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6 Switching Between the PDE, the Expectation and the Binomial Replication Approaches

6 Switching Between the PDE, the Expectation and the Binomial Replication Approaches

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Chapter 3



The Building Blocks

3.1



Introduction and Plan of the Chapter



The purpose of this chapter is threefold. First, I intend to define precisely the quantities

(implied, spot, instantaneous, forward, future volatility, etc.) used in the rest of the book.

Then, by placing myself in a Black-and-Scholes world, I show that the ‘implied’ volatility

can be linked to a function(al) of the instantaneous (spot) volatility of the underlying, and

that, for a given maturity, no smiles are observed. I also explain the fundamental difference

between the concepts of volatilities and correlations in the case of equities and FX on the

one hand, and of interest rates on the other.

Second, I intend to show the link between the set of current implied volatilities and

the future (spot) volatility of the underlying. In order to do so I will discuss in some detail

the case of ‘forward-starting’ options. I will also explain why, in most cases, working in

a forward-based (Black) framework is both simpler and conceptually more satisfactory

than working in a spot-based (Black-and-Scholes) setting.

Lastly, I will give a first introduction of the concept of quadratic variation. This quantity

is central to the treatment presented in the whole book, and will be revisited in the context

of smile-producing models.



3.2



Definition of Market Terms



The analysis of those aspects of option pricing and risk management that are connected

with the stochastic behaviour of financial quantities is plagued by imprecise, confusing

and often contradictory terminology. The terms ‘forward volatility’, ‘future volatility’,

‘forward-forward volatility’, ‘term structure of volatility’, ‘volatility of a forward’, ‘instantaneous volatility’, to name just a few, often mean different things to different practitioners

(and, sometimes, mean very little tout court). Even the apparently uncontroversial term

‘correlation’ can give rise of ambiguity, since the fundamental distinction between instantaneous and terminal correlation is often overlooked.

This confusion is of more-than-academic relevance: it sometimes prompts some practitioners to claim, for instance, that multi-factor models must be employed in order to

75



76



CHAPTER 3 THE BUILDING BLOCKS



account at the same time for the observed prices of caps and swaptions; or, more generally, to argue that, for a model to capture imperfect correlation between rates, it cannot

be single-factor. In reality, imperfect instantaneous correlation is but one of the mechanisms that can produce terminal de-correlation amongst rates or prices, and the relative

merits and shortcomings of using multi-factor models against employing time-dependent

volatilities must be carefully weighed and compared. Similarly, confusion often exists

about the current volatility of a forward quantity, the future volatility of a spot price or

rate and the future volatility of a forward quantity.

One reason for this rather confused state of affairs is that the underlying traded quantities in the equity and FX markets on the one hand, and in the interest-rate market on

the other are fundamentally different. Not surprisingly, this gives rise to different links

between the volatilities of the corresponding spot or forward quantities. In order to clarify

these issues is it necessary to define the ‘tools of the trade’ as precisely as possible. We

shall therefore start by defining:

1. the current volatility of a forward rate or price;

2. the future volatility of a spot rate or price; and

3. the future volatility of a forward rate or price.

In order to define their meaning, one must, in turn, define the concept of forward (rate

or price). This is done as follows:

Definition 1 The T -maturity forward price today in a given currency of a given security

is the strike that gives zero value to a forward contract for delivery at time T of the same

security in exchange for strike units of the same currency.

Definition 2 The T -maturity forward exchange rate (expressed as units of currency A per

unit of currency B) is the strike that gives zero value today to the forward contract for

delivery at time T of strike units of currency A against one unit of currency B.

Definition 3 The T -expiry–(T + τ )-maturity forward rate today is the strike that gives

zero value to a forward contract paying at time T + τ the difference between the reset at

time T of the rate spanning the period [T T + τ ] and the strike itself.

Note that in the definitions there is no mention of expectations (either in a mathematical

or subjective sense) about the underlying. As a consequence, distributional assumptions

about the underlying in general, and volatilities and correlations in particular, do not

enter the definition of a forward. When it is possible to take long and short positions in

the underlying without frictions and/or restrictions, forward prices and rates are uniquely

determined by no-arbitrage considerations. (See Hull (1993) or Rebonato (1998a) for an

example in the interest-rate case.) Any introductory text (see, for example, Hull (1993))

shows that, by no-arbitrage, the forward rates and prices are related to the spot quantities

by

Equity case

(3.1)

F S(t, T ) = S(t) exp[(rt,T − d)(T − t)]

d

f

F F X(t, T ) = F X(t) exp (rt,T

− rt,T

)(T − t)



F R(t, T1 , T2 ) =



1

P (t, T1 )

−1

P (t, T2 )

T2 − T1



FX case



(3.2)



Interest-rate case



(3.3)



3.3 HEDGING FORWARD CONTRACTS USING SPOT QUANTITIES



77



where

• S(t) is the spot price of the equity stock at time t

• F S(t, T ) is the time-t forward price of the equity stock for delivery at time T

• rt,T is the (T − t)-period spot interest rate from time t to time T

• d is the (constant) dividend yield

d (r f ) is the (T − t)-period spot domestic (foreign) interest rate from time t to

• rt,T

t,T

time T



• F X(t) is the spot FX rate expressed in units of domestic currency per units of

foreign currency at time t

• F F X(t, T ) is the time-t forward FX rate expressed in units of domestic currency

per units of foreign currency for delivery at time T

• F R(t, T1 , T2 ) is the time-t value of a forward rate expiring at time T1 and spanning

the period [T1 T2 ]

• P (t, T ) is the time-t value of a discount bond maturing at time T .

Note that, in the definitions above, rt,T is not the short rate, but the yield of the

T -maturity bond, i.e.

exp[−rt,T (T − t)] ≡ P (t, T )



(3.4)



Definitions 1 and 2 are basically identical in the equity and in the FX cases (as one

can readily see by thinking of one unit of currency B in Definition 2 as ‘the security’ in

Definition 1). Definition 3, however, describes a type of contract that is fundamentally

different. In order to appreciate the nature of this difference, in the next section we shall

contrast the case of spot hedging forward contracts in equities with the case of hedging

forward contracts with spot quantities in the interest-rate case. The analysis will provide

a useful framework to treat options in the following section, and to explore the added

complexity in the specification of the volatility structure as one moves from equities/FX

to interest-rate products.



3.3



Hedging Forward Contracts Using Spot Quantities



Let us begin by considering how to hedge with spot transactions1 a series of equity

forward contracts on the one hand, and a series of interest-rate forward contracts on the

other. My goal is to show that solving these two apparently similar hedging tasks requires

approaches of an intrinsically different nature. In the equity case a single net position in

the spot underlying asset is sufficient. In the interest-rate case delta hedging will require

taking a position in a number of different assets. The implications of this will become

apparent when we deal with options rather than forward contracts.

1A



spot transaction is buying or selling a non-derivative security today at the current market price.



78



CHAPTER 3 THE BUILDING BLOCKS



3.3.1 Hedging Equity Forward Contracts

Let us begin with the first case, and assume that we have entered at time t0 N forward contracts, struck at X1 , X2 , . . . , XN , for delivery at times T1 , T2 , . . . , TN of A1 , A2 , . . . , An

amounts of the same given stock. To lighten the notation I will set the dividend yield to

zero. The present value of the ith contract, P Vi , is given by

P Vi = Ai [F S(t0 , Ti ) − Xi ]P (t0 , Ti )

= Ai [S(t0 ) exp[rt0 ,Ti (Ti − t0 )] − Xi ]P (t0 , Ti )

= Ai



S(t0 )

− Xi P (t0 , Ti )

P (t0 , Ti )



= Ai [S(t0 ) − Xi P (t0 , Ti )]



(3.5)



In going from the second to the third line, use has been made of Equation (3.4). In

order to spot hedge this series of positions one can work out the net sensitivity of the

forward contracts to the spot price and to the discount bonds. For each contract one can

in fact write

∂P Vi

= Ai

∂S(t0 )



(3.6)



∂P Vi

= Ai Xi

∂P (t0 , Ti )



(3.7)



and



Therefore the net position, Ahedge , to spot hedge the exposure to the stock price of all the

forward contracts is simply given by

Ahedge =



Ai



(3.8)



i=1,N



This is not surprising, since, in the equity stock case, different equity forward prices are

the strikes that give zero value to a series of contracts for delivery of the same underlying

at different points in time. Therefore, holding the amount of stock Ai will certainly allow

the trader to fulfil her delivery obligation for the ith forward contract, and the net quantity

of stock Ahedge will simultaneously perfectly hedge all her forward contracts. Since an

FX rate can be thought of as the price of a unit of foreign currency in terms of domestic

currency, the same reasoning also applies with little change to the FX case.2

Summarizing: if one enters a simultaneous position in a series of equity- (or FX-)

based forward contracts, in order to execute a spot hedge (i.e. in order to hedge against

movements in the underlying by dealing in the stock today) one simply has to take a

2 In the ‘equity-stock example’ just presented we assumed that we were dealing with a non-dividend paying

stock. For non-zero dividend, d, the ‘growth rate’ of the stock becomes r − d. In the FX case the no-arbitrage

growth rate of the domestic currency is given by the difference between the domestic rate, rd , and the foreign

rate, rf .



3.3 HEDGING FORWARD CONTRACTS USING SPOT QUANTITIES



79



suitable single net position in the same spot underlying. If one wanted to be fully hedged,

one would also have to carry out the hedging transactions required to neutralize the

uncertainty from the discounting. This would entail taking positions in different discount

bonds. For the moment, however, we are just focusing on hedging the exposure to the

underlying (‘delta’ hedging).



3.3.2 Hedging Interest-Rate Forward Contracts

Let us now consider the apparently similar situation where N forward contracts, struck at

X1 , X2 , . . . , XN , and with notional amounts A1 , A2 , . . . , AN , are entered at time t0 on a

series of forward rates spanning the periods [T1 , T1 +τ1 ], [T2 , T2 +τ2 ], . . . , [TN , TN +τN ].

(Note that we are not assuming that Ti +τi = Ti+1 , i.e. the payment time of the ith forward

contract does not necessarily coincide with the expiry of the (i + 1)th; in other words,

the forward rates are not necessarily ‘spanning’ in the sense of Rebonato (1998a, 2002).)

The present value of each contract, P Vi , is then given by

P Vi = Ai [F R(t0 , Ti , Ti + τi ) − Xi ]τi P (t0 , Ti + τi )

= Ai



P (t0 , Ti )

1

− Xi τi P (t0 , Ti + τi )

−1

P (t0 , Ti + τi )

τi



= Ai [P (t0 , Ti ) − P (t0 , Ti + τi ) − Xi τi P (t0 , Ti + τi )]

= Ai [P (t0 , Ti ) − P (t0 , Ti + τi )(1 + Xi τi )]



(3.9)



In working out the spot hedges for each of the contracts in the series, one therefore finds

∂P Vi

= Ai

∂P (t0 , Ti )



(3.10)



and

∂P Vi

= −Ai (1 + Xi τi )

∂P (t0 , Ti + τi )



(3.11)



Despite the superficial similarity between Equation (3.10) and Equation (3.6), note that,

unlike the equity/FX case, the partial derivatives are now taken with respect to different

spot discount bond prices, and the quantity Ai denotes the required amount of the ith bond.

Therefore, the netting of the quantities Ai as in Equation (3.8) is no longer possible, and,

in order to spot hedge the whole series of N forward contracts, the trader must in general

take a spot position in 2N assets (discount bonds): different interest-rate forwards are the

strikes that give zero value to a series of contracts for delivery of different underlying

instruments at different points in time. Even neglecting the hedging of the uncertainty

due to discounting (i.e. even if we concentrate on delta hedging), in order to hedge

a simultaneous position in N interest-rate forwards contracts one must in general take

positions in 2N underlying spot assets (bonds).3

3 If the forward rates underlying the different forward contracts were spanning (i.e. if the pay-time of one

coincided with the expiry time of the next), the number of underlying bonds would be less than 2N .



80



CHAPTER 3 THE BUILDING BLOCKS



The message from the analysis carried out in this section is that, in order to spot hedge

forward contracts, the trader has to take positions in one, or several spot underlying

assets. Whether the hedging position is in one or many assets depends on the nature

of the underlying. The statement appears (and is) rather obvious. However, when one

moves from the hedging of forward contracts to the hedging of non-linear derivatives

(e.g. options), the choice of hedging instrument is no longer so obvious. One can carry

out, in fact, two very similar treatments that give as a result the notionals either of the spot

transactions or of the forward contracts that the trader has to enter in order to be deltahedged. I will argue below that, while mathematically equivalent, the two approaches are

financially very different, and that, whenever possible, hedging of options should always

be carried out using forward contracts rather than spot transactions. This way of looking

at the hedging problem will also clarify my view that it is more fruitful to regard options

as being written on forwards rather than on future spot quantities (in other words, the

Black, rather than the Black-and-Scholes, framework is, in my opinion, more useful and

conceptually more satisfying).



3.4



Hedging Options: Volatility of Spot

and Forward Processes



So far we have considered simple forward contracts, for the pricing of which volatilities

and correlations are irrelevant. In moving from a series of forward contracts to a series of

options on equity or FX forwards, volatilities and correlation do matter, but the reasoning

remains otherwise similar. The main change, as far as the argument above is concerned,

will be that the notional of the spot hedge will become a volatility-dependent delta amount

of the underlying. The exact amount of delta hedge will in turn depend on the model

used to price the option, but, at least as long as the discounting is deterministic, only the

volatility of the same spot process should matter for equities or FX rates.

Another important result that we obtain in this section is the following. Let us accept

for the moment that, in a Black world, given a series of options on forwards with different expiries, the average (root-mean-squared) volatilities out to different horizons will

determine the correct hedges (this will be shown to be the case in the next chapter). As

the spacing between the maturities for which prices can be observed becomes finer and

finer, these root-mean-squared volatilities will in turn be shown to give information about

the future instantaneous volatility of the underlying process. Therefore, in order to price

and hedge, in a deterministic-discounting regime, N options on equity stock or FX rates

all that matters is the time-dependent volatility of the spot process.

The situation is different in the case of interest rates. If the task before us is the

pricing and hedging of N interest-rate derivatives, it is the volatilities of (and, possibly,

depending on the type of option, the correlations among) 2N processes that determine

the required delta amounts of spot discount bonds. Specifying volatilities for interest-rate

problems is therefore intrinsically more complex, and I will show that, in general, it

involves specification of the time dependence of the covariance matrix.

With the definitions and observations above clearly in mind, let us extend the analysis

presented above to the simplest possible setting required for option pricing. To do so, let us

consider first an (equity or FX) spot process, with time-dependent volatility (no smiles) and

deterministic rates. We shall assume that we are given a series of plain-vanilla European



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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