A note on the worst case approach for a market with a stochastic interest rate
aa r X i v : . [ q -f i n . M F ] J a n A NOTE ON THE WORST CASE APPROACH FOR AMARKET WITH A STOCHASTIC INTEREST RATE
DARIUSZ ZAWISZA
Abstract.
We solve robust optimization problem and show the ex-ample of the market model for which the worst case measure is nota martingale measure. In our model the instantaneous interest rateis determined by the Hull-White model and the investor employs theHARA utility to measure his satisfaction.To protect against the modeluncertainty he uses the worst case measure approach. The problem isformulated as a stochastic game between the investor and the marketfrom the other side. PDE methods are used to find the saddle point andthe precise verification argument is provided.
Published in Appl. Math. (Warsaw), 45 (2018) 151–160, https://doi.org/10.4064/am2348-2-2018 Introduction
We consider a portfolio problem embedded into a game theoretic prob-lem. We assume that the investor does not trust his model much and be-lieves it is only the best guess based on existing data. In such situation wesay that the investor faces the model uncertainty (or the model ambiguity).In this work we would like to put more light into the portfolio optimizationproblem under the assumption that the short term interest rate exhibitssome stochastic nature. We consider a financial market consisting of n as-sets and a bank account. The interest rate on the bank account follows theHull–White model, which is extended version of the Vasicek model. Theinvestor chooses between holding cash in a bank account and holding riskyassets. The same model has been considered first by Korn and Kraft [4] butwithout the model uncertainty assumption. Instead of supposing that wehave the exact model, we assume here the whole family of equivalent mod-els, which will be described later. To determine robust investment controlsthe investor maximizes the total expected HARA utility of the final wealthafter taking the infimum over all possible models. The robust optimizationin the diffusion setting has been popularized especially by A. Schied andhis coauthors (e.g. Schied [10] and references therein). The model ambigu-ity in the Vasicek model and its extensions has been considered already byFlor and Larsen [2], Sun et al. [11], Munk and Rubtsov [6], Wang and Li[12]. However, their objective function is different, because it includes theexpression (along the lines of Maenhout [5]) which penalize the expected Date : January 8, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Robust optimization, stochastic differential games, model un-certainty, portfolio optimization, martingale measure, Vasicek model, Hull-White model. utility for divergence from the reference probability measure. Our model isin fact their limiting model, when their ambiguity coefficients are passingto + ∞ (0 respectively). In the current paper the problem is formulated asa theoretic stochastic game between the market and the investor and thesaddle point of this game is determined, despite of the fact we do not includethe penalizing term into the objective function. Moreover, in addition toaferomentioned papers we provide correct and precise verification reason-ing. First, we consider the full problem, without any constraints on the setof uncertainty measures. Further, we investigate what are the propertiesof the restricted model. To solve the game, we use the Hamilton-Jacobi-Bellman-Isaacs equation. After several substitutions we are able to solvethe equation and use suitable version of the verification theorem to justifythe method. Previously the same method has been used by Zawisza [13],[14], but in the model with a deterministic interest rate and with a differ-ent objective function. The major motivation for considering such modelis to provide an example in which results of Oksendal and Sulem [7],[8]do not hold. In the papers they have considered the jump diffusion modelbut without assuming the stochastic nature of the interest rate, and havediscovered that in that game the investor should always choose to investonly in the bank account and at the same time optimal market strategy isto choose a martingale measure. It is interesting because the martingalemeasure plays prominent role in derivative pricing. Our paper proves thatin our framework the worst case measure is different from the martingalemeasure. 2. Model description
Let (Ω , F , P ) be a probability space with filtration ( F t , ≤ t ≤ T )(possibly enlarged to satisfy usual assumptions) spanned by n -dimensionalBrownian motion ( W t = ( W t , W t , . . . W nt ) T , ≤ t ≤ T ). We have theinitial measure P , but our investor concerns model uncertainty, so the mea-sure should be treated only as a proxy for the real life measure. Further,we will consider a whole class of equivalent measures, which will describethe model uncertainty. Our agent has an access to the market with a bankaccount ( B t , ≤ t ≤ T ) and risky assets ( S t = ( S t , S t , . . . , S nt ) , ≤ t ≤ T ).Under the measure P the system is given by(2.1) dB t = r t B t dt,dS t = diag ( S t )[( r t e + Σ t λ Tt ) dt + Σ t dW t ] ,dr t = ( b t − κ t r t ) dt + a t dW t . We assume that e = (1 , , . . . , κ t , b t , λ t = ( λ t , λ t , . . . , λ nt ), a t = ( a t , a t , . . . , a nt ), Σ t = [ σ i,jt ] i,j =1 ...n are continuous deterministic func-tions, and in addition Σ t is invertible. For notational convenience we omitthe term a t λ Tt dt in the dynamics for r , and we assume it is already includedin b t dt term. The representative example for the process ( S t , t ∈ [0 , T ]) is NOTE ON THE WORST CASE APPROACH 3 the mixed stock-bond model (e.g. Korn and Kraft [4, Section 2.2]): dS t = ( r t + λ t σ , t + λ t σ , t ) S t dt + σ , t S t dW t + σ , t S t dW t ,dS t = ( r t + λ t σ , t ) S t dt + σ , t S t dW t ,dr t = ( b t − κr t ) dt + a t dW t . Here S t is the price of the bond in the Vasicek model with the maturity T ′ > T , which means that σ , t = − aκ (1 − e − κ ( T ′ − t ) ).The portfolio process evolves according to dX πt = r t X πt dt + π t Σ t λ Tt X πt dt + X πt π t Σ t dW t . The symbol A t denotes the class of progressively measurable processes π =( π , π , . . . , π n ) such that Z T t | π s | ds < + ∞ a.s.To describe the model uncertainty or model ambiguity issues we assume thatthe probability measure is not precisely known and the investor considers awhole class of possible measures. We follow the approach of Oksendal andSulem [7] or Schied [10] in defining the set(2.2) Q T := (cid:26) Q η T ∼ P | dQ η T dP = E (cid:18)Z η t dW t (cid:19) T , η ∈ M (cid:27) , where E ( · ) t denotes the Doleans-Dade exponential and M denotes the setof all, progressively measurable processes η = ( η , η , . . . , η n ), such that E (cid:20) dQ η T dP (cid:21) < + ∞ . In the latter part of the paper we assume that the process η takes his valuesin a fixed compact and convex set Γ. It is convenient to use the Q η T dynamicsof the stochastic system ( X t , r t ) i.e.(2.3) ( dX πt = r t X πt dt + π t Σ t ( λ Tt + η Tt ) X πt dt + π t Σ t X πt dW ηt ,dr t = [( b t − κ t r t ) + a t η Tt ] dt + a t dW ηt . Our investor takes into account the model ambiguity and has worst casepreferences (Gilboa and Schmeidler [3] ), i.e. his aim is to maximize(2.4) J π,η ( x, r, t ) = inf η ∈M E ηx,r,t U ( X π T ) . The symbol E ηx,r,t is used to denote the expected value under the measure Q η T when system starts at ( x, r, t ). Here we assume that U ( x ) = x γ γ with0 < γ <
1. The solution for γ < U has negative values, it is needed to use few more restrictions andtechnicalities to complete the proof.Here we are interested not only in the optimal portfolio π ∗ , but alsoin the measure Q η ∗ T for which the infimum is attained. Therefore, we arelooking for a saddle point ( π ∗ , η ∗ ) i.e. J π,η ∗ ( x, r, t ) ≤ J π ∗ ,η ∗ ( x, r, t ) ≤ J π ∗ ,η ( x, r, t ) . D.ZAWISZA The solution
To solve the problem we will use the Hamilton-Jacobi-Bellman-Isaacsoperator given by L π,η V ( x, r, t ) := V t + 12 a t V rr + 12 π Σ t Σ Tt π T x V xx + π Σ t a t xV xr (3.1) + π Σ t ( λ Tt + η T ) xV x + ηa Tt V r + ( b t − κ t r ) V r + rxV x . It should be considered together with the verification theorem. Thereasoning behind its proof is of standard type (see for instance Zawisza [13,Theorem 3.1]). Here we present only short sketch, just to emphasis someminor differences.
Theorem 3.1 (Verification Theorem) . Suppose there exists a positive func-tion V ∈ C , , ((0 , + ∞ ) × R × [0 , T )) ∩ C ([0 , + ∞ ) × R × [0 , T ]) and a Markov control ( π ∗ ( x, r, t ) , η ∗ ( x, r, t )) ∈ A t × M , such that L π ∗ ( x,r,t ) ,η V ( x, r, t ) ≥ , (3.2) L π,η ∗ ( x,r,t ) V ( x, r, t ) ≤ , (3.3) L π ∗ ( x,r,t ) ,η ∗ ( x,r,t ) V ( x, r, t ) = 0 , (3.4) V ( x, r, T ) = x γ γ (3.5) for all η ∈ R , π ∈ R , ( x, r, t ) ∈ (0 , + ∞ ) × R × [0 , T ) ,and (3.6) E ηx,r,t (cid:20) sup t ≤ s ≤T (cid:12)(cid:12) V ( X π ∗ s , r s , s ) (cid:12)(cid:12)(cid:21) < + ∞ for all ( x, r, t ) ∈ [0 , + ∞ ) × R × [0 , T ] , π ∈ A t , η ∈ M .Then J π,η ∗ ( x, r, t ) ≤ V ( x, r, t ) ≤ J π ∗ ,η ( x, r, t ) for all π ∈ A t , η ∈ M ,and V ( x, r, t ) = J π ∗ ,η ∗ ( x, r, t ) . Proof.
Let us fix first π ∈ A t . Consider Q η ∗ T - dynamics of the system ( X t , r t )and apply the Itˆo formula using the function V . By using inequality (3.3)and taking the expectation from both sides, we obtain V ( x, r, t ) ≥ E η ∗ V ( X ( T − ε ) ∧ τ n , r ( T − ε ) ∧ τ n , ( T − ε ) ∧ τ n ) , where ( τ n , n ≥
0) is a localizing sequence of stopping times. The function V is positive, thus the Fatou Lemma implies V ( x, r, t ) ≥ E η ∗ x,r,t V ( X π T , r T , T ) = E η ∗ x,r,t U ( X π T ) = J π,η ∗ ( x, r, t ) . NOTE ON THE WORST CASE APPROACH 5
To prove the reverse inequality we fix η ∈ M and consider Q η T - dynamicsof the system ( X t , r t ). After applying the Itˆo rule we get V ( x, r, t ) ≤ E ηx,r,t V ( X π ∗ ( T − ε ) ∧ τ n , r ( T − ε ) ∧ τ n , ( T − ε ) ∧ τ n )and the same is true with the equality V ( x, r, t ) = E η ∗ x,r,t V ( X π ∗ ( T − ε ) ∧ τ n , r ( T − ε ) ∧ τ n , ( T − ε ) ∧ τ n ) . Property (3.6) and the dominated convergence theorem finish the proof. (cid:3)
Following Korn and Kraft [4] we predict that conditions (3.2) – (3.6) aresatisfied by the function of the form V ( x, r, t ) = x γ γ e f ( t ) r + g ( t ) , f ( T ) = 0 , g ( T ) = 0 . Substituting it into (3.2)-(3.4) and dividing the expression by x γ γ e f ( t ) r + g ( t ) ,we get H ( π,η ∗ ) ( r, t ) ≤ H ( π ∗ ,η ∗ ) ( r, t ) = 0 ≤ H ( π ∗ ,η ) ( r, t ) , π, η ∈ R n . where H ( π,η ) ( r, t ) := f ′ ( t ) r + g ′ ( t ) + 12 a t f ( t ) + 12 γ ( γ − π Σ t Σ Tt π T + π Σ t a Tt γf ( t )+ γπ Σ t ( λ Tt + η T ) + ηa Tt f ( t ) + ( b t − κ t r ) f ( t ) + γr. Now, it is possible to determine the saddle point. Suppose first that wealready have the saddle point ( π ∗ , η ∗ ). Therefore, H ( π,η ∗ ) ( r, t ) ≤ H ( π ∗ ,η ∗ ) ( r, t ) , π, η ∈ R n and consequently π ∗ t = 1(1 − γ ) ( λ t + η ∗ + a t f ( t ))Σ − t . On the other hand, H ( π ∗ ,η ∗ ) ( r, t ) ≤ H ( π ∗ ,η ) ( r, t ) , η ∈ R n . We should notice first that H forms a linear function in η . In that case, theonly chance to find η ∗ is to delete the expression with η i.e. γπ ∗ Σ t + a t f ( t ) = 0 . This means that π ∗ = − f ( t ) γ a t Σ − t . So, we should have f ( t )(1 − γ ) a t Σ − t + λ t + η ∗ t (1 − γ ) Σ − t = − a t f ( t ) γ Σ − t , which yields η ∗ t = − λ t − f ( t ) γ a t . D.ZAWISZA
Substituting π ∗ and η ∗ into the equation and using the fact that the expres-sion with η is equal to 0, we get f ′ ( t ) r + g ′ ( t ) + 12 | a t | f ( t ) + 12 | a t | f ( t ) ( γ − γ − | a t | f ( t ) − λ t a Tt f ( t ) + ( b t − κ t r ) f ( t ) + γr = 0 . Thus, f ′ ( t ) − κ t f ( t ) + γ = 0 ,g ′ ( t ) + 12 | a t | f ( t ) + 12 | a t | f ( t ) ( γ − γ − | a t | f ( t ) − λ t a Tt f ( t ) + b t f ( t ) = 0 . More explicit forms are: f ( t ) = γe − R T t κ s ds Z T t e R T k κ s ds dk,g ( t ) = Z T t (cid:20) f ( s ) | a s | + 12 | a s | f ( s ) ( γ − γ − | a s | f ( s ) − λ s a Ts f ( s ) + b s f ( s ) (cid:21) ds. We can now summarize our preparatory calculations.
Proposition 3.2.
The pair ( π ∗ , η ∗ ) given by π ∗ t = − f ( t ) γ a t Σ − t , η ∗ t = − λ t − f ( t ) γ a t is a saddle point for problem (2.4) .Proof. Note that π ∗ t and Σ t are deterministic functions. To complete theproof we need only to verify that E ηx,r,t (cid:20) sup t ≤ s ≤ T (cid:12)(cid:12) V ( X π ∗ s , r s , s ) (cid:12)(cid:12)(cid:21) < + ∞ , η ∈ M . We have E ηx,r,t (cid:20) sup t ≤ s ≤T (cid:12)(cid:12) V ( X π ∗ s , r s , s ) (cid:12)(cid:12)(cid:21) = E x,r,t dQ η dP (cid:20) sup t ≤ s ≤T V ( X π ∗ s , r s , s ) (cid:21) . By the Cauchy - Schwarz inequality E x,r,t dQ η dP (cid:20) sup t ≤ s ≤T V ( X π ∗ s , r s , s ) (cid:21) ≤ " E (cid:20) dQ η dP (cid:21) (cid:20) E x,r,t (cid:20) sup t ≤ s ≤ T V ( X π ∗ s , r s , s ) (cid:21)(cid:21) . The explicit formula for the function V leads to V ( X π ∗ s , r s , s ) = 1 γ (cid:2) X π ∗ s (cid:3) γ e f ( s ) r s + g ( s ) . The portfolio process X t is a solution to the linear equation, so X π ∗ s = xe R st [ r l + π ∗ l Σ l λ Tl − ( π ∗ l Σ l Σ Tl π T ∗ l )] dl + R st π ∗ l Σ l dW l . NOTE ON THE WORST CASE APPROACH 7
Note that the process ζ s = e R st κ l dl r s has the dynamics dζ s = e R st κ l dl b s ds + e R st κ l dl a s dW s . We have r s = e − R st κ l dl (cid:20) r + Z st b l dl + Z st a l dW l (cid:21) . By the stochastic Fubini theorem, the expression V ( X π ∗ s , r s , s ) can berewritten in the form V ( X π ∗ s , r s , s ) = xZ s e β ( s ) r s + ξ ( s ) , where the process Z s is a square integrable martingale, β , ξ are boundedand deterministic functions.After repeating the Cauchy - Schwarz inequality once more it is nowsufficient to prove that for any bounded deterministic function ˆ β we have(3.7) E r,t sup t ≤ s ≤ T e ˆ β ( s ) ζ s < + ∞ . Note that the following inequality is true e ˆ β ( s ) ζ s ≤ e ˆ β max ζ s + e ˆ β min ζ s , where ˆ β max = max t ≤ s ≤ T ˆ β ( s ) , ˆ β min = min t ≤ s ≤ T ˆ β ( s ) . Both processes e ˆ β max ζ s , e ˆ β min ζ s are solutions to linear equations with boundedcoefficients and thus usual Lipschitz and linear growth conditions are satis-fied. Property (3.7) follows from standard estimates for stochastic differen-tial equations (see Pham [9, Theorem 1.3.16]). (cid:3) Concluding remarks
To conclude the result we show that the measure Q η ∗ T is not a martingalemeasure i.e. the process S t e − R t r s ds is not a Q η ∗ T - martingale. To see this,it is sufficient to write Q η ∗ T dynamics of S t : dS t = diag ( S t ) (cid:20)(cid:20) r t e − f ( t ) γ Σ t a Tt (cid:21) dt + Σ t dW t (cid:21) . At the end, it is worth to compare the robust investment strategy π ∗ t = 1(1 − γ ) ( λ t + η ∗ t + f ( t ) a t )Σ − t , η ∗ t = − λ t − f ( t ) γ a t with the solution to the traditional utility maximization problem π ∗ t = 1(1 − γ ) ( λ t + f ( t ) a t )Σ − t . It is worth noticing as well that π ∗ can be rewritten as π ∗ t = − f ( t ) γ a t Σ − t = − e − R T t κ s ds Z T t e R T k κ s ds dk (cid:2) a t Σ − t (cid:3) . and it does not depend on the risk aversion coefficient γ . The same propertyis true for η ∗ . D.ZAWISZA Model uncertainty with restrictions
From the practitioner’s point of view, it might be interesting to solvethe problem with restrictions imposed on the uncertainty set M . In thissection we assume that the class M consists of all progressively measurableprocesses taking values in a compact and convex fixed set Γ ⊂ R n .We can use the same function HH ( π,η ) ( r, t ) = f ′ ( t ) r + g ′ ( t ) + 12 a t f ( t ) + 12 γ ( γ − π Σ t Σ Tt π T + π Σ t a Tt γf ( t )+ γπ Σ t ( λ Tt + η T ) + ηa Tt f ( t ) + ( b t − κ t r ) f ( t ) + γr. To find the explicit saddle point for the function H , we start with solvingthe upper Isaacs equation(4.1) min η ∈ Γ max π ∈ R n H ( π,η ) ( r, t ) = 0 . Furthermore, we use known results on max-min theorems (Fan [1, Theorem2]) to verify min η ∈ Γ max π ∈ R n H ( π,η ) ( r, t ) = max π ∈ R n min η ∈ Γ H ( π,η ) ( r, t ) . We can determine a saddle point candidate ( π ∗ , η ∗ ) by finding a Borel mea-surable function η ∗ , such thatmin η ∈ Γ max π ∈ R H ( π,η ) ( r, t ) = max π ∈ R H ( π,η ∗ ) ( r, t )and a Borel measurable function π ∗ , such thatmin η ∈ Γ max π ∈ R H ( π,η ) ( r, t ) = min η ∈ Γ H ( π ∗ ,η ) ( r, t ) . Because the variable η is separated from r , equation (4.1) can be split intotwo equations (the first one has already been solved): f ( t ) = γe − R T t κ s ds Z T t e R T k κ s ds dk, and g ′ ( t ) + 12 | a t | f ( t ) + b t f ( t )+ min η ∈ Γ (cid:20) − γ − γ | λ t + η + f ( t ) a t | + γ − γ ( λ t + η + f ( t ) a t )( λ t + η ) T + f ( t ) a t η T (cid:21) = 0 . Therefore, to find η ∗ , it is sufficient to determine any Borel measurableminimizer to the expression(4.2) − γ − γ | λ t + η + f ( t ) a t | + γ − γ ( λ t + η + f ( t ) a t )( λ t + η ) T + f ( t ) a t η T . Now, let π ∗ be a Borel measurable maximizer of the functionmin η ∈ Γ H ( π,η ) ( r, t ) . NOTE ON THE WORST CASE APPROACH 9
Then, ( π ∗ , η ∗ ) is a saddle point for the function H ( π,η ) ( r, t ). In particular, H ( π,η ∗ ) ( r, t ) ≤ H ( π ∗ ,η ∗ ) ( r, t ) , π ∈ R n . The unique function π ∗ which satisfy the above condition is given by π ∗ t = 1(1 − γ ) ( λ t + η ∗ t + f ( t ) a t )Σ − t . Proposition 4.1.
Suppose that η ∗ is a minimizer of (4.2) and π ∗ t = 1(1 − γ ) ( λ t + η ∗ t + f ( t ) a t )Σ − t . Then the pair ( π ∗ , η ∗ ) is a saddle point for problem (2.4) with the restrictionsimposed by the set Γ . The proof is omitted because it is the repetition of the steps from theproof of Proposition 3.2.
Acknowledgements:
I would like to express my gratitude to the Ref-eree for helping me to improve the first version of the paper.
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Dariusz Zawisza,Faculty of Mathematics and Computer ScienceJagiellonian University in Krakow Lojasiewicza 630-348 Krak´ow, Poland
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