Price European put option on bonds using Black model
This example shows how to price European put options on bonds using the Black model. Consider a European put option on a bond maturing in 10 years. The underlying bond has a clean price of $122.82, a face value of $100, and pays 8% semiannual coupons. Also, assume that the annualized volatility of the forward bond yield is 20%. Furthermore, suppose the option expires in 2.25 years and has a strike price of $115, and that the annualized continuously compounded risk free zero (spot) curve is flat at 5%. For a hypothetical settlement date of March 15, 2004, the following code illustrates the use of Black's model to duplicate the put prices in Example 22.2 of the Hull reference. In particular, it illustrates how to convert a broker's yield volatility to a price volatility suitable for Black's model.
% Specify the option information. Settle = '15-Mar-2004'; Expiry = '15-Jun-2006'; % 2.25 years from settlement Strike = 115; YieldSigma = 0.2; Convention = [0; 1]; % Specify the interest-rate environment. Since the % zero curve is flat, interpolation into the curve always returns % 0.05. Thus, the following curve is not unique to the solution. ZeroData = [datenum('15-Jun-2004') 0.05 -1; datenum('15-Dec-2004') 0.05 -1; datenum(Expiry) 0.05 -1]; % Specify the bond information. CleanPrice = 122.82; CouponRate = 0.08; Maturity = '15-Mar-2014'; % 10 years from settlement Face = 100; BondData = [CleanPrice CouponRate datenum(Maturity) Face]; Period = 2; % semiannual coupons Basis = 1; % 30/360 day-count basis % Convert a broker's yield volatility quote to a price volatility % required by Black's model. To duplicate Example 22.2 in Hull, % first compute the periodic (semiannual) yield to maturity from % the clean bond price. Yield = bndyield(CleanPrice, CouponRate, Settle, Maturity,... Period, Basis); % Compute the duration of the bond at option expiration. Most % fixed-income sensitivity analyses use the modified duration % statistic to examine the impact of small changes in periodic % yields on bond prices. However, Hull's example operates in % continuous time (annualized instantaneous volatilities and % continuously compounded zero yields for discounting coupons). % To duplicate Hull's results, use the second output of BNDDURY, % the Macaulay duration. [Modified, Macaulay] = bnddury(Yield, CouponRate, Expiry,... Maturity, Period, Basis); % Convert the yield-to-maturity from a periodic to a % continuous yield. Yield = Period .* log(1 + Yield./Period); % Convert the yield volatility to a price volatility via % Hull's Equation 22.6 (page 514). PriceSigma = Macaulay .* Yield .* YieldSigma; % Finally, call Black's model. PutPrices = bkput(Strike, ZeroData, PriceSigma, BondData,... Settle, Expiry, Period, Basis, , , Convention)
PutPrices = 2×1 1.7838 2.4071
When the strike price is the dirty price (
0), the call option value is $1.78. When the strike price is the clean price (
1), the call option value is $2.41.
Strike— Strike price
Strike price, specified as a scalar numeric or an
1 vector of strike
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 occur after
Settle (dates must correspond to future
investment horizons) and must be in ascending order.
Column 3 — (optional) Annual compounding frequency. Values are
2 (semiannual, default),
3 (three times per year),
12 (monthly), and
If cash flows occur beyond the dates spanned by
ZeroData, the 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
property to create a vector of dates and data acceptable for
bkput. 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 an
BondData— Characteristics of underlying bonds
Characteristics of underlying bonds, specified as a row vector with three
(optionally four) columns or
specifying characteristics of underlying bonds in the form:
[CleanPrice CouponRate Maturity Face]
CleanPrice is the price excluding accrued
CouponRate is the decimal coupon
Maturity is the bond maturity date in
serial date number format.
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
Settle also represents the starting reference
date for the input zero curve.
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
1 vector using the
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
0) nearest, (
1) linear, and
2) cubic. For more information on interpolation
StrikeConvention— Option contract strike price convention
0(default) | integer with value
(Optional) Option contract strike price convention, specified as a scalar
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.
PutPrice— Price for European put option on bonds derived from Black model
Price for European put 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.