Bond indifference prices and indifference yield curves
aa r X i v : . [ q -f i n . C P ] J u l Bond indifference prices and indifference yield curves
Matthew Lorig (cid:3)
This version: July 21, 2020
Abstract
In a market with stochastic interest rates, we consider an investor who can either (i) invest all if hismoney in a savings account or (ii) purchase zero-coupon bonds and invest the remainder of his wealth ina savings account. The indifference price of the bond is the price for which the investor could achieve thesame expected utility under both scenarios. In an affine term structure setting, under the assumptionthat an investor has a utility function in either exponential or power form, we show that the indifferenceprice of a zero-coupon bond is the root of an integral expression. As an example, we compute bondindifference prices and the corresponding indifference yield curves in the Vasicek setting and interpretthe results.
When the short-rate is modeled as a stochastic process, the market is incomplete and there is not a uniqueno-arbitrage price for a zero-coupon bond. Utility indifference pricing (see Carmona (2009) for an overview)provides a framework for deducing bond prices uniquely. A number of authors have analyzed indifferenceprices of corporate bonds under the assumption of constant interest rates and stochastic default intensi-ties; see Bielecki and Jeanblanc (2006); Sigloch (2009); Houssou and Besson (2010); Jaimungal and Sigloch(2012). By contrast, in this note, we deduce indifference prices of government bonds under the assumptionof a stochastic short-rate and no default. Notably, unlike the vast majority of papers on indifference pricing,which present results for exponential utility only, in this note we consider both exponential and power utility.Although government bonds are sufficiently liquid that there is no practical need to price them, our analysisis still of interest, as it provides an answer to the question: how much would a risk-averse investor be willingto pay for a bond?The rest of this paper proceeds as follows: in Section 2 we describe the dynamics of a money market accountunder the physical (i.e., real-world) probability measure. We consider an investor, who wishes to maximizehis expected utility of wealth at a future date. We define the investor’s indifference price for a zero-couponbond and the corresponding indifference yield curve. In Section 3 we compute the Laplace transform of thefuture value of the money market account. Our main results appear in Section 4, where we show that the (cid:3)
Department of Applied Mathematics, University of Washington. e-mail : [email protected] Let us fix a time horizon T < ∞ and probability space ( Ω , F , P ) with a filtration F = ( F t ) ≥ t ≤ T . Themeasure P represents the physical (i.e., real-world) probability measure and the filtration F represents thehistory of the market. Suppose an investor in this market can invest in two assets (i) a money market account,which grows at the short-rate R = (R t ) ≤ t ≤ T , and (ii) a T -maturity zero-coupon bond P T = (P T t ) ≤ t ≤ T ,which pays one unit of currency at the maturity date T. We wish to answer the following question: Howmuch would this investor be willing to pay for ν zero-coupon bonds ?To answer this question, we suppose that the dynamics of the short-rate R = (R t ) ≤ t ≤ T are of the formdR t = µ ( t , R t )d t + σ ( t , R t )dW t ,where W is a ( P , F )-Brownian motion. We shall assume the µ and σ are such that the short rate belongs tothe class of one-factor affine term-structure (ATS) models µ ( t , r ) = µ ( t ) + µ ( t ) r , σ ( t , r ) = σ ( t ) + σ ( t ) r . (2.1)See (Filipovic, 2009, Chapter 5) for an overview. Let X = (X t ) ≤ t ≤ T denote the value of a portfolio investedentirely in the money market account. The dynamics of X aredX t = R t X t d t .Consider now, an investor who, at time t , has x units of currency invested in the money market accountand also owns ν zero-coupon bonds. The investor’s expected utility at time T isV( t , x , r ; T, ν , γ ) := E t , x , r U (cid:16) X T + ν P TT P TT ; γ (cid:17) = E t , x , r U(X T + ν ; γ ),where U is the investor’s utility function and γ is the investor’s risk aversion . Observe that the investor isusing the bond P T as numéraire. This is a logical choice as, once purchased, the bond provides a known rateof return over the interval [ t , T], whereas the return of the money market account over the interval [ t , T] isstochastic. For the above investor, the money market account is a riskier investment than the bond.Now, consider an investor who has two investment options: he can either (i) invest all of his money in themoney market account, or (ii) purchase ν bonds for p units of currency each, and invest the rest of hiswealth in the money market account. Assuming a time t total wealth x , both options would yield the sameexpected utility if the price p per bond satisfiedV( t , x , r ; T, 0, γ ) = V( t , x – ν p , r ; T, ν , γ ). (2.2)2hus, we define the investor’s indifference price for ν zero-coupon bonds as the solution p ≡ p ( t , x , r ; T, ν , γ )of (2.2). If the market price of a bond were higher than the indifference price, then the investor would preferto invest his entire wealth in the money market account. If the market price of a bond were lower than theindifference price, then the investor would prefer to purchase ν bonds and invest the rest of his wealth inthe money market account.Frequently, bond prices are described in terms of their yield Y T = (Y T t ) ≤ t ≤ T , whereY T t := –1T – t log P T t .Accordingly, we define the investor’s indifference yield y ( t , x , r ; T, ν , γ ) as follows y ( t , x , r ; T, ν , γ ) := –1T – t log p ( t , x , r ; T, ν , γ ). (2.3)If the market yield of a bond were lower than the indifference yield, then the investor would prefer toinvest his entire wealth in the money market account. If the market yield of a bond were higher than theindifference yield, then the investor would prefer purchase ν bonds an invest the rest of his wealth in themoney market account. X T In order to deduce the indifference price of a bond, we shall need the following proposition.
Proposition 3.1.
Let L denote the Laplace transform of X T L( t , x , r ; T, z ) := E t , x , r e – z X T . (3.1) Then the function L satisfies L( t , x , r ; T, z ) = 12 π Z R d ω r z i ω Γ (– i ω )e A( t , ω ;T)+ r B( t , ω ;T)+ i ω log x , ω i > 0, (3.2) where Γ denotes a Gamma function, ω = ω r + i ω i and A and B satisfy ∂ t A + µ B + σ B , A(T, ω ; T) = 0, (3.3)0 = ∂ t B + µ B + σ B + i ω , B(T, ω ; T) = 0. (3.4) Proof.
The function L satisfies the following partial differential equation (PDE)( ∂ t + L )L = 0, L(T, x , r ; T, z ) = e – zx ,where L is the generator of (X, R) and is given explicitly by L = rx ∂ x + µ ( t , r ) ∂ r + σ ( t , r ) ∂ r .Consider the following change of variablesL( t , x , r ; T, z ) = M( t , q ( x ), r ; T, z ), q ( x ) = log x . (3.5)3t is easy to show that the function M satisfies( ∂ t + M )M = 0, M(T, q , r ; T, z ) = exp (– z e q ),where the operator M is given by M = r ∂ q + µ ( t , r ) ∂ r + σ ( t , r ) ∂ r .Now, let b M denote the generalized Fourier transform of M with respect to the q variable b M( t , ω , r ; T, z ) := F [M( t , · , r ; T, z )]( ω ) = Z R d q M( t , q , r ; T, z )e – i ω q .Then, for any ω = ω r + i ω i such that ω i > 0, the function b M satisfies( ∂ t + b M ) b M = 0, b M(T, ω , r ; T, z ) = z i ω Γ (– i ω ), (3.6)where the operator b M is given by b M = i ω r + µ ( t , r ) ∂ r + σ ( t , r ) ∂ r .Now, consider the following Ansatz b M( t , ω , r ; T, z ) = z i ω Γ (– i ω )e A( t , ω ;T)+ r B( t , ω ;T) , A(T, ω ; T) = 0, B(T, ω ; T) = 0. (3.7)Observe that the Ansatz b M in (3.7) satisfies the terminal condition in (3.6). Inserting the Ansatz (3.7) intothe PDE (3.6), recalling that µ are σ are of the form (2.1) and collecting terms of like order in r we findthat A and B satisfy the system of coupled ordinary differential equations (ODEs) given by (3.3) and (3.4).Assuming A and B can be computed either analytically or numerically, then the function M is obtained bytaking the inverse Fourier transform of b M. We haveM( t , q , r ; T, z ) = F –1 [ b M( t , · , r ; T, z )]( q ) = 12 π Z R d ω r b M( t , ω , r ; T, z )e i ω q = 12 π Z R d ω r z i ω Γ (– i ω )e A( t , ω ;T)+ r B( t , ω ;T)+ i ω q .Finally, undoing the variable change (3.5), we find that the Laplace transform of X T is given by (3.2). Having obtained the Laplace transform of the money market account X T , we are now in a position to showthat the indifference price of a bond is the root of an integral expression. We will consider two cases (i) theinvestor has a utility function of exponential form, and (ii) the investor has a utility function in the powerform. Theorem 4.1 (Exponential utility) . Suppose the investor’s utility function U is of the exponential form U( x ; γ ) = –1 γ e – γ x , γ > 0, (4.1)4 hen the indifference price p ≡ p ( t , x , r ; T, ν , γ ) satisfies Z R d ω r γ i ω Γ (– i ω )e A( t , ω ;T)+ r B( t , ω ;T) (cid:16) e i ω log x – e i ω log ( x – ν p )– γν (cid:17) , ω i > 0, (4.2) and the indifference yield y ≡ y ( t , x , r ; T, ν , γ ) satisfies Z R d ω r γ i ω Γ (– i ω )e A( t , ω ;T)+ r B( t , ω ;T) (cid:16) e i ω log x – e i ω log ( x – ν e –(T– t ) y )– γν (cid:17) , ω i > 0, (4.3) where A and B are the solutions of (3.3) and (3.4) , respectivelyProof. Using (3.1), (3.2) and (4.1) we haveV( t , x , r ; T, ν , γ ) = –1 γ E t , x , r e – γ (X T + ν , γ ) = –e – γν γ L( t , x , r ; T, γ )= –e – γν πγ Z R d ω r γ i ω Γ (– i ω )e A( t , ω ;T)+ r B( t , ω ;T)+ i ω log x . (4.4)From (2.2) we find that the indifference price p ≡ p ( t , x , r ; T, ν , γ ) satisfies0 = –2 πγ (cid:16) V( t , x , r ; T, 0, γ ) – V( t , x – ν p , r ; T, ν , γ ) (cid:17) . (4.5)Inserting (4.4) into (4.5) and simplifying yields (4.2). The expression (4.3) for the indifference yield y ≡ y ( t , x , r ; T, ν , γ ) follows from (2.3) and (4.2). Theorem 4.2 (Power utility) . Suppose the investor’s utility function U is of the power form U( x ; γ ) = x γ γ , γ ∈ (0, 1) ∪ (1, ∞ ). (4.6) Then the indifference price p satisfies Z ∞ d zz γ Z R d ω r z i ω Γ (– i ω )e A( t , ω ;T)+ r B( t , ω ;T) (cid:16) e i ω log x – e i ω log ( x – ν p )– z ν (cid:17) , ω i > 0, (4.7) and the indifference yield y ≡ y ( t , x , r ; T, ν , γ ) satisfies Z ∞ d zz γ Z R d ω r z i ω Γ (– i ω )e A( t , ω ;T)+ r B( t , ω ;T) (cid:16) e i ω log x – e i ω log ( x – ν e –(T– t ) y )– z ν (cid:17) , ω i > 0,(4.8) where A and B are the solutions of (3.3) and (3.4) , respectively.Proof. In the case that γ ∈ (0, 1), using the following identity from from Schürger (2002) x α = α Γ (1 – α ) Z ∞ d zz α (cid:16) – zx (cid:17) , α ∈ (0, 1),we obtain from (3.1), (3.2) and (4.6) thatV( t , x , r ; T, ν , γ ) = 1 Γ ( γ ) Z ∞ d zz γ (cid:16) E t , x , r e – z (X T + ν , γ ) (cid:17) = 1 Γ ( γ ) Z ∞ d zz γ (cid:16) – z ν L( t , x , r ; T, z ) (cid:17) Γ ( γ ) Z ∞ d zz γ (cid:16) – z ν π Z R d ω r z i ω Γ (– i ω )e A( t , ω ;T)+ r B( t , ω ;T)+ i ω log x (cid:17) . (4.9)And, in the case that γ ∈ (1, ∞ ), using the following identity from from Schürger (2002)1 x α = 1 Γ ( α ) Z ∞ d z z α –1 e – zx , α > 0,we obtain from (3.1), (3.2) and (4.6) thatV( t , x , r ; T, ν , γ ) = 1(1 – γ ) Γ ( γ – 1) Z ∞ d zz γ E t , x , r e – z (X T + ν , γ ) = –1 Γ ( γ ) Z ∞ d zz γ e – ν z L( t , x , r ; T, z )= –1 Γ ( γ ) Z ∞ d zz γ e – z ν π Z R d ω r z i ω Γ (– i ω )e A( t , ω ;T)+ r B( t , ω ;T)+ i ω log x . (4.10)From (2.2) we have 0 = –2 π Γ ( γ ) (cid:16) V( t , x , r ; T, 0, γ ) – V( t , x – ν p , r ; T, ν , γ ) (cid:17) . (4.11)Inserting either (4.9) or (4.10) into (4.11) we find that, for both γ ∈ (0, 1) and γ ∈ (1, ∞ ), the indifferenceprice p satisfies (4.7). The expression (4.8) for the indifference yield y ≡ y ( t , x , r ; T, ν , γ ) follows from (2.3)and (4.7).Indifference bond prices and yields can be easily found using a numerical root-finding algorithm such as FindRoot[] in Wolfram’s Mathematica.
In this section, we assume that the short-rate R has Vasicek dynamics Vasicek (1977)dR t = κ ( θ – R t )d t + δ dW t .Note that the Vasicek model is an ATS model with µ ( t ) = κθ , µ ( t ) = – κ , σ ( t ) = δ , σ ( t ) = 0. (5.1)Inserting (5.1) into (3.3) and (3.4) and solving the ODEs we obtainA( t , ω ; T) = i ωθκ (cid:16) e – κ (T– t ) – 1 + κ (T – t ) (cid:17) + ω δ κ (cid:16) e –2 κ (T– t ) – 4e – κ (T– t ) + (cid:0) κ (T – t ) (cid:1)(cid:17) ,B( t , ω , T) = i ωκ (cid:16) – κ (T– t ) (cid:17) .Now, suppose that, with the money market account as numéraire, the market were pricing bonds under a risk-neutral (i.e., martingale) measure e P ( λ ) , which is defined by a constant market price of interest raterisk λ d e P ( λ ) d P = exp (cid:16) – 12 λ T – λ W T (cid:17) .6hen the dynamics of R under e P ( λ ) would bedR t = κ ( e θ ( λ ) – R t )d t + δ d f W ( λ ) t , e θ ( λ ) := θ – δλ / κ ,where the process f W ( λ ) = ( f W ( λ ) t ) ≤ t ≤ T , defined by f W ( λ ) t = W t + λ t , is a ( e P ( λ ) , F )-Brownian motion. market bond prices e P T,( λ ) = ( e P T,( λ ) t ) ≤ t ≤ T and the corresponding market yields e Y T,( λ ) = ( e Y T,( λ ) t ) ≤ t ≤ T would be given by e P T,( λ ) t = e E ( λ ) (cid:16) e – R T t R s d s (cid:12)(cid:12)(cid:12) F t (cid:17) =: e p ( t , R t ; T, λ ), e Y T,( λ ) t = – log e P T,( λ ) t T – t = – log e p ( t , R t ; T, λ )T – t =: e y ( t , R t ; T, λ ).The function e p satisfies the following PDE( ∂ t – r + e A ( λ ) ) e p = 0, e p (T, r ; T, λ ) = 1,where the operator e A ( λ ) is the generator of R under e P ( λ ) and is given explicitly by e A ( λ ) = κ ( e θ ( λ ) – r ) ∂ r + δ ∂ r .As R has ATS dynamics under e P ( λ ) , one can easily verify that e p is given by e p ( t , r ; T, λ ) = e e A( t ;T, λ )+ r e B( t ;T, λ ) , (5.2)where the functions e A and e B are defined as follows e A( t ; T, λ ) := A( t , i ; T), e B( t ; T, λ ) := B( t , i ; T), θ → e θ ( λ ).Now, let us consider an investor with an exponential utility function, as described in Theorem 4.1. Fix thefollowing parameters κ = 0.05, θ = 0.03, δ = √ κ θ , t = 0, x = 5, r = 0.01.On the left side of Figure 1 we plot the indifference yield curve y ( t , x , r ; T, ν , γ ) (i.e., the solution of (4.3))as a function of T with the number of bonds purchased fixed at ν = 1 and with the risk-aversion parametervarying from γ = 0.1 (blue) to γ = 0.2 ( magenta ). For comparison, we also plot the market yield curve e y ( t , x , r ; T, λ ) (dashed, black) assuming the market prices of risk is zero λ = 0. Observe that as the risk-aversion parameter γ increases the corresponding indifference yield curves decrease. This is consistent withthe intuition that, the more risk-averse an investor is, the more willing he would be to accept a low butknown rate of return of a bond rather than a potentially higher but uncertain rate of return of a moneymarket account.On the right side of Figure 1 we plot the indifference yield curve y ( t , x , r ; T, ν , γ ) as a function of T withthe risk-aversion parameter γ = 0.15 fixed and with the number of bonds purchased varying from ν = –4.5(blue) to ν = 4.5 ( magenta ). Note that a negative ν indicates selling bonds. For comparison, we also7lot the market yield curve e y ( t , x , r ; T, λ ) (dashed, black) assuming the market prices of risk is zero λ = 0.Observe that as the number of bonds purchased increases the corresponding indifference yield curves alsoincrease. This is consistent with the intuition that a risk-averse seller of a bond will ask a higher price thana risk-averse buyer of a bond is willing to bid (thereby resulting in a bid-ask spread).Using (5.2), a direct computation shows that the dynamics of the market bond price e P T,( λ ) under thephysical probability measure P satisfiesd (cid:16) e P T,( λ ) t X t (cid:17) = Q T,( λ ) t (cid:16) e P T,( λ ) t X t (cid:17) d t + S T,( λ ) t (cid:16) e P T,( λ ) t X t (cid:17) dW t ,where the drift Q T,( λ ) and volatility S T,( λ ) are given byQ T,( λ ) t = 1 e p ( t , R T ; T, λ ) (cid:16) ∂ t – R t + κ ( θ – R t ) ∂ r + δ ∂ r (cid:17)e p ( t , R T ; T, λ ), (5.3)S T,( λ ) t = 1 e p ( t , R T ; T, λ ) δ∂ r e p ( t , R T ; T, λ ). (5.4)One can show from (5.2) that Q T,( λ ) t S T,( λ ) t = λ ,which allows us to interpret the market price of risk λ as the return Q T,( λ ) that the market pays an investorin order to bear the risk of holding a bond with volatility S T,( λ ) . Using the above, we can (and we do) definethe investor’s indifference price of interest rate risk as the solution λ ≡ λ ( t , x , r ; T, ν , γ ) of p ( t , x , r ; T, ν , γ ) = e p ( t , r ; T, λ ).On the left side of Figure 2, we plot the market yield e y ( t , r ; T, λ ) as a function of T with the market priceof risk λ varying from λ = –0.05 (blue) to λ = 0.05 ( magenta ). For reference, we also plot e y ( t , r ; T, 0)(dashed black). In the figure, we see that, as the market price of risk λ increases, the corresponding yieldcurve decreases. This is to be expected, as the long-run mean e θ ( λ ) of R under e P ( λ ) is a decreasing functionof λ . A lower long-run mean e θ ( λ ) results in a higher bond price and therefore a lower yield.On the right-hand side of Figure 2 we plot the investor’s indifference price of interest rate risk λ ( t , x , r ; T, ν , γ )as a function of T with the investor’s risk aversion varying from γ = 0.1 (blue) to γ = 0.2 ( magenta ). Tounderstand the plot, consider the following: the investor described in Section 2 has chosen the bond asnuméraire. For this investor, it is the money market account, rather than the bond, that is the risky asset.A routine application of Itô’s Lemma shows that X/P T,( λ ) satisfiesd (cid:16) X t P T,( λ ) t (cid:17) = (cid:16) (S T,( λ ) t ) – Q T,( λ ) t (cid:17)(cid:16) X t P T,( λ ) t (cid:17) d t + S T,( λ ) t (cid:16) X t P T,( λ ) t (cid:17) (–dW t ).One can show from (5.2), (5.3) and (5.4) that(S T,( λ ) t ) – Q T,( λ ) t S T,( λ ) t = – λ – δκ (cid:16) – κ (T– t ) (cid:17) .8he higher investor’s indifference price of interest rate risk λ , the lower the reward the investor perceivesfor bearing the risk of holding the money market account. Thus, as the investor’s risk-aversion γ goes up,we expect the investor’s implied price of interest rate risk to go up as well, and this is precisely what we seein Figure 2. Although we have not focused on it, one could have repeated the analysis of this manuscript assuming theinvestor had used a money market account M = (M t ) ≤ t ≤ T , rather than the bond, as his numéraire. In thiscase, if the investor at x units of currency invested in the moneny market account at time t and additionallyowned ν bonds, his expected utility would beV( t , x , m , r ; T, ν , γ ) := E t , x , m , r U (cid:16) X T + ν P TT M T ; γ (cid:17) = E t , x , m , r U (cid:16) X T + ν M T ; γ (cid:17) ,where dM t = R t M t d t . In this scenario, the bond P T would be considered the risky asset and the moneymarket account M would be considered the safe asset. Rather than repeat the entire analysis here, we simplynote that, if the investor has exponential utility, the indifference prices of a bond can be computed explicitlyas a numerical integral p ( t , m , r ; T, ν , γ ) = – m γν log π Z R d ω r Γ ( i ω )e A( t , ω ;T)+ r B( t , ω ;T)+ i ω log m γν , ω i < 0,and if the investor has power utility, the indifference price of a bond p ≡ p ( t , x , m , r ; T, ν , γ ) satisfies0 = Z ∞ d zz γ e – zx / m (cid:16) z ν p / m π Z R d ω r Γ ( i ω )e A( t , ω ;T)+ r B( t , ω ;T)+ i ω log mz ν (cid:17) , ω i < 0,where A and B are the solutions to (3.3) and (3.4), respectively.9 eferences Bielecki, T. and M. Jeanblanc (2006). Indifference pricing of defaultable claims.
Indifference Pricing,Theory and Applications , 211–240.Carmona, R. (2009).
Indifference Pricing: Theory and Applications . Princeton Series in FinancialEngineering. Princeton University Press.Filipovic, D. (2009).
Term-Structure Models: A Graduate Course . Springer Finance. Springer BerlinHeidelberg.Houssou, R. and O. Besson (2010). Indifference of defaultable bonds with stochastic intensity models. arXivpreprint arXiv:1003.4118 .Jaimungal, S. and G. Sigloch (2012). Incorporating risk and ambiguity aversion into a hybrid model ofdefault.
Mathematical Finance 22 (1), 57–81.Schürger, K. (2002). Laplace transforms and suprema of stochastic processes. In
Advances in Finance andStochastics , pp. 285–294. Springer.Sigloch, G. (2009).
Utility Indifference Pricing of Credit Instruments . University of Toronto.Vasicek, O. (1977). An equilibrium characterization of the term structure.
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Effect of γ on y Effect of ν on y Figure 1:
Left : Indifference yields y ( t , x , r ; T, ν , γ ) as a function of T with ν = 1 fixed and risk aversionvarying from γ = 0.1 (blue) to γ = 0.2 ( magenta ). Right : Indifference yields y ( t , x , r ; T, ν , γ ) as a functionof T with γ = 0.15 and ν varying from ν = –4.5 (blue) to ν = 4.5 ( magenta ). Right and Left : The dashedblack line if the risk-neutral yield curve e y ( t , x , r ; T, λ ) with λ = 0 fixed. - - Effect λ on e y Effect of γ on λ Figure 2:
Left : Risk-neutral yields e y ( t , r ; T, λ ) as a function of T with λ varying from λ = –0.05 (blue)to λ = 0.05 ( magenta ). The black-dashed line corresponds to λ = 0. Right : Investor’s indifference priceof interest rate risk λ ( t , x , r ; T, ν , γ ) as a function of T with ν = 1 fixed and the risk aversion varying from γ = 0.1 (blue) to γ = 0.2 ( magentamagenta