Valuing Bonds with Embedded Options (August, 2009)

Article Summary:

This article explains how bonds with embedded options are priced using the Black-Derman-Toy (BDT) interest rate tree in AXIS.

In AXIS 12.5, a more robust algorithm has been implemented to model embedded put and embedded call options in bonds.

Model Review

Black-Derman-Toy (BDT) is a one-factor model describing the future evolution of short-term interest rates. Under this model, the rate behaves as a lognormal process with constant volatility s and varying drift m(t):

dr / r = m(t) dt + s dW. (1)

Note: There are modifications to this model allowing variable volatility, but they require either a variable time step tree to model, or additional drift term, which sometimes causes mean reverting behavior.

The "state of the world" in this model is completely defined by the value of instantaneous rate r(t) at time t.

Discrete approximation of this model assumes that:

• The state of the world is discrete, so instantaneous rate r(t) can take one of a finite number of values { r1(t) , r2(t) ,… rM(t) } (the set depending on time t);
• Time is discrete, i.e. takes values t0 < t1 < t2< …< tN (for fixed time step tk+1 = tk+dt with constant dt).

It is assumed that the state of the world r(t) remains constant during each time interval (tk, tk+1) between tree nodes.

A tree representation of this framework consists of a tree with vertexes (nodes) R(i, j) corresponding to the state ri (tj) of the world at time tj. The tree structure is defined by the probabilities of transitions from a state R(i, j) at time ti to a state R(i+1, k) at time ti+1; an edge links them if this probability is positive. Values ri (tj) and transition probabilities are set in such a way that ri (tj) represent an approximation to the solution of (1) above.

A binomial representation of the short-term interest rate assumes that there are only two states at time tk+1 where the world can transit from a state at time tk. Then, if a state at time t0 is known, it suffices to consider N states only at time tN.

A binomial tree representation for the BDT model can be constructed such that:

• Probabilities of transition to next time world states are 0.5 for two neighbor states and 0 for all others;

• The bond prices computed based on the tree exactly matched with the set of zero coupon prices directly observable in the market.

This model can be used to calculate risk-neutral expectations of present values for simple dedicated cash flows – those with payments appearing at time steps tk exactly. If the cash flow is not dedicated to this time structure (some payments may appear between tk and tk+1) they should be shifted to the appropriate time tk. This should be performed with proper discount depending on the world state at which a payment occurs and time interval to the nearest preceding tree node. Then at the tree node time all the calculations should take into account the order in which payment occurred during the time interval.

This is the case with a callable/puttable bond: coupon payments, exercise date(s) and tree nodes may not align in the same pattern. In this case, the routine equivalent to the one described above is used in AXIS.

Some implementation peculiarities are that:

• If two payments or a payment and a tree node coincide in time some order of operation should be prefixed. In AXIS, it is assumed that the earliest event is Coupon Payment, then Option Exercise, then the World state change (tree node recalculation). As a result, in the Option report the data values are displayed immediately after the change of the world state and before any other event.
• The tree size (number of time steps in the tree) is fixed by the user and remains constant throughout the calculation. This leads to shortening of the tree time step as the valuation date becomes nearer to the bond maturity / interest reset date.

Product Information

A call/put option on a bond gives the right to purchase/sell that bond at a fixed strike or exercise price. If the option is European in style, the right may be exercised on one particular date called the expiration or exercise date. If the option is Bermudan in style, the right may be exercised on some fixed set of dates. A typical Bermudan option on a bond would allow the option holder to exercise on any coupon payment date. An American call or put option allows for exercise at any time on or before the expiration date.

The new method allows arbitrary relations between coupon payment, exercise and tree node dates. Therefore, with the new method, no adjustment is made to these dates and to the tree size (i.e. the number of tree steps). The only restrictions left deal with the financial instrument itself: the option end date cannot be later than the bond maturity/interest reset date.

The new algorithm induced some changes to the Option price report. Since an exercise (and even several of them) may now happen between tree nodes, the calculated option price at the node no longer presents a mean to follow the Exercise boundary. Therefore the early exercise boundary is omitted from the option pricing report.

The parameters of the BDT callable bond model used if the Market value method is set to "9 - PV of cashflows to next reset date with BDT option model", "14 - BDT model for bond option solve for volatility" or "16 - BDT model for bond option solve for spread", are the same as with the old method. A specific example of setting up a Bermudan callable bond is described below. If the old method is preferable, it can still be switched on via Feature code 167 in the "Custom codes" tab of dataset parameters in the set of "Features to be removed". To obtain the activation code, please call GGY: (416) 250–6777.

Sample Setup in AXIS

These parameters can be specified in the Asset Features section of the asset cell screen.

 Purpose Field Example To specify underlying asset type Calculation Type 0 - Bond To specify reporting type Reporting Type 0 - Bond To specify coupon rate Income rate Flat 12% To specify coupon frequency Payment mode 2 - Quarterly To specify coupon rate type Income rate type 3 - Nominal annual rate compounded quarterly To specify dates of bond Date of issue 30-12-1990 Date of purchase 30-12-1991 Date of maturity 30-12-1995 To specify option type Option type 0 - Call To specify exercising frequency Months between exercise 3 To specify dates of embedded option Embedded option timing 1 – Use embedded option dates below to calculate option value Embedded option start date 12-1993 Embedded option end date 12-1994 To specify strike price Default to be 100 N/A

In AXIS, users can specify the BDT assumptions in the Market Value section of the asset cell screen.

 Purpose Field Example To specify market value method Market value method 9 – PV of cashflows to next rest date with BDT option model To specify calibration basis Scenario reference: Yield curve 3 - Reference processed scenario YC (adj for spread from current date) To specify tree size BDT assumptions: Tree size 10 To specify the option report date BDT assumptions: Report date 06-1993

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