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Price European put option on bonds using Black model



PutPrice = bkput(Strike,ZeroData,Sigma,BondData,Settle,Expiry) computes prices of European put options using a Black model.


PutPrice = bkput(___,Period,Basis,EndMonthRule,InterpMethod,StrikeConvention) adds optional input arguments for Period, Basis, EndMonthRule, InterpMethod, and StrikeConvention.


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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


When the strike price is the dirty price (Convention = 0), the call option value is $1.78. When the strike price is the clean price (Convention = 1), the call option value is $2.41.

Input Arguments

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Strike price, specified as a scalar numeric or an NOPT-by-1 vector of strike prices.

Data Types: double

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 1 (annual), 2 (semiannual, default), 3 (three times per year), 4 (quarterly), 6 (bimonthly), 12 (monthly), and -1 (continuous).

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 assumed).

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 bkput. For more information, see Converting an IRDataCurve or IRFunctionCurve Object.

Data Types: double

Annualized price volatilities required by the Black model, specified as a scalar or an NOPT-by-1 vector.

Data Types: struct

Characteristics of underlying bonds, specified as a row vector with three (optionally four) columns or NOPT-by-3 (optionally NOPT-by-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 using a serial date number, date character vector, or string.

  • Face is the face value of the bond. If unspecified, the face value is assumed to be 100.

Data Types: double | char | string

Settlement date, specified as a scalar string, date character vector, or serial date number. Settle also represents the starting reference date for the input zero curve.

Data Types: char | double | string

Option maturity date, specified as an NOPT-by-1 vector using a string array, date character vectors, or serial date numbers.

Data Types: char | string | double

(Optional) Number of coupons per year for the underlying bond, specified as an integer with supported values of 0, 1, 2, 3, 4, 6, and 12.

Data Types: double

(Optional) Day-count basis of underlying bonds, specified as a scalar or an NOPT-by-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.

Data Types: double

(Optional) End-of-month rule flag, specified as a scalar or an NOPT-by-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.

Data Types: logical

(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 methods, see interp1.

Data Types: double

(Optional) Option contract strike price convention, specified as a scalar or an NOPT-by-1 vector.

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.

Data Types: double

Output Arguments

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Price for European put option on bonds derived from the Black model, returned as a NOPT-by-1 vector.


[1] Hull, John C. Options, Futures, and Other Derivatives. 5th Edition, Prentice Hall, 2003, pp. 287–288, 508–515.

Version History

Introduced before R2006a