Pricing Interest Rate Derivatives under Volatility Uncertainty
aa r X i v : . [ q -f i n . P R ] M a r Pricing Interest Rate Derivatives under VolatilityUncertainty
Julian H¨olzermann ∗ March 3, 2021
Abstract
We study the pricing of contracts in fixed income markets in the presence of volatil-ity uncertainty. The starting point is an arbitrage-free bond market under volatilityuncertainty. The uncertainty about the volatility is modeled by a G -Brownian mo-tion, which drives the forward rate dynamics. The absence of arbitrage is ensuredby a drift condition. Such a setting leads to a sublinear pricing measure for ad-ditional contracts, which yields either a single price or a range of prices. Similarto the forward measure approach, we define the forward sublinear expectation tosimplify the pricing of cashflows. Under the forward sublinear expectation, weobtain a robust version of the expectations hypothesis and we show how to pricebond options. In addition, we develop pricing methods for contracts consisting ofa stream of cashflows, since the nonlinearity of the pricing measure implies thatwe cannot price a stream of cashflows by pricing each cashflow separately. Withthese tools, we derive robust pricing formulas for all major interest rate derivatives.The pricing formulas provide a link to the pricing formulas of traditional modelswithout volatility uncertainty and show that volatility uncertainty naturally leadsto unspanned stochastic volatility. Keywords:
Robust Finance, Model Uncertainty, Fixed Income Markets
JEL Classification:
G12, G13
MSC2010:
The present paper deals with the pricing of interest rate derivatives in the presence ofvolatility uncertainty. Due to the assumption of a single, known probability measure, tra-ditional models in finance are subject to model uncertainty, since it is not always possibleto specify the probabilistic law of the model. Therefore, a new stream of research, called robust finance , emerged in the literature, examining financial markets in the presence of afamily of probability measures or none at all. The most frequently studied type of model ∗ Center for Mathematical Economics, Bielefeld University. Email: [email protected]. The author thanks Frank Riedel for valuable advice and Wolfgang Runggaldier and theparticipants of the “13th European Summer School in Financial Mathematics” in Vienna for interestingremarks. The author gratefully acknowledges financial support by the German Research Foundation(Deutsche Forschungsgemeinschaft) via Collaborative Research Center 1283. set of beliefs , consisting of all beliefs about the volatility. This framework naturally leadsto a sublinear expectation and a G -Brownian motion. A G -Brownian motion, which wasinvented by Peng [40], is basically a standard Brownian motion with a volatility that iscompletely uncertain but bounded by two extremes. We model the bond market in thespirit of Heath, Jarrow, and Morton [28] (HJM), that is, we model the instantaneousforward rate as a diffusion process, driven by a G -Brownian motion. The absence of arbi-trage is ensured by a suitable drift condition. Additionally, we assume that the diffusioncoefficient of the forward rate is deterministic, which results in analytical pricing formulas.This corresponds to an HJM model in which the foward rate is normal distributed.Such a setting leads to a sublinear pricing measure for additional contracts we add tothe bond market. Representing the uncertainty about the volatility by a family of prob-ability measures naturally leads to a sublinear expectation. For notational simplicity, wemodel the forward rate directly in a risk-neutral way. Thus, we refer to the sublinearexpectation as the risk-neutral sublinear expectation . We can use the risk-neutral sublin-ear expectation to determine prices of contracts, i.e., as a pricing measure. Due to thesublinearity, the pricing measure yields either a single price or a range of prices.To simplify the pricing of cashflows, we introduce the so-called forward sublinearexpectation . The forward sublinear expectation is defined by a G -backward stochasticdifferential equation and corresponds to the expectation under the forward measure. Theforward measure, invented by Geman [26], is used for pricing contracts in classical modelswithout volatility uncertainty [11, 27, 32]. Similar to the forward measure, the forwardsublinear expectation has the advantage that computing the sublinear expectation of adiscounted payoff reduces to computing the forward sublinear expectation of the payoff,discounted with the bond price. Under the forward sublinear expectation, we obtainseveral results needed for pricing cashflows of typical fixed income products. Moreover,we obtain a robust version of the expectations hypothesis under the forward sublinearexpectation and show how to price bond options. In many typical cases, prices of bondoptions are characterized by pricing formulas of models without volatility uncertainty.In addition, we develop pricing methods for contracts consisting of several cashflows.In traditional models without volatility uncertainty, there is no distinction between pric-ing single cashflows and pricing a stream of cashflows, since the pricing measure is linear.However, when there is uncertainty about the volatility, the nonlinearity of the pricingmeasure implies that we cannot generally price a stream of cashflows by pricing eachcashflow separately. Therefore, we provide different schemes for pricing a family of cash-flows. If the cashflows of a contract are sufficiently simple, we can price the contractas in the classical case. In general, we use a backward induction procedure to find theprice of a contract, which we can use to price a stream of bond options, for example.In typical situations, the price of the latter is determined by the pricing formulas frommodels without volatility uncertainty.With the tools mentioned above, we derive robust pricing formulas for all majorinterest rate derivatives. We consider typical linear contracts, such as fixed coupon bonds,floating rate notes, and interest rate swaps, and nonlinear contracts, such as swaptions,2aps and floors, and in-arrears contracts. Due to the linearity of the payoff, we obtaina single price for fixed coupon bonds, floating rate notes, and interest rate swaps. Thepricing formula is the same as the one from classical models without volatility uncertainty.Due to the nonlinearity of the payoff, we obtain a range of prices for swaptions, caps andfloors, and in-arrears contracts. The range is bounded from above, respectively below, bythe price from classical models with the highest, respectively lowest, possible volatility.Therefore, the pricing of common interest rate derivatives under volatility uncertaintyreduces to computing prices in models without volatility uncertainty.The pricing formulas show that volatility uncertainty naturally leads to unspannedstochastic volatility. According to empirical evidence, volatility risk in fixed incomemarkets cannot be hedged by trading solely bonds, which is termed unspanned stochasticvolatility . Collin-Dufresne and Goldstein [16] empirically showed that interest rate deriva-tives exposed to volatility risk are driven by factors that do not affect bond prices. Thesefindings contradict traditional term structure models. The empirical investigation hasled to the development of models that are able to exhibit unspanned stochastic volatility[14, 24, 25]. Since the presence of volatility uncertainty naturally leads to market in-completeness, the pricing formulas mentioned above show that it is no longer possible tohedge volatility risk in fixed income markets with a portfolio consisting solely of bondswhen there is uncertainty about the volatility. Moreover, the pricing formulas are in linewith the empirical findings of Collin-Dufresne and Goldstein [16].The literature on model uncertainty and, especially, volatility uncertainty in financialmarkets or, primarily, asset markets is very extensive. The first to apply the concept ofvolatility uncertainty to asset markets were Avellaneda, Levy, and Par´as [3] and Lyons[34]. Over a decade afterwards, the topic gained a lot of interest [21, 49]. The inter-esting fact about volatility uncertainty is that it is represented by a nondominated setof probability measures. Hence, traditional results from mathematical finance like thefundamental theorem of asset pricing break down. There are various attempts to extendthe theorem to a multiprior setting [6, 9, 10]. In some situations the theorem can be evenextended to a model-free setting, that is, without any reference measure at all [1, 12, 42].Most of these works also deal with the problem of pricing and hedging derivatives in thepresence of model uncertainty. The topic has been studied separately in the presence ofvolatility uncertainty [49], in the presence of a general set of priors [2, 13, 41], and in amodel-free setting [5, 7, 43]. The most similar setting is the one of Vorbrink [49], since itfocuses on volatility uncertainty modeled by a G -Brownian motion. However, the focus,as in most of the studies from above, lies on asset markets.In addition, there is an increasing number of articles dealing with interest rate modelsor related credit risk under model uncertainty [4, 8, 20, 22, 23, 29, 30]. Among these, thereare also articles focusing on volatility uncertainty in interest rate models [4, 22, 29, 30].The only one working in a general HJM framework is a companion paper [30]. Theremaining articles on model uncertainty in interest rate models either correspond to shortrate models or do not study volatility uncertainty. The main result of the accompanyingarticle [30] is a drift condition, which shows how to obtain an arbitrage-free term structurein the presence of volatility uncertainty. Starting from an arbitrage-free term structure,the aim of the present paper is to study the pricing of derivatives in fixed income marketsunder volatility uncertainty.There are several ways to describe volatility uncertainty from a mathematical point ofview. The classical approach is the one of Denis and Martini [19] and Peng [40]. Actually,these are two different approaches, but they are equivalent as it was shown by Denis, Hu,3nd Peng [18]. The difference is that Denis and Martini [19] start from a probabilisticsetting, whereas the calculus of G -Brownian motion from Peng [40] relies on nonlinearpartial differential equations. Moreover, there are various extensions and generalizations[36, 37, 38]. Additional results and a different approach to volatility uncertainty weredeveloped by Soner, Touzi, and Zhang [44, 45, 46, 47]. They also related the topic tosecond-order backward stochastic differential equations, introduced by Cheridito, Soner,Touzi, and Victoir [15]. In addition, there are many attempts to a pathwise stochasticcalculus, which works without any reference measure [17, and references therein]. In thispaper, we use the calculus of G -Brownian motion, since the literature on G -Brownianmotion contains a lot of results. In particular, the results of Hu, Ji, Peng, and Song [31]are of fundamental importance for the results derived in this paper.The remainder of this paper is organized as follows. Section 2 introduces the overallsetting of the model: an arbitrage-free bond market under volatility uncertainty. InSection 3, we show that we can use the risk-neutral sublinear expectation as a pricingmeasure. In Section 4, we define the forward sublinear expectation and discuss relatedresults for the pricing of single cashflows. Section 5 provides schemes for pricing contractsconsisting of a stream of cashflows. In Section 6, we derive pricing formulas for the mostcommon interest rate derivatives. In Section 7, we discuss market incompleteness andshow that volatility uncertainty leads to unspanned stochastic volatility. Section 8 givesa conclusion. The proofs of Section 3, Section 4, and Section 5 are given in Section A,Section B, and Section C of the appendix, respectively. We represent the uncertainty about the volatility by a familiy of probability measures suchthat each measure corresponds to a specific belief about the volatility. Let us considera probability space (Ω , F , P ) such that the canonical process B = ( B t , ..., B dt ) t ≥ is a d -dimensional standard Brownian motion under P . Furthermore, let F = ( F t ) t ≥ be thefiltration generated by B completed by all P -null sets. The state space of the uncertainvolatility is given byΣ := (cid:8) σ ∈ R d × d (cid:12)(cid:12) σ = diag( σ , ..., σ d ) , σ i ∈ [ σ i , σ i ] (cid:9) , where σ i ≥ σ i > i = 1 , ..., d . That means, we consider all scenarios in whichthere is no correlation and the volatility is bounded by two extremes: σ = diag( σ , ..., σ d )and σ = diag( σ , ..., σ d ). For each Σ-valued, F -adapted process σ = ( σ t ) t ≥ , we define theprocess B σ = ( B σt ) t ≥ by B σt := Z t σ u dB u and the measure P σ to be the probabilistic law of the process B σ , that is, P σ := P ◦ ( B σ ) − . We denote the collection of all such measures by P , which is termed the set of beliefs ,since it contains all beliefs about the volatility. Now the canonical process has a differentvolatility under each measure in the set of beliefs.4olatility uncertainty naturally leads to a G -expectation and a G -Brownian motion.If we define the sublinear expectation ˆ E byˆ E [ ξ ] := sup P ∈P E P [ ξ ]for all random variables ξ such that E P [ ξ ] exists for all P ∈ P , then ˆ E corresponds to the G -expectation on L G (Ω) and B is a G -Brownian motion under ˆ E [18, Theorem 54]. Theletter G refers to the sublinear function G : S d → R , defined by G ( A ) := sup σ ∈ Σ tr( σσ ′ A ) , where S d is the space of all symmetric d × d matrices and · ′ denotes the transpose of amatrix. G is the generator of the nonlinear partial differential equation which defines the G -expectation and characterizes the distribution and the uncertainty of a G -Brownianmotion. L G (Ω) is the space of random variables for which the G -expectation is defined.We identify random variables in L G (Ω) if they are equal quasi-surely , that is, P -almostsurely for all P ∈ P . For further details, the reader may refer to the book of Peng [40].We model the forward rate as a diffusion process in the spirit of Heath, Jarrow, andMorton [28] (HJM). We denote by f t ( T ) the forward rate with maturity T at time t for t ≤ T ≤ τ , where τ < ∞ is a fixed terminal time. We assume that the forward rateprocess f ( T ) = ( f t ( T )) ≤ t ≤ T , for all T ≤ τ , evolves according to the dynamics f t ( T ) = f ( T ) + Z t α u ( T ) du + d X i =1 Z t β iu ( T ) dB iu + d X i =1 Z t γ iu ( T ) d h B i i u for some initial integrable forward curve f : [0 , τ ] → R and sufficiently regular processes α ( T ) = ( α t ( T )) ≤ t ≤ τ , β i ( T ) = ( β it ( T )) ≤ t ≤ τ , and γ i ( T ) = ( γ it ( T )) ≤ t ≤ τ to be specified.The difference compared to the classical HJM model without volatility uncertainty isthat there are additional drift terms depending on the quadratic variation of the G -Brownian motion. We need the additional drift terms in order to obtain an arbitrage-freemodel as it is described below. However, due to the uncertainty about the volatility, thequadratic variation is an uncertain process, which cannot be included in the first driftterm. Thus, we add additional drift terms to the dynamics of the forward rate. Moredetails on this can be found in a companion paper [30, Section 2].The forward rate determines the remaining quantities on the bond market. Thebond market consists of zero-coupon bonds for all maturities in the time horizon andthe money-market account. The zero-coupon bonds P ( T ) = ( P t ( T )) ≤ t ≤ T for T ≤ τ aredefined by P t ( T ) := exp (cid:16) − Z Tt f t ( s ) ds (cid:17) and the money-market account M = ( M t ) ≤ t ≤ τ is given by M t := exp (cid:16) Z t r s ds (cid:17) , where r = ( r t ) ≤ t ≤ τ denotes the short rate process, defined by r t := f t ( t ). We usethe money-market account as a num´eraire, that is, we focus on the discounted bonds˜ P ( T ) = ( ˜ P t ( T )) ≤ t ≤ T for T ≤ τ , given by˜ P t ( T ) := M − t P t ( T ) .
5e model the forward rate in such a way that the related bond market is arbitrage-free. That means, we assume that the forward rate satisfies a suitable drift condition,which implies the absence of arbitrage. In particular, we directly model the forward ratein a risk-neutral way in order to avoid technical difficulties due to a migration to a risk-neutral framework. More specifically, we assume that the drift terms of the forward rateare defined by α t ( T ) := 0 ,γ it ( T ) := β it ( T ) b it ( T )for all i , where the process b i ( T ) = ( b it ( T )) ≤ t ≤ τ , for all i , is defined by b it ( T ) := Z Tt β it ( s ) ds. Under suitable regularity assumptions on T β i ( T ), we can then show that the dis-counted bonds are symmetric G -martingales under ˆ E , which implies that the bond mar-ket is arbitrage-free [30, Theorem 3.1]. Therefore, we call ˆ E the risk-neutral sublinearexpectation . This shows that we need the additional drift terms in the forward rate dy-namics to obtain an arbitrage-free model. Furthermore, we assume that β i , for all i , isa continuous function, mapping from [0 , τ ] × [0 , τ ] into R . Then, for each T , β i ( T ) and b i ( T ), for all i , are bounded processes in M pG (0 , τ ) for all p < ∞ . M pG (0 , τ ) is the spaceof admissible integrands for stochastic integrals related to a G -Brownian motion. Theassumption ensures that the forward rate is sufficiently regular to apply the result fromabove. In addition, it enables us to obtain specific pricing formulas for common interestrate derivatives. This is similar to the classical case without volatility uncertainty, inwhich we obtain analytical pricing formulas by assuming that the diffusion coefficientis deterministic. So the present model corresponds to an HJM model with a normaldistributed forward rate. Now we extend the bond market to an additional contract for which we want to find aprice. A typical contract in fixed income markets consists of a stream of cashflows. Sowe consider a contract X which has a payoff of ξ i at each time τ i for all i = 0 , , ..., N ,where 0 < τ < τ < ... < τ N = τ is the tenor structure. The price at time t of sucha contract is denoted by X t for all t ≤ τ . As for the bonds, we consider the discountedpayoff ˜ X , defined by ˜ X := N X i =0 M − τ i ξ i , and the discounted price ˜ X t for t ≤ τ , which is defined by˜ X t := M − t X t . We assume that M − τ i ξ i ∈ L G (Ω τ i ) for all i = 0 , , ..., N for X to be regular enough.The pricing of contracts in the presence of volatility uncertainty differs from the tra-ditional approach. Classical arbitrage pricing theory suggests that prices are determined6y computing the expected discounted payoff under the risk-neutral measure. The im-portant difference in the case of volatility uncertainty is that the risk-neutral sublinearexpectation is nonlinear. In particular, this impliesˆ E [ ˜ X ] ≥ − ˆ E [ − ˜ X ] , (3.1)that is, the upper expectation does not necessarily coincide with the lower expectation.Thus, we distinguish between symmetric and asymmetric contracts. We consider twocontracts: a contract X S , which has a symmetric payoff, and a contract X A , which hasan asymmetric payoff. Strictly speaking, this means that ˜ X S satisfies (3.1) with equalityand for ˜ X A the inequality (3.1) is strict. Of course, the discounted payoffs ˜ X S and ˜ X A are defined as above by considering different payoffs ξ Si and ξ Ai for all i , respectively. Therelated prices are denoted by X St and ˜ X St and X At and ˜ X At for all t , respectively.We determine the prices of contracts by using the risk-neutral sublinear expectationto either obtain the price of a contract or the upper and the lower bound for the price.In the case of a symmetric payoff, we proceed as in the classical case without volatilityuncertainty and choose the expected discounted payoff as the price for the contract. Inthe case of an asymmetric payoff, we use the upper and the lower expectation as boundsfor the price, which is a typical approach in the literature on model uncertainty. Hence,we assume that ˜ X St = ˆ E t [ ˜ X S ]for all t , where ˆ E t denotes the conditional G -expectation, andˆ E [ ˜ X A ] > ˜ X A > − ˆ E [ − ˜ X A ] . Since X S has a symmetric payoff, by the martingale representation theorem for sym-metric G -martingales [48, Theorem 4.8], there exists a process H = ( H t , ..., H dt ) ≤ t ≤ τ in M G (0 , τ ; R d ) such that, for all t ,˜ X St = ˜ X S + d X i =1 Z t H iu dB iu . The reason why we only impose assumptions on the price of the asymmetric contract attime 0 is described below.In order to show that this pricing procedure yields no-arbitrage prices, we introducethe notion of trading strategies related to the extended bond market and a suitable notionof arbitrage. We allow the agents in the market to trade a finite number of bonds. Thesymmetric contract can be traded dynamically, but we only allow static trading strategiesfor the asymmetric contract. Therefore, we do not impose assumptions on ˜ X At for t > Definition 3.1.
An admissible market strategy is a quadruple ( π, π S , π A , T ) consisting ofa bounded process π = ( π t , ..., π nt ) ≤ t ≤ τ in M G (0 , τ ; R n ) , a bounded process π S = ( π St ) ≤ t ≤ τ in M G (0 , τ ) , a constant π A ∈ R , and a vector T = ( T , ..., T n ) ∈ [0 , τ ] n for some n ∈ N .The corresponding portfolio value at terminal time is given by ˜ v τ ( π, π S , π A , T ) := n X i =1 Z T i π it d ˜ P t ( T i ) + Z τ π St d ˜ X St + π A ( ˜ X A − ˜ X A ) . (3.2)7he three terms on the right-hand side of (3.2) correspond to the gains from trading a fi-nite number of bonds, the symmetric contract, and the asymmetric contract, respectively.The assumptions on the processes ensure that the integrals in (3.2) are well-defined. Inaddition, we use the quasi-sure definition of arbitrage, which is commonly used in theliterature on model uncertainty [9, 10, 49]. Definition 3.2.
An admissible market strategy ( π, π S , π A , T ) is an arbitrage strategy if ˜ v τ ( π, π S , π A , T ) ≥ quasi-surely ,P (cid:0) ˜ v τ ( π, π S , π A , T ) > (cid:1) > for at least one P ∈ P . We say that the extended bond market is arbitrage-free if there is no arbitrage strategy.
The following proposition shows that we can use the risk-neutral sublinear expectationas a pricing measure as described above. Therefore, we can reduce the problem of pricingderivatives to evaluating the upper and the lower expectation of the discounted payoff.
Proposition 3.1.
The extended bond market is arbitrage-free.
The proof is based on the following observations. The bond market itself is arbitrage-freedue to the assumptions stated in the previous section. Adding an additional symmetric G -martingale to the market does not change the result. Thus, the market can be enlargedby the symmetric contract without admitting arbitrage opportunities. In general, theliterature on model uncertainty shows that prices bounded by the upper and the lowerexpectation do not admit arbitrage. Hence, we can also add the asymmetric contract tothe market without admitting arbitrage, since its price is bounded by the upper and thelower expectation of its payoff. In the classical case without volatility uncertainty, discounted cashflows are priced underthe forward measure. Evaluating the expectation of a discounted cashflow related to aninterest rate derivative can be very elaborate. This is due to the fact that the discountfactor, in addition to the cashflows, is stochastic. The common way to avoid this issue isthe forward measure approach. The forward measure, which was introduced by Geman[26], is equivalent to the pricing measure and defined by choosing a particular densityprocess. The density process is defined in such a way that the expectation of a discountedcashflow under the risk-neutral measure can be rewritten as the expectation of the cash-flow under the forward measure, discounted by a zero-coupon bond. Thus, by changingthe measure, we can replace the stochastic discount factor by the current bond price,which is already determined by the model.In the presence of volatility uncertainty, we define the so-called forward sublinearexpectation to simplify the pricing of discounted cashflows. In contrast to the forwardmeasure approach, we define the forward sublinear expectation by a G -backward stochas-tic differential equation. Definition 4.1.
For ξ ∈ L pG (Ω T ) with p > and T ≤ τ , we define the T -forward sub-linear expectation ˆ E T by ˆ E Tt [ ξ ] := Y T,ξt , where Y T,ξ = ( Y T,ξt ) ≤ t ≤ T solves the G -backwardstochastic differential equation Y T,ξt = ξ − d X i =1 Z Tt b iu ( T ) Z iu d h B i i u − d X i =1 Z Tt Z iu dB iu − ( K T − K t ) .
8y Theorem 5.1 of Hu, Ji, Peng, and Song [31], the forward sublinear expectation is atime consistent sublinear expectation. We refer to the paper of Hu, Ji, Peng, and Song[31] for further details related to G -backward stochastic differential equations.The forward sublinear expectation corresponds to the expectation under the forwardmeasure. This can be deduced from the explicit solution to the G -backward stochasticdifferential equation defining the forward sublinear expectation. For T ≤ τ , we definethe process X T = ( X Tt ) ≤ t ≤ T by X Tt := ˜ P t ( T ) P ( T ) . The process X T is the density used to define the forward measure. One can verify that X T satisfies the G -stochastic differential equation X Tt = 1 − d X i =1 Z t b iu ( T ) X Tu dB iu [30, Proposition 3.1]. By Theorem 3.2 of Hu, Ji, Peng, and Song [31], Y T,ξ is given by Y T,ξt = ( X Tt ) − ˆ E t [ X TT ξ ] . Thus, we basically arrive at the same expression as in the classical definition of theforward measure.We obtain the following preliminary results related to the forward sublinear expecta-tion, which simplify the pricing of discounted cashflows. Similar to the classical case, wefind that the valuation of a discounted cashflow reduces to determining the forward sub-linear expectation of the cashflow, which is then discounted with the bond price. Further-more, there is a relation between forward sublinear expectations with different maturitiesand the forward rate process f ( T ) and the forward price process X T,T ′ = ( X T,T ′ t ) ≤ t ≤ T ∧ T ′ ,defined by X T,T ′ t := P t ( T ′ ) P t ( T ) , for T, T ′ ≤ τ , are symmetric G -martingales under the T -forward sublinear expectation. Proposition 4.1.
Let ξ ∈ L pG (Ω T ) with p > and t ≤ T, T ′ ≤ τ . Then we have(i) it holds M t ˆ E t [ M − T ξ ] = P t ( T ) ˆ E Tt [ ξ ] , (ii) for T ≤ T ′ , it holds P t ( T ′ ) ˆ E T ′ t [ ξ ] = P t ( T ) ˆ E Tt [ P T ( T ′ ) ξ ] , (iii) the forward rate process f ( T ) is a symmetric G -martingale under ˆ E T ,(iv) the forward price process X T,T ′ satisfies X T,T ′ t ∈ L pG (Ω t ) for all p < ∞ and X T,T ′ t = X T,T ′ − d X i =1 Z t σ iu ( T, T ′ ) X T,T ′ u dB iu − d X i =1 Z t σ iu ( T, T ′ ) X T,T ′ u b iu ( T ) d h B i i u , where σ i ( T, T ′ ) = ( σ it ( T, T ′ )) ≤ t ≤ T ∧ T ′ , for all i , is defined by σ it ( T, T ′ ) := b it ( T ′ ) − b it ( T ) , and is a symmetric G -martingale under ˆ E T . iii ), we obtain a robust version of the expectations hypothesis.The traditional expectations hypothesis states that forward rates reflect the expectationof future short rates. In the classical case without volatility uncertainty, we know thatthe forward rate is a martingale under the forward measure. Therefore, the expectationshypothesis holds true under the forward measure. In our case, we obtain a much strongerversion, called robust expectations hypothesis . This is because the forward rate is a sym-metric G -martingale under the forward sublinear expectation. Thus, the forward ratereflects the upper expectation of the short rate and the lower expectation of the shortrate. In particular, it implies that the forward rate reflects the expectation of the shortrate in each possible scenario for the volatility. Corollary 4.1.
The forward rate satisfies the robust expectations hypothesis under theforward sublinear expectation, that is, for t ≤ T ≤ τ , it holds ˆ E Tt [ r T ] = f t ( T ) = − ˆ E Tt [ − r T ] . For convex bond options, the upper, respectively lower, bound for the price is givenby the price in the corresponding HJM model without volatility uncertainty with thehighest, respectively lowest, possible volatility. If we consider a bond option, the payoffis a function depending on a selection of bond prices for different maturities. We considerthe more general case when the payoff is a function depending on a selection of forwardprices, since we can express every bond option as an option on forward prices. If the payofffunction is convex and satisfies a suitable growth condition, we can use the nonlinearFeynman-Kac formula from Hu, Ji, Peng, and Song [31] to show that the range of pricesis bounded from above, respectively below, by the price from the classical model when thedynamics of the forward price are driven by a standard Brownian motion with constantvolatility σ , respectively σ . Proposition 4.2.
For n ∈ N , let ϕ : R n → R be a convex function such that | ϕ ( x ) − ϕ ( y ) | ≤ C (1 + | x | m + | y | m ) | x − y | (4.1) for a positive integer m and a constant C > and let < t < ... < t n ≤ τ . Then ˆ E Tt [ ϕ ( X T,t t , ..., X T,t n t )] = u σ ( t, X T,t t , ..., X T,t n t ) , − ˆ E Tt [ − ϕ ( X T,t t , ..., X T,t n t )] = u σ ( t, X T,t t , ..., X T,t n t ) for t ≤ t ≤ T ≤ τ , where the function u σ : [0 , t ] × R n → R , for σ ∈ Σ , is defined by u σ ( t, x , ..., x n ) := E P [ ϕ ( X t , ..., X nt )] and the process X i = ( X is ) t ≤ s ≤ t , for all i = 1 , ..., n , is given by X is = x i − d X j =1 Z st σ ju ( T, t i ) X iu σ j dB ju . For bond options that are neither convex nor concave, we generally need to use nu-merical procedures to obtain the pricing bounds. If we deal with a bond option having aconcave (instead of a convex) payoff function, we can use the same approach as in Propo-sition 4.2 to find the pricing bounds by simply interchanging σ and σ . The convexity10r the concavity of the payoff function reduces the nonlinear partial differential equationwhich results from the nonlinear Feynman-Kac formula of Hu, Ji, Peng, and Song [31] anddetermines the pricing bounds to a linear partial differential equation. Then the pricingbounds coincide with the price of traditional models when the underlying is driven by astandard Brownian motion with volatility σ and σ , respectively. When the payoff func-tion is neither convex nor concave, we can still use the nonlinear Feynman-Kac formulato obtain the pricing bounds, but then we need to solve the nonlinear partial differentialequation, since it does not reduce to a linear one. One can find the solution, for example,by using numerical schemes similar to the ones of Nendel [35, Section 5]. Due to the nonlinearity of the pricing measure, in general, we cannot price interest ratederivatives by pricing each cashflow separately. As in Section 3, we consider a contract X consisting of a stream of cashflows. Then the discounted payoff is given by˜ X = N X i =0 M − T i ξ i for a tenor structure 0 < T < T < ... < T N = τ and ξ i ∈ L pG (Ω T i ) with p > i . In order to price the contract, we are interested in ˆ E [ ˜ X ] and − ˆ E [ − ˜ X ]. When thereis no volatility uncertainty, we can simply price the contract by pricing each cashflowindividually, since the pricing measure is linear in that case. However, in the presence ofvolatility uncertainty, the pricing measure ˆ E is sublinear, which impliesˆ E [ ˜ X ] ≤ N X i =0 ˆ E [ M − T i ξ i ] , − ˆ E [ − ˜ X ] ≥ N X i =0 − ˆ E [ − M − T i ξ i ] . Therefore, if we price each cashflow separately, we possibly only obtain an upper, respec-tively lower, bound for the upper, respectively lower, bound of the price, which does notyield much information about the price of the contract.For contracts with symmetric cashflows, we can still determine the price of the contractby pricing each of its cashflows individually. If each cashflow has a symmetric payoff underthe forward sublinear expectation, there is a single price for the contract, which coincideswith the sum of the prices of the cashflows. Hence, the pricing measure is linear on thesubspace of contracts with symmetric cashflows.
Lemma 5.1. If ξ i , for all i , satisfies ˆ E T i t [ ξ i ] = − ˆ E T i t [ − ξ i ] for t ≤ T , then it holds M t ˆ E t [ ˜ X ] = N X i =0 P t ( T i ) ˆ E T i t [ ξ i ] = − M t ˆ E t [ − ˜ X ] . For general contracts, we can use a backward induction procedure to obtain the upperand the lower expectation of the discounted payoff. The procedure works as follows.First, we compute the forward sublinear expectation of the last cashflow conditioned on11he second last payoff time and discount it with the bond price. Next, we compute theforward sublinear expectation of the second last cashflow and the previous expressionconditioned on the third last payoff time and discount it with the bond price. Thenwe recursively repeat this procedure until we arrive at the first payoff. This gives useventually the upper expectation of the discounted payoff. The procedure for the lowerexpectation is similar.
Lemma 5.2.
It holds ˆ E [ ˜ X ] = ˜ Y and − ˆ E [ − ˜ X ] = − ˜ Z , where ˜ Y i and ˜ Z i are defined by ˜ Y i := P T i − ( T i ) ˆ E T i T i − [ ξ i + ˜ Y i +1 ] , ˜ Z i := P T i − ( T i ) ˆ E T i T i − [ − ξ i + ˜ Z i +1 ] , respectively, for all i = 0 , , ..., N and T − := 0 and ˜ Y N +1 := 0 and ˜ Z N +1 := 0 . If the contract can be written as a stream of convex bond options, the upper, respec-tively lower, bound for the price is given by the price from the classical model withoutvolatility uncertainty with the highest, respectively lowest, possible volatility. Similarto Proposition 4.2, if the cashflows can be written as convex functions of forward pricessatisfying a suitable growth condition, we can show that the upper, respectively lower,expectation of the discounted payoff is given by its linear expectation when the dynamicsof the forward price are driven by a standard Brownian motion with constant volatility σ , respectively σ . We show this by using the backward induction procedure of Lemma5.2 and recursively applying the nonlinear Feynman-Kac formula of Hu, Ji, Peng, andSong [31]. Proposition 5.1.
For m, n ∈ N such that m = n , let ¯ Y i and ¯ Z i be defined by ¯ Y i := X t i − n ,t i + n t i ˆ E t i + n t i [ ϕ i ( X t i + n ,t i + m t i +1 ) + ¯ Y i +1 ] , ¯ Z i := X t i − n ,t i + n t i ˆ E t i + n t i [ − ϕ i ( X t i + n ,t i + m t i +1 ) + ¯ Z i +1 ] , respectively, for all i = 1 , ..., N , where ϕ i : R → R is a convex function such that (4.1) holds and t < ... < t N +( m ∨ n ) ≤ τ , and ¯ Y N +1 := 0 and ¯ Z N +1 := 0 . Then ¯ Y = N X i =1 X t n ,t i + n u iσ (0 , X t i + n ,t i + m ) , − ¯ Z = N X i =1 X t n ,t i + n u iσ (0 , X t i + n ,t i + m ) , where the function u iσ : [0 , t i +1 ] × R → R , for all i = 1 , ..., N and σ ∈ Σ , is defined by u iσ ( t, x i ) := E P [ ϕ i ( X it i +1 )] and the process X i = ( X is ) t ≤ s ≤ t i +1 is given by X is = x i − d X j =1 Z st σ ju ( t i + n , t i + m ) X iu σ j dB ju .
12f the stream of cashflows consists of bond options that are neither convex nor con-cave, we need to use a numerical scheme to apply the backward induction procedure fromLemma 5.2. As in the previous section, we can price a stream of concave bond optionsin the same way as in Proposition 5.1 by interchanging σ and σ . However, when thebond options are neither convex nor concave, we can use the general backward inductionprocedure from Lemma 5.2 and recursively solve the nonlinear partial differential equa-tions arising due to the nonlinear Feynman-Kac formula of Hu, Ji, Peng, and Song [31]by numerical procedures as mentioned at the end of Section 4. With the tools from the preceeding sections we can price all major derivatives traded infixed income markets. We consider typical linear contracts, such as fixed coupon bonds,floating rate notes, and interest rate swaps, and nonlinear contracts, such as swaptions,caps and floors, and in-arrears contracts. Using the general pricing techniques for wholecontracts from Section 5 and the valuation methods for single cashflows from Section 4,we show how to derive robust pricing formulas for all these contracts. That means, weconsider a contract with discounted payoff˜ X = N X i =0 M − T i ξ i for 0 < T < T < ... < T N = τ and specifically given cashflows and then we show howto find ˆ E [ ˜ X ] and − ˆ E [ − ˜ X ] or M t ˆ E t [ ˜ X ] and − M t ˆ E t [ − ˜ X ] for t ≤ T if the contract has asymmetric payoff. We can price fixed coupon bonds as in the classical case without volatility uncertainty.A fixed coupon bond is a contract which pays a fixed rate of interest
K > T i for all i = 1 , ..., N and thenominal value at the last payment date T N . Hence, the cashflows are given by ξ i = 1 { N } ( i ) + 1 { ,...,N } ( i )( T i − T i − ) K (6.1)for all i = 0 , , ..., N . Due to its simple payoff structure, the contract has a symmetricpayoff and its price is given by the same expression as the one obtained in traditionalterm structure models. Proposition 6.1.
Let ξ i be given by (6.1) for all i = 0 , , ..., N . Then, for t ≤ T , M t ˆ E t [ ˜ X ] = P t ( T N ) + N X i =1 P t ( T i )( T i − T i − ) K = − M t ˆ E t [ − ˜ X ] . Proof.
Since the cashflows are constants, the assertion follows by Lemma 5.1.13 .2 Floating Rate Notes
We can also price floating rate notes as in the classical case without volatility uncertainty.A floating rate note is a fixed coupon bond in which the fixed rate K , at each paymentdate T i for all i = 1 , ..., N , is replaced by a floating rate: the simply compounded spotrate L T i − ( T i ). For t ≤ T ≤ τ , the simply compounded spot rate with maturity T at time t is defined by L t ( T ) := T − t ( P t ( T ) − . The cashflows are then given by ξ i = 1 { N } ( i ) + 1 { ,...,N } ( i )( T i − T i − ) L T i − ( T i ) (6.2)for all i = 0 , , ..., N . Although the cashflows are not constant, the contract yet has asymmetric payoff. As in the classical case, the price is simply given by the price of azero-coupon bond with maturity T . Proposition 6.2.
Let ξ i be given by (6.2) for all i = 0 , , ..., N . Then, for t ≤ T , M t ˆ E t [ ˜ X ] = P t ( T ) = − M t ˆ E t [ − ˜ X ] . Proof.
We show that the cashflows have a symmetric payoff and apply Lemma 5.1. Dueto Proposition 4.1 ( ii ) and ( iv ), we have P t ( T i ) ˆ E T i t [( T i − T i − ) L T i − ( T i )] = P t ( T i − ) ˆ E T i − t [1 − P T i − ( T i )] = P t ( T i − ) − P t ( T i )for all i = 1 , ..., N . In a similar fashion we can show that − P t ( T i ) ˆ E T i t [ − ( T i − T i − ) L T i − ( T i )] = P t ( T i − ) − P t ( T i )for all i = 1 , ..., N . The result follows by Lemma 5.1 and summation. The pricing formula for interest rate swaps is the same as in traditional models. Aninterest rate swap exchanges the floating rate L T i − ( T i ) with a fixed rate K at eachpayment date T i for all i = 1 , ..., N . Without loss of generality we consider a payerinterest rate swap, that is, we pay the fixed rate K and receive the floating rate L T i − ( T i ).Hence, the cashflows are given by ξ i = 1 { ,...,N } ( i )( T i − T i − ) (cid:0) L T i − ( T i ) − K (cid:1) (6.3)for all i = 0 , , ..., N . Since the payoff is the difference of a zero-coupon bond and afloating rate note, the contract is symmetric. As in traditional term structure models,the price is given by a linear combination of zero-coupon bonds with different maturities.In particular, this implies that the swap rate, i.e., the fixed rate K which makes thevalue of the contract zero, is uniquely determined and does not differ from the expressionobtained by standard models. Proposition 6.3.
Let ξ i be given by (6.3) for all i = 0 , , ..., N . Then, for t ≤ T , M t ˆ E t [ ˜ X ] = P t ( T ) − P t ( T N ) − N X i =1 P t ( T i )( T i − T i − ) K = − M t ˆ E t [ − ˜ X ] . roof. Again, we show that the cashflows have a symmetric payoff and use Lemma 5.1to obtain the result. As in the proof of Proposition 6.2, we can show that P t ( T i ) ˆ E T i t (cid:2) ( T i − T i − ) (cid:0) L T i − ( T i ) − K (cid:1)(cid:3) = P t ( T i − ) − P t ( T i ) − P t ( T i )( T i − T i − ) K, − P t ( T i ) ˆ E T i t (cid:2) − ( T i − T i − ) (cid:0) L T i − ( T i ) − K (cid:1)(cid:3) = P t ( T i − ) − P t ( T i ) − P t ( T i )( T i − T i − ) K for all i = 1 , ..., N . Then the assertion follows by Lemma 5.1 and summation. We can price swaptions by using the pricing formulas from traditional models to computethe upper and the lower bound for the price. A swaption gives the buyer the right to enteran interest rate swap at the first payment date T . Hence, there is only one cashflow,which is determined by Proposition 6.3, i.e., ξ i = 1 { } ( i ) (cid:16) − P T ( T n ) − N X j =1 P T ( T j )( T j − T j − ) K (cid:17) + (6.4)for all i = 0 , , ..., N . Due to the nonlinearity of the payoff function, the upper and thelower expectation of the discounted payoff do not necessarily coincide. Thus, the contracthas an asymmetric payoff. The related pricing bounds are given by the price from theclassical case with the highest and the lowest possible volatility, respectively. Theorem 6.1.
Let ξ i be given by (6.4) for all i = 0 , , ..., N . Then it holds ˆ E [ ˜ X ] = P ( T ) u σ (cid:0) , P ( T ) P ( T ) , ..., P ( T N ) P ( T ) (cid:1) , − ˆ E [ − ˜ X ] = P ( T ) u σ (cid:0) , P ( T ) P ( T ) , ..., P ( T N ) P ( T ) (cid:1) , where the function u σ : [0 , T ] × R N → R , for σ ∈ Σ , is defined by u σ ( t, x , ..., x N ) := E P h(cid:16) − X NT − N X i =1 X iT ( T i − T i − ) K (cid:17) + i and the process X i = ( X is ) t ≤ s ≤ T , for all i = 1 , ..., N , is given by X is = x i − d X j =1 Z st σ ju ( T , T i ) X iu σ j dB ju . Proof.
We prove the claim by using Proposition 4.2. By Proposition 4.1 ( i ), we haveˆ E [ ˜ X ] = P ( T ) ˆ E T h(cid:16) − X T ,T N T − N X i =1 X T ,T i T ( T i − T i − ) K (cid:17) + i , − ˆ E [ − ˜ X ] = − P ( T ) ˆ E T h − (cid:16) − X T ,T N T − N X i =1 X T ,T i T ( T i − T i − ) K (cid:17) + i . Hence, the assertion follows by Proposition 4.2, since the payoff of a swaption is convexand satisfies (4.1), which is shown in Section D.2 of the appendix.15 .5 Caps and Floors
Similar to swaptions, we can compute the upper and the lower bound for the price ofa cap by using the pricing formulas from traditional models. A cap gives the buyer theright to exchange the floating rate L T i − ( T i ) with a fixed rate K at each payment date T i for all i = 1 , ..., N . The cashflows are called caplets and are given by ξ i = 1 { ,...,N } ( i )( T i − T i − ) (cid:0) L T i − ( T i ) − K (cid:1) + (6.5)for all i = 0 , , ..., N . The upper and the lower bound for the price of the contract aregiven by the price from the classical case without volatility uncertainty with the highestand the lowest possible volatility, respectively. We obtain the latter by computing pricesof put options on the forward price. Theorem 6.2.
Let ξ i be given by (6.5) for all i = 0 , , ..., N . Then it holds ˆ E [ ˜ X ] = N X i =1 P ( T i − ) u iσ (cid:0) , P ( T i ) P ( T i − ) (cid:1) , − ˆ E [ − ˜ X ] = N X i =1 P ( T i − ) u iσ (cid:0) , P ( T i ) P ( T i − ) (cid:1) , where the function u iσ : [0 , T i − ] × R → R , for all i = 1 , ..., N and σ ∈ Σ , is defined by u iσ ( t, x i ) := K i E P [( K i − X iT i − ) + ] for K i := T i − T i − ) K and the process X i = ( X is ) t ≤ s ≤ T i − is given by X is = x i − d X j =1 Z st σ ju ( T i − , T i ) X iu σ j dB ju . Proof.
According to Lemma 5.2, we need to determine ˜ Y and ˜ Z in order to obtain ˆ E [ ˜ X ]and ˆ E [ − ˜ X ], respectively. We only show how to obtain ˜ Y . We can compute ˜ Z in thesame way.We compute ˜ Y by using Proposition 5.1. For this purpose, we need to rewrite ˜ Y i for all i = 0 , , ..., N and define a sequence of random variables to which we can applyProposition 5.1. For all i = 0 , , ..., N , we have˜ Y i = P T i − ( T i ) ˆ E T i T i − [ ξ i + ˜ Y i +1 ] , where ξ i is given by (6.5), and ˜ Y N +1 = 0. Since ξ i ∈ L G (Ω T i − ) for all i = 1 , ..., N and ξ = 0, we can show that˜ Y i = K i ( K i − X T i − ,T i T i − ) + + X T i − ,T i T i − ˆ E T i T i − [ ˜ Y i +1 ]for all i = 1 , ..., N and ˜ Y = X ,T ˆ E T [ ˜ Y ]. Now we define ¯ Y i := X T i − ,T i − T i − ˆ E T i − T i − [ ˜ Y i ] for all i = 1 , ..., N + 1. Then we have ˜ Y = ¯ Y and¯ Y i = X T i − ,T i − T i − ˆ E T i − T i − [ K i ( K i − X T i − ,T i T i − ) + + ¯ Y i +1 ]16or all i = 1 , ..., N , where ¯ Y N +1 = 0. Moreover, we define t i := T i − for all i = 1 , ..., N + 2.Then it holds 0 = t < ... < t N +2 ≤ τ and¯ Y i = X t i ,t i +1 t i ˆ E t i +1 t i [ K i ( K i − X t i +1 ,t i +2 t i +1 ) + + ¯ Y i +1 ]for all i = 1 , ..., N . Thus, we can apply Proposition 5.1, to obtain¯ Y = N X i =1 X ,t i +1 u iσ (0 , X t i +1 ,t i +2 ) , which proves the assertion.Floors can be priced in the same manner as caps. A floor gives the buyer the right toexchange a fixed rate K with the floating rate L T i − ( T i ) at each payment date T i for all i = 1 , ..., N . The cashflows are called floorlets and are given by ξ i = 1 { ,...,N } ( i )( T i − T i − ) (cid:0) K − L T i − ( T i ) (cid:1) + (6.6)for all i = 0 , , ..., N . Since the cashflows are very similar to caplets, we obtain similarpricing bounds compared to Theorem 6.2. The only difference is that we need to computeprices of call options on the forward price instead of put options to obtain the pricingbounds. It is remarkable that we can show this with the put-call parity, since the non-linearity of the pricing measure implies that the put-call parity, in general, does not holdin the presence of volatility uncertainty. Theorem 6.3.
Let ξ i be given by (6.6) for all i = 0 , , ..., N . Then it holds ˆ E [ ˜ X ] = N X i =1 P ( T i − ) u iσ (cid:0) , P ( T i ) P ( T i − ) (cid:1) , − ˆ E [ − ˜ X ] = N X i =1 P ( T i − ) u iσ (cid:0) , P ( T i ) P ( T i − ) (cid:1) , where the function u iσ : [0 , T i − ] × R → R , for all i = 1 , ..., N and σ ∈ Σ , is defined by u iσ ( t, x i ) := K i E P [( X iT i − − K i ) + ] and K i and the process X i = ( X is ) t ≤ s ≤ T i − are given as in Theorem 6.2.Proof. Although ˆ E is sublinear, we can still use the put-call parity to prove the claim,since interest rate swaps have a symmetric payoff. For all i = 1 , ..., N , we have ξ i = ( T i − T i − ) (cid:0) L T i − ( T i ) − K (cid:1) + − ( T i − T i − ) (cid:0) L T i − ( T i ) − K (cid:1) . Thus, we get ˜ X = ˜ Y − ˜ Z , where ˜ Y , respectively ˜ Z , denotes the discounted payoff of acap, respectively interest rate swap, that is˜ Y := N X i =1 M − T i ( T i − T i − ) (cid:0) L T i − ( T i ) − K (cid:1) + , ˜ Z := N X i =1 M − T i ( T i − T i − ) (cid:0) L T i − ( T i ) − K (cid:1) . Due to the sublinearity of ˆ E , we get ˆ E [ ˜ X ] ≤ ˆ E [ ˜ Y ] + ˆ E [ − ˜ Z ] and ˆ E [ ˜ X ] ≥ ˆ E [ ˜ Y ] − ˆ E [ ˜ Z ].Hence, by Proposition 6.3, we obtain ˆ E [ ˜ X ] = ˆ E [ ˜ Y ] − ˆ E [ ˜ Z ]. In a similar fashion, we canshow that − ˆ E [ − ˜ X ] = − ˆ E [ − ˜ Y ] − ˆ E [ ˜ Z ]. Therefore, the assertion follows by the classicalput-call parity. 17 .6 In-Arrears Contracts The pricing procedure from the previous subsection also works for contracts in whichthe floating rate is settled in arrears. The difference between the contracts from aboveand in-arrears contracts is that the simply compounded forward rate L T i ( T i +1 ) is reseteach time T i when the contract pays off for all i = 0 , , ..., N −
1. As a representativecontract, we show how to price in-arrears swaps. Other contracts, such as in-arrears capsand floors, can be priced in a similar way. In contrast to a plain vanilla interest rateswap, the cashflows are now given by ξ i = 1 { , ,...,N − } ( i )( T i +1 − T i ) (cid:0) L T i ( T i +1 ) − K (cid:1) (6.7)for all i = 0 , , ..., N . Then the contract is not necessarily symmetric and the pricingbounds are given by the price from traditional models with the highest and the lowestpossible volatility, respectively. As a consequence, there is not a unique swap rate forin-arrears swaps. Theorem 6.4.
Let ξ i be given by (6.7) for all i = 0 , , ..., N . Then it holds ˆ E [ ˜ X ] = N X i =1 P ( T i ) u iσ (cid:0) , P ( T i − ) P ( T i ) (cid:1) , − ˆ E [ − ˜ X ] = N X i =1 P ( T i ) u iσ (cid:0) , P ( T i − ) P ( T i ) (cid:1) , where the function u iσ : [0 , T i − ] × R → R , for all i = 1 , ..., N and σ ∈ Σ , is defined by u iσ ( t, x i ) := E P [ X iT i − ( X iT i − − K i )] , for K i as in Theorem 6.2 and the process X i = ( X is ) t ≤ s ≤ T i − is given by X is = x i − d X j =1 Z st σ ju ( T i , T i − ) X iu σ j dB ju . Proof.
As in the proof of Theorem 6.2, by Lemma 5.2, we need to compute ˜ Y and ˜ Z tofind ˆ E [ ˜ X ] and ˆ E [ − ˜ X ], respectively. We only show how to obtain ˜ Y . We can find ˜ Z inthe same way.In order to find ˜ Y , we rewrite ˜ Y i for all i = 0 , , ..., N and define a sequence of randomvariables to which we can apply Proposition 5.1. For all i = 0 , , ..., N , we have˜ Y i = P T i − ( T i ) ˆ E T i T i − [ ξ i + ˜ Y i +1 ] , where ξ i is given by (6.7) and ˜ Y N +1 = 0. Since ξ N = 0, we get ˜ Y N = 0. For all i = 0 , , ..., N −
1, we obtain, by Proposition 4.1 ( ii ),˜ Y i = X T i − ,T i +1 T i − ˆ E T i +1 T i − [ X T i +1 ,T i T i ( X T i +1 ,T i T i − K i +1 ) + X T i +1 ,T i T i ˜ Y i +1 ] . We define ¯ Y i := X T i − ,T i − T i − ˜ Y i − for all i = 1 , ..., N + 1. Then it holds ˜ Y = X ,T ¯ Y and¯ Y i = X T i − ,T i T i − ˆ E T i T i − [ X T i ,T i − T i − ( X T i ,T i − T i − − K i ) + ¯ Y i +1 ]18or all i = 1 , ..., N , where ¯ Y N +1 = 0. Furthermore, we set t i := T i − for all i = 1 , ..., N + 2.Then we get 0 = t < ... < t N +2 ≤ τ and¯ Y i = X t i +1 ,t i +2 t i ˆ E t i +2 t i [ X t i +2 ,t i +1 t i +1 ( X t i +2 ,t i +1 t i +1 − K i ) + ¯ Y i +1 ]for all i = 1 , ..., N . Therefore, by Proposition 5.1, it holds¯ Y = N X i =1 X t ,t i +2 u iσ (0 , X t i +2 ,t i +1 ) , which proves the assertion. Empirical evidence shows that volatility risk in fixed income markets cannot be hedged bytrading solely bonds, which is referred to as unspanned stochastic volatility and contradictsmany traditional term structure models. By using data on interest rate swaps, caps, andfloors, Collin-Dufresne and Goldstein [16] showed that interest rate derivatives exposedto volatility risk are driven by factors which do not affect the term structure. Therefore,derivatives exposed to volatility risk, such as caps and floors, cannot be replicated by aportfolio consisting solely of bonds, which implies that it is not possible to hedge volatilityrisk in fixed income markets. The empirical findings of Collin-Dufresne and Goldstein[16] contradict many traditional term structure models, since bond prices are typicallyfunctions depending on all risk factors driving the model and bonds can typically be usedto hedge caps and floors. As a consequence, Collin-Dufresne and Goldstein [16] examinedwhich term structure models exhibit unspanned stochastic volatility. This led to thedevelopment of new models displaying unspanned stochastic volatility [14, 24, 25].In the presence of volatility uncertainty, term structure models naturally exhibit un-spanned stochastic volatility, since volatility uncertainty naturally leads to market incom-pleteness. A classical result in the literature on robust finance is that model uncertaintyleads to market incompleteness. Instead of perfectly hedging derivatives, one has to su-perhedge the payoff of most derivatives, which can be inferred from the pricing-hedgingduality. Similar to the pricing-hedging duality in the presence of volatility uncertainty[49, Theorem 3.6], we can show that it is not possible to hedge a contract with an asym-metric payoff with a portfolio of bonds. From Theorem 6.2 and Theorem 6.3, we candeduce that caps and floors have an asymmetric payoff if σ > σ . Therefore, derivativesexposed to volatility risk cannot be hedged by trading solely bonds when there is volatilityuncertainty.Moreover, the uncertain volatility affects prices of nonlinear contracts while pricesof linear contracts and the term structure are robust with respect to the volatility, con-firming the empirical findings of Collin-Dufresne and Goldstein [16]. In simple modelspecifications, bond prices have an affine structure with respect to the short rate and anadditional factor [30, Theorems 4.1, 4.2]. However, they are completely unaffected by theuncertain volatility and its bounds. The same holds for the swap rate, since the price ofan interest rate swap, by Proposition 6.3, is a linear combination of bond prices as in theclassical case without volatility uncertainty. On the other hand, the uncertain volatilityinfluences prices of caps and floors, by Theorem 6.2 and Theorem 6.3, since they dependon the bounds for the volatility. Therefore, the prices of caps and floors are driven by anadditional factor that does not influence term structure movements and thus, changes inswap rates. 19
Conclusion
In the present paper, we deal with the pricing of contracts in fixed income markets undervolatility uncertainty. The starting point is an arbitrage-free bond market under volatilityuncertainty. Such a framework leads to a sublinear pricing measure, which we can use todetermine either the price of a contract or its pricing bounds. To simplify the pricing ofcashflows, we introduce the forward sublinear expectation, under which the expectationshypothesis holds in a robust sense. We can use the forward sublinear expectation to pricebond options. Due to the nonlinearity of the pricing measure, we additionally derivemethods to price contracts consisting of a collection of cashflows, which differs from thecase without volatility uncertainty. We can price contracts with a simple payoff structureas in the classical case. For more general contracts, we need to use a backward inductionprocedure to find the price. We can use this procedure to price contracts consistingof a stream of bond options. These results enable us to price all major interest ratederivatives, including linear contracts, such as fixed coupon bonds, floating rate notes,and interest rate swaps, and nonlinear contracts, such as swaptions, caps and floors, andin-arrears contracts. We obtain a single price for linear contracts, which is the same asthe one obtained by traditional term structure models, and a range of prices for nonlinearcontracts, which is bounded by the price from traditional models with the highest andthe lowest possible volatility, respectively. Therefore, the pricing of typical interest ratederivatives reduces to computing prices in the corresponding model without volatilityuncertainty. Since volatility uncertainty leads to market incompleteness, we can showthat term structure models in the presence of volatility uncertainty naturally displayunspanned stochastic volatility.
AppendixA Proof of Proposition 3.1
We assume that there exists an arbitrage strategy ( π, π S , π A , T ) and show that this yieldsa contradiction. We only examine the case in which X A is traded, i.e., π A = 0. If π A = 0,the proof is similar to the proof of Proposition 4.1 from H¨olzermann [29]. By the definitionof arbitrage, it holds ˜ v τ ( π, π S , π A , T ) ≥
0. Then the monotonicity of ˆ E implies thatˆ E h n X i =1 Z T i π it d ˜ P t ( T i ) + Z τ π St d ˜ X St i ≥ ˆ E [ − π A ( ˜ X A − ˜ X A )] . Due to the sublinearity of ˆ E and the fact that the discounted bonds are symmetric G -martingales under ˆ E and X S has a symmetric payoff, we haveˆ E h n X i =1 Z T i π it d ˜ P t ( T i ) + Z τ π St d ˜ X St i ≤ G -martingales [48, Theorem 4.8].Furthermore, if we use the properties of ˆ E and the assumption on ˜ X A , we getˆ E [ − π A ( ˜ X A − ˜ X A )] = ( π A ) + ( ˆ E [ − ˜ X A ] + ˜ X A ) + ( π A ) − ( ˆ E [ ˜ X A ] − ˜ X A ) > . Combining the previous steps, we obtain a contradiction.20
Proofs of Section 4
B.1 Proof of Proposition 4.1
Part ( i ) follows by a simple calculation. We have M t ˆ E t [ M − T ξ ] = P t ( T ) M t P ( T ) P t ( T ) ˆ E t [ M − T P T ( T ) P ( T ) ξ ] = P t ( T )( X Tt ) − ˆ E t [ X TT ξ ] = P t ( T ) ˆ E Tt [ ξ ] . To show part ( ii ), we use some properties of G -backward stochastic differential equa-tions. By Definition 4.1, we have ˆ E T ′ t [ ξ ] = Y T ′ ,ξt , where Y T ′ ,ξ solves Y T ′ ,ξt = ξ − d X i =1 Z T ′ t b iu ( T ′ ) Z iu d h B i i u − d X i =1 Z T ′ t Z iu dB iu − ( K T ′ − K t ) . Since ξ ∈ L pG (Ω T ), Y T ′ ,ξ also solves the G -backward stochastic differential equation Y T ′ ,ξt = ξ − d X i =1 Z Tt b iu ( T ′ ) Z iu d h B i i u − d X i =1 Z Tt Z iu dB iu − ( K T − K t ) . By Theorem 3.2 of Hu, Ji, Peng, and Song [31], the solution to the latter is given by Y T ′ ,ξt = ( X T ′ t ) − ˆ E t [ X T ′ T ξ ] . Moreover, for each t ≤ T , we have X T ′ t = X T,T ′ t X T ′ ,T X Tt . Hence, we obtainˆ E T ′ t [ ξ ] = X T ′ ,Tt X T,T ′ ( X Tt ) − ˆ E t [ X T,T ′ T X T ′ ,T X TT ξ ] = X T ′ ,Tt ˆ E Tt [ X T,T ′ T ξ ] , which proves part ( ii ).For part ( iii ), we use the Girsanov transformation for G -Brownian motion from Hu,Ji, Peng, and Song [31]. We define the process B T = ( B ,Tt , ..., B d,Tt ) ≤ t ≤ T by B i,Tt := B it + Z t b iu ( T ) d h B i i u . Then B T is a G -Brownian motion under ˆ E T [31, Theorems 5.2, 5.4]. Since the dynamicsof the forward rate are given by f t ( T ) = f ( T ) + d X i =1 Z t β iu ( T ) dB iu + d X i =1 Z t β iu ( T ) b it ( T ) d h B i i u , the forward rate is a symmetric G -martingale under ˆ E T .To obtain part ( iv ), we first show that X T,T ′ t ∈ L pG (Ω t ) for all p < ∞ by using therepresentation of the space L pG (Ω t ) from Denis, Hu, and Peng [18] and a proof similarto the proof of Proposition 5.10 from Osuka [39]. The space L pG (Ω t ) consists of allBorel measurable random variables X which have a quasi-continuous version and satisfylim n →∞ ˆ E [ | X | p {| X | >n } ] = 0 [40, Proposition 6.3.2]. One can show that X T,T ′ t = X T,T ′ exp (cid:16) − d X i =1 Z t σ iu ( T, T ′ ) dB iu − d X i =1 Z t (cid:0) σ iu ( T, T ′ ) + σ iu ( T, T ′ ) b iu ( T ) (cid:1) d h B i i u (cid:17) σ i ( T, T ′ ) and b i ( T ), for all i , are bounded processes in M pG (0 , τ ) forall p < ∞ , we already know that X T,T ′ t is measurable and has a quasi-continuous version.Now we show that ˆ E [ | X T,T ′ t | p ′ ] < ∞ for p ′ > p , which implies lim n →∞ ˆ E [ | X | p {| X | >n } ] = 0.By H¨older’s inequality, for p ′ > p and q ′ >
1, we have ˆ E [ | X T,T ′ t | p ′ ] ≤ X T,T ′ ˆ E h exp (cid:16) − p ′ q ′ d X i =1 Z t σ iu ( T, T ′ ) dB iu − ( p ′ q ′ ) d X i =1 Z t σ iu ( T, T ′ ) d h B i i u (cid:17)i q ′ ˆ E h exp (cid:16) p ′ q ′ q ′ − d X i =1 Z t (cid:0) ( p ′ q ′ − σ iu ( T, T ′ ) − σ iu ( T, T ′ ) b iu ( T ) (cid:1) d h B i i u (cid:17)i q ′− q ′ . The two terms on the right-hand side are finite. The second term is finite, since σ i ( T, T ′ )and b i ( T ) are bounded for all i . By the same argument, we haveˆ E h exp (cid:16) ( p ′ q ′ ) d X i =1 Z t σ iu ( T, T ′ ) d h B i i u (cid:17)i < ∞ . Then we can use Novikov’s condition to show that the first term is finite, since theexponential inside the sublinear expectation is a martingale under each P ∈ P .Using Itˆo’s formula for G -Brownian motion from Li and Peng [33] and the Girsanovtransformation of Hu, Ji, Peng, and Song [31] completes the proof. We have X T,T ′ t = X T,T ′ − d X i =1 Z t σ iu ( T, T ′ ) X T,T ′ u dB iu − d X i =1 Z t σ iu ( T, T ′ ) X T,T ′ u b iu ( T ) d h B i i u by Itˆo’s formula [33, Theorem 5.4]. Moreover, since σ i ( T, T ′ ) and b i ( T ), for all i , arebounded processes in M pG (0 , τ ) for all p < ∞ , one can then show that X T,T ′ belongs to M pG (0 , τ ) for all p < ∞ [30, Proposition B.1]. As in the proof of part ( iii ), the Girsanovtransformation then implies that X T,T ′ is a symmetric G -martingale under ˆ E T . B.2 Proof of Proposition 4.2
First, we characterize the forward sublinear expectation in the first equation as the so-lution to a nonlinear partial differential equation by using the nonlinear Feynman-Kacformula of Hu, Ji, Peng, and Song [31]. With Proposition 4.1 ( iv ) and inequality (4.1)one can show that ξ := ϕ ( X T,t t , ..., X T,t n t ) belongs to L pG (Ω t ) ⊂ L pG (Ω T ) with p >
1. ByDefinition 4.1, we have ˆ E Tt [ ξ ] = Y T,ξt , where Y T,ξ = ( Y T,ξt ) ≤ t ≤ T solves the G -backwardstochastic differential equation Y T,ξt = ϕ ( X T,t t , ..., X T,t n t ) − d X i =1 Z Tt b iu ( T ) Z iu d h B i i u − d X i =1 Z Tt Z iu dB iu − ( K T − K t ) . Since ξ ∈ L pG (Ω t ), Y T,ξ also solves the G -backward stochastic differential equation Y T,ξt = ϕ ( X T,t t , ..., X T,t n t ) − d X i =1 Z t t b iu ( T ) Z iu d h B i i u − d X i =1 Z t t Z iu dB iu − ( K t − K t ) , where ϕ satisfies (4.1). From Proposition 4.1 ( iv ) we deduce the dynamics and theregularity of X T,t i for all i = 1 , ..., n . Then, by Theorem 4.4 and Theorem 4.5 of Hu, Ji,22eng, and Song [31], we have Y T,ξt = u ( t, X T,t t , ..., X T,t n t ), where u : [0 , t ] × R n → R isthe unique viscosity solution to the nonlinear partial differential equation ∂ t u + G (cid:16)(cid:16) n X k,l =1 σ it ( T, t k ) x k σ jt ( T, t l ) x l ∂ x k x l u (cid:17) i,j =1 ,...,d (cid:17) = 0 ,u ( t , x , ..., x n ) = ϕ ( x , ..., x n ) . Now we show that u σ solves the nonlinear partial differential equation. By the classicalFeynman-Kac formula, we know that u σ , for σ ∈ Σ, satisfies ∂ t u σ + tr (cid:16) σσ ′ (cid:16) n X k,l =1 σ it ( T, t k ) x k σ jt ( T, t l ) x l ∂ x k x l u σ (cid:17) i,j =1 ,...,d (cid:17) = 0 ,u σ ( t , x , ..., x n ) = ϕ ( x , ..., x n ) . In addition, the convexity of ϕ implies that u σ ( t, · ) is convex for each σ and t . Thus, n X k,l =1 σ it ( T, t k ) x k σ jt ( T, t l ) x l ∂ x k x l u σ ≥ i, j = 1 , ..., d . Therefore, one can verify that u σ solves the nonlinear partialdifferential equation from above, which proves the first assertion.In order to prove the second assertion, we repeat the procedure from above. Dueto the nonlinear Feynman-Kac formula, we have ˆ E Tt [ − ξ ] = u ( t, X T,t t , ..., X T,t n t ), where u : [0 , t ] × R n → R is the unique viscosity solution to the nonlinear partial differentialequation ∂ t u + G (cid:16)(cid:16) n X k,l =1 σ it ( T, t k ) x k σ jt ( T, t l ) x l ∂ x k x l u (cid:17) i,j =1 ,...,d (cid:17) = 0 ,u ( t , x , ..., x n ) = − ϕ ( x , ..., x n ) . Then we can use the concavity of − u σ ( t, · ) to show that − u σ solves the nonlinear partialdifferential equation from above. C Proofs of Section 5
C.1 Proof of Lemma 5.1
We derive an upper, respectively lower, bound for the upper, respectively lower, expecta-tion of ˜ X and show that they coincide. Using the sublinearity of ˆ E and Proposition 4.1( i ), for t ≤ T , we get M t ˆ E t [ ˜ X ] ≤ N X i =0 M t ˆ E t [ M − T i ξ i ] = N X i =0 P t ( T i ) ˆ E T i t [ ξ i ] . By the same arguments, for t ≤ T , we obtain − M t ˆ E t [ − ˜ X ] ≥ N X i =0 − M t ˆ E t [ − M − T i ξ i ] = N X i =0 − P t ( T i ) ˆ E T i t [ − ξ i ] . For t ≤ T , it holds ˆ E t [ ˜ X ] ≥ − ˆ E t [ − ˜ X ] and ˆ E T i t [ ξ i ] = − ˆ E T i t [ − ξ i ] for all i . Therefore, allexpressions from above are equal. 23 .2 Proof of Lemma 5.2 First, we exclude the last cashflow from the sum and write it in terms of ˜ Y N . Due to thetime consistency of the G -expectation, we haveˆ E [ ˜ X ] = ˆ E h N − X i =0 M − T i ξ i + ˆ E T N − [ M − T N ξ N ] i . By Proposition 4.1 ( i ), we obtainˆ E T N − [ M − T N ξ N ] = M − T N − P T N − ( T N ) ˆ E T N T N − [ ξ N ] = M − T N − ˜ Y N . Second, we exclude the second last cashflow from the sum and repeat the calculationfrom above. Using the time consistency of ˆ E , we getˆ E [ ˜ X ] = ˆ E h N − X i =0 M − T i ξ i + ˆ E T N − [ M − T N − ( ξ N − + ˜ Y N )] i . Due to Proposition 4.1 ( i ), we haveˆ E T N − [ M − T N − ( ξ N − + ˜ Y N )] = M − T N − P T N − ( T N − ) ˆ E T N − T N − [ ξ N − + ˜ Y N ] = M − T N − ˜ Y N − . Now we work recursively backwards until we arrive at the last cashflow. Repeatingthe step from above, we finally obtainˆ E [ ˜ X ] = ˆ E [ M − T ξ + M − T ˜ Y ] = P ( T ) ˆ E T [ ξ + ˜ Y ] = ˜ Y . By replacing ˜ X , ξ i , and ˜ Y i , for all i = 0 , , ..., N , by − ˜ X , − ξ i , and ˜ Z i , respectively,we get ˆ E [ − ˜ X ] = ˜ Z , which completes the proof. C.3 Proof of Proposition 5.1
We only compute ¯ Y . The derivation of ¯ Z can be carried out in the same way, which issimilar to the proof of Proposition 4.2.In order to determine ¯ Y , we show, by induction, that¯ Y i = N X j = i X t i − n ,t j + n t i u jσ ( t i , X t j + n ,t j + m t i )for all i = 1 , ..., N . We start by computing ¯ Y N . We have¯ Y N = X t N − n ,t N + n t N ˆ E t N + n t N [ ϕ N ( X t N + n ,t N + m t N +1 )] . Since ϕ N is convex and satisfies (4.1), by Proposition 4.2, we obtain¯ Y N = X t N − n ,t N + n t N u Nσ ( t N , X t N + n ,t N + m t N )] . Next, we do the inductive step. For 1 ≤ i ≤ N −
1, let us suppose that¯ Y i +1 = N X j = i +1 X t i + n ,t j + n t i +1 u jσ ( t i +1 , X t j + n ,t j + m t i +1 ) . Y i = X t i − n ,t i + n t i ˆ E t i + n t i h ϕ i ( X t i + n ,t i + m t i +1 ) + N X j = i +1 X t i + n ,t j + n t i +1 u jσ ( t i +1 , X t j + n ,t j + m t i +1 ) i . We use the nonlinear Feynman-Kac formula of Hu, Ji, Peng, and Song [31] to computethe expectation, since we cannot apply Proposition 4.2. Thus, we obtain¯ Y i = X t i − n ,t i + n t i u ( t i , X t i + n ,t i + m t i , ..., X t N + n ,t N + m t i , X t i + n ,t i +1+ n t i , ..., X t i + n ,t N + n t i ) . The function u : [0 , t i +1 ] × R N − i )+1 → R is the unique viscosity solution to the nonlinearpartial differential equation ∂ t u + G (cid:16)(cid:0) H κ,λ ( t, x, D x u, D x u ) (cid:1) κ,λ =1 ,...,d (cid:17) = 0 ,u ( t i +1 , x ) = ϕ ( x ) , (C.1)where x = (ˆ x i , ..., ˆ x N , ˜ x i +1 , ..., ˜ x N ) ∈ R N − i )+1 , D x , respectively D xx , denotes the gradi-ent, respectively Hessian, with respect to x , and H κ,λ ( t, x, D x u, D x u ) := N X j,k = i σ κt ( t j + n , t j + m )ˆ x j σ λt ( t k + n , t k + m )ˆ x k ∂ x j ˆ x k u + 2 N X j = i N X k = i +1 σ κt ( t j + n , t j + m )ˆ x j σ λt ( t i + n , t k + n )˜ x k ∂ x j ˜ x k u + N X j,k = i +1 σ κt ( t i + n , t j + n )˜ x j σ λt ( t i + n , t k + n )˜ x k ∂ x j ˜ x k u + 1 { κ = λ } ( κ, λ )2 N X j = i +1 σ κt ( t j + n , t j + m )ˆ x j σ κt ( t j + n , t i + n ) ∂ ˆ x j u,ϕ ( x ) := ϕ i (ˆ x i ) + N X j = i +1 ˜ x j u jσ ( t i +1 , ˆ x j ) . It is feasible to apply the nonlinear Feynman-Kac formula, since ϕ satisfies (4.1), whichis shown in Section D.1 of the appendix. To show that ϕ satisfies (4.1), we use estimatesfor u jσ , for all j = i + 1 , ..., N , which follow from the nonlinear Feynman-Kac formula [31,Proposition 4.2]. In order to solve (C.1), we define u ∗ : [0 , t i +1 ] × R N − i )+1 → R by u ∗ ( t, x ) := u iσ ( t, ˆ x i ) + N X j = i +1 ˜ x j u jσ ( t, ˆ x j ) . Then ∂ x j ˆ x k u ∗ = 0 and ∂ x j ˜ x k u ∗ = 0 for all j, k such that j = k , ˜ x k ∂ x j ˜ x k u ∗ = ∂ ˆ x j u ∗ for all j, k such that j = k , and ∂ x j ˜ x k u ∗ = 0 for all j, k . Since u jσ ( t, · ) is convex for each t and u jσ satisfies ∂ t u jσ + tr (cid:16) σσ ′ (cid:0) σ kt ( t j + n , t j + m ) σ lt ( t j + n , t j + m )ˆ x j ∂ x j ˆ x j u jσ (cid:1) k,l =1 ,...,d (cid:17) = 0 ,u jσ ( t j +1 , ˆ x j ) = ϕ j (ˆ x j )25or all j = i, ..., N and σ ∈ Σ, one can verify that u ∗ solves (C.1) on [0 , t i +1 ] × R N − i )+1+ .We are only interested in a solution for positive x , since the forward prices are positive.Hence, we obtain ¯ Y i = N X j = i X t i − n ,t j + n t i u jσ ( t i , X t j + n ,t j + m t i )and the proof is complete. D Estimates for the Proofs
D.1 Estimate for the Proof of Proposition 5.1
Let us define the function ϕ : R n +1 → R by ϕ ( x ) := f (ˆ x ) + n X i =1 ˜ x i u i (ˆ x i ) , where f : R → R , for a positive integer m and a constant C >
0, satisfies | f (ˆ x ) − f (ˆ x ′ ) | ≤ C (1 + | ˆ x | m + | ˆ x ′ | m ) | ˆ x − ˆ x ′ | and u i : R → R , for all i = 1 , ..., n , satisfies | u i (ˆ x i ) | ≤ C (1 + | ˆ x i | m +1 ) , | u i (ˆ x i ) − u i (ˆ x ′ i ) | ≤ C (1 + | ˆ x i | m + | ˆ x ′ i | m ) | ˆ x i − ˆ x ′ i | . Then we have | ϕ ( x ) − ϕ ( y ) | = (cid:12)(cid:12)(cid:12) f (ˆ x ) − f (ˆ x ′ ) + n X i =1 ˜ x i u i (ˆ x i ) − ˜ x ′ i u i (ˆ x ′ i ) (cid:12)(cid:12)(cid:12) ≤ | f (ˆ x ) − f (ˆ x ′ ) | + n X i =1 | u i (ˆ x i ) || ˜ x i − ˜ x ′ i | + | ˜ x ′ i || u i (ˆ x i ) − u i (ˆ x ′ i ) |≤ C (1 + | ˆ x | m + | ˆ x ′ | m ) | ˆ x − ˆ x ′ | + n X i =1 C (1 + | ˆ x i | m +1 ) | ˜ x i − ˜ x ′ i | + | ˜ x ′ i | C (1 + | ˆ x i | m + | ˆ x ′ i | m ) | ˆ x i − ˆ x ′ i |≤ C (2 + | ˆ x | m +1 + | ˆ x ′ | m +1 ) | ˆ x − ˆ x ′ | + n X i =1 C (1 + | ˆ x i | m +1 ) | ˜ x i − ˜ x ′ i | + 2 C (1 + | ˜ x ′ i | m +1 + | ˆ x i | m +1 + | ˆ x ′ i | m +1 ) | ˆ x i − ˆ x ′ i |≤ C (cid:16) | ˆ x | m +1 + | ˆ x ′ | m +1 + n X i =1 | ˆ x i | m +1 + | ˜ x i | m +1 + | ˆ x ′ i | m +1 + | ˜ x ′ i | m +1 (cid:17)(cid:16) | ˆ x − ˆ x ′ | + n X i =1 | ˆ x i − ˆ x ′ i | + | ˜ x i − ˜ x ′ i | (cid:17) ≤ CC ′ (1 + | x | m +1 + | x ′ | m +1 ) | x − x ′ | . .2 Estimates for the Proof of Theorem 6.1 Let us define the function ϕ : R N → R by ϕ ( x ) := (cid:16) − x N − N X i =1 x i ( T i − T i − ) K (cid:17) + . Then, for λ ∈ (0 , ϕ (cid:0) λx + (1 − λ ) y (cid:1) = (cid:16) − (cid:0) λx N + (1 − λ ) y N (cid:1) − N X i =1 (cid:0) λx i + (1 − λ ) y i (cid:1) ( T i − T i − ) K (cid:17) + ≤ λ (cid:16) − x N − N X i =1 x i ( T i − T i − ) K (cid:17) + + (1 − λ ) (cid:16) − y N − N X i =1 y i ( T i − T i − ) K (cid:17) + = λϕ ( x ) + (1 − λ ) ϕ ( y ) . Moreover, it holds | ϕ ( x ) − ϕ ( y ) | = (cid:12)(cid:12)(cid:12)(cid:16) − x N − N X i =1 x i ( T i − T i − ) K (cid:17) + − (cid:16) − y N − N X i =1 y i ( T i − T i − ) K (cid:17) + (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ( x N − y N ) + N X i =1 ( x i − y i )( T i − T i − ) K (cid:12)(cid:12)(cid:12) ≤ N X i =1 (cid:0) T i − T i − ) K (cid:1) | x i − y i |≤ (cid:16) N X i =1 (cid:0) T i − T i − ) K (cid:1) (cid:17) | x − y | . References [1] Acciaio, B., M. Beiglb¨ock, F. Penkner, and W. Schachermayer (2016). A model-freeversion of the fundamental theorem of asset pricing and the super-replication theorem.
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