# Pricing and Hedging Bond Power Exchange Options in a Stochastic String Term-Structure Model

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## Abstract

**:**

## 1. Introduction

- Recalibration is unnecessary (Goldstein (2000); McDonald and Beard (2002) and Kimmel (2004)). The HJM models allow a perfect fit to the current TSIR but they are inconsistent with the innovations in the forward curve as, in general, the realizations from the N Brownian motions in a N-factor HJM model are incompatible with the possible innovations along the whole TSIR. Thus, HJM models require continuous recalibration to fit the current curve. For stochastic string models, such recalibration is not necessary as we can always find a path for the stochastic string shock to go from the initial to the final forward curve.
- The best instrument to hedge a bond is another bond with a close maturity (Goldstein (2000); Carmona and Tehranchi (2004) and Cont (2005)). The N-factor HJM models have the property that any interest rate derivative can be hedged with N bonds with arbitrary maturities chosen a priori and independently from the bonds underlying the derivative. This fact is inconsistent with the usual practice of market participants who hedge interest rate derivatives using bonds of similar maturities. This practice suggests the existence of a specific risk at maturity, not considered in the factor models. String models incorporate this risk in the stochastic string shock and predict that the best instrument to hedge a bond is another bond with a close maturity.
- We do not need to include the error term when estimating the model (Santa-Clara and Sornette (2001); Bester (2004)). In the N-factor models, any sample of L(>N) forward rates has a covariance matrix whose rank is not greater than N. In this case, error terms must be introduced in the econometric specification of the model. In stochastic string models, we can always find a realization from the shock on a time interval to go from the initial forward curve to the final one. Thus, these models are compatible with any sample of forward rates and there is no need to include error terms in econometric specifications.
- Stochastic string models are more parsimonious than factor models (Goldstein (2000); Santa-Clara and Sornette (2001)). The number of extra parameters in string models with respect to a one-factor model depends on the parameter specification of the correlation function between shocks. If we choose a one-parameter specification for such a function, there is just one more parameter to estimate than in the one-factor HJM model. Thus, string models are more parsimonious than the corresponding N-factor models, that consider a large number of factors (and, then, of parameters) to obtain realistic correlations.

## 2. Preliminary Results

- (a)
- The stochastic processes $Z\left(\xb7,x\right)$ and $Z\left(t,\xb7\right)$ are continuous for each $x\ge 0$ and for each $t\in \left[0,\mathrm{Y}\right]$, respectively.
- (b)
- The process $Z\left(\xb7,x\right)$ is a martingale for each $x\ge 0$.
- (c)
- The process $Z\left(t,\xb7\right)$ is differentiable for each $t\in \left[0,\mathrm{Y}\right]$.
- (d)
- For each $x,y\ge 0$, it is the case that$$d{\left[Z\left(\xb7,x\right),Z\left(\xb7,y\right)\right]}_{t}=c\left(t,x,y\right)dt$$

**Lemma**

**1.**

**Definition**

**1.**

**portfolio**in the bond market is a pair $\left\{{g}_{t},h\left(t,\xb7\right)\right\}$ where

- (a)
- g is a predictable process.
- (b)
- For each ω, t, $h\left(\omega ,t,\xb7\right)$ is a generalized function in $\left(t,\infty \right)$.
- (c)
- For each T, the process $h\left(t,T\right)$ is predictable.

**Definition**

**2.**

**value process**, V, of a portfolio $\left\{g,h\right\}$ is defined by

**Definition**

**3.**

**self-financing**if its value process satisfies

**Definition**

**4.**

**replicated**or that we can

**hedge**against $\overline{X}$ if there exists a self-financing portfolio with bounded, discounted value process $\overline{V}$, such that ${\overline{V}}_{{T}_{0}}=\overline{X}$.

**Theorem**

**1.**

**Theorem**

**2.**

**Proposition**

**1.**

## 3. Pricing Bond Power Exchange Options

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

#### 3.1. American Power Exchange Options

**Theorem**

**4.**

**Proof.**

**Corollary**

**4.**

#### 3.2. Put-Call Parity

**Proposition**

**2.**

**Proof.**

**Corollary**

**5.**

## 4. Hedging Bond Power Exchange Options

**Theorem**

**5.**

**Proof.**

**Corollary**

**6.**

## 5. Applications

#### 5.1. Pricing Extendable/Accelerable Maturity Zero-Coupon Bonds

#### 5.2. Option to Price a Zero-Coupon Bond off of a Shifted Term-Structure

#### 5.3. Bond Option Duration and Convexity

#### 5.4. Options on Interest Rates

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BPE | Bond Power Exchange Option |

$Con{v}_{BPE}$ | Convexity of the bond power exchange option |

CP | Call Power Option |

Cpl | Caplet |

CUES | Constant Underlying Elasticity in Strikes Option |

$Du{r}_{BPE}$ | Duration of the bond power exchange option |

EAM | Extendable/Accelerable Maturity zero-coupon bond |

Frl | Floorlet |

HJM | Heath, Jarrow, and Morton Model |

STS | Option to price a zero-coupon bond off of a Shifted Term-Structure |

TSIR | Term Structure of Interest Rates |

## Notes

1 | By admissibility we mean that for each t, $c\left(t,\xb7,\xb7\right)$ is symmetric, positive semidefinite and satisfies $\left|c\left(t,x,y\right)\right|\le 1$ and $c\left(t,x,x\right)=1,\phantom{\rule{0.277778em}{0ex}}\forall x,y\ge 0$ (Santa-Clara and Sornette (2001)). |

2 | We refer the reader to Nualart (2006) for the general theory of Malliavin calculus and to Bueno-Guerrero et al. (2017) for the specific issues related to stochastic strings. |

3 | The duration of an interest rate sensitive asset is $-\frac{(1+\frac{y}{2})}{P}\frac{\partial P}{\partial y}$ where y is the rate or yield to maturity to which it is exposed. For assets or portfolios of assets with exposures to multiple rates in a term structure, duration is a measure of price sensitivity to a parallel shift of the term structure. |

4 | Bueno-Guerrero et al. (2020) study the valuation of caplets, caps, and swaptions under a stochastic string model, however they do not consider caplet pricing as an application of bond power exchange options. |

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**MDPI and ACS Style**

Blenman, L.P.; Bueno-Guerrero, A.; Clark, S.P.
Pricing and Hedging Bond Power Exchange Options in a Stochastic String Term-Structure Model. *Risks* **2022**, *10*, 188.
https://doi.org/10.3390/risks10100188

**AMA Style**

Blenman LP, Bueno-Guerrero A, Clark SP.
Pricing and Hedging Bond Power Exchange Options in a Stochastic String Term-Structure Model. *Risks*. 2022; 10(10):188.
https://doi.org/10.3390/risks10100188

**Chicago/Turabian Style**

Blenman, Lloyd P., Alberto Bueno-Guerrero, and Steven P. Clark.
2022. "Pricing and Hedging Bond Power Exchange Options in a Stochastic String Term-Structure Model" *Risks* 10, no. 10: 188.
https://doi.org/10.3390/risks10100188