RE S E A R C H PA P E R
Q U A N T I T A T I V E F I N A N C E V O L U M E 2 (2002) 31–44 INSTITUTE O F PHYSICS PUBLISHING
quant.iop.org
Deterministic implied volatility models P Balland Merrill Lynch Financial Centre, 2 King Edward Street, London EC1, UK E-mail: philippe
[email protected] Received 20 September 2001 Published 4 February 2002 Online at stacks.iop.org/Quant/2/31
Abstract In this paper, we characterize two deterministic implied volatility models, defined by assuming that either the per-delta or the per-strike implied volatility surface has a deterministic evolution. Practitioners have recently proposed these two models to describe two regimes of implied volatility (see Derman (1999 Risk 4 55–9)). In an arbitrage-free sticky-delta model, we show that the underlying asset price is the exponential of a process with independent increments under the unique risk neutral measure and that any square-integrable claim can be replicated up to a vanishing risk by trading portfolios of vanilla options. This latter result is similar in nature to the quasi-completeness result obtained by Bjork et al (1997 Finance Stochastics 1 141–74) for interest rate models driven by Levy processes. Finally, we show that the only arbitrage-free sticky-strike model is the standard Black–Scholes model. dvt /vt = αt dt + ξ dBt
1. Introduction In classic extensions of the Black–Scholes (1973) model that accounts for the smile effect, the underlying asset price S is driven by one or two Brownian motions. These models are referred as local volatility models as they differ from the Black– Scholes model by simply allowing the local volatility σt of the underlying asset price to be stochastic: dSt /St = µt dt + σt dWt . Two types of local volatility models have been proposed: the so-called deterministic and stochastic local volatility models. In the deterministic local volatility models, the local volatility satisfies σt = σ (t, St ) as in Dupire (1994) and Derman and Kani (1994). Deterministic local volatility models are complete and complex options are replicated using the underlying S alone! In the stochastic local volatility models, νt = σt2 is a stochastic process characterized by its meanreversion rate, its volatility and its correlation with St as in Hull and White (1987) and Heston (1993): √ dvt = (αt − κvt ) dt + ξ vt dBt 1469-7688/02/010031+14$30.00
(Heston)
© 2002 IOP Publishing Ltd
(Hull–White)
The processes B and W are Brownian motions with correlation ρ and are defined under the risk-neutral measure. This riskneutral measure is unique and the stochastic local volatility models are complete if call options are traded instruments. The parameters αt , κ, ξ and ρ are implied by calibration to the initial smile surface. When inferred from historical data, these parameters are typically much lower. The reason for this can be understood if we observe that in stochastic models, d ln St has a normal conditional distribution with variance σt2 dt as in the Black–Scholes model. Consequently, the calibration of a stochastic local volatility model to a short-dated smile curve, which typically has large variations near the at-themoney strike, results in unrealistically large correlation ρ and volatility ξ . To compensate for these large parameters, the calibration to a long-dated smile curve, which is typically flatter, implies a large mean-reversion rate κ. Deterministic local volatility models have similar problems despite accurate calibrations. The local volatility function inferred from the initial smile surface, has typically ‘unrealistically’ large slope and convexity for small maturity while it is almost flat for large maturity.
PII: S1469-7688(02)32504-1
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It is important to understand that a complex option will be hedged efficiently using a smile model only if the model implies a dynamic of the smile that is sufficiently ‘realistic’ and ‘stationary’ for the model to not require frequent re-calibration. Local volatility models are, in this sense, unsatisfactory. The dynamic of the implied volatility surface is entirely specified once the model has been calibrated with the initial smile surface, leaving thus no control on this dynamic. To illustrate further this point, consider the foreignexchange market, where implied volatilities are quoted per delta. The underlying S is typically traded more frequently than vanilla options and thus, the implied volatilities do not change as frequently as S does. In a deterministic local volatility model such as the Dupire model, the implied volatilities are however functions of S. Hence, over a short time interval during which implied volatilities do not change, the model will still need re-calibration every time S changes! These frequent re-calibrations cause the hedging strategies implied by the model to be ‘inefficient’ as reported by Dumas et al (1998). In a stochastic local volatility model, implied volatilities depend on t, the local volatility σt and on the percentage-in-the-money St /K. This last dependence implies a greater stability of the corresponding hedging strategies. However, to be consistent with market smiles, such a model over-estimates, as previously explained, the volatility of volatility and the correlation of volatility with S and, as a consequence, the cost of hedging is not accurately accounted for. The need for smile models that are calibrated to an initial smile surface and that imply a ‘realistic’ evolution of implied volatility explains the recent interest in the so-called implied volatility models. These models are defined by direct assumptions on the stochastic evolution of the smile surface from an initial surface, as in Schonbucher (1999) and in Cont and da Fonseca (2001). It is important to stress, however, that these models need severe restrictions to be arbitrage-free. In this paper, we characterize two recently proposed deterministic implied volatility models, defined by assuming that either the per-delta or the per-strike implied volatility surface has a deterministic evolution. In arbitrage-free stickydelta models, we show that the underlying asset price is the exponential of a process with independent increments under the risk neutral measure and that any square-integrable claim can be replicated up to a vanishing risk by trading portfolios of vanilla options. Finally, we show that the only arbitrage-free sticky-strike model is the Black–Scholes model.
We assume that only short-dated vanilla options are traded assets because in typical option markets, only short-dated options have liquid enough prices to be regarded as traded instruments. In typical foreign exchange markets for example, only the vanilla options with maturity less than a couple of years have liquid prices. We denote by St > 0, Ct (x, K) > 0 and Pt (x, K) > 0 the price at time t of one unit of S, of one call and of one put option on S with maturity t + x ∈ (t, t + xm ) and with strike K > 0. We assume no transaction costs. We assume deterministic interest rates and we denote by Ptx the price at time t of a discount bond with maturity t + x and by Bt ≡ 1/P0t the money-market account. We denote by Ftx the forward price at time t of receiving one unit of S at time t + x t. We assume that there exists a real bounded function µ(t) such that t+x µ(s) ds ≡ mtx . Ftx /St = exp
2. The option market and the implied volatility models
Definition 2.2. An implied volatility model is regular if (i) limx→0 E[f (St+x )|St = S] = f (S), (f ∈ C0 ), (ii) supx ∂x E[(St+x /Ftx )2 |St ] is locally bounded, (iii) supt E[supx |∂x Et [(St+x /Ftx )±2 ]|] < ∞.
We consider a continuous trading economy on a finite horizon [0, T ] with traded assets at time t, a financial asset S, the money market account B, the call and the put options on S with strike K > 0 and maturity in (t, t + xm ). We also assume that static portfolios consisting of a continuum of traded vanilla options are traded instruments as in Breeden and Litzenberger (1978) and Carr and Madan (1998).
32
t
We assume that there exists a stochastic basis (, = T , {t : t ∈ [0, T ]}, P ) with a right-continuous, complete, increasing filtration with respect to which St and 1/St are cadlag quasi-left continuous square-integrable processes and sup{E[St2 + St−2 ] : 0 t T } < ∞, Ct (x, K) = Ptx Et [(St+x − K)+ ], Pt (x, K) = Ptx Et [(K − St+x )+ ]. We assume that the map (x, K) → Ct (x, K) from (0, xm ) × [0, +∞) into (0, +∞) is of class C 1 (resp. C 2 ) in the first (resp. second) variable for all t ∈ (0, T ). The market filtration {t : t ∈ [0, T ]} is the filtration generated by all primary traded asset prices. Definition 2.1. A family {Ct (x, K), Pt (x, K), St : (x, K) ∈ (0, xm ) × (0, +∞)} satisfying the above assumptions is called an implied volatility model. We use the terminology of implied volatility models instead of option price models because implied volatilities rather than option prices are the financial observable (see section 1). It is clear, however, that a process St and a two-parameter family of implied volatility processes define an implied volatility model providing that this specification is arbitrage-free. Our definition of implied volatility models is based on option prices to ensure our implied volatility models are arbitrage-free. We shall need the following technical restriction to ensure that S has bounded local characteristics. This restriction is fairly mild and met by most local volatility models.
We write C0 for the space of continuous functions vanishing at zero and infinity. We refer to Derman and Kani (1998), Schonbucher (1999) and Cont and da Fonseca (2001) for examples of stochastic implied volatility models. We now restrict our attention to two deterministic implied volatility models.
Q UANTITATIVE F I N A N C E
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3. The sticky-delta and the sticky-strike implied volatility models
4. Characterization of the sticky-delta implied volatility models
The Black–Scholes function is denoted C BS (V , F, K) with √ C BS (V , F, K) = F N (d) − KN(d − V ),
For a sticky-delta implied volatility model, we define the following deterministic function:
√ √ d = ln(F /K)/ V + 21 V . We denote #BS (V , F, K) = ∂F C BS (V , F, K), $ BS (V , F, K) = ∂F2 C BS (V , F, K). We recall that ∂V C BS (V , F, K) = 21 F 2 $ BS (V , F, K). Definition 3.1. Let (x, I ) ∈ (0, xm ) × (0, +∞). The per-delta implied volatility denoted σt (x, I ) where I stands for the forward moneyness ratio, is the unique positive solution of the equation Ptx C (xσt (x, I ) , Ftx , Ftx I ) = Ct (x, Ftx I ). BS
2
The per-strike implied volatility denoted &t (x, K) is the unique positive solution of the equation Ptx C BS (x&t (x, K)2 , Ftx , K) = Ct (x, K). Since the function C BS (V , F, K) is strictly increasing with respect to V , we conclude that &t (x, K) = σt (x, K/Ftx ), lim xσt (x, I ) = lim x&t (x, K) = 0. 2
x↓0
c(t, x, I ) = C BS (xσt (x, I )2 , 1, I ).
Since y 2 < 2(ey + e−y ) for all real y and St , 1/St are square-integrable, ln St is square-integrable. Finally, we have the following result. Theorem 4.1. In a sticky-delta implied volatility model defined on [0, T ], the market filtration {t : 0 t T } coincides with the filtration generated by S, all risk-neutral probability measures are equal to the probability measure P on the σ -algebra T and the stochastic process ln St has independent increments under P . Proof. We note that P is a risk-neutral measure. Consider a forward-start call option with fixing date t and maturity t + x with x < xm . This option pays at maturity the quantity (St+x − kFtx )+ /Ftx . At time t, this forward-start option is a regular call option on S with strike kFtx , time-to-maturity x and notional 1/Ftx . At time t, the cost of replicating this option is Ptx c(t, x, k). This value is deterministic. Hence at time s t, the forward value of this option is c(t, x, k) and thus independent of s. Finally, we have proved that for any risk-neutral measure Q and any s t, we have EsQ [(St+x − kFtx )+ /Ftx ] = c(t, x, k).
At a given time t, the volatility functions (x, I ) → σt (x, I ) and (x, K) → &t (x, K) are of class C 1 in the first variable and C 2 in the second variable in (0, xm ) × (0, +∞). For (x, I ) ∈ (0, xm ) × (0, +∞), the implied volatility processes σt (x, I ) and &t (x, I ) are cadlag quasi-left continuous processes adapted to the market filtration. Following Derman (1999) and Reiner (1999), we propose the following definition of sticky-delta and sticky-strike implied volatility models. Definition 3.2. An implied volatility model is sticky-delta if the per-delta volatility process σt (x, I ) is deterministic on [0, T ] for all (x, I ) in (0, xm ) × (0, +∞). An implied volatility model is sticky-strike if the per-strike volatility process &t (x, K) is deterministic on [0, T ] for all (x, K) in (0, xm ) × (0, +∞). We observe that the Black–Scholes model is an implied volatility model defined by σt (x, I ) = &t (x, K) = σt (x, 1).
(4.2)
Next, we note that
2
x↓0
(4.1)
0−
1 ∂k (St+x − kFtx )+ 1. Ftx
(4.3)
We can thus permute differentiation with respect to k and Q-expectation in (4.2): Pr Q {St+x < kFtx | s } = 1 + ∂k c(t, x, k).
(4.4)
The cumulative distribution of ln St+x − ln St conditional on s is thus independent of s t, Ss and Q. Let u1 = ln Sr − ln Ss and u2 = ln Su − ln St with s < r < t < u < T , 0 < r − u < xm and u − t < xm . Equation (4.4) implies that for any risk-neutral measure Q, we have E Q [eiαu1 +iβu2 ] = E Q [eiαu1 ] E Q [eiβu2 ]. (4.5) The variables u1 and u2 are thus independent and have the same joint distribution under any risk-neutral measure Q. This result is extended by convolution to n arbitrary nonoverlapping increments of ln St in [0, T ]. It thus follows that ln St has independent increment in [0, T ] under any riskneutral measure.
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Finally, we show by induction that for any integer n, any sequence {Ti } ∈ [0, T ]n and any real Ui : Q{ST1 < U1 , . . . , STn < Un } = P {ST1 < U1 , . . . , STn < Un }. All risk-neutral probability measures Q coincide with P on the algebra A of cylindrical sets {ω ∈ : S(T1 , ω) < U1 , . . . , S(Tn , ω) < Un }. A direct application of the monotone class theorem as in Jacod and Protter (1991, p 32) implies that these probability measures coincide on σ (A) and thus on T . We note that the uniqueness of a risk-neutral measure does not, however, imply completeness of the model since the number of traded assets is infinite (see Jarrow and Madan 1999). We have proved that in a sticky-delta implied volatility model, there exists a cadlag square-integrable P -martingale X with P -independent increments such that St = S0 m0t exp(Xt )/E[exp(Xt )]. Examples of processes with independent increments are Brownian motions, Poisson processes, Levy processes and jump-diffusion processes. We refer to Protter (1995) for the construction of stochastic integrals with respect to cadlag square-integrable martingales. In this paper, the stochastic integrals from a 0 to b are integrals on (a, b]. We recall that the quadratic variation of a cadlag square-integrable martingale Z is an increasing cadlag adapted process defined by t [Z, Z]t = Zt2 − 2 Zu− dZu . 0
To ease notation, this process will also be denoted [Z]t or [Zt ]. Restricting our attention to regular sticky-delta models will ensure that X has finite moments and satisfy the representation property of Nualart and Schoutens. This will allow us to prove the quasi-completeness of regular sticky-delta models. But first, we need some notation and some preliminary results. Since S is assumed quasi-left continuous, the martingale X has no fixed points of discontinuity and can be decomposed as in Jacod and Shiryaev (1987, p 77) Xt = (σ • W )t + (x ∗ (N − n))t , t xN(dx × ds), #Xt = (x ∗ N )t = 0
0 0.
Proposition 5.1. Implied volatility models are arbitrage-free. Proof. Suppose that λ is an arbitrage strategy then E[1{VT (λ) < 0}] = 0 and thus, we have E[VT (λ) 1{VT (λ) < 0}] = 0. It follows that E[VˆT (λ)] = E[|VˆT (λ)|] > 0 = Vˆ0 (λ). This contradicts the fact that Vˆt (λ) is a P -martingale. Remark 5.1. This result is not surprising since we have assumed the existence of at least one risk-neutral measure P . We can now define attainability and attainability up to a vanishing risk. Definition 5.1. A claim 8 ∈ L2 (, ) with maturity t T is attainable or can be replicated if there is a self-financing trading strategy λ such that 8 = Vt (λ), i.e. 8 ∈ V (&). A claim 8 ∈ L2 (, ) with maturity t T is attainable up to a vanishing risk or can be replicated up to a vanishing risk if there is a sequence of self-financing strategies {λn } such that limn→∞ E[(8 − Vt (λn ))2 ] = 0, i.e.8 ∈ V¯ (&).
Q UANTITATIVE F I N A N C E
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A claim 8 ∈ L2 (, ) with maturity t is thus attainable if there are some static trading strategies {φi = (t0i , t1i , φiP , φiC , pi , ci , αi ) : 0 i n}, some predictable processes {λit : 0 i n} satisfying (5.4) and (5.5) and a real V0 = E[P0t 8] which is the cost of replication, such that n t1i αi P0t 8 = V0 + λis gi (K)pi (dK) ds Pˆs (t1i − s, K) +
n
i=0 t0i t1i +∞
t0i
i=0
αi
0
λis fi (K)ci (dK) ds Cˆ s (t1i − s, K).
We say that a claim 8 ∈ L2 (, ) with maturity t is attainable at time s if there exists a self-financing strategy λ and Cs ∈ s such that P0t 8 = Vˆt (λ) − Vˆs (λ) + Cs . Similarly, we define attainable up to a vanishing risk, at time s. With the above notations, we define quasi-completeness as in Jarrow and Madan (1999) and in Bjork et al (1997). Definition 5.2. An implied volatility model is complete up to a vanishing risk or quasi-complete if any 8 ∈ L2 (, ) is attainable up to a vanishing risk i.e. L2 (, ) − V¯ (&). We now prove the following classic result to be used later. Proposition 5.2. Let a ∈ (0, xm ), b = t + a ∈ [0, T ] and let f be a C 2 (0, +∞) function with f (Sb ) ∈ L2 (, ) such that there is 0 < α < ∞ satisfying α Sb ∨α 2 2 2 2 E f (K) K dK + f (K) Sb dK < ∞. Sb ∧α
α
(5.7) Then the claim with payoff f (Sb ) at time b can be replicated at time t at a cost Ptb Et [f (Sb )]. Proof. Following Carr and Madan (1998), we decompose the payoff into four components α f (Sb ) = (K − Sb )+ f (K) dK 0 +∞ + (Sb − K)+ f (K) dK + f (α) α
+f (α)(Sb − α) ≡ A + B + C + D. (5.8) Thanks to our assumptions on f , we derive by Cauchy– Schwartz: α +∞ P0bK |f (K)|dK + C0bK |f (K)|dK < ∞. (5.9) α
0
|f (K)|(K − S)+ 1{K < α} + |f (K)|(S − K)+ 1{K > α} is thus P {Sb ∈ dS}×dK integrable on (0, +∞)2 and the Fubini– Lebesgue theorem implies (Malliavin and Airault (1993, p 46)) α +∞ Ptb Et [A+B] = f (z)Pt (a, z) dz+ f (z)Ct (a, z) dz. α
0
Thanks to our assumption on f , we conclude that the claim A + B is square-integrable and attainable at t by the static trading strategy (t, b, f , f , dK, dK, α): b α Ptb (A + B) = Ptb Et [A + B] + f (z) dz ds Pˆs (b − s, z)
b
t
+∞
+ t
α
0
f (z) dz ds Cˆ s (b − s, z).
The third component is trivially replicated by taking position in the money market account. The fourth component is replicated by buying a call option and selling a put option since it is the terminal value of a forward contract. Corollary 5.1. The above result still holds if we replace in (5.7) the squares by absolute values and replicated by replicating up to a vanishing risk. Proof. Consider the sequence of smooth functions fn defined by replacing in (5.8) f (K) by f (K)1{|f (K)| < n}. By applying the Lebesgue dominated convergence theorem, we obtain limn→∞ E[(fn − f )2 ] = 0. By application of proposition 5.2 to fn (Sb ), we conclude that fn (Sb ) is attainable and thus at time t, the claim f (Sb ) ∈ L2 (, ) can be replicated up to a vanishing risk at a cost Ptb Et [f (Sb )]. Proposition 5.3. Let Ut be a square-integrable martingale such that the claim UT with maturity T is attainable and let γt be a predictable bounded a process. Then the claim with payoff at time a, Za = 0 γt dUt ∈ L2 (, ), can be replicated at zero cost. Proof. The claim with payoff UT /P0T is attained by a self-financing trading strategy λ = (λis , φi ). We note that Vˆt (λ) = Et [UT ] = Ut satisfies n t λis dVˆs (φi ). Ut = U0 + i=1
0
Since γt is a predictable bounded process, (γs λis , φi ) defines a self-financing trading strategy and we have 0
a
γt dUt =
n i=1
0
a
γt λit dVˆt (φi ).
Therefore, the claim Za with maturity a is attainable at zero cost. We conclude this section with a first illustration of how hedging in a sticky-delta model works in practice (see remark 7.2). Proposition 5.4. All call and put options with maturity in [0, T ] are attainable. Proof. Consider the long-dated call option with strike K > 0 and maturity t = kxm /2 + x in [0, T ] where k is an integer and 0 < x xm /2. Given the set of dates {ti = (ixm /2) ∧ t : 1 i k + 1}, we shall construct a sequence of static trading strategies φi having trading intervals (ti , ti+1 ] such that (1, φi ) replicates the long-dated call option. At time tk = kxm /2, the call option is a short-dated option and thus can be replicated by purchasing the call option itself. The discounted value process associated with this static
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strategy is Vˆu = Cˆ k (t − u, K) for u ∈ [tk , t]. Given the stickydelta assumption, we obtain Vˆtk = P0t E[(St − K)+ |Stk ] = P0t Stk c(t − tk , K/Stk ), where c(x, y) = E[(St /St−x − y)+ ] is a C 2 function which is decreasing and convex with respect to the second argument. The density of St /St−x is ∂y2 c(x, y) and y 2 ∂y2 c(x, y) is uniformly bounded in (0, +∞). We note that Vˆ ∈ L2 (, ) and ∂S2t Vˆtk = K 2 ∂y2 c(#tk , K/Stk )/St3k 0. We derive k 1 2 2 6 6 E ∂y c(#tk , K/Sk ) K /Sk dK < ∞, Sk ∧1 Sk ∨1
E
1
Ht(i) =
t
0
The deterministic functions aij (s) are bounded in [0, T ] and are such that the martingales Ht(i) are square-integrable and pairwise strongly orthogonal. We show in the appendix that the [ ]-orthogonal family {H (i) : i 1} forms a complete basis of L2 (, ). More precisely, we have the following representation property. Proposition 6.1. Let F ∈ L2 (, ) then there are predictable processes {φt(i) : i 1} such that
∂y2 c(#tk , K/Sk )2 K 4 /Sk4 dK < ∞.
We can thus apply proposition 5.2 and replicate the claim Vˆtk between time tk−1 and tk using a static portfolio with positive holdings in short-dated call and put options having maturity tk , at a discounted cost Vˆtk−1 = P0t E[(St − K)+ |Stk−1 ] = P0t Stk−1 c(t − tk−1 , K/Stk−1 ). By repeating the above argument, we obtain a set of static trading strategy {φi : 0 i k} with positive holdings such that the self-financing trading strategy (1, φi ) replicates the long-dated call option at the following initial cost: Vˆ0 = P0t E[(St − K)+ ]. We obtain the result for the long-dated put options by put– call parity.
dYs(i) + ai,i−1 (s) dYs(i−1) + . . . + ai,1 (s) dYs(1) .
F = E[F ] +
+∞ i=1
0
T
φs(i) dHs(i) ,
2 where {φt(i) : i 1} belongs to ⊕+∞ i=1 LH (i) .
Proof. See appendix.
Using this representation property, we prove in the next section that in regular sticky-delta models, all square integrable claims can be replicated up to a vanishing residual risk by trading portfolios of vanilla options. We cannot typically replicate, in the classic sense, contingent claims because the family of martingales underlying the representation property of proposition 6.1 has in general an infinite dimension and thus exact replication would be possible only if strategies based on an infinite number of traded instruments were admissible.
In order to construct similar hedging strategies for squareintegrable claims, we need a representation property for X.
6. Representation property for regular martingales with independent increments In this section and in the appendix, we extend the representation property obtained by Nualart and Schoutens (2000) for Levy processes, to regular martingales with independent increments. We adapt the notation and the approach taken by Nualart and Schoutens to our purpose. We define the Teugels martingales on [0, T ] as in Nualart and Schoutens (2000):
Yt(k)
Yt(1) = Xt , t k ≡ (#Xs ) − mk (s) ds, 0 b − xm using the static strategy: b f0b i i Xb = Ea [(Xb ) ] + x (i) (K)K −2 dKdu Pˆu (b − u, K)
b
a
+∞
+ a
f0b
0
x (i) (K)K −2 dK du Cˆ u (b − u, K),
x (i) (K) = {i(i − 1)(ln{K/f0b })i−2 − i(ln{K/f0b })i−1 }/P0b .
Proof. Using equation (4.10) and proposition 5.2 with f (S) = ln(S/f0b )i , we derive the result.
Q UANTITATIVE F I N A N C E
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Proposition 7.1. For i 1 and b ∈ (a, a + xm ), Hb(i) − Ha(i) satisfies b f0b (i) (i) −2 ˆ h(i) Hb − H a = u (K)K dKdu Pu (b − u, K)
a
b
+∞
+ a
f0b
0
−2 ˆ h(i) u (K)K dK du Cu (b − u, K)
b
+
0j ki−1
gu(i)
=
k k gu,i Xu− .
0ki−1 jk hu,i
k gu,i
and are deterministic and uniformly The coefficients bounded on [0, T ]. For all i 1, the claim HT(i) is attainable, i.e. HT(i) ∈ V (&). Proof. Yt(1) = ln(St /f0t ) is a square-integrable martingale. By proposition 5.2, we conclude that Yb(1) is attainable using the static trading strategy defined by f0b (1) Pˆt (b − t, K)K −2 dK P0b Yt = − 0 +∞ − Cˆ t (b − t, K)K −2 dK f0b
+(Cˆ t (b − t, f0b ) − Pˆt (b − t, f0b ))/f0b . This equation implies for t ∈ [a, b] t f0b yu(1) (K)K −2 dK du Pˆu (b − u, K) Yt(1) − Ya(1) = 0 a t +∞ + yu(1) (K)K −2 dK du Cˆ u (b − u, K) a
0
0
