In complete market models such as the Black-Scholes model, probability does not really matter: the “objective” evolution of the asset is only there to define the set of “impossible” events and serves to specify the class of equivalent measures. Thus, two statistical models P_{1} ∼ P_{2} with equivalent measures lead to the same option prices in a complete market setting.

This is not true anymore in incomplete markets: probabilities matter and model specification has to be taken seriously since it will affect hedging decisions. This situation is more realistic but also more challenging and calls for an integrated approach between option pricing methods and statistical modeling. In incomplete markets, not only does probability matter but attitudes to risk also matter: utility based methods explicitly incorporate these into the hedging problem via utility functions. While these methods are focused on hedging with the underlying asset, common practice is to use liquid call/put options to hedge exotic options. In incomplete markets, options are not redundant assets; therefore, if options are available as hedging instruments they can and should be used to improve hedging performance.

While the lack of liquidity in the options market prevents in practice from using dynamic hedges involving options, options are commonly used for static hedging: call options are frequently used for dealing with volatility or convexity exposures and for hedging barrier options.

What are the implications of hedging with options for the choice of a pricing rule? Consider a contingent claim H and assume that we have as hedging instruments a set of benchmark options with prices C_{i}^{∗}, i = 1 . . . n and terminal payoffs H_{i}, i = 1 . . . n. A static hedge of H is a portfolio composed from the options H_{i}, i = 1 . . . n and the numeraire, in order to match as closely as possible the terminal payoff of H:

H = V_{0} + ∑_{i=1}^{n} x_{i}H_{i} + ∫_{0}^{T} φdS + ε —– (1)

where ε is an hedging error representing the nonhedgeable risk. Typically H_{i} are payoffs of call or put options and are not possible to replicate using the underlying so adding them to the hedge portfolio increases the span of hedgeable claims and reduces residual risk.

Consider a pricing rule Q. Assume that E_{Q}[ε] = 0 (otherwise E_{Q}[ε] can be added to V_{0}). Then the claim H is valued under Q as:

e^{-rT}E_{Q}[H] = V_{0} ∑_{i=1}^{n} x_{i }e^{-rT}E_{Q}[H_{i}] —– (2)

since the stochastic integral term, being a Q-martingale, has zero expectation. On the other hand, the cost of setting up the hedging portfolio is:

V_{0} + ∑_{i=1}^{n} x_{i }C_{i}^{∗} —– (3)

So the value of the claim given by the pricing rule Q corresponds to the cost of the hedging portfolio if the model prices of the benchmark options H_{i} correspond to their market prices C_{i}^{∗}:

∀i = 1, …, n

e^{-rT}E_{Q}[H_{i}] = C_{i}^{∗ }—– (4)

This condition is called calibration, where a pricing rule verifies the calibration of the option prices C_{i}^{∗}, i = 1, . . . , n. This condition is necessary to guarantee the coherence between model prices and the cost of hedging with portfolios and if the model is not calibrated then the model price for a claim H may have no relation with the effective cost of hedging it using the available options H_{i}. If a pricing rule Q is specified in an *ad hoc* way, the calibration conditions will not be verified, and thus one way to ensure them is to incorporate them as constraints in the choice of the pricing measure Q.

As an alternative to the riskless hedging approach, * Robert Merton* derived the option pricing equation via the construction of a self-financing and dynamically hedged portfolio containing the risky asset, option and riskless asset (in the form of money market account). Let Q

Π(t) = M_{S}(t) + M_{V}(t) + M(t) = Q_{S}(t)S + Q_{V}(t)V + M(t) —– (1)

where M(t) is the currency value of the riskless asset invested in a riskless money market account. Suppose the asset price process is governed by the stochastic differential equation * (1) in here*, we apply the Ito lemma to obtain the differential of the option value V as:

dV = ∂V/∂t dt + ∂V/∂S dS + σ^{2}/2 S^{2} ∂^{2}V/∂S^{2} dt = (∂V/∂t + μS ∂V/∂S σ^{2}/2 S^{2} ∂^{2}V/∂S^{2})dt + σS ∂V/∂S dZ —– (2)

If we formally write the stochastic dynamics of V as

dV/V = μ_{V} dt + σ_{V} dZ —– (3)

then μ_{V} and σ_{V} are given by

μ_{V} = (∂V/∂t + ρS ∂V/∂S + σ^{2}/2 S^{2} ∂^{2}V/∂S^{2})/V —– (4)

and

σ_{V} = (σS ∂V/∂S)/V —– (5)

The instantaneous currency return dΠ(t) of the above portfolio is attributed to the differential price changes of asset and option and interest accrued, and the differential changes in the amount of asset, option and money market account held. The differential of Π(t) is computed as:

dΠ(t) = [Q_{S}(t) dS + Q_{V}(t) dV + rM(t) dt] + [S dQ_{S}(t) + V dQ_{V}(t) + dM(t)] —– (6)

where rM(t)dt gives the interest amount earned from the money market account over dt and dM(t) represents the change in the money market account held due to net currency gained/lost from the sale of the underlying asset and option in the portfolio. And if the portfolio is self-financing, the sum of the last three terms in the above equation is zero. The instantaneous portfolio return dΠ(t) can then be expressed as:

dΠ(t) = Q_{S}(t) dS + Q_{V}(t) dV + rM(t) dt = M_{S}(t) dS/S + M_{V}(t) dV/V + rM(t) dt —– (7)

Eliminating M(t) between (1) and (7) and expressing dS/S and dV/V in terms of their stochastic dynamics, we obtain

dΠ(t) = [(μ − r)M_{S}(t) + (μ_{V} − r)M_{V}(t)]dt + [σM_{S}(t) + σ_{V} M_{V}(t)]dZ —– (8)

How can we make the above self-financing portfolio instantaneously riskless so that its return is non-stochastic? This can be achieved by choosing an appropriate proportion of asset and option according to

σM_{S}(t) + σ_{V} M_{V}(t) = σS Q_{S}(t) + σS ∂V/∂S Q_{V}(t) = 0

that is, the number of units of asset and option in the self-financing portfolio must be in the ratio

Q_{S}(t)/Q_{V}(t) = -∂V/∂S —– (9)

at all times. The above ratio is time dependent, so continuous readjustment of the portfolio is necessary. We now have a dynamic replicating portfolio that is riskless and requires zero initial net investment, so the non-stochastic portfolio return dΠ(t) must be zero.

(8) becomes

0 = [(μ − r)M_{S}(t) + (μ_{V} − r)M_{V}(t)]dt

substituting the ratio factor in the above equation, we get

(μ − r)S ∂V/∂S = (μ_{V} − r)V —– (10)

Now substituting μ_{V }from (4) into the above equation, we get the black-Scholes equation for V,

∂V/∂t + σ^{2}/2 S^{2} ∂^{2}V/∂S^{2} + rS ∂V/∂S – rV = 0

Suppose we take Q_{V}(t) = −1 in the above dynamically hedged self-financing portfolio, that is, the portfolio always shorts one unit of the option. By the ratio factor, the number of units of risky asset held is always kept at the level of ∂V/∂S units, which is changing continuously over time. To maintain a self-financing hedged portfolio that constantly keeps shorting one unit of the option, we need to have both the underlying asset and the riskfree asset (money market account) in the portfolio. The net cash flow resulting in the buying/selling of the risky asset in the dynamic procedure of maintaining ∂V/∂S units of the risky asset is siphoned to the money market account.

* Fischer Black and Myron Scholes* revolutionized the pricing theory of options by showing how to hedge continuously the exposure on the short position of an option. Consider the writer of a call option on a risky asset. S/he is exposed to the risk of unlimited liability if the asset price rises above the strike price. To protect the writer’s short position in the call option, s/he should consider purchasing a certain amount of the underlying asset so that the loss in the short position in the call option is offset by the long position in the asset. In this way, the writer is adopting the hedging procedure. A hedged position combines an option with its underlying asset so as to achieve the goal that either the asset compensates the option against loss or otherwise. By adjusting the proportion of the underlying asset and option continuously in a portfolio, Black and Scholes demonstrated that investors can create a riskless hedging portfolio where the risk exposure associated with the stochastic asset price is eliminated. In an efficient market with no riskless arbitrage opportunity, a riskless portfolio must earn an expected rate of return equal to the riskless interest rate.

Black and Scholes made the following assumptions on the financial market.

- Trading takes place continuously in time.
- The riskless interest rate r is known and constant over time.
- The asset pays no dividend.
- There are no transaction costs in buying or selling the asset or the option, and no taxes.
- The assets are perfectly divisible.
- There are no penalties to short selling and the full use of proceeds is permitted.
- There are no riskless arbitrage opportunities.

The stochastic process of the asset price S_{t} is assumed to follow the geometric Brownian motion

dS_{t}/S_{t} = μ dt + σ dZ_{t} —– (1)

where μ is the expected rate of return, σ is the volatility and Z_{t} is the standard Brownian process. Both μ and σ are assumed to be constant. Consider a portfolio that involves short selling of one unit of a call option and long holding of Δ_{t} units of the underlying asset. The portfolio value Π (S_{t}, t) at time t is given by

Π = −c + Δ_{t} S_{t} —– (2)

where c = c(S_{t}, t) denotes the call price. Note that Δ_{t} changes with time t, reflecting the dynamic nature of hedging. Since c is a stochastic function of S_{t}, we apply the * Ito lemma* to compute its differential as follows:

dc = ∂c/∂t dt + ∂c/∂S_{t} dS_{t} + σ^{2}/2 S_{t}^{2} ∂^{2}c/∂S_{t}^{2} dt

such that

-dc + Δ_{t} dS_{t }= (-∂c/∂t – σ^{2}/2 S_{t}^{2} ∂^{2}c/∂S_{t}^{2})dt + (Δ_{t }– ∂c/∂S_{t})dS_{t}

= [-∂c/∂t – σ^{2}/2 S_{t}^{2} ∂^{2}c/∂S_{t}^{2 }+ (Δ_{t }– ∂c/∂S_{t})μS_{t}]dt + (Δ_{t }– ∂c/∂S_{t})σS_{t }dZ_{t}

The cumulative financial gain on the portfolio at time t is given by

G(Π (S_{t}, t )) = ∫_{0}^{t} -dc + ∫_{0}^{t} Δ_{u} dS_{u}

= ∫_{0}^{t} [-∂c/∂u – σ^{2}/2 S_{u}^{2} ∂^{2}c/∂S_{u}^{2} + (Δ_{u }– ∂c/∂S_{u})μS_{u}]du + ∫_{0}^{t} (Δ_{u }– ∂c/∂S_{u})σS_{u }dZ_{u }—– (3)

The stochastic component of the portfolio gain stems from the last term, ∫_{0}^{t} (Δ_{u }– ∂c/∂S_{u})σS_{u }dZ_{u}. Suppose we adopt the dynamic hedging strategy by choosing Δ_{u} = ∂c/∂S_{u}_{ }at all times u < t, then the financial gain becomes deterministic at all times. By virtue of no arbitrage, the financial gain should be the same as the gain from investing on the risk free asset with dynamic position whose value equals -c + S_{u}∂c/∂S_{u}. The deterministic gain from this dynamic position of riskless asset is given by

M_{t} = ∫_{0}^{t}r(-c + S_{u}∂c/∂S_{u})du —– (4)

By equating these two deterministic gains, G(Π (S_{t}, t)) and M_{t}, we have

-∂c/∂u – σ^{2}/2 S_{u}^{2} ∂^{2}c/∂S_{u}^{2} = r(-c + S_{u}∂c/∂S_{u}), 0 < u < t

which is satisfied for any asset price S if c(S, t) satisfies the equation

∂c/∂t + σ^{2}/2 S^{2} ∂^{2}c/∂S^{2 }+ rS∂c/∂S – rc = 0 —– (5)

This parabolic partial differential equation is called the Black–Scholes equation. Strangely, the parameter μ, which is the expected rate of return of the asset, does not appear in the equation.

To complete the formulation of the option pricing model, let’s prescribe the auxiliary condition. The terminal payoff at time T of the call with strike price X is translated into the following terminal condition:

c(S, T ) = max(S − X, 0) —– (6)

for the differential equation.

Since both the equation and the auxiliary condition do not contain ρ, one concludes that the call price does not depend on the actual expected rate of return of the asset price. The option pricing model involves five parameters: S, T, X, r and σ. Except for the volatility σ, all others are directly observable parameters. The independence of the pricing model on μ is related to the concept of risk neutrality. In a risk neutral world, investors do not demand extra returns above the riskless interest rate for bearing risks. This is in contrast to usual risk averse investors who would demand extra returns above r for risks borne in their investment portfolios. Apparently, the option is priced as if the rates of return on the underlying asset and the option are both equal to the riskless interest rate. This risk neutral valuation approach is viable if the risks from holding the underlying asset and option are hedgeable.

The governing equation for a put option can be derived similarly and the same Black–Scholes equation is obtained. Let V (S, t) denote the price of a derivative security with dependence on S and t, it can be shown that V is governed by

∂V/∂t + σ^{2}/2 S^{2} ∂^{2}V/∂S^{2 }+ rS∂V/∂S – rV = 0 —– (7)

The price of a particular derivative security is obtained by solving the Black–Scholes equation subject to an appropriate set of auxiliary conditions that model the corresponding contractual specifications in the derivative security.

The original derivation of the governing partial differential equation by Black and Scholes focuses on the financial notion of riskless hedging but misses the precise analysis of the dynamic change in the value of the hedged portfolio. The inconsistencies in their derivation stem from the assumption of keeping the number of units of the underlying asset in the hedged portfolio to be instantaneously constant. They take the differential change of portfolio value Π to be

dΠ =−dc + Δ_{t} dS_{t},

which misses the effect arising from the differential change in Δ_{t}. The ability to construct a perfectly hedged portfolio relies on the assumption of continuous trading and continuous asset price path. It has been commonly agreed that the assumed Geometric Brownian process of the asset price may not truly reflect the actual behavior of the asset price process. The asset price may exhibit jumps upon the arrival of a sudden news in the financial market. The interest rate is widely recognized to be fluctuating over time in an irregular manner rather than being constant. For an option on a risky asset, the interest rate appears only in the discount factor so that the assumption of constant/deterministic interest rate is quite acceptable for a short-lived option. The Black–Scholes pricing approach assumes continuous hedging at all times. In the real world of trading with transaction costs, this would lead to infinite transaction costs in the hedging procedure.

In d-dimensional topological field theory one begins with a category S whose objects are oriented (d − 1)-manifolds and whose morphisms are oriented cobordisms. Physicists say that a theory admits a group G as a global symmetry group if G acts on the vector space associated to each (d−1)-manifold, and the linear operator associated to each cobordism is a G-equivariant map. When we have such a “global” symmetry group G we can ask whether the symmetry can be “gauged”, i.e., whether elements of G can be applied “independently” – in some sense – at each point of space-time. Mathematically the process of “gauging” has a very elegant description: it amounts to extending the field theory functor from the category S to the category S_{G} whose objects are (d − 1)-manifolds equipped with a principal G-bundle, and whose morphisms are cobordisms with a G-bundle. We regard S as a subcategory of S_{G} by equipping each (d − 1)-manifold S with the trivial G-bundle S × G. In S_{G} the group of automorphisms of the trivial bundle S × G contains G, and so in a gauged theory G acts on the state space H(S): this should be the original “global” action of G. But the gauged theory has a state space H(S,P) for each G-bundle P on S: if P is non-trivial one calls H(S,P) a “twisted sector” of the theory. In the case d = 2, when S = S^{1} we have the bundle P_{g} → S^{1} obtained by attaching the ends of [0,2π] × G via multiplication by g. Any bundle is isomorphic to one of these, and P_{g} is isomorphic to P_{g‘} iff g′ is conjugate to g. But note that the state space depends on the bundle and not just its isomorphism class, so we have a twisted sector state space C_{g} = H(S,P_{g}) labelled by a group element g rather than by a conjugacy class.

We shall call a theory defined on the category S_{G} a G-equivariant Topological Field Theory (TFT). It is important to distinguish the equivariant theory from the corresponding “gauged theory”. In physics, the equivariant theory is obtained by coupling to nondynamical background gauge fields, while the gauged theory is obtained by “summing” over those gauge fields in the path integral.

An alternative and equivalent viewpoint which is especially useful in the two-dimensional case is that S_{G} is the category whose objects are oriented (d − 1)-manifolds S equipped with a map p : S → BG, where BG is the classifying space of G. In this viewpoint we have a bundle over the space Map(S,BG) whose fibre at p is H_{p}. To say that H_{p} depends only on the G-bundle p^{∗}EG on S pulled back from the universal G-bundle EG on BG by p is the same as to say that the bundle on Map(S,BG) is equipped with a flat connection allowing us to identify the fibres at points in the same connected component by parallel transport; for the set of bundle isomorphisms p^{∗}_{0}EG → p^{∗}_{1}EG is the same as the set of homotopy classes of paths from p_{0} to p_{1}. When S = S^{1} the connected components of the space of maps correspond to the conjugacy classes in G: each bundle P_{g} corresponds to a specific point p_{g} in the mapping space, and a group element h defines a specific path from p_{g} to p_{hgh−1} .

G-equivariant topological field theories are examples of “homotopy topological field theories”. Using * Vladimir Turaev*‘s two main results: first, an attractive generalization of the theorem that a two-dimensional TFT “is” a commutative Frobenius algebra, and, secondly, a classification of the ways of gauging a given global G-symmetry of a semisimple TFT.

*Definition of the product in the G-equivariant closed theory. The heavy dot is the basepoint on S ^{1}. To specify the morphism unambiguously we must indicate consistent holonomies along a set of curves whose complement consists of simply connected pieces. These holonomies are always along paths between points where by definition the fibre is G. This means that the product is not commutative. We need to fix a convention for holonomies of a composition of curves, i.e., whether we are using left or right path-ordering. We will take h(γ_{1} ◦ γ_{2}) = h(γ_{1}) · h(γ_{2}).*

A G-equivariant TFT gives us for each element g ∈ G a vector space C_{g}, associated to the circle equipped with the bundle p_{g} whose holonomy is g. The usual pair-of-pants cobordism, equipped with the evident G-bundle which restricts to p_{g1} and p_{g2} on the two incoming circles, and to p_{g1g2} on the outgoing circle, induces a product

C_{g1} ⊗ C_{g2} → C_{g1g2} —– (1)

making C := ⊕_{g∈G}C_{g} into a G-graded algebra. Also there is a trace θ: C1 → C defined by the disk diagram with one ingoing circle. The holonomy around the boundary of the disk must be 1. Making the standard assumption that the cylinder corresponds to the unit operator we obtain a non-degenerate pairing

C_{g} ⊗ C_{g−1} → C

A new element in the equivariant theory is that G acts as an automorphism group on C. That is, there is a homomorphism α : G → Aut(C) such that

α_{h} : C_{g} → C_{hgh−1} —– (2)

Diagramatically, α_{h} is defined by the surface in the immediately above figure. Now let us note some properties of α. First, if φ ∈ C_{h} then α_{h}(φ) = φ. The reason for this is diagrammatically in the below figure.

*If the holonomy along path P _{2} is h then the holonomy along path P_{1} is 1. However, a Dehn twist around the inner circle maps P_{1} into P_{2}. Therefore, α_{h}(φ) = α_{1}(φ) = φ, if φ ∈ C_{h}.*

Next, while C is not commutative, it is “twisted-commutative” in the following sense. If φ_{1} ∈ C_{g1} and φ_{2} ∈ C_{g2} then

α_{g2} (φ_{1})φ_{2} = φ_{2}φ_{1} —– (3)

The necessity of this condition is illustrated in the figure below.

The trace of the identity map of C_{g} is the partition function of the theory on a torus with the bundle with holonomy (g,1). Cutting the torus the other way, we see that this is the trace of α_{g} on C_{1}. Similarly, by considering the torus with a bundle with holonomy (g,h), where g and h are two commuting elements of G, we see that the trace of α_{g} on C_{h} is the trace of α_{h} on C_{g−1}. But we need a strengthening of this property. Even when g and h do not commute we can form a bundle with holonomy (g,h) on a torus with one hole, around which the holonomy will be c = hgh^{−1}g^{−1}. We can cut this torus along either of its generating circles to get a cobordism operator from C_{c} ⊗ C_{h} to C_{h} or from C_{g−1} ⊗ C_{c} to C_{g−1}. If ψ ∈ C_{hgh−1g−1}. Let us introduce two linear transformations L_{ψ}, R_{ψ} associated to left- and right-multiplication by ψ. On the one hand, L_{ψ}α_{g} : φ ↦ ψ_{α}g(φ) is a map C_{h} → C_{h}. On the other hand R_{ψ}α_{h} : φ ↦ α_{h}(φ)ψ is a map C_{g−1} → C_{g−1}. The last sewing condition states that these two endomorphisms must have equal traces:

TrC_{h} L_{ψ}α_{g} = TrC_{g−1} R_{ψ}α_{h} —– (4)

(4) was taken by Turaev as one of his axioms. It can, however, be reexpressed in a way that we shall find more convenient. Let ∆_{g} ∈ C_{g} ⊗ C_{g−1} be the “duality” element corresponding to the identity cobordism of (S^{1},P_{g}) with both ends regarded as outgoing. We have ∆_{g} = ∑ξ_{i} ⊗ ξ^{i}, where ξ_{i} and ξ^{i} run through dual bases of C_{g} and C_{g−1}. Let us also write

∆_{h} = ∑η_{i} ⊗ η^{i} ∈ C_{h} ⊗ C_{h−1}. Then (4) is easily seen to be equivalent to

∑α_{h}(ξ_{i})ξ^{i} = ∑η_{i}α_{g}(η^{i}) —– (5)

in which both sides are elements of C_{hgh−1g−1}.

The “architecture of the Internet also lends itself to vulnerabilities and makes it more difficult to wiretap” on a manageable scale. Expanding surveillance programs like CALEA (Commission on Accreditation for Law Enforcement Agencies) to the Internet would consequently “require a different and more complicated protocol, which would create serious security problems.” Furthermore, because “[t]he Internet is easier to undermine than a telephone network due to its ‘flexibility and dynamism,'” incorporating means for surveying its use would “build security vulnerabilities into the communication protocols.” Attempts to add similar features in the past have “resulted in new, easily exploited security flaws rather than better law enforcement access.”

Moreover, Internet surveillance would likely cost a significant amount of money, much of which would be foisted upon online companies themselves. Consequently, not only would expanded surveillance lead to a “technology and security headache,” but the “hassles of implementation” and “the investigative burden and costs will shift to providers.”

When we have an open and closed Topological Field Theory (TFT) each element ξ of the closed algebra C defines an endomorphism ξ_{a} = i_{a}(ξ) ∈ O_{aa} of each object a of B, and η ◦ ξ_{a} = ξ_{b} ◦ η for each morphism η ∈ O_{ba} from a to b. The family {ξ_{a}} thus constitutes a natural transformation from the identity functor 1_{B} : B → B to itself.

For any C-linear category B we can consider the ring E of natural transformations of 1_{B}. It is automatically commutative, for if {ξ_{a}}, {η_{a}} ∈ E then ξ_{a} ◦ η_{a} = η_{a} ◦ ξ_{a} by the definition of naturality. (A natural transformation from 1_{B} to 1_{B} is a collection of elements {ξ_{a} ∈ O_{aa}} such that ξ_{a} ◦ f = f ◦ ξ_{b} for each morphism f ∈ O_{ab} from b to a. But we can take a = b and f = η_{a}.) If B is a Frobenius category then there is a map π_{a}^{b} : O_{bb} → O_{aa} for each pair of objects a, b, and we can define j^{b} : O_{bb} → E by j^{b}(η)_{a} = π_{a}^{b}(η) for η ∈ O_{bb}. In other words, j^{b} is defined so that the * Cardy condition* ι

θ_{a}(ι_{a}(ξ)η) = θ(ξj^{a}(η)) —– (1)

∀ ξ ∈ E and η ∈ O_{aa}. This is certainly true if B is a semisimple * Frobenius* category with finitely many simple objects, for then E is just the ring of complex-valued functions on the set of classes of these simple elements, and we can readily define θ : E → C by θ(ε

The commutative algebra E of natural endomorphisms of the identity functor of a linear category B is called the Hochschild cohomology HH^{0}(B) of B in degree 0. The groups HH^{p}(B) for p > 0, vanish if B is semisimple, but in the general case they appear to be relevant to the construction of a closed string algebra from B. For any Frobenius category B there is a natural homomorphism K(B) → HH^{0}(B) from the Grothendieck group of B, which assigns to an object a the transformation whose value on b is π_{b}^{a}(1_{a}) ∈ O_{bb}. In the semisimple case this homomorphism induces an isomorphism K(B) ⊗ C → HH^{0}(B).

For any additive category B the Hochschild cohomology is defined as the cohomology of the cochain complex in which a k-cochain F is a rule that to each composable k-tuple of morphisms

Y_{0} →^{φ1} Y_{1} →^{φ2} ··· →^{φk} Y_{k} —– (2)

assigns F(φ_{1},…,φ_{k}) ∈ Hom(Y_{0},Y_{k}). The differential in the complex is defined by

(dF)(φ_{1},…,φ_{k+1}) = F(φ_{2},…,φ_{k+1}) ◦ φ_{1} + ∑_{i=1}^{k}(−1)^{i} F(φ_{1},…,φ_{i+1} ◦ φ_{i},…,φ_{k+1}) + (−1)^{k+1}φ_{k+1} ◦ F(φ_{1},…,φ_{k}) —– (3)

(Notice, in particular, that a 0-cochain assigns an endomorphism F_{Y} to each object Y, and is a cocycle if the endomorphisms form a natural transformation. Similarly, a 2-cochain F gives a possible infinitesimal deformation F(φ_{1}, φ_{2}) of the composition law (φ_{1}, φ_{2}) ↦ φ_{2} ◦ φ_{1} of the category, and the deformation preserves the associativity of composition iff F is a cocycle.)

In the case of a category B with a single object whose algebra of endomorphisms is O the cohomology just described is usually called the Hochschild cohomology of the algebra O with coefficients in O regarded as a O-bimodule. This must be carefully distinguished from the Hochschild cohomology with coefficients in the dual O-bimodule O^{∗}. But if O is a Frobenius algebra it is isomorphic as a bimodule to O^{∗}, and the two notions of Hochschild cohomology need not be distinguished. The same applies to a Frobenius category B: because Hom(Y_{k}, Y_{0}) is the dual space of Hom(Y_{0}, Y_{k}) we can think of a k-cochain as a rule which associates to each composable k-tuple of morphisms a linear function of an element φ_{0} ∈ Hom(Y_{k}, Y_{0}). In other words, a k-cochain is a rule which to each “circle” of k + 1 morphisms

···→^{φ0} Y_{0} →^{φ1} Y1 →^{φ2}···→^{φk} Y_{k} →^{φ0}··· —– (4)

assigns a complex number F(φ_{0},φ_{1},…,φ_{k}).

If in this description we restrict ourselves to cochains which are cyclically invariant under rotating the circle of morphisms (φ_{0},φ_{1},…,φ_{k}) then we obtain a sub-cochain complex of the Hochschild complex whose cohomology is called the cyclic cohomology HC^{∗}(B) of the category B. The cyclic cohomology, which evidently maps to the Hochschild cohomology is a more natural candidate for the closed string algebra associated to B than is the Hochschild cohomology. A very natural Frobenius category on which to test these ideas is the category of holomorphic vector bundles on a compact Calabi-Yau manifold.

So far all astrophysical evidence supports the cosmological constant idea, but there is some wiggle room in the measurements. Upcoming experiments such as Europe’s * Euclid space telescope*, NASA’s

Quintessence is not the only other option. In the wake of Vafa’s papers, Ulf Danielsson, a physicist at Uppsala University and colleagues proposed * another way of fitting dark energy into string theory*. In their vision our universe is the three-dimensional surface of a bubble expanding within a larger-dimensional space. “The physics within this surface can mimic the physics of a cosmological constant,” Danielsson says. “This is a different way of realizing dark energy compared to what we’ve been thinking so far.”

The most general target category we can consider is a symmetric tensor category: clearly we need a tensor product, and the axiom H_{Y1⊔Y2} ≅ H_{Y1} ⊗ H_{Y2} only makes sense if there is an involutory canonical isomorphism H_{Y1} ⊗ H_{Y2} ≅ H_{Y2} ⊗ H_{Y1} .

A very common choice in physics is the category of super vector spaces, i.e., vector spaces V with a mod 2 grading V = V^{0} ⊕ V^{1}, where the canonical isomorphism V ⊗ W ≅ W ⊗ V is v ⊗ w ↦ (−1)^{deg v deg w}w ⊗ v. One can also consider the category of Z-graded vector spaces, with the same sign convention for the tensor product.

In either case the closed string algebra is a graded-commutative algebra C with a trace θ : C → C. In principle the trace should have degree zero, but in fact the commonly encountered theories have a grading anomaly which makes the trace have degree −n for some integer n.

We define topological-spin^{c} theories, which model 2d theories with N = 2 supersymmetry, by replacing “manifolds” with “manifolds with spin^{c} structure”.

A spin^{c} structure on a surface with a conformal structure is a pair of holomorphic line bundles L_{1}, L_{2} with an isomorphism L_{1} ⊗ L_{2} ≅ TΣ of holomorphic line bundles. A spin structure is the particular case when L_{1} = L_{2}. On a 1-manifold S a spin^{c} structure means a spin^{c} structure on a ribbon neighbourhood of S in a surface with conformal structure. An N = 2 superconformal theory assigns a vector space H_{S;L1,L2} to each 1-manifold S with spin^{c} structure, and an operator

U_{S0;L1,L2}: H_{S0;L1,L2} → H_{S1;L1,L2}

to each spin^{c}-cobordism from S_{0} to S_{1}. To explain the rest of the structure we need to define the N = 2 Lie superalgebra associated to a spin^{c }1-manifold (S;L_{1},L_{2}). Let G = Aut(L_{1}) denote the group of bundle isomorphisms L_{1} → L_{1} which cover diffeomorphisms of S. (We can identify this group with Aut(L_{2}).) It has a homomorphism onto the group Diff^{+}(S) of orientation-preserving diffeomorphisms of S, and the kernel is the group of fibrewise automorphisms of L_{1}, which can be identified with the group of smooth maps from S to C^{×}. The Lie algebra Lie(G) is therefore an extension of the Lie algebra Vect(S) of Diff^{+}(S) by the commutative Lie algebra Ω^{0}(S) of smooth real-valued functions on S. Let Λ^{0}_{S;L1,L2} denote the complex Lie algebra obtained from Lie(G) by complexifying Vect(S). This is the even part of a Lie super algebra whose odd part is Λ^{1}_{S;L1,L2} = Γ(L_{1}) ⊕ Γ(L_{2}). The bracket Λ^{1} ⊗ Λ^{1} → Λ^{0} is completely determined by the property that elements of Γ(L_{1}) and of Γ(L_{2}) anticommute among themselves, while the composite

Γ(L_{1}) ⊗ Γ(L_{2}) → Λ^{0} → Vect_{C}(S)

takes (λ_{1},λ_{2}) to λ_{1}λ_{2} ∈ Γ(TS).

In an N = 2 theory we require the superalgebra Λ(S;L_{1},L_{2}) to act on the vector space H_{S;L1,L2}, compatibly with the action of the group G, and with a similar intertwining property with the cobordism operators to that of the N = 1 case. For an N = 2 theory the state space always has an action of the circle group coming from its embedding in G as the group of fibrewise multiplications on L_{1} and L_{2}. Equivalently, the state space is always Z-graded.

An N = 2 theory always gives rise to two ordinary conformal field theories by equipping a surface Σ with the spin^{c} structures (C,TΣ) and (TΣ,C). These are called the “A-model” and the “B-model” associated to the N = 2 theory. In each case the state spaces are cochain complexes in which the differential is the action of the constant section of the trivial component of the spin^{c}-structure.

A spin structure on a surface means a double covering of its space of non-zero tangent vectors which is non-trivial on each individual tangent space. On an oriented 1-dimensional manifold S it means a double covering of the space of positively-oriented tangent vectors. For purposes of gluing, this is the same thing as a spin structure on a ribbon neighbourhood of S in an orientable surface. Each spin structure has an automorphism which interchanges its sheets, and this will induce an involution T on any vector space which is naturally associated to a 1-manifold with spin structure, giving the vector space a mod 2 grading by its ±1-eigenspaces. A topological-spin theory is a functor from the cobordism category of manifolds with spin structures to the category of super vector spaces with its graded tensor structure. The functor is required to take disjoint unions to super tensor products, and additionally it is required that the automorphism of the spin structure of a 1-manifold induces the grading automorphism T = (−1)^{degree} of the super vector space. This choice of the supersymmetry of the tensor product rather than the naive symmetry which ignores the grading is forced by the geometry of spin structures if the possibility of a semisimple category of boundary conditions is to be allowed. There are two non-isomorphic circles with spin structure: S^{1}_{ns}, with the * Möbius or “Neveu-Schwarz” structure, and S^{1}_{r}, with the trivial or “Ramond” structure*. A topological-spin theory gives us state spaces C

There are four cobordisms with spin structures which cover the standard annulus. The double covering can be identified with its incoming end times the interval [0,1], but then one has a binary choice when one identifies the outgoing end of the double covering over the annulus with the chosen structure on the outgoing boundary circle. In other words, alongside the cylinders A^{+}_{ns,r} = S^{1}_{ns,r} × [0,1] which induce the identity maps of C_{ns,r} there are also cylinders A^{−}_{ns,r} which connect S^{1}_{ns,r} to itself while interchanging the sheets. These cylinders A^{−}_{ns,r} induce the grading automorphism on the state spaces. But because A^{−}_{ns} ≅ A^{+}_{ns} by an isomorphism which is the identity on the boundary circles – the * Dehn twist* which “rotates one end of the cylinder by 2π” – the grading on C

There is a unique spin structure on the pair-of-pants cobordism in the figure below, which restricts to S^{1}_{ns} on each boundary circle, and it makes C_{ns} into a commutative * Frobenius algebra* in the usual way.

If one incoming circle is S^{1}_{ns} and the other is S^{1}_{r} then the outgoing circle is S^{1}_{r}, and there are two possible spin structures, but the one obtained by removing a disc from the cylinder A^{+}_{r} is preferred: it makes C_{r} into a graded module over C_{ns}. The chosen U-shaped cobordism P, with two incoming circles S^{1}_{r}, can be punctured to give us a pair of pants with an outgoing S^{1}_{ns}, and it induces a graded bilinear map C_{r} × C_{r} → C_{ns} which, composing with the trace on C_{ns}, gives a non-degenerate inner product on C_{r}. At this point the choice of symmetry of the tensor product becomes important. Let us consider the diffeomorphism of the pair of pants which shows us in the usual case that the Frobenius algebra is commutative. When we lift it to the spin structure, this diffeomorphism induces the identity on one incoming circle but reverses the sheets over the other incoming circle, and this proves that the cobordism must have the same output when we change the input from S(φ_{1} ⊗ φ_{2}) to T(φ_{1}) ⊗ φ_{2}, where T is the grading involution and S : C_{r} ⊗ C_{r} → C_{r} ⊗ C_{r} is the symmetry of the tensor category. If we take S to be the symmetry of the tensor category of vector spaces which ignores the grading, this shows that the product on the graded vector space C_{r} is graded-symmetric with the usual sign; but if S is the graded symmetry then we see that the product on C_{r} is symmetric in the naive sense.

There is an analogue for spin theories of the theorem which tells us that a two-dimensional topological field theory “is” a commutative Frobenius algebra. It asserts that a spin-topological theory “is” a Frobenius algebra C = (C_{ns} ⊕ C_{r},θ_{C}) with the following property. Let {φ_{k}} be a basis for C_{ns}, with dual basis {φ^{k}} such that θ_{C}(φ_{k}φ^{m}) = δ^{m}_{k}, and let β_{k} and β^{k} be similar dual bases for C_{r}. Then the Euler elements χ_{ns} := ∑ φ_{k}φ^{k} and χ_{r} = ∑ β_{k}β^{k} are independent of the choices of bases, and the condition we need on the algebra C is that χ_{ns} = χ_{r}. In particular, this condition implies that the vector spaces C_{ns} and C_{r} have the same dimension. In fact, the Euler elements can be obtained from cutting a hole out of the torus. There are actually four spin structures on the torus. The output state is necessarily in C_{ns}. The Euler elements for the three even spin structures are equal to χ_{e} = χ_{ns} = χ_{r}. The Euler element χ_{o} corresponding to the odd spin structure, on the other hand, is given by χ_{o} = ∑(−1)^{degβk}β_{k}β^{k}.

A spin theory is very similar to a Z/2-equivariant theory, which is the structure obtained when the surfaces are equipped with principal Z/2-bundles (i.e., double coverings) rather than spin structures.

It seems reasonable to call a spin theory semisimple if the algebra C_{ns} is semisimple, i.e., is the algebra of functions on a finite set X. Then C_{r} is the space of sections of a vector bundle E on X, and it follows from the condition χ_{ns} = χ_{r} that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra C_{ns} together with a binary choice at each point x ∈ X of the grading of the fibre E_{x} of the line bundle E at x.

We can now see that if we had not used the graded symmetry in defining the tensor category we should have forced the grading of C_{r} to be purely even. For on the odd part the inner product would have had to be skew, and that is impossible on a 1-dimensional space. And if both C_{ns} and C_{r} are purely even then the theory is in fact completely independent of the spin structures on the surfaces.

A concrete example of a two-dimensional topological-spin theory is given by C = C ⊕ C_{η} where η^{2} = 1 and η is odd. The Euler elements are χ_{e} = 1 and χ_{o} = −1. It follows that the partition function of a closed surface with spin structure is ±1 according as the spin structure is even or odd.

The most common theories defined on surfaces with spin structure are not topological: they are 2-dimensional conformal field theories with N = 1 supersymmetry. It should be noticed that if the theory is not topological then one does not expect the grading on C_{ns} to be purely even: states can change sign on rotation by 2π. If a surface Σ has a conformal structure then a double covering of the non-zero tangent vectors is the complement of the zero-section in a two-dimensional real vector bundle L on Σ which is called the spin bundle. The covering map then extends to a symmetric pairing of vector bundles L ⊗ L → TΣ which, if we regard L and TΣ as complex line bundles in the natural way, induces an isomorphism L ⊗_{C} L ≅ TΣ. An N = 1 * superconformal field theory* is a conformal-spin theory which assigns a vector space H

Γ(S,L) ⊗ H_{S,L} → H_{S,L}

(σ,ψ) ↦ G_{σ}ψ,

where Γ(S,L) is the space of smooth sections of L, such that G_{σ} is real-linear in the section σ, and satisfies G^{2}_{σ} = D_{σ2}, where D_{σ2} is the * Virasoro action* of the vector field σ

G_{σ1} ◦ U_{Σ,L} = U_{Σ,L} ◦ G_{σ0}

….

From * Lowenstein*‘s

The real culprit in 1994 was leverage. If you aren’t in debt, you can’t go broke and can’t be made to sell, in which case “liquidity” is irrelevant. but, a leveraged firm may be forced to sell, lest fast accumulating losses put it out of business. Leverage always gives rise to this same brutal dynamic, and its dangers cannot be stressed too often…

One of * LTCM*‘s first trades involved the thirty-year Treasury bond, which are issued by the US Government to finance the federal budget. Some $170 billion of them trade everyday, and are considered the least risky investments in the world. but a funny thing happens to thirty-year Treasurys six months or so after they are issued: they are kept in safes and drawers for long-term keeps. with fewer left in the circulation, the bonds become harder to trade. Meanwhile, the Treasury issues new thirty-year bond, which has its day in the sun. On Wall Street, the older bond, which has about 29-and-a-half years left to mature, is known as off the run; while the shiny new one is on the run. Being less liquid, the older one is considered less desirable, and begins to trade at a slight discount. And as arbitrageurs would say, a spread opens.

LTCM with its trademark precision calculated that owning one bond and shorting another was twenty-fifth as risky as owning either outright. Thus, it reckoned, it would prudently leverage this long/short arbitrage twenty-five times. This multiplied its potential for profit, but also its potential for loss. In any case, borrow it did. It paid for the cheaper off the run bonds with money it had borrowed from a Wall Street bank, or from several banks. And the other bonds, the ones it sold short, it obtained through a loan, as well. Actually, the transaction was more involved, though it was among the simplest in LTCM’s repertoire. No sooner than LTCM buy off the run bonds than it loaned them to some other Wall street firm, which then wired cash to LTCM as collateral. Then LTCM turned around and used this cash as a collateral on the bonds it borrowed. On Wall street, such short-term, collateralized loans are known as “repo financing”. The beauty of the trade was that LTCM’s cash transactions were in perfect balance. The money that LTCM spent going long matched the money that it collected going short. The collateral it paid equalled the collateral it collected. In other words, LTCM pulled off the entire transaction without using a single dime of its own cash. Maintaining the position wasn’t completely cost free, however. Though, a simple trade, it actually entailed four different payment streams. LTCM collected interest on the collateral it paid out and paid interest at a slightly higher-rate on the collateral it took in. It made some of this deficit back because of the difference in the initial margin, or the slightly higher coupon on the bond it owned as compared to the bond it shorted. This, overall cost a few basis points to LTCM each month.

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