Pricing of
Callable Bonds.

Understanding the pricing problem.

The objective of this first project was therefore to understand and implement the different components required to price a callable bond, from the valuation of the underlying vanilla bond to the modelling of interest rates and the optimal exercise decision. In particular, I studied the Hull–White one-factor interest rate model (HW1F) and the optimal stopping problem associated with the issuer’s call decision. Depending on the exercise features of the bond, different pricing approaches were considered from analytical methods for European-style exercise to numerical methods based on Monte Carlo simulation.

Depending on the exercise structure of the callable bond, different analytical and numerical approaches were considered.

Underlying fixed-income instrument.

A callable bond has the same basic structure as a vanilla bond, but includes additional callability dates. The underlying vanilla bond consists of a series of contractual cash flows. If the payment dates are \(T_1,\ldots,T_n\), and the corresponding cash flows are \(c_1,\ldots,c_n\), the value of the bond at time \(t\) is:

Here, \(P(t,T_i)\) denotes the price at time \(t\) of a zero-coupon bond paying one unit of currency at \(T_i\).

Equivalently, when separating coupons from the final principal repayment, the bond value can be written as:

This representation is particularly useful for callable bonds because zero-coupon bond prices provide the building blocks for the fixed-income valuation.

Bond + embedded call option.

A callable bond gives the issuer the right to redeem the bond before its maturity. The dates on which this right can be exercised, together with the corresponding redemption prices, are specified in the product's term sheet.

In the framework considered in the project, the bond was assumed to be callable at par. From the investor's perspective, the callable bond can be decomposed into the vanilla bond minus the value of the embedded call option:

The main pricing problem is therefore to determine the value of the embedded call option. This value depends on the evolution of interest rates and, for multiple exercise dates, on the optimal exercise strategy of the issuer.

European call feature.

For a European callable bond with a single call date \(T_c\), the issuer can exercise the call only at this predefined date.

If \(K\) denotes the call price, the payoff of the embedded call option from the issuer's perspective is:

Since the vanilla bond itself is a portfolio of future cash flows, its value at the call date can be expressed as:

Therefore:

Hull–White One-Factor Model.

The pricing team uses the Hull–White one-factor model to model the evolution of the short-term interest rate.

Under the risk-neutral measure \(\mathbb{Q}\), the short rate \(r_t\) follows:

The parameter \(a\) controls the speed at which the short rate tends to return toward its time-dependent long-term level, while \(\sigma\) controls its instantaneous volatility.

Market-consistent calibration.

The coefficients of the model are calibrated using market data. This calibration is carried out daily through the Numerix system used by the pricing team. This subject will be study shortly in an other project about the calibration of HW1F model.

The initial instantaneous forward rate is defined from the zero-coupon bond curve by:

The deterministic drift adjustment \(\theta(t)\) is calibrated such that the model exactly reproduces the initial zero-coupon bond curve.

This calibration property makes the model consistent with the market term structure observed at the valuation date.

Affine representation.

Under the Hull–White one-factor model, the price of a zero-coupon bond admits an affine representation in the short rate:

The function \(B(t,T)\) is given by:

The function \(A(t,T)\) is:

This affine representation is central to the analytical treatment of the European callable bond.

The issuer's exercise decision.

The decision to call the bond can be formulated as an optimal stopping problem.

Let \(V(t,r_t)\) denote the value of the callable bond at time \(t\), and let \(K(t)\) denote the redemption price paid by the issuer when the call is exercised.

The continuation value can be represented by the conditional expectation:

Here, \(C(s)\) represents the contractual coupon cash flow and \(Z(\tau)\) represents the cash flow received at the stopping time \(\tau\).

The payoff at the stopping time is:

The optimal exercise time is the stopping time at which the issuer chooses to exercise rather than continue the contract:

Economically, the issuer exercises when the benefit associated with redeeming the bond is greater than the value associated with continuing the contract.

Calculation of the forward rate.

During the forward induction, the main objective is to calculate all the future rates

Exercise versus continuation.

At each exercise date \(t_i\), the issuer compares the immediate redemption value \(K_i\) with the continuation value \(C(t_i)\).

From the investor's perspective, the callable bond value at an exercise date is:

If the redemption price is lower than the continuation value, the issuer has an economic incentive to exercise the call. Conversely, if the continuation value is lower, the issuer prefers not to exercise.

The value obtained at each exercise date is then used to determine the continuation value at the previous exercise date. The procedure is repeated backwards until the initial valuation date.

Analytical pricing with Jamshidian.

A European callable bond has a single predefined call date \(T_c\). There is therefore no optimization over several exercise dates.

In the Hull–White framework, the problem can be transformed into a portfolio of European options and solved analytically. This provides a useful benchmark for the numerical methods considered for Bermudan and American callable bonds.

Jamshidian decomposition

Consider a bond with contractual cash flows \(c_i\) paid at dates \(T_1,\ldots,T_n\), with \(T_c \leq T_1 < \cdots < T_n\).

At the call date, the value of the underlying bond is:

Using the affine Hull–White representation:

The critical short rate \(r^*\) is defined as the solution of:

For each cash flow \(c_i\), the corresponding zero-coupon bond strike is then defined by:

The callable feature can consequently be decomposed into a portfolio of options on zero-coupon bonds.

Monte Carlo and Longstaff–Schwartz.

A Bermudan callable bond allows the issuer to exercise the call at a finite set of predetermined dates:

Unlike the European case, the issuer must determine whether exercising at a given date is preferable to continuing the contract.

The approach described in the project uses Monte Carlo simulation combined with the Longstaff–Schwartz algorithm to estimate the continuation value.

Starting from the last exercise date, the callable bond value is determined by comparing the redemption price with the value of keeping the bond outstanding. This information is then propagated backwards through the exercise dates.

Continuous exercise.

For an American callable bond, the issuer can exercise the call at any time during a predefined period. The set of possible stopping times is therefore continuous:

This leads to a continuous-time optimal stopping problem, which is approximated numerically by discretizing time and applying a Monte Carlo regression methodology.

Increasing the number of exercise dates in the time grid provides a finer approximation of the continuous exercise feature.

The estimated American callable bond price is obtained as the average of the discounted cash flows generated by the optimal exercise strategy over the simulated paths:

Here, \(\Pi^{(m)}_0\) represents the discounted cash flows associated with the optimal exercise strategy on simulation path \(m\).

The accuracy of the approximation depends on both the number of simulated paths and the fineness of the time discretization.

The American case therefore introduces several sources of numerical error, including Monte Carlo sampling error, time discretization error and regression error.

Three exercise structures.

Implementation in Numerix.

The Pricing Team uses Numerix as its pricing software. The callable bond work was developed within this pricing environment.

The project covered the different components involved in the valuation framework, including the Hull–White model, calibration, the optimal stopping formulation and the different pricing methodologies associated with the exercise structure.

For the European case, the analytical formulation provides a benchmark for the numerical approaches. The implementation of the exact analytical formula in Numerix could not, however, be completed during the first stage of the project.

Another practical aspect of the work involved checking the information contained in the term sheets provided through the Trade Blotter, as these could differ from the actual term sheet of the product.

From interest rates to optimal exercise.

The project connects several components of quantitative fixed-income pricing: