Arbitrage bounds for weighted variance swap prices Mark Davis - - PowerPoint PPT Presentation

Analysis, Stochastics and Applications Walter Schachermayer, 60th Birthday Vienna, 12-16 July 2010

• Arbitrage bounds for

weighted variance swap prices

• Mark Davis

Department of Mathematics Imperial College London www.ma.ic.ac.uk/∼mdavis Joint work with Vimal Raval and Jan Ob l´

• j.

1

AGENDA

• Introduction
• Model-free bounds for puts and calls
• Variance swap + one put option
• General formulation
• Weighted realized variance and convex payoffs
• Lower and upper arbitrage bounds for convex payoffs
• Application to weighted variance swaps
• Example
• Empirical results
• Concluding remarks
• Left-wing information

2

Model-free bounds on European option prices Given a set of traded option prices, is there an arbitrage opportunity? Note: no model is given a priori. There are three possibilities

• There is a model-independent arbitrage. We can realize a profit by trading

at time zero. Example: butterfly spread with negative price.

• The prices are consistent with absence of arbitrage. There is a model such

that for all options (pi, Hi) we have pi = DE[Hi], where Hi is the possibly path-dependent exercise value. A model for an asset price St is simply a filtered probability space (Ω, Ft, Q) carrying an adapted process St such that St = FtMt where Ft is the forward price and Mt is a Q-martingale with M0 = 1. Example: a put option has model price pK = DE[(K − ST)+]. In normalized units r = pK/DF, k = K/F this is r = E[(k − MT)+].

3

• There is a model-dependent arbitrage.

Prices are inconsistent with any model but more information is needed to determine the arbitrage strategy. Example: two call options with different strikes but the same price. We say there is weak arbitrage if the 1st or 3rd cases hold. We then have a dichotomy between ‘weak arbitrage’ and ‘consistency with absence of arbitrage’. Standing assumptions

• Liquid market in underlying asset St, t ∈ [0, T].
• No interest rate volatility; time-0 discount factors are p(0, t) = Dt (usually

D ≡ DT)

• Uniquely determined forward price Ft (e.g. deterministic dividend yield q)

Usually F = FT.

• Options are traded at time 0 at quoted prices. In this talk all options are

European with the same exercise time T.

4

Put and call bounds

DF K1 F K2 K3 K p1 p2 p3

Call options. Normalized prices (ki, ci) (with (k0, c0) = (0, 1))are consistent with absence of arbitrage if the linear interpolant is strictly decreasing and convex and lies above the line c = 1 − k. Model-dependent arbitrage if not strictly decreasing. Else model-free arbitrage.

5

Put options By put-call parity, the conditions are as shown in the figure. In particular D(K − F)+ ≤ P < DK

• r in normalized units

(k − 1)+ ≤ p < k.

1 kn kn' k3 k2 k1

The slope is < 1 unless there is some n′ such that pn′ = kn′ − 1. (In this case P[ST/FT > kn′] = 0 in any model.)

6

Adding a Variance swap From now on we make—at least—the standing assumption (A) The price St is a positive continuous function of t ∈ [0, T]. Important point: In considering puts and calls, we essentially determine a prob- ability law µ for MT = ST/FT. To create a ‘model’ satisfying (A), we need to specify a continuous martingale whose law at time T is µ. Let Bt be Brownian motion with B0 = S0. By Skorohod embedding, for any law µ there is a stop- ping time τ such that Bτ ∼ µ and Bt∧τ is a u.i. martingale. We can now take Mt = B(

t T−t ∧ τ).

The key point is that imposing (A) does not change the arbitrage conditions for plain-vanilla puts and calls.

7

Variance swap By standard convention, a variance swap is deemed to be a forward contract in which a cash payment of pvs is exchanged at time T for the realized quadratic variation of returns < log S >T. By Ito < log S >t= −2 log(ST/S0) + 2 T dSt St , and hence in any model pvs = −2E[log(ST)] + 2 log(S0). However, without further assumptions we have no model-free definition of the variance swap contract (other than the actual market definition!) For this we need calcul d’Ito sans probabilit´

• es. Strengthen (A) to

(A′) The price St is a positive continuous function on [0, T] having the quadratic variation property.

8

Definition: A partition π of [0, T] is a finite sequence 0 ≤ t0 < t1 < ∙ ∙ ∙ < tk ≤ T. The mesh size is max1≤j≤k(tj−tj−1). A function S : [0, T] → R has the quadratic variation property (QVP) if there is some sequence πn of partitions such that the mesh size converges to zero and the sequence of measures µn =

• tj∈πn

(S(tj+1) − S(tj))2δtj converges weakly to a measure on [0, T] whose distribution function is denoted < S >t. If S has the QVP and Xt = F(St) for F ∈ C1 then X has the quadratic variation property and < X >t= t (F ′(Su))2d < S >u .

9

In particular if S satisfies (A′) and F = log we have < log S >t= t 1 S2

u

d < S >u (1) Applying the F¨

• llmer-Ito formula to log St and using (1) we obtain

< log S >t= −2 log(St/S0) + 2 t 1 Su dSu.

• Remark. If St is a continuous semimartingale on some probability space then

almost all paths have the QVP. Hence if the sample paths do not have the QVP then there is an arbitrage opportunity with or without options—no equivalent martingale measure. So strengthening (A) to (A′) is ‘harmless’.

10

Next question: What’s the relation between the prices of the log option and

• ther traded options?

Start with one put option, strike K. We obtain bounds by considering super- replicating strategies.

log x x1 K b x0

We can superhedge the log option by a static portfolio containing cash, the underlying asset and a short position in the put option with strike K. For minimum cost the payoff profile of the superhedging portfolio consists of two lines tangent to the log curve as shown in the figure.

11

The two lines have slopes 1/x0 and 1/x1, so that b + 1 x1 x1 = 1 + b = log x1 and hence x1 = e1+b. For x0 we have log x0 + 1 x0 (K − x0) = b + 1 x1 K = b + Ke−(1+b), which implies that log x0 + K x0 = 1 + b + K x1 = log x1 + K x1 = 1 + b + Ke−(1+b). (2) Thus x0 and x1 are the two solutions of f(x) = z where f(x) = log x+K/x and z = z(b) is the expression on the right of (2). We find that z(b) ≥ 1 + log K for all b.

12

K x1 x0 1+log K z log x + K/x x

The value of the superhedging portfolio at time T is vT = b + 1 x1 ST − 1 x0 − 1 x1

• (K − ST)+

so its value at time 0 is v0 = Db + 1 x1 DF − 1 x0 − 1 x1

• pK

where pK is the time-0 put option price. Since the portfolio superhedges, there is an arbitrage opportunity unless vlog ≤ v0. We obtain the tightest bound by minimizing v0 over the one remaining free parameter b, or equivalently z.

13

Writing y0 = 1/x0, y1 = 1/x1 we have, since y1 = e−(1+b), v0 + D = D(Fy1 − log y1) − (y0 − y1)pK = D((F − K)y1 + z) − (y0 − y1)pK. Now dyj/dz = 1/(K − 1/yj) = 1/(K − xj), j = 0, 1 so d dz(v0 + D) = 1 K − x1

• (F − x1) − x0 − x1

K − x0 pK

• .

At the minimum point the derivative is zero, i.e. pK = D x1 − F x1 − x0 (K − x0). (3) Letting Q be the distribution of ST given by the two-point probability measure Q = qδx0 + (1 − q)δx1 with q = (x1 − F)/(x1 − x0) we have

• EQ[ST] = F
• pK = DEQ[K − ST]+, from (3)
• v0 = DEQ[log ST], since vT = log ST a.s.

14

S 105 F 107.12 D 0.95123 K 100 T 1

Table 1: Parameter values

Example Parameter values shown correspond to an interest rate of 5% and a dividend yield of 3%. We find that if the put option price is pK = 88.02 then v0 is minimized at b = 14.407 and the minimum value is 1.912. The values of x0 and x1 are 7.465 and 4.91 × 106. We conclude that if there is a quoted log-option price of 1.912 in the market (this corresponds roughly to the Black-Scholes value with σ = 25%) then the put price cannot be more than 88.02, since the minimum v0 decreases with pK. This contrasts with the maximum put value DK = 95.13 in the absence of the log option.

15

Interpretation The above problem is a semi-infinite linear program: Find vP = inf{c′z|z ∈ Z} where Z = {z ∈ R3 : a(s)z ≥ b(s) ∀s ∈ R+}. Here a(s) is the vector of exercise values a(s) = (1, s, (K − s)+), b(s) = log s and c is the vector of asset prices c = (D, DF, pK) Formally the LP dual is Find vD = sup

• R+ b(s)µ(ds), where the supremum is taken over positive

measures µ satisfying the equality constraints c =

• a(s)µ(ds), i.e.

(D, DF, pK) =

• 1 dµ,
• s dµ,
• (K − s)+dµ
• ≡ (
• a1(s)µ(ds),
• a2(s)µ(ds),
• a3(s)µ(ds)).

Our calculation shows there is no duality gap.

16

The Karlin-Isii Theorem Define the moment cone M = {c = (c1, . . . , cm) : ci =

• ai(s)µ(ds), µ ∈ M}

where M is the set of positive measures. Suppose

• 1. (a1, . . . , am) are linearly independent.
• 2. c is an interior point of M.
• 3. vD is finite.

Then vP = vD and the primal problem has a solution.

17

General Formulation Additional assumptions

• The asset S pays dividends at a deterministic yield; let Γ(t) denote the

number of shares that would be owned by time t if dividend income is fully re-invested in shares. (If the dividend yield is constant, then Γ(t) = exp(qt).

• Suppose a finite number, n ∈ N, of European put options on S are traded

at time 0, maturing at time T for strikes 0 < K1 < . . . < Kn < ∞. Let pi denote the price of the put struck at Ki. Further, a w-weighted variance swap maturing at T is also traded on the asset S, with swap rate kw, i.e. a forward contract that exchanges fixed kw for floating V T

w = 1

T T w(St/Ft)d < log S >t .

18

Let P denote the prices of traded securities at time 0. Operator P represents the market input and is specified as follows: P(1t) = Dt, P(Γ(t)St1t) = S0, P(Ki − ST)+ = pi, i = 1, . . . , n, P(V w

T − kw) = 0,

and P acts linearly on the combinations of the above, (4) Let X denote the set of traded options on S in which the options are identified by the payoffs, i.e. X = {(Ki − ST)+; i = 1 . . . n, V w

T − kw}

We can assume that a put with strike K0 = 0 is traded at price p0 = P(0 − ST)+ = 0 .

19

Portfolios – pre-definition A portfolio is a triple (π, φ, ψ), where:

• π = (π1, . . . , πn+1) ∈ Rn+1, in which πi denotes the number of units of the

put option with strike Ki that is held at time 0, for i = 1, . . . , n and πn+1 denotes the number of units of the weighted variance swap held at time

• zero. This forms the static component of the portfolio.
• For t ∈ [0, T], φt ∈ R denotes the number of units of the asset S held at

time t and is determined by market observations up to time t.

• Finally, ψt ∈ R denotes the number of units of the risk-free asset held at

time t for t ∈ [0, T] and is dependent on market observations up time t. The payoff of a portfolio (π, φ, ψ) at time T is denoted by X(π,φ,ψ)

T

. A portfolio (π, φ, ψ) is termed simple if the position in the underlying asset, φ, is modified only at a finite number of times in [0, T]. A simple portfolio (π, φ, ψ) is called self-financing if no further funds are required after the initial investment to follow the portfolio’s strategy.

20

At time T, the payoff from a portfolio (π, φ, ψ) is given by X(π,φ,ψ)

T

=

n

• i=1

πi(Ki − ST)+ + πn+1 (V w

T − kw) + φTST + ψTD−1 T .

The price of entering the portfolio at time 0 is specified from the market data as: X(π,φ,ψ) =

n

• i=1

πipi + φ0S0 + ψ0. If the portfolio is self-financing, then PX(π,φ,ψ)

T

= X(π,φ,ψ) . Definition: A model-independent arbitrage is a simple self-financing portfolio (π, φ, ψ) with PX(π,φ,ψ)

T

< 0 and X(π,φ,ψ)

T

≥ 0.

21

Models A model, M, for the asset price is a filtered probability space (Ω, F, (Ft)t∈[0,T], Q) with a positive and continuous (Ft, Q)-semimartingale (St)t∈[0,T] with S0 almost surely equal to the spot price. The filtration satisfies the usual hypotheses and F0 is trivial. A model is called a (P, X)-market model if (St/Ft) is an (Ft, Q)-martingale and PX = DTEQ[X] for all market quoted options X ∈ X. Given a model M ∈ M, a portfolio (π, φ, ψ) is M-admissible if: φ, ψ : [0, T] × Ω → R are predictable processes satisfying T φ2

tdSt/Ftt < ∞ a.s,

T |φtStΓ−1

t |dΓt < ∞ a.s,

T |ψt|d(D−1

t ) < ∞ a.s.

and there exists1Y ∈ L1(M) and constant c ∈ R such that t (φu − c)Γ−1

u d(Su/Fu) ≥ Y

(5)

22

Moreover, (π, φ, ψ) is an M-admissible self-financing strategy if almost surely for all t ∈ [0, T] φtSt + ψtD−1

t

− φ0S0 − ψ0 = t φudSu + t φuSuΓ−1

u dΓu +

t ψud(D−1

u ),

• r equivalently : Dt(φtSt + ψtD−1

t ) = φ0S0 + ψ0 + S0

t Γ−1

u φud(Su/Fu).

(6) The role of the constant c appearing in the admissibility conditions is to ensure that the strategy φt = cΓt, or such a component of a strategy, in which c units of the stock is held with dividends re-invested in to the stock is admissible. Definition The market prices (P, X) admit a weak-arbitrage (WA) if in any model M ∈ M, there exists an M-admissible self-financing strategy (π, φ, ψ) satisfying: X(π,φ,ψ)

T

≥ 0 almost surely, Q[X(π,φ,ψ)

T

> 0] > 0 and PX(π,φ,ψ)

T

≤ 0. The WA is simple if the strategy can always be chosen simple. Lemma: Suppose P is finite on X. If there exists a (P, X)–market model then (P, X) do not admit a weak arbitrage.

23

Recall that X is the set of traded options on S and let XE denote the set of traded vanilla European options and cash, i.e. XE = {(Ki − ST)+ : i = 0, . . . , n}, X = XE ∪ T w(St/Ft)dln St − kw

• .

(7) The set of market models for European puts is ME := {M ∈ M : M is a (P, XE)–market model}. (8) For M ∈ ME, ST ≤ Kn′ Q-a.s. if rn′ = kn′ −1. A market model has to match

• nly the first n′ ≤ n put prices to match all n put prices.

We study swap rates kw in (4) under which there exists a (P, X)–market model and compute the bounds LB := inf

M∈ME EM

T w( ˜ St)dln St

• and UB := sup

M∈ME EM

T w( ˜ St)dln St

• ,

(9) and show that there exists a (P, X)–market model if kw ∈ (LB, UB) and there is a weak arbitrage if kw / ∈ [LB, UB]. We settle the boundary cases.

24

Weighted realised variance The weighting function w : (0, ∞) → [0, ∞) is a measurable map satisfying

• A

w(a) a2 da < ∞ for all compact A ⊂ (0, ∞). (10) Lemma For w satisfying Assumption (10) and λw a convex function defined by λ′′

w(a) = w(a) a2

we have, in any model M ∈ M and stopping time τ, τ w( ˜ Su)dln Su = 2λw( ˜ St) − 2λw(1) − 2 τ λ′

w( ˜

Su)d ˜ Su a.s . (11) (The right hand side of (11) is zero for affine functions so any normalisation can be chosen.) This lemma is an application of the Meyer-Itˆ

• formula, see Rogers & Williams

Section IV.45.

25

For three common specifications of the weight w, all of which satisfy Assump- tion (10), we specify functions λw explicitly as:

• 1. Realised variance swap - w ≡ 1:

λw(x) = − ln(x).

• 2. Corridor variance swap - w(x) = 1(0,a)(x) or w(x) = 1(a,∞)(x), where 0 <

a < ∞: λw(x) =

• − ln

x a

• + x

a − 1

• w(x).
• 3. Gamma swap - w(x) = x:

λw(x) = x ln(x) − x.

26

The lemma shows that half the realised variance, with weight w, at time T is replicated by the following strategy:

• holding a European claim paying λw( ˜

ST),

• dynamic trading in the underlying with the strategy, which using (6), has

−Γtλ′

w( ˜

St) units of the stock at time t,

• and −λw(1) cash at time T.

For no arbitrage then, the price kw of a weighted variance swap is the forward price of the above portfolio. The task of determining no-arbitrage bounds for the price of the w-variance swap is reduced to doing so for the price of a European

• ption with payoff λw. We need this technical result, proved in the paper.

Proposition Let M ∈ M. For w satisfying (10), if E T w( ˜ St)dln St

• < ∞,

(12) then the strategy (Γtλ′

w( ˜

St))t∈[0,T] is admissible in this model.

27

• Proposition. Let ( ˜

St) be a continuous martingale on a filtered probability space (Ω, F, (Ft), P) satisfying the usual assumptions, and let σ be a bounded stop- ping time. For w satisfying (10) it holds E σ w( ˜ St)dln ˜ St

• = 2E[λw( ˜

Sσ)] − 2λw(1) (13) and in particular E σ w( ˜ St)dln ˜ St

• < ∞ ⇐

⇒ E[|λw( ˜ Sσ)|] < ∞.

• Proof. Let τn be a localising sequence for

t

0 λ′ w( ˜

Su)d ˜ Su and such that ˜ Su ≤ n, u ≤ τn. Stopping (11) at σ ∧ τn and taking expectations we see that E σ∧τn w( ˜ Su)dln Su = 2Eλw( ˜ Sσ∧τn) − 2λw(1). (14)

28

As λw is convex, its negative part λ−

w(s) is bounded above by an affine function

and hence Eλ−

w( ˜

St) < ∞ for all t ≤ σ ∧ τn. The positive part λ+

w(s) is also a

convex function and Jensen’s inequality together with Optional Sampling and Fatou’s lemma yield lim sup

n

Eλ+

w( ˜

Sσ∧τn) ≤ Eλ+

w( ˜

Sσ) ≤ lim inf

n

Eλ+

w( ˜

Sσ∧τn). In consequence Eλ+

w( ˜

Sσ) = lim Eλ+

w( ˜

Sσ∧τn) and the lemma follows by taking limits as n → ∞ in (14).

29

General Results Hedging of w-weighted realised variance is achieved by dynamically trading the underlying asset and holding a claim paying λw( ˜ ST) at time T, the key point being that the claim depends only on ST, the asset price at time T, which is the time at which the traded European options also expire. In view of the previous propositions, inf

M∈ME EM

T w( ˜ St)dln St

• =

2 inf

M∈ME EM

• λw( ˜

ST)

• − 2λw(1)

and sup

M∈ME EM

T w( ˜ St)dln St

• =

2 sup

M∈ME EM

• λw( ˜

ST)

• − 2λw(1).

Consequently, the problem of determining no-arbitrage bounds for the price of the weighted variance swap reduces to proving the bounds for the corresponding European claim.

30

Arbitrage bounds for convex payoffs We suppose that a European claim is written on S paying λ( ˜ ST) at maturity T, where the payoff function λ : (0, ∞) → R is convex. The forward price of this claim, under any market model, is determined by the law of ˜

• ST. Recalling ( ˜

St) is a martingale with ˜ S0 = 1 and that the normalised put price for moneyness- strike ki is ri, the set of feasible laws of ˜ ST is given by ME

T =

• µ : µ([0, ∞)) = 1,
• xµ(dx) = 1,
• (ki − x)+µ(dx) = ri for i = 1, . . . , n
• =
• LM( ˜

ST) : M ∈ ME , (15) where LM( ˜ ST) is the law of ˜ ST under a model M. We prove below that the lower bound price is determined by the solution of a dynamic programming recursion.

31

• Theorem. Assume that prices of European options (P, XE) do not admit weak
• arbitrage. Then for a differentiable convex function λ : (0, ∞) → R,

∞ > inf

µ∈ME

T

• λ(x)µ(dx) =
• λ(x)µ†(dx) ≥ λ(1),

in which recall ME

T , defined in (15), denotes the set of calibrated marginals for

˜ ST and µ† is an atomic probability measure with at most n + 1 atoms. Moreover, there exists a static portfolio (π†, φ†, ψ†) in the n European options, the underlying asset S and cash respectively, with π†

n+1 = 0, such that

X(π†,φ†,ψ†)

T

=

n

• i=1

π†

i(Ki − ST)+ + φ†ST + ψ† ≤ λ(ST/FT)

and the price of which satisfies P

• X(π†,φ†,ψ†)

T

• =

n

• i=1

π†

ipi + DTφ†FT + DTψ† = DT

• λ(x)µ†(dx).

(16) In particular, if Pλ(ST/FT) < DT

• λ(x)µ†(dx) then market prices (P, XE ∪

{λ(ST/FT)}) admit a model independent arbitrage.

32

• Proof uses the Karlin-Isii semi-infinite programming theorem.
• The lower bound is computed by a dynamic programming algorithm.
• There is also (in some cases) an upper bound—simpler argument.

33

Summary of main general result. Let (P, XE) be the market input, and assume the European prices (P, XE) do not admit a weak arbitrage. For a differentiable convex function λ : (0, ∞) → R let LBλ := DT inf

µ∈ME

T

• λ(x)µ(dx)

and UBλ := DT sup

µ∈ME

T

• λ(x)µ(dx).

(17) (i) If Pλ(ST/FT) ∈ (LBλ, UBλ) then there exists a (P, XE ∪ {λ(ST/FT)})– market model. (ii) If Pλ(ST/FT) = LBλ then there exists a (P, XE ∪ {λ(ST/FT)})–market model if and only if rn = kn − 1 or if rn > kn − 1 but µ†([0, kn]) < 1, where µ† is the minimising probability measure such that LBλ = DT

• λ(x)µ†(dx).

Otherwise (P, XE ∪ {λ(ST/FT)}) admit a weak arbitrage. (iii) If Pλ(ST/FT) = UBλ < ∞ then there exists a (P, XE ∪ {λ(ST/FT)})– market model if and only if rn = kn − 1. Otherwise (P, XE ∪ {λ(ST/FT)}) admit a weak arbitrage. (iv) If Pλ(ST/FT) / ∈ [LBλ, UBλ] then (P, XE ∪ {λ(ST/FT)}) admit a model- independent arbitrage.

34

Arbitrage bounds for Weighted Variance Swaps We now return to the original problem: determining if the market trading the underlying asset, the put options and the variance swap admits arbitrage, given that the European put prices (P, XE) admit no weak arbitrage. The w-weighted variance swap maturing at T has payoff T w( ˜ St)dln(S)t − kw, where kw is fixed so that (4) holds, i.e. there is no time zero cost of entering the contract.

35

Theorem Let (P, X) be the market input, and suppose (P, XE) do not admit a weak arbitrage. Let λw denote the convex function corresponding to w. Then the following are equivalent:

• 1. There exists a (P, X)–market model.
• 2. The market prices (P, X) do not admit a weak-arbitrage.
• 3. The market prices (P, XE∪{λw(ST/FT)}), with Pλw(ST/FT) := DT
• λw(1) + kw

2

• ,

do not admit a weak-arbitrage. In particular if kw ∈

• 2D−1

T LBλw − 2λw(1), 2D−1 T UBλw − 2λw(1)

• then there ex-

ists a (P, X)–market model and if kw / ∈

• 2D−1

T LBλw − 2λw(1)2D−1 T UBλw − 2λw(1)

• then market prices admit weak arbitrage.

36

Example To illustrate an application of our results, consider a fictional market with an index S that trades three European put options maturing in 1 year. The data are S0 = 100, FT = 105, DT = exp(−0.03), Ki = 50, 100 and 150, p1 = 1.127, p2 = 18.006 and p3 = 53.326. The range of (weak) arbitrage-free prices for a vanilla variance swap, corridor variance swap and gamma swap have been determined and summarised as follows.

Variance Swap w(x) λw(x) Arbitrage bounds for kw Type (£) Vanilla Swap 1 − ln(x) [0.224, ∞) Corridor Variance Swap 1

75 FT ,∞

(x)

[− ln( xFT

75 ) + FT x 75 − 1]w(x)

(0.038, 0.340) Gamma Swap x x ln(x) − x (0.125, ∞)

37

Log contract: The figure shows the log contract payoff − ln(ST/FT) (blue line) and the consequent sub-hedging portfolio (black line). The portfolio is given by π∗

1 = 0.01706, π∗ 2 = 0.00472, π∗ 3 = 0.00259, φ† = −0.00536 and ψ† = 0.42517.

50 100 150 S

38

Corridor variance swap: The figure shows the payoff [− ln(ST

75 ) + ST 75 −

1]1

75 FT ,∞

(ST/FT) (blue line), the consequent sub-hedging portfolio (black line)

and super-hedging portfolio (red line). The portfolios are given by π†

1 = 0,

π†

2 = 0.00606, π† 3 = −0.00606, φ† = 0 and ψ† = 0.30293 for the sub-hedge, and

for the super-hedge: π∗

1 = 0.00091, π∗ 2 = 0.00431, π∗ 3 = 0.00811, φ∗ = 0.01333

and ψ∗ = −1.69315.

50 100 150 75 S

39

Gamma swap: Here we see the payoff ST

FT ln

• ST

FT

• − ST

FT (blue line) and the

consequent sub-hedging portfolio (black line). The portfolio is given by π†

1 =

0.00772, π†

2 = 0.00571, π† 3 = −0.00225, φ† = 0 and ψ† = −0.92899.

50 100 150 S

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Comment

• We have p3 = 53.326 > 43.670 = DT(K3 − FT).
• The interval for the vanilla variance swap is half closed and unbounded.
• For the corridor variance swap, the sub-hedging strategy does not hold the

underlying index and so indicating the lower bound is not attained. Note that for the choice of corridor in the weight of the corridor variance swap, the swap price has a finite upper bound. In particular the upper bound is not attained due to the inequality p3 > DT(K3 − FT).

• The Gamma swap has no finite upper bound and the lower bound is not an

admissible price, which again is indicated by the corresponding sub-hedge for the convex payoff not holding the index.

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The Variance Swap (w ≡ 1) Here we demonstrate that for the vanilla vari- ance swap, given by weight w ≡ 1 and corresponding payoff ln ST − kvs, the interval of arbitrage-free swap rates is never bounded above and half closed, irrespective of the solution to the dynamic program µ†. The model independent relation (11) reads − ln ˜ ST = − T d ˜ St ˜ St + 1 2ln ST. The interval of weak arbitrage-free variance swap prices is 2 n

• i=1

π†

iFTri + φ†FT + ψ†, ∞

• .

(18) Since − ln(x) is not bounded near x = 0 there is no upper bound. On the other hand the lower bound is an admissible price. When rn = kn − 1 this is always the case. Otherwise we have to show that µ†([0, kn]) < 1. See paper.

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Empirical results Variance swaps on S&P500 Index Term Quote date VS quote LB

• No. of puts

2M 20/04/2008 21.78 18.73 58 2M 19/07/2008 23.6 21.18 51 2M 19/10/2008 57.97 57.07 101 2M 20/01/2008 52.88 47.68 82 3M 20/03/2008 27.22 26.33 48 3M 19/06/2008 22.33 19.24 40 3M 19/09/2008 26.78 26.02 58 3M 20/12/2008 45.93 65.81 137 6M 19/03/2008 25.63 22.97 25 6M 19/06/2008 22.88 21.76 28

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Concluding remarks To get a finite upper bound for the plain-vanilla variance swap requires more ‘left-wing’ information. See paper. There are several issues that remain unaddressed, for example:

• The situation with multiple exercise times.
• The effect of jumps.
• Quantifying the error due to discrete sampling (there are some calculations

in Gatheral’s book).

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Finally .. Alles gute zum Geburtstag Walter!

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