## _wp03201

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---

### Introduction — Key points and arguments
- Puttable and extendible bonds:
  - Give investors the right to sell bonds back at par at a predetermined date (puttable) or to extend bond term beyond a specified date (extendible).
  - Are common among private sector issuers in developed countries and in international financial markets, but not common among public sector issuers in both developed and emerging markets.
- Paper objectives:
  - Analyze key characteristics: price stabilizing properties, potential to start interest rate derivative markets, and use by governments.
- Main analytical insight:
  - Merton (1995) shows puttable bonds are economically equivalent to a portfolio of discount bonds plus a risk-free short-term bond.
  - When discount bond prices fall, the put option limits investor interest rate exposure; when prices rise, the put option is worthless and exposure increases.
  - Extension: the result holds for coupon bonds and bonds subject to credit risk — prices of puttable and extendible bonds fall less when underlying bond prices fall.
- Policy and market implications:
  - Puttable and extendible bonds can capture convexity gains from volatility in emerging markets where interest rate derivative markets (short selling, futures) are not available.
  - They can act as cushions replicating trading gains that would arise from hedging long-term bonds by short selling short-term bonds and futures.
  - Suggested role for governments: include puttable and extendible bonds as part of debt management strategy alongside development of interest rate derivative markets for government bonds.
- Historical context and cautions:
  - OECD evidence: some countries used puttable and extendible bonds in the 1980s when short-term rates were high and volatile and when futures and short selling were uncommon (example: Italy in the 1980s).
  - Risks:
    - Traders may have incentives to increase underlying bond volatility to raise probability of option exercise.
    - If all bonds were puttable/extendible, widespread exercise could create serious liquidity problems for governments.
  - Fiscal implications:
    - Early redemption risk is not fundamentally different from a government buying back option-free debt before maturity.
    - Embedded options make explicit an implicit guarantee the government might otherwise provide, potentially reducing market jitters and overshooting.
  - Signaling:
    - Issuing such bonds signals government commitment to macroeconomic stability and fiscal discipline because it is not in the government’s interest for holders to exercise embedded options.

### The Stabilizing Properties of Puttable and Extendible Bonds — Analytical findings
- Market constraint motivating analysis:
  - Markets for short selling bonds and futures contracts are not available to investors in emerging markets, depriving them of convexity gains.
- Hedging framework and key equations (notation preserved):
  - Portfolio value: P_t dollars invested in a portfolio of a long-term bond B_t^L and a short-term bond B_t^S:
    - 12 LS tt t PBBθθ = +  (equation (1) in source)
  - Portfolio change:
    - 12 LS tt t dPdBdBθθ = +  (equation (2) in source)
  - One-factor interest rate model assumption: single source of randomness, same Wiener process diffusion terms for short- and long-term bonds.
  - Hedge ratios that eliminate randomness (equation (3) in source), with σ_L and σ_S as diffusion parameters:
    - θ1 = ( ) S t L SLt P B σ σ σ =
    - θ2 = ( ) L t S SLt P B σ σ σ = − −
- Technical implication:
  - Selecting appropriate quantities of long and short bonds (dynamic hedging) eliminates randomness driven by the single interest-rate source of risk.
  - Puttable and extendible bonds, via embedded options, provide payoff structures that limit downside when underlying bond prices fall, stabilizing prices relative to plain-vanilla bonds.
- Credit risk extension:
  - The stabilizing result applies when underlying bonds are coupon-paying or subject to credit risk — puttable/extendible bond prices fall less than the underlying when those prices decline.

### Appendix I — Hedging long-term bonds with short-term instruments (one-factor framework)
- Delta-hedged dynamics and hedge ratios:
  - No-arbitrage condition (paper notation):
    - (4)  () 121   2 LSL   S ttttt rB   Bdt dB   dBθθ   θ θ+=+,
  - Dividing by θ1 leads to equation (5); using hedge-ratio approximation θ2/θ1 ≈ B_y^L gives delta-hedged dynamics:
    - (6)  ()  LLSLLS ttyttyt r   BB  BdtdBB  dB+=+,
  - Hedge ratio B_y^L equals the first derivative of the long bond price with respect to its yield-to-maturity and is negative; it changes as yields change, requiring dynamic rebalancing.
- Practical constraints in emerging markets:
  - Only 10 out of 22 emerging market countries allow short selling bonds (IOSCO (2002) compilation).
  - Securities borrowing and lending availability is similar in frequency to short-selling permissions.
  - Where repos/reverse repos exist but are thin or impaired by tax/other impediments, short squeezes can make short selling impractical.
  - Bond futures are rare in emerging markets, limiting alternative delta-hedging routes.
- Puttable/extendible bond definitions and replication:
  - Put payoff at T for strike E: (7)  () TT FMax  B   E=.
  - Puttable bond = straight bond + put option; extendible bond = short-term bond + call option to buy a long-term bond at price E at T.
  - Replication/replication-based pricing:
    - Ft = F(Bt, t). (8)
    - Portfolio P_t = θ1 Ft + θ2 Bt. (9)
    - Eliminating randomness requires θ1 = 1 and θ2 = −F1 (F1 = ∂F/∂B). (11)
    - No-arbitrage return condition → (13) () 11 ttttt dFF dBr   FF B   dt=+−.
  - Economic equivalence: a puttable bond is equivalent to F1 units of the discount bond and (F_t − B_t F1) units of the short-term bond.
- Stabilizing (cushion) mechanism:
  - 0 ≤ F1 ≤ 1 and F11 > 0; when B_t falls, F1 falls (government effectively repurchases discount bonds), reducing interest-rate exposure of investors. When B_t rises, F1 rises.
  - When embedded put is in-the-money, a puttable bond behaves like a short-term bond.
  - Coupon-paying underlyings introduce an extra coupon term; analogous economic equivalence holds.

### Credit risk adjustments for puttable bonds
- Default-adjusted pricing:
  - Price of a risky underlying and put payoff priced under risk-neutral expectation discounted at (r_t + h_t L_t), where h_t is hazard rate and L_t = 1 − R_t is loss process. (16)
  - No-arbitrage dynamics with credit risk:
    - (17) () 11 tttttttt dFF dBrh LFF B   dt=++  −.
  - Economic equivalence persists: puttable bonds correspond to F1 units of underlying risky bond and (F_t − F1 B_t) units of a short-term bond adjusted for credit-risk premium.
- Conclusion:
  - Puttable bonds still mimic markets for short selling and futures when underlying incorporates credit risk.

### Empirical illustration — The Brazilian experience (1997–2002)
- Context and policy actions:
  - Brazil’s 1999 program to lengthen term, reduce financing costs: more than twenty measures, partial implementation improved average maturity and share of discount bonds in 2000.
  - After Argentine crisis and U.S. recession in 2001, authorities reduced share of discount bonds due to liquidity risk.
  - May 2002 mark-to-market accounting and election uncertainty triggered high discounts on indexed debt and large volatility in discount bond prices; mutual funds experienced withdrawals ~ R$ 55 billion from May 2002 to August 2002 (National Treasury figures).
  - Authorities responded by shortening debt maturity and exempting assets with maturity < one year from fair-value accounting; central bank considered selling put options on bonds.
- Modeling convexity gains for Brazil (one-factor setup):
  - Discount bond price PDE and boundary conditions: (18) and (19).
  - Convexity (Gamma) gains arise from the 1/2 B11 b(r,t)^2 term; increased interest-rate volatility can increase long-bond prices ceteris paribus (if credit risk unchanged).
  - Vasicek model used to simulate C-Bond discount curve:
    - Short-rate dynamics: (22) d r_t = a(d − r_t) dt + b r_t dW.
    - Parameter values used for Brazilian C-bond: 0.50,0.90,0.11bad===, evaluated at mean interest rate 11 percent. (23)
    - Two volatilities used: 5 percent a year (absolute) and 15 percent.
  - Simulation findings:
    - Increased volatility raises long-bond prices substantially: a 30-year bond price ~ 30 percent higher in the higher-volatility environment; a 10-year bond price increases by about 10 percent.
  - Implication:
    - Lack of liquid interest-rate derivatives and the tendency for volatility to increase credit risk prevent traders from extracting convexity gains, leading to adverse portfolio reactions and debt-maturity shortening.
- Puttable bond effect in Brazil simulation:
  - Using daily C-Bond returns (January 1, 1997–October 30, 2002), a 10-year discount bond experienced explosive price volatility (bond decreased by about 50 percent during election/mark-to-market period).
  - Adding a 1-year put option (strike 40) via Black-Scholes approximation would have stabilized portfolios and led to higher overall bond values in the critical period.
  - Note: a fuller joint sovereign-credit-and-bond simulation would be more complex and was beyond scope.

### Historical evidence and issuance patterns
- OECD experience:
  - Puttable/extendible issues documented in Austria, Belgium, Canada, France, Germany, Italy, Netherlands, UK, and U.S.; many issues in 1980s when rates were high and volatile.
- Belgium case (State notes, extendible):
  - State notes increased from 0.64 percent of long-term debt and 0.50 percent of total domestic government debt in 1996 to 3.31 percent and 2.85 percent in 2001.
  - Outstanding state notes (millions of Euros): 1996: 1,106; 1997: 2,286; 1998: 3,880; 1999: 4,913; 2000: 6,119; 2001: 7,122. (Table 2)
  - Descriptive statistics (September 14, 2000–October 25, 2002) comparing 4.2% state note (1998) and 4 3/4% linear bond (1999):
    - Mean: 0.01 vs 0.01
    - Median: 0.00 vs 0.01
    - Maximum: 0.43 vs 1.71
    - Minimum: -0.36 vs -1.72
    - Std. Dev.: 0.10 vs 0.20
    - Skewness: -0.03 vs -0.19
    - Kurtosis: 2.38 vs 18.80
    - Observations: 552 each (Table 3)
- Emerging market sovereign issuance (millions of U.S. dollars):
  - 1997: 3,052 (of which loans 555); number of transactions 22
  - 1998: 4,064 (of which loans 358); number of transactions 21
  - 1999: 2,543 (of which loans 942); number of transactions 12
  - 2000: 1,295 (of which loans 650); number of transactions 7
  - 2001: 2,062 (table totals)

### Key findings and policy recommendations
- Key findings:
  - Puttable and extendible bonds are less price sensitive to increases in short-term interest rates than the underlying bonds; when underlying prices fall, puttable/extendible bond prices fall less.
  - These instruments embed a self-adjusting hedge (F1 between 0 and 1, F11 > 0) that reduces investor exposure automatically when underlying prices fall.
  - Puttable/extendible bonds can mimic markets for short-selling short-term bonds and bond futures and thereby provide access to convexity gains in markets lacking derivatives.
  - In many emerging markets, absence of liquid derivatives markets (short selling, securities lending, bond futures, interest-rate swaps) prevents extraction of convexity gains, exacerbating volatility and incentivizing shorter maturities.
- Policy recommendations and suggested uses:
  - Governments could issue puttable or extendible bonds as a transitory financing source until liquid and transparent interest-rate derivative markets are established.
  - As a debt-management tool, puttable/extendible bonds can facilitate a move from indexed debt to fixed-coupon debt and serve as an alternative to issuing indexed debt during market turbulence.
  - Puttable/extendible bonds offer a cheaper source of financing (lower coupon) because the government charges investors for the embedded insurance (put option).
  - Governments issuing such bonds should ensure ex ante fiscal credibility and sufficient resources to honor potential early redemptions.
  - Monitor incentives that may induce market participants to increase underlying bond volatility.
  - Avoid universal issuance of puttable/extendible bonds to prevent liquidity strains from mass option exercise.
- Historical precedent:
  - OECD usage during periods of high volatility suggests feasibility and precedent for such issuance.

*Source: IMF working paper content unit _wp03201 (references and appendices excerpt as supplied).*

### References..............................................................................................................

### _wp03201 - References..............................................................................................................

### Figures and Tables Inventory
- Figures listed:
  - 1. Use of Interest Rate Derivatives by Emerging Markets ................................................6
  - 2. Brazil: Composition of Government Securities, June 1994–December 2002 .............13
  - 3. Brazil: Average Maturity of Outstanding Discount Bonds, July 1996–December 2002 ....................................................................................13
  - 4. Discount Bond Prices as a Function of Interest Rate Volatility ..................................17
  - 5. Brazil: C-Bond Price, January 1, 1997–October 30, 2002 ..........................................18
  - 6. Brazil: Puttable Bond Price, January 1, 1997–October 30, 2002  ...............................18
- Tables listed:
  - 1. Puttable and Extendible Bonds in Selected OECD Countries .....................................19
  - 2. Belgium: Outstanding State Notes, 1996–2001...........................................................20
  - 3. Belgium: Descriptive Statistics for Price Changes of 1998 State Note and 1999 Linear Bond, September 14, 2000–October 25, 2002 ....................................20
  - 4. Sovereign Puttable Bonds and Loans by Emerging Markets, 1997–2001...................21

### I. Introduction — Key points and arguments
- Puttable and extendible bonds:
  - Give investors the right to sell bonds back at par at a predetermined date (puttable) or to extend bond term beyond a specified date (extendible).
  - Are common among private sector issuers in developed countries and in international financial markets, but not common among public sector issuers in both developed and emerging markets.
- Paper objectives:
  - Analyze key characteristics: price stabilizing properties, potential to start interest rate derivative markets, and use by governments.
- Main analytical insight:
  - Merton (1995) shows puttable bonds are economically equivalent to a portfolio of discount bonds plus a risk-free short-term bond.
  - When discount bond prices fall, the put option limits investor interest rate exposure; when prices rise, the put option is worthless and exposure increases.
  - Extension: the result holds for coupon bonds and bonds subject to credit risk — prices of puttable and extendible bonds fall less when underlying bond prices fall.
- Policy and market implications:
  - Puttable and extendible bonds can capture convexity gains from volatility in emerging markets where interest rate derivative markets (short selling, futures) are not available.
  - They can act as cushions replicating trading gains that would arise from hedging long-term bonds by short selling short-term bonds and futures.
  - Suggested role for governments: include puttable and extendible bonds as part of debt management strategy alongside development of interest rate derivative markets for government bonds.
- Historical context and cautions:
  - Informal OECD evidence: some countries used puttable and extendible bonds in the 1980s when short-term rates were high and volatile and when futures and short selling were uncommon.
  - Examples cited: Italy in the 1980s; suggested for lengthening debt maturity in Brazil (Favero and Giavazzi (2002)).
  - Risks:
    - Traders may have incentives to increase underlying bond volatility to raise probability of option exercise.
    - If all bonds were puttable/extendible, widespread exercise could create serious liquidity problems for governments.
  - Fiscal implications:
    - Early redemption risk is not fundamentally different from a government buying back option-free debt before maturity.
    - Embedded options make explicit an implicit guarantee the government might otherwise provide, potentially reducing market jitters and overshooting.
  - Signaling:
    - Issuing such bonds signals government commitment to macroeconomic stability and fiscal discipline because it is not in the government’s interest for holders to exercise embedded options.

### II. The Stabilizing Properties of Puttable and Extendible Bonds — Analytical findings
- Market constraint motivating analysis:
  - Markets for short selling bonds and futures contracts are not available to investors in emerging markets, depriving them of convexity gains.
- Main analytical question:
  - How can puttable and extendible bonds replicate convexity gains that arise when investors hedge long-term bond exposure by short selling short-term bonds and futures?
- Approach:
  - Analyze dynamic hedging strategies and stabilizing properties; incorporate credit risk aspects of bonds.
- Hedging setup (notation and structure preserved):
  - Portfolio value: Pt dollars invested in a portfolio of a long-term bond BtL and a short-term bond BtS:
    - 12 LS tt t PBBθθ = + (equation (1) in source)
  - Portfolio change:
    - 12 LS tt t dPdBdBθθ = + (equation (2) in source)
  - Assumption: a one-factor interest rate model describes short-term interest rate dynamics, implying a single source of randomness and the same Wiener process diffusion terms for short- and long-term bonds.
  - Hedge ratios that eliminate randomness (equation (3) in source), with σL and σS as diffusion parameters for long-term and short-term bonds:
    - θ1 = ( ) S t L SLt P B σ σ σ =
    - θ2 = ( ) L t S SLt P B σ σ σ = − −
- Key technical implication:
  - By selecting appropriate quantities of long and short bonds (dynamic hedging), investors can eliminate randomness in the portfolio driven by the single interest-rate source of risk.
  - Puttable and extendible bonds, via their embedded options, provide payoff structures that limit downside when underlying bond prices fall, thus stabilizing prices relative to plain-vanilla bonds.
- Credit risk extension:
  - The stabilizing result for puttable and extendible bonds applies when underlying bonds are coupon-paying or subject to credit risk — they fall less in price than the underlying when those prices decline.

### Policy recommendations and strategic uses
- Short- to medium-term market development:
  - Use puttable and extendible bonds to temporarily overcome obstacles in developing long-term government bond markets where derivative instruments (futures, short selling) are not available.
- Debt management strategy:
  - Consider incorporating puttable and extendible bonds in a debt management program that simultaneously pursues the development of interest rate derivative markets for government bonds.
- Fiscal and signaling considerations:
  - Governments issuing such bonds should ensure ex ante fiscal credibility and sufficient resources to honor potential early redemptions.
  - Issuance can serve as a commitment device signaling macroeconomic stability and fiscal discipline, since option exercise is disadvantageous to the issuer.
- Cautionary measures:
  - Monitor incentives that may induce market participants to increase underlying bond volatility.
  - Avoid universal issuance of puttable/extendible bonds to prevent liquidity strains from mass option exercise.

*Italic: Content derived solely from the supplied IMF chapter/section excerpt.*

### Appendix I shows how the parameters describing the dynamics for discount bonds are a

### _wp03201 - Appendix I shows how the parameters describing the dynamics for discount bonds are a

### Hedging long-term bonds with short-term instruments (one-factor framework)
- Portfolio construction:
  - Long one unit of a long-term bond B_L and short θ2/θ1 units of a short-term bond yields a risk-free portfolio. Key no-arbitrage condition (in the paper notation):
    - (4)  () 121   2 LSL   S ttttt rB   Bdt dB   dBθθ   θ θ+=+,
  - Dividing by θ1 leads to equation (5) and, using the hedge-ratio approximation θ2/θ1 ≈ B_y^L, gives the delta-hedged dynamics:
    - (6)  ()  LLSLLS ttyttyt r   BB  BdtdBB  dB+=+,
  - Hedge ratio B_y^L equals the first derivative of the long bond price with respect to its yield-to-maturity and is negative; it changes as yields change, requiring dynamic rebalancing (dynamic hedging).
- Practical constraints in emerging markets:
  - Only 10 out of 22 emerging market countries allow short selling bonds (IOSCO (2002) compilation).
  - Securities borrowing and lending availability is similar in frequency to short-selling permissions.
  - Where repos/reverse repos exist but are thin or impaired by tax/other impediments, short squeezes can make short selling impractical.
  - Bond futures are rare in emerging markets, limiting alternative delta-hedging routes.

### Puttable and extendible bonds as market-mimicking instruments
- Definitions and equivalence:
  - Puttable bond holder has the right to sell the bond back at strike price E at time T ≤ U. At T the puttable bond price equals:
    - (7)  () TT FMax  B   E=.
  - A puttable bond = straight bond + put option on the underlying bond; higher price / lower coupon than straight bond.
  - Extendible bond = short-term bond + call option to buy a long-term bond at price E at T. Put-call parity (European options) links valuation to puttable bonds.
- Replicating/replication-based pricing and hedging:
  - Puttable bond price Ft depends on Bt: Ft = F(Bt, t). (8)
  - Portfolio P_t = θ1 Ft + θ2 Bt. (9)
  - Eliminating randomness requires θ1 = 1 and θ2 = -F1 (F1 = ∂F/∂B, the hedge ratio). (11)
  - No-arbitrage return condition leads to economic equivalence:
    - (13) () 11 ttttt dFF dBr   FF B   dt=+−.
  - Interpretation: a puttable bond is equivalent to F1 units of the discount bond and (Ft − B_t F1) units of the short-term bond.
- Stabilizing (cushion) mechanism:
  - 0 ≤ F1 ≤ 1 and F11 > 0; when Bt falls, F1 falls (government effectively repurchases discount bonds), reducing interest-rate exposure of investors. When Bt rises, F1 rises.
  - When the embedded put becomes in-the-money, a puttable bond behaves like a short-term bond.
  - For coupon bonds, an extra coupon term appears; analogous economic equivalence holds. (Equations (14), (15))
- Policy implication:
  - Puttable/extendible bonds can mimic short-selling and bond-futures markets, enabling investors to obtain convexity gains without actual short selling or futures.

### Credit risk adjustments for puttable bonds
- Using default-adjusted discounting:
  - Price of a risky underlying and put payoff priced under risk-neutral expectation discounted at (r_t + h_t L_t), where h_t is hazard rate and L_t = 1 − R_t is loss process. (16)
  - No-arbitrage dynamics with credit risk:
    - (17) () 11 tttttttt dFF dBrh LFF B   dt=++  −.
  - Economic equivalence persists: puttable bonds correspond to F1 units of underlying risky bond and (Ft − F1 B_t) units of a short-term bond adjusted for credit-risk premium.
- Conclusion:
  - Puttable bonds still mimic markets for short selling and futures when underlying incorporates credit risk.

### Empirical illustration and the Brazilian experience (1997–2002)
- Context and policy actions:
  - Brazil’s 1999 program to lengthen term, reduce financing costs: more than twenty measures, partial implementation improved average maturity and share of discount bonds in 2000.
  - After Argentine crisis and U.S. recession in 2001, authorities reduced share of discount bonds due to liquidity risk.
  - May 2002 mark-to-market accounting and election uncertainty triggered high discounts on indexed debt and large volatility in discount bond prices; mutual funds experienced withdrawals ~ R$ 55 billion from May 2002 to August 2002 (National Treasury figures).
  - Authorities responded by shortening debt maturity and exempting assets with maturity < one year from fair-value accounting; central bank considered selling put options on bonds.
- Modeling convexity gains for Brazil:
  - Discount bond price partial differential equation under one-factor models: (18) and boundary (19).
  - Convexity (Gamma) gains arise from the 1/2 B11 b(r,t)^2 term; increased interest-rate volatility can increase long-bond prices ceteris paribus (if credit risk unchanged).
  - Vasicek model used to simulate C-Bond discount curve:
    - Short-rate dynamics: (22) d r_t = a(d − r_t) dt + b r_t dW.
    - Parameter values used for Brazilian C-bond: 0.50,0.90,0.11bad===, evaluated at mean interest rate 11 percent. (23)
    - Two volatilities used: 5 percent a year (absolute) and 15 percent.
  - Simulation findings:
    - Increased volatility raises long-bond prices substantially: a 30-year bond price ~ 30 percent higher in the higher-volatility environment; a 10-year bond price increases by about 10 percent.
  - Implication for emerging markets:
    - Lack of liquid interest-rate derivatives and the tendency for volatility to increase credit risk prevent traders from extracting convexity gains, leading to adverse portfolio reactions and debt-maturity shortening.
- Puttable bond effect in Brazil simulation:
  - Using daily C-Bond returns (Jan 1, 1997–Oct 30, 2002), a 10-year discount bond experienced explosive price volatility (bond decreased by about 50 percent during election/mark-to-market period).
  - Adding a 1-year put option (strike 40) via Black-Scholes approximation would have stabilized portfolios and led to higher overall bond values in the critical period (Figure 6 discussion).
  - Authors note a fuller joint sovereign-credit-and-bond simulation would be more complex and was beyond scope.

### Historical evidence and issuance patterns
- OECD experience:
  - Table 1 lists puttable/extendible issues in Austria, Belgium, Canada, France, Germany, Italy, Netherlands, UK, and U.S.; many issues in 1980s when rates were high and volatile.
- Belgium case:
  - State notes (extendible) increased from 0.64 percent of long-term debt and 0.50 percent of total domestic government debt in 1996 to 3.31 percent and 2.85 percent in 2001.
  - Outstanding state notes (millions of Euros): 1996: 1,106; 1997: 2,286; 1998: 3,880; 1999: 4,913; 2000: 6,119; 2001: 7,122. (Table 2)
  - Descriptive statistics (Sep 14, 2000–Oct 25, 2002) comparing 4.2% state note (1998) and 4 3/4% linear bond (1999):
    - Mean: 0.01 vs 0.01
    - Median: 0.00 vs 0.01
    - Maximum: 0.43 vs 1.71
    - Minimum: -0.36 vs -1.72
    - Std. Dev.: 0.10 vs 0.20
    - Skewness: -0.03 vs -0.19
    - Kurtosis: 2.38 vs 18.80
    - Observations: 552 each (Table 3)
- Emerging market sovereign issuance:
  - Sovereign puttable bonds and loans by emerging markets (millions of U.S. dollars):
    - 1997: 3,052 (of which loans 555); number of transactions 22
    - 1998: 4,064 (of which loans 358); number of transactions 21
    - 1999: 2,543 (of which loans 942); number of transactions 12
    - 2000: 1,295 (of which loans 650); number of transactions 7
    - 2001: 2,062 (table totals)
  - Source: IMF (2002). (Table 4)

### Key findings and policy recommendations
- Key findings:
  - Puttable and extendible bonds are less price sensitive to increases in short-term interest rates than the underlying bonds; when underlying prices fall, puttable/extendible bond prices fall less.
  - These instruments embed a self-adjusting hedge (F1 between 0 and 1, F11 > 0) that reduces investor exposure automatically when underlying prices fall.
  - Puttable/extendible bonds can mimic markets for short-selling short-term bonds and bond futures and thereby provide access to convexity gains in markets lacking derivatives.
  - In many emerging markets, absence of liquid derivatives markets (short selling, securities lending, bond futures, interest-rate swaps) prevents extraction of convexity gains, exacerbating volatility and incentivizing shorter maturities.
- Policy recommendations and suggested uses:
  - Governments could issue puttable or extendible bonds as a transitory financing source until liquid and transparent interest-rate derivative markets are established.
  - As a debt-management tool, puttable/extendible bonds can facilitate a move from indexed debt to fixed-coupon debt and serve as an alternative to issuing indexed debt during market turbulence.
  - Puttable/extendible bonds offer a cheaper source of financing (lower coupon) because the government charges investors for the embedded insurance (put option).
  - Historical OECD usage during periods of high volatility suggests feasibility and precedent for such issuance.

### Appendix — pricing and stochastic dynamics (selected equations and parameters)
- Short-rate dynamics assumed (one-factor):
  - (25) () (, ) t t t dr a r t dt b r t dW=+,
- Bond price as function B(r_t, t; U). Ito application yields:
  - (26) () () () 2 12111 1 ,, , 2 ttttt dBBart   BBbrt    dt  BbrtdW = + ++,
  - Rewritten as (27): dB_t = αB_t dt + σB_t dW_t with
    - (28) αB_t = ()() 2 1211 1 , , 2 tt BBa r tBB b r t+ +,
    - (29) σB_t = ()1 , tt BBb r t.
- Market price of risk:
  - (20') () t t r α λ σ − = , and relation (30) () () 1 , tt t t Br brt B α λ − =,
- Hedging positions:
  - θ1 and θ2 chosen to eliminate randomness leading to θ2/θ1 = −B1^L / B1^S and, approximating B1^S = -1 and B1^L ≈ B_y^L, yields θ2/θ1 = B_y^L and equation (6') reproduced.
- Puttable bond PDEs:
  - Discount underlying, Black-Scholes-Merton-like: (40) and boundary conditions (41).
  - Coupon-paying underlying: PDE (44).
  - With credit risk (exogenous h_t and L_t): stochastic eqn (45) and PDE (46).

*Source: IMF working paper content unit as supplied.*

### REFERENCES

### _wp03201 - REFERENCES

### Academic articles and journal pieces
- Ananthanarayanan, A. L., and Eduardo S. Schwartz, 1980, "Retractable and Extendible Bonds: The Canadian Experience," Journal of Finance, Vol. 35, No. 1, pp. 31–47.
- Athanassakos, George, 1996, "On the Application of the Black and Scholes Formula to Valuing Bonds with Embedded Options: The Case of Extendible Bonds," Applied Financial Economics, Vol. 6, No. 1, pp. 37–48.
- Bliss, Robert R., and Ehud I. Ronn, 1995, "To Call or Not to Call? Optimal Call Policies for Callable U.S. Treasury Bonds," Federal Reserve Bank of Atlanta Economic Review, November-December, pp. 1–14.
- Buetow Jr., Gerald W., Frank J. Fabozzi, and Bernd Hanke, 2001, “Impact of Different Interest Rate Models on Bond Value Measures,” Journal of Fixed Income, Vol. 11, No. 3, pp. 41–53.
- Crabbe, Leland E., and Panos Nikoulis, 1997, “The Puttable Bond Market: Structure, Historical Experience, and Strategies,” Journal of Fixed Income, Vol. 7, No. 3, pp. 47–60.
- David, Alexander, 2001, “Pricing the Strategic Value of Puttable Securities in Liquidity Crises,” Journal of Financial Economics, Vol. 59, No. 1, pp. 63–99.
- Duffie, Darrell, Lasse Heje Pedersen, and Kenneth Singleton, 2003, “Modeling Sovereign Yield Spreads: A Case Study of Russian Debt,” Journal of Finance, Vol. 28, No. 1, pp. 119–59.
- Duffie, Darrell, and Kenneth Singleton, 1999, “Modeling Term Structures of Defaultable Bonds,” Review of Financial Studies, Vol. 12, No. 4, pp. 687–720.
- Kalotay, Andrew J., and Leslie A. Abreo, 1999, “Puttable/Callable/Reset Bonds: Intermarket Arbitrage with Unpleasant Side Effects,” Journal of Derivatives, Vol. 6, No. 4, pp. 88–93.
- Tufano, Peter, 1996, “Who Manages Risk? An Empirical Examination of Risk Management Practices in the Gold Mining Industry,” Journal of Finance, Vol. 51 (September), pp. 1097–1137.
- Neftci, Salih N., 2000b, “Domestic Currency Emerging Market Bonds Pricing and Risk Management Aspects,” Revista de Análisis Económico, Vol. 15, No. 1, pp. 47–59.
- Missale, Alessandro, 1999, Public Debt Management (Oxford: Oxford University Press). [Note: listed here as it addresses public debt management issues in journal/book context.]

### Books, handbooks, and monographs
- Fabozzi, Frank J., ed., 2001, The Handbook of Fixed Income Securities (New York: McGraw-Hill, 6th edition).
- Hull, John C., 2000, Options, Futures, and Other Derivatives (Upper Saddle River: Prentice Hall, 4th edition).
- Merton, Robert C., 1990, Continuous-time Finance (Cambridge: Blackwell Publishers).
- Merton, Robert C., 1995, “Financial Innovation and the Management and Regulation of Financial Institutions,” Journal of Banking and Finance, Vol. 19, No. 3–4, pp. 461–81.
- McLean, Stuart K., ed., 1993, The European Bond Markets: An Overview and Analysis for Money Managers and Traders (Cambridge: Probus Publishing Company, 5th edition).
- Neftci, Salih N., 2000a, An Introduction to the Mathematics of Financial Derivatives (New York: Academic Press, 2nd edition).
- Rebonato, Riccardo, 1998, Interest-Rate Option Models (New York: John Wiley & Sons, 2nd edition).
- Urich, Thomas J., 1991, “U.K., German and Japanese Government Bond Markets,” Monograph Series in Finance and Economics (New York: New York University, Salomon Center).
- International Bank for Reconstruction and Development, and International Monetary Fund, 2001, Developing Government Bond Markets: A Handbook (Washington: International Bank for Reconstruction and Development and International Monetary Fund).

### Institutional reports, manuals, and working papers
- Board of Governors of the Federal Reserve System, 1998, Trading and Capital-Markets Activities Manual (Washington: Federal Reserve System).
- Central Bank of Brazil, 1998, Annual Report 1997, Vol. 34 (Brasília: Central Bank of Brazil).
- Central Bank of Brazil, 1999, Resolução 2675 (Brasília: Central Bank of Brazil).
- Central Bank of Brazil, 2001, Annual Report 2000, Vol. 36 (Brasília: Central Bank of Brazil).
- Central Bank of Brazil, 2002a, Annual Report 2001, Vol. 37 (Brasília: Central Bank of Brazil).
- Central Bank of Brazil, 2002b, Circular 3139 (Brasília: Central Bank of Brazil).
- Central Bank of Brazil, 2002c, Financial Stability Report, Vol. 1, No. 1 (Brasília: Central Bank of Brazil).
- Central Bank of Brazil, 2002d, Política Monetária – Relatório de Atividades 1999 a 2002 (Brasília: Central Bank of Brazil).
- Central Bank of Brazil, 2002e, Resolução 3006 (Brasília: Central Bank of Brazil).
- Financial Stability Forum (2000), Report of the Working Group on Capital Flows (Basel: Bank of International Settlements).
- International Monetary Fund, 2002, Global Financial Stability Report, March 2002: A Quarterly Report by the International Capital Markets Department on Market Developments and Issues, World Economic and Financial Surveys (Washington: International Monetary Fund).
- De Broeck, Mark, Dominique Guillaume, and Emmanuel Van der Stichele, 1998, “Structural Reforms in Government Bond Markets,” IMF Working Paper No. 98/108 (Washington: International Monetary Fund).
- Favero, Carlo A. and Francesco Giavazzi, 2002, “Why Are Brazil’s Interest Rates so High?” (unpublished; Milan: University of Milan).
- National Treasury of Brazil (2001), Public Debt: Annual Borrowing Plan 2001, No. 1 (Brasília: National Treasury of Brazil).
- National Treasury of Brazil (2002), Public Debt: Annual Borrowing Plan 2002, No. 2 (Brasília: National Treasury of Brazil).
- Securities and Exchange Commission of Brazil, 2002, Instrução No. 375 (Brasília: Securities and Exchange Commission).
- International Organization of Securities Commissions (IOSCO), 1999, Collective Investment Schemes Unit Pricing (London: International Organization of Securities Commissions).
- International Organization of Securities Commissions (IOSCO), 2002, The Development of Corporate Bond Markets in Emerging Market Countries (London: International Organization of Securities Commissions).

### Technical and methodological works
- Duffie, Darrell, and Kenneth Singleton, 1999, “Modeling Term Structures of Defaultable Bonds,” Review of Financial Studies, Vol. 12, No. 4, pp. 687–720.
- Duffie, Darrell, Lasse Heje Pedersen, and Kenneth Singleton, 2003, “Modeling Sovereign Yield Spreads: A Case Study of Russian Debt,” Journal of Finance, Vol. 28, No. 1, pp. 119–59.
- Buetow Jr., Gerald W., Frank J. Fabozzi, and Bernd Hanke, 2001, “Impact of Different Interest Rate Models on Bond Value Measures,” Journal of Fixed Income, Vol. 11, No. 3, pp. 41–53.
- Kalotay, Andrew J., and Leslie A. Abreo, 1999, “Puttable/Callable/Reset Bonds: Intermarket Arbitrage with Unpleasant Side Effects,” Journal of Derivatives, Vol. 6, No. 4, pp. 88–93.
- Hull, John C., 2000, Options, Futures, and Other Derivatives (Upper Saddle River: Prentice Hall, 4th edition).
- Neftci, Salih N., 2000a, An Introduction to the Mathematics of Financial Derivatives (New York: Academic Press, 2nd edition).

*Source: _wp03201 - REFERENCES*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2003/_wp03201.pdf_