Price European call option on bonds using Black model
This example shows how to price a European call option on bonds using the Black model. Consider a European call option on a bond maturing in 9.75 years. The underlying bond has a clean price of $935, a face value of $1000, and pays 10% semiannual coupons. Since the bond matures in 9.75 years, a $50 coupon will be paid in 3 months and again in 9 months. Also, assume that the annualized volatility of the forward bond price is 9%. Furthermore, suppose the option expires in 10 months and has a strike price of $1000, and that the annualized continuously compounded risk-free discount rates for maturities of 3, 9, and 10 months are 9%, 9.5%, and 10%, respectively.
% specify the option information Settle = '15-Mar-2004'; Expiry = '15-Jan-2005'; % 10 months from settlement Strike = 1000; Sigma = 0.09; Convention = [0 1]'; % specify the interest-rate environment ZeroData = [datenum('15-Jun-2004') 0.09 -1; % 3 months datenum('15-Dec-2004') 0.095 -1; % 9 months datenum(Expiry) 0.10 -1]; % 10 months % specify the bond information CleanPrice = 935; CouponRate = 0.1; Maturity = '15-Dec-2013'; % 9.75 years from settlement Face = 1000; BondData = [CleanPrice CouponRate datenum(Maturity) Face]; Period = 2; Basis = 1; % call Black's model CallPrices = bkcall(Strike, ZeroData, Sigma, BondData, Settle,... Expiry, Period, Basis, , , Convention)
CallPrices = 2×1 9.4873 7.9686
When the strike price is the dirty price (
0), the call option value is $9.49. When the strike price is the clean price (
1), the call option value is $7.97.
Strike— Strike price
Strike price, specified as a scalar numeric or an
1 vector of strike prices.
ZeroData— Zero rate information used to discount future cash flows
Zero rate information used to discount future cash flows, specified using a two-column (optionally three-column) matrix containing zero (spot) rate information used to discount future cash flows.
Column 1 — Serial maturity date associated with the zero rate in the second column.
Column 2 — Annualized zero rates, in decimal form, appropriate for discounting
cash flows occurring on the date specified in the first column. All dates must
Settle (dates must correspond to future
investment horizons) and must be in ascending order.
Column 3 — (optional) Annual compounding frequency. Values are
3 (three times per year),
12 (monthly), and
If cash flows occur beyond the dates spanned by
input zero curve, the appropriate zero rate for discounting such cash flows is obtained
by extrapolating the nearest rate on the curve (that is, if a cash flow occurs before
the first or after the last date on the input zero curve, a flat curve is
In addition, you can use the method
getZeroRates for an
IRDataCurve object with a
Dates property to create a vector of dates and data acceptable for
bkcall. For more information, see Converting an IRDataCurve or IRFunctionCurve Object.
Sigma— Annualized price volatilities required by Black model
Annualized price volatilities required by the Black model, specified as a scalar or
BondData— Characteristics of underlying bonds
Characteristics of underlying bonds, specified as a row vector with three
(optionally four) columns or
4) matrix specifying characteristics of
underlying bonds in the form:
[CleanPrice CouponRate Maturity Face]
CleanPrice is the price excluding accrued interest.
CouponRate is the decimal coupon rate.
Maturity is the bond maturity date in serial date number
Face is the face value of the bond. If unspecified, the
face value is assumed to be 100.
Settle— Settlement date
Settlement date, specified as a serial date number or date character vector.
Settle also represents the starting reference date for the input
Expiry— Option maturity date
Option maturity date, specified as a scalar or an
1 vector of serial date numbers or cell
array of date character vectors.
Period— Number of coupons per year for underlying bond
2(semiannual) (default) | integer with value
(Optional) Number of coupons per year for the underlying bond, specified as an
integer with supported values of
Basis— Day-count basis of underlying bonds
0(actual/actual) (default) | integer from
(Optional) Day-count basis of underlying bonds, specified as a scalar or an
1 vector using the following values:
0 = actual/actual
1 = 30/360 (SIA)
2 = actual/360
3 = actual/365
4 = 30/360 (PSA)
5 = 30/360 (ISDA)
6 = 30/360 (European)
7 = actual/365 (Japanese)
8 = actual/actual (ICMA)
9 = actual/360 (ICMA)
10 = actual/365 (ICMA)
11 = 30/360E (ICMA)
12 = actual/365 (ISDA)
13 = BUS/252
For more information, see Basis.
EndMonthRule— End-of-month rule flag
1(in effect) (default) | nonnegative integer
(Optional) End-of-month rule flag, specified as a scalar or an
1 vector of end-of-month rules.
0 = Ignore rule, meaning that a bond coupon payment date is
always the same numerical day of the month.
1 = Set rule on, meaning that a bond coupon payment date is
always the last actual day of the month.
InterpMethod— Zero curve interpolation method
1(linear interpolation) (default) | integer with value
(Optional) Zero curve interpolation method for cash flows that do not fall on a date
found in the
ZeroData spot curve, specified as a scalar integer.
InterpMethod is used to interpolate the appropriate zero discount
rate. Available interpolation methods are (
1) linear, and (
2) cubic. For more information
on interpolation methods, see
StrikeConvention— Option contract strike price convention
0(default) | integer with value
(Optional) Option contract strike price convention, specified as a scalar or an
StrikeConvention = 0 (default) defines the strike price as the
cash (dirty) price paid for the underlying bond.
StrikeConvention = 1 defines the strike price as the quoted
(clean) price paid for the underlying bond. When evaluating Black's model, the accrued
interest of the bond at option expiration is added to the input strike price.
CallPrice— Price for European call option on bonds derived from Black model
Price for European call option on bonds derived from the Black model, returned as a
 Hull, John C. Options, Futures, and Other Derivatives. 5th Edition, Prentice Hall, 2003, pp. 287–288, 508–515.