6 Switching Between the PDE, the Expectation and the Binomial Replication Approaches
Tải bản đầy đủ - 0trang
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)