of a spot price = σ (S, t).



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CHAPTER 3 THE BUILDING BLOCKS



In the equities and FX cases, when the future or present volatility of the spot price

depends on the future value of the underlying, it is often referred to in the literature as

the ‘local volatility’. We shall make frequent use of this term when dealing with smiles.

The term ‘instantaneous volatility’ can also be used, but, in this book, it will mainly be

used in the context of interest rates. See the definition in point 5 below.

2. When, at time t, we consider a forward-price process (typically, for equities or FX)

described by a diffusive equation of the form

dF S(t, T )

= µ(F S, t, T ) dt + σ (F S, t, T ) dz(t)

F S(t, T )



(3.36)



the volatility that appears in the equation above is called the present (if t = t0 ) or

future (if t > t0 ) volatility of the forward-price process. Note that we are not using

the (meaningless and confusing, but none the less common) expression ‘forward

volatility’: a volatility is not a traded asset, and therefore can be either present or

future, but not forward; nor shall we ever use the term ‘forward-forward volatility’,

which probably is used to mean, if anything, the future volatility of a forward quantity. Note also that an additional argument, T , enters the definition of the volatility

of a forward price in order to emphasize that forwards of different maturities will

not, in general, have the same volatility at time t. Therefore:

Definition 5 The future or present volatility of a forward price = σ (F S, t, T ).

3. Just as in the case of spot processes, another term that is sometimes used in the

context of forward price processes is ‘instantaneous volatility’: it is simply another

way to denote the time-dependent volatilities in Equation (3.36). See point 5 below

for its more common usage.

4. If the volatility is at most time-dependent, then the time integral of the square of

the volatility of a forward or spot price between time T1 and time T2 is often called

the total variance, var(T1 , T2 ), of that forward or spot price:

var(T1 , T2 ) =



T2

T1



σ (u)2 du



(3.37)



The term ‘total variance’ is common, but only appropriate if the drift of the spot or

forward price process is at most dependent on time. For most choices of numeraires,

this is indeed the case for price processes, but it is not automatically true for interest

rates. See Chapters 4 and 19.

5. The square root of the total variance from time 0 to time T of a forward price or

rate of expiration T divided by T is the root-mean-squared volatility, σ (T ), of the

forward price or rate itself:

σ (T ) =



1

T



T

0



σ (u)2 du



(3.38)



3.8 HEDGING AN OPTION WITH A FORWARD-SETTING STRIKE



89



This quantity is also called the implied volatility or the Black volatility. Note,

however, that the term ‘implied volatility’ can be used in a wider context (i.e. even

when the volatility is not deterministic, or the underlying process is not a diffusion),

and simply indicates the number that must be input in the Black formula to obtain

the correct price. Implied volatilities are discussed at length in Part II, which deals

with smiles.

Finally, note carefully that the term ‘average volatility’ is often used in the

literature or in traders’ jargon. It is somewhat misleading, since the quantity defined

T

above does not in general coincide with the value T1 0 σ (u) du, as one would

expect from the name. Despite being imprecise, this usage is very common. This

leads us to:

Definition 6 The root-mean-squared volatility to maturity T = σ (T ).

6. Whenever, in what follows, we deal with a forward rate, described by a stochastic

differential equation of the form

dF R(t, T , T + τ )

= µ(F R, t) dt + σ (F R, t) dzt

F R(t, T , T + τ )



(3.39)



the volatility experienced at time t by this forward rate of expiry T is always referred

to as the present (if t = 0) or future (if t > 0) instantaneous volatility of the forward

rate. In the market the term serial volatility is sometimes encountered, because of

the importance of the instantaneous volatility in the pricing of serial options. I shall

not use this terminology. Therefore:

Definition 7 The time-t instantaneous volatility of a forward rate, FR σ (FR, t).

7. The function that associates to a forward rate of a given maturity its own Black

market-implied volatility is called the term structure of volatilities. Notice that the

term sometimes has a different meaning in the equity and FX world, where it is

used to indicate the time dependence of the volatility of the spot or forward price

process. This usage is widespread, but, to avoid confusion, I will never use the

term ‘term structure of volatilities’ in this second meaning. For equities (and spot

processes in general) I will talk instead about ‘time dependence of volatility’ or

‘implied volatility as a function of expiry’, as appropriate.

Having clarified these concepts, the next two sections will present in detail a couple of

case studies that illustrate some of the more subtle points in the definitions just presented.



3.8



Hedging an Option with a Forward-Setting Strike



Let us place ourselves again in a perfect Black world, and let us consider an option with

a forward-setting strike (often referred to as a ‘forward option’). To be more specific, the

contract specification is such that the reset of the stock price at time T1 will determine

the strike for the remaining life of the option from T1 to T2 . At time T2 the value of the



90



CHAPTER 3 THE BUILDING BLOCKS



stock price will then be compared with the strike determined at time T1 , and the payoff

of the option at time T2 will be given by

PAYOFF(T2 ) = max



S(T2 ) − S(T1 )

,0

S(T1 )



(3.40)



3.8.1 Why Is This Option Important? (And Why Is it Difficult

to Hedge?)

Forward-starting options of the type (3.40) are important because they guarantee the

holder the percentage increase in an index, S, if positive. Therefore they have had great

appeal with retail investors, who would like to receive a principal-protected equity return.

When, in the mid-1990s, interest rates were relatively high and equity volatility relatively

low it was easy to offer the product in a very plain-vanilla form: the investor would

invest $100 for, say, five years. $P (0, 5) × 100 would be invested in a zero-coupon

bond expiring in five years’ time, thereby guaranteeing the principal at maturity. The

rest, $100 × [1 − P (0, 5)], would be used to buy five forward-setting calls on the yearly

percentage increase of an equity index over years 1, 2,. . .,5. Low volatilities made the

‘forward-starting’ options ‘cheap’, and, because of the relatively high rates, the discount

to par of the bond would provide enough upfront cash to buy all the ‘cheap’ options. With

the fall in interest rates and the increase in equity volatility in the years between 1998

and 2002, this simple strategy has become more complex, because the discount to par of

the five-year bond does not provide enough upfront cash to purchase the now-expensive

set of forward-setting options. As a response, variations such as limited-equity-upside

structures have been introduced. Therefore, the first plain-vanilla products produced for

the selling banks an exposure to the future realization of the at-the-money volatility, while

the later call–spread-like structures gave an exposure to the relative volatility level of an

at-the-money and an out-of-the-money option (i.e. a forward risk reversal). Since in Part I

we do not deal with smiles, in this chapter I will only consider the plain-vanilla structure.

It is easy to see why the option is difficult to hedge. In the simple case of constant

volatility, σ , the present value of the forward-setting option, can be written as

P V (T0 ) = E

=E

=E



S(T2 ) − S(T1 )

S(T1 )



+



P (0, T2 ) = E



S(T2 )

−1

S(T1 )



+





+

S0 exp[rT2 − 12 σ 2 T2 + σ T2 2 ]

−1

P (0, T2 )



S0 exp[rT1 − 12 σ 2 T1 + σ T1 1 ]



+

exp[rT2 − 12 σ 2 T2 + σ T2 2 ]



1

P (0, T2 )



exp[rT1 − 12 σ 2 T1 + σ T1 1 ]



P (0, T2 )



(3.41)



(The expression (a)+ means max(a, 0).) Note how the dependence on S0 has disappeared:

as the spot moves, the future strike moves as well, and therefore the option will always

be at-the-money spot when it comes to life. Therefore the option has zero delta, and

zero gamma. However, it does have a vega exposure (see the discussion below). As a

consequence, in order to hedge it, we must find a combination of plain-vanilla options



3.8 HEDGING AN OPTION WITH A FORWARD-SETTING STRIKE



91



with zero delta, zero gamma and positive vega. To do so exactly is clearly impossible,

but a (locally) approximating portfolio can be built. Note also that 1 and 2 are not

independent standard Gaussian draws. I discuss this point in Chapter 5.



3.8.2 Valuing a Forward-Setting Option

How can we value such an option? A rather wasteful and ‘brute-force’ approach (useful,

however, to appreciate the mechanics of the product) is to do a Monte Carlo simulation

of the spot process, S(t). One can first evolve the price from time T0 to time T1 in one

single move (see Chapter 5 for a discussion and justification of the procedure) using the

formula:

2

S(T1 , z1 ) = S(0) exp[rT1 − 12 σ0→T

T + σ0→T1 T1 z1 ]

1 1



(3.42)



where σ0→T1 is the root-mean-squared volatility of S from time T0 to time T1 . This is

obtainable from the price of the option expiring at time T1 . Given this realization one can

then evolve the stock price from time T1 to time T2 . In order to do this the simulation will

need the root-mean-squared volatility of the spot price from time T1 to time T2 , σT21 →T2 :

σT21 →T2 =



1

T2 − T1



T2

T1



σ (u)2 du



(3.43)



Since we have assumed a perfect Black world and deterministic discounting, we can

obtain this future volatility from the quoted prices of plain-vanilla options expiring at

the two times, as shown in Section 3.5. Armed with the knowledge of this quantity, and

making use of the same formula employed above, one can then obtain for the realization

of the stock price at time T2 :



S(T2 , z1 , z2 ) = S(T1 ) exp[rτ − 12 σT21 →T2 τ + σT1 →T2 τ z2 ]



(3.44)



with τ = T2 − T1 . Note that the Brownian shocks z2 and z1 used in the simulations

have been obtained from independent draws, since the increments of a Brownian motion

should display no serial correlation.

Having reached time T2 , and having kept track of the realization of the stock price at

time T1 , one can easily construct the option payoff, discount and average as usual. By

the way it has been presented, one can easily see that the simulation effectively carries

out a two-dimensional integration.

While correct, the approach can be made much simpler (and more elegant) as follows.

Looking at the payoff formula one can rewrite it as

PAYOFF(T2 ) = max

= max



S(T2 ) − S(T1 )

S(T2 )

, 0 = max

− 1, 0

S(T1 )

S(T1 )

F S(T2 , T2 )

F S(T2 , T2 )

− 1, 0 = max

− 1, 0

F S(T1 , T1 )

F S(T2 , T1 )



(3.45)



In obtaining the last two lines use has been made of the fact that F S(T , T ) = S(T ),

and, more subtly, we have extended the definition of the time-t, expiry-T1 forward price



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4 Hedging Options: Volatility of Spot and Forward Processes

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