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European bond options

The standalone contract

Section titled “The standalone contract”

A European bond call gives the holder the right, but not the obligation, to buy a named bond at option-expiry-time. A put gives the right to sell. This option is not the issuer’s call feature inside a callable bond. The two rights are held by different parties and belong to different instruments.

Put the bond value and the strike on one price basis

Section titled “Put the bond value and the strike on one price basis”

The local bond-option-expiry-value BToptdirty\explain{bond-option-expiry-value}{B_{\explain{option-expiry-time}{T}_{\mathrm{opt}}}^{\mathrm{dirty}}} and option-strike-price K\explain{option-strike-price}{K} must both be dirty cash prices. If a contract states a clean strike, add the accrued-interest at the exercise date to the strike before the comparison.

In this lesson, exercise occurs immediately after any coupon paid on the expiry date. The call payoff and the put payoff are:

XToptB,call=max⁡(BToptdirty−K,0)\explain{bond-call-expiry-payoff}{X^{B,\mathrm{call}}_{T_{\mathrm{opt}}}} =\max\left( \explain{bond-option-expiry-value}{B_{T_{\mathrm{opt}}}^{\mathrm{dirty}}} -\explain{option-strike-price}{K},0\right)
(5.1.1)
XToptB,put=max⁡(K−BToptdirty,0)\explain{bond-put-expiry-payoff}{X^{B,\mathrm{put}}_{T_{\mathrm{opt}}}} =\max\left( \explain{option-strike-price}{K} -\explain{bond-option-expiry-value}{B_{T_{\mathrm{opt}}}^{\mathrm{dirty}}},0\right)
(5.1.2)

The lattice first gives the bond values at expiry. Apply (5.1.1) or (5.1.2) at the expiry nodes. Then apply backward induction to calculate the lattice-node-value at time zero.

Value the bond to option expiry
Calculate the expiry payoff
Discount expected option values backward
Diagram 5.1.1European bond-option valuation flow. First value the bond at every option-expiry node, then apply the call or put payoff, then discount the risk-neutral expected payoffs backward through the input lattice.

Worked example

Section titled “Worked example”

Suppose a one-period input lattice gives dirty bond values at expiry of USD 90 and USD 110. The strike is USD 100, the risk-neutral up weight is 0.60, and the discount factor is 0.95. Applying (5.1.1) and one backward step gives the call value:

XToptB,call,down=max⁡(90−100,0)=0XToptB,call,up=max⁡(110−100,0)=10V0=0.95[(1−0.60)(0)+0.60(10)]=5.70 USD per option\begin{aligned} \explain{bond-call-expiry-payoff}{X_{\explain{option-expiry-time}{T_{\mathrm{opt}}}}^{B,\mathrm{call},\mathrm{down}}} &=\max(90-100,0)=0\\ \explain{bond-call-expiry-payoff}{X_{\explain{option-expiry-time}{T_{\mathrm{opt}}}}^{B,\mathrm{call},\mathrm{up}}} &=\max(110-100,0)=10\\ \explain{lattice-node-value}{V_0} &=0.95\left[\left(1-0.60\right)\left(0\right)+0.60\left(10\right)\right]\\ &=5.70\ \text{USD per option} \end{aligned}
(5.1.3)

The tested domain code does the same two steps: europeanOptionPayoff calculates the expiry payoffs, and backwardInductionValue discounts them.

Coupon and exercise ordering

Section titled “Coupon and exercise ordering”

In this lesson, a coupon on the expiry date is paid before exercise. So the bond value at expiry is ex-coupon. A contract that is exercised before the coupon, or that transfers the right to the coupon, requires different terminal inputs.

Knowledge check 5.1.1 European bond option

Link to Knowledge check 5.1.1: European bond option

Sources

Section titled “Sources”

The bond-option definition and the need to distinguish a cash (dirty) strike from a quoted (clean) strike follow Hull, Chapter 28 §28.1 [1]. This lesson uses an input lattice, not Black’s market model, so that the next lesson can model the issuer-default state explicitly.

References

Section titled “References”
  1. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). draft ↩
Notation used on this page (7)
XToptB,callX^{B,\mathrm{call}}_{T_{\mathrm{opt}}}Bond call expiry payoffdraft

Non-negative holder payoff from buying the bond at the dirty strike when economically beneficial.

Units: stated currency per option at expiry

BToptdirtyB_{T_{\mathrm{opt}}}^{\mathrm{dirty}}Bond option expiry valuedraft

Ex-coupon dirty cash value of the named bond at the European exercise instant.

Units: stated currency per bond at option expiry

XToptB,putX^{B,\mathrm{put}}_{T_{\mathrm{opt}}}Bond put expiry payoffdraft

Non-negative holder payoff from selling the bond at the dirty strike when economically beneficial.

Units: stated currency per option at expiry

AIAIAccrued interestdraft

Coupon amount attributed to the interval from the previous coupon date through settlement under the stated day-count convention; a positive amount added to the clean price to obtain the dirty invoice price.

Units: stated currency at settlement

Vi,jV_{i,j}Lattice node valuedraft

Claim value at time row i and state node j obtained by one-period backward induction from its successor nodes, conditional on reaching that node under the input pricing lattice.

Units: stated currency at the node time

ToptT_{\mathrm{opt}}Option expiry timedraft

Future model time when a European option's exercise decision and payoff are determined; a contract date shared by holder and writer, distinct from a bond maturity.

Units: model-years from the stated valuation time

KKOption strike pricedraft

Contractual price per unit of underlying used to determine the option's exercise payoff; a positive contractual amount, with payoff signs depending on call or put and holder or writer perspective.

Units: expiry-time currency per unit of underlying