nag_lookback_fls_greeks (s30bbc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_lookback_fls_greeks (s30bbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_lookback_fls_greeks (s30bbc) computes the price of a floating-strike lookback option together with its sensitivities (Greeks).

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_lookback_fls_greeks (Nag_OrderType order, Nag_CallPut option, Integer m, Integer n, const double sm[], double s, const double t[], double sigma, double r, double q, double p[], double delta[], double gamma[], double vega[], double theta[], double rho[], double crho[], double vanna[], double charm[], double speed[], double colour[], double zomma[], double vomma[], NagError *fail)

3  Description

nag_lookback_fls_greeks (s30bbc) computes the price of a floating-strike lookback call or put option, together with the Greeks or sensitivities, which are the partial derivatives of the option price with respect to certain of the other input parameters. A call option of this type confers the right to buy the underlying asset at the lowest price, Smin, observed during the lifetime of the contract. A put option gives the holder the right to sell the underlying asset at the maximum price, Smax, observed during the lifetime of the contract. Thus, at expiry, the payoff for a call option is S-Smin, and for a put, Smax-S.
For a given minimum value the price of a floating-strike lookback call with underlying asset price, S, and time to expiry, T, is
Pcall = S e-qT Φa1 - Smin e-rT Φa2 + S e-rT   σ2 2b S Smin -2b / σ2 Φ -a1 + 2b σ T -e bT Φ -a1 ,
where b=r-q0. The volatility, σ, risk-free interest rate, r, and annualised dividend yield, q, are constants.
The corresponding put price is
Pput = Smax e-rT Φ -a2 - S e-qT Φ -a1 + S e-rT   σ2 2b - S Smax -2b / σ2 Φ a1 - 2b σ T + ebT Φ a1 .
In the above, Φ denotes the cumulative Normal distribution function,
Φx = - x ϕy dy
where ϕ denotes the standard Normal probability density function
ϕy = 12π exp -y2/2
and
a1 = ln S / Sm + b + σ2 / 2 T σT a2=a1-σT
where Sm is taken to be the minimum price attained by the underlying asset, Smin, for a call and the maximum price, Smax, for a put.

4  References

Goldman B M, Sosin H B and Gatto M A (1979) Path dependent options: buy at the low, sell at the high Journal of Finance 34 1111–1127

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     optionNag_CallPutInput
On entry: determines whether the option is a call or a put.
option=Nag_Call
A call. The holder has a right to buy.
option=Nag_Put
A put. The holder has a right to sell.
Constraint: option=Nag_Call or Nag_Put.
3:     mIntegerInput
On entry: the number of minimum or maximum prices to be used.
Constraint: m1.
4:     nIntegerInput
On entry: the number of times to expiry to be used.
Constraint: n1.
5:     sm[m]const doubleInput
On entry: sm[i-1] must contain Smin i , the ith minimum observed price of the underlying asset when option=Nag_Call, or Smax i , the maximum observed price when option=Nag_Put, for i=1,2,,m.
Constraints:
  • sm[i-1]z ​ and ​ sm[i-1] 1 / z , where z = nag_real_safe_small_number , the safe range parameter, for i=1,2,,m;
  • if option=Nag_Call, sm[i-1]S, for i=1,2,,m;
  • if option=Nag_Put, sm[i-1]S, for i=1,2,,m.
6:     sdoubleInput
On entry: S, the price of the underlying asset.
Constraint: sz ​ and ​s1.0/z, where z=nag_real_safe_small_number, the safe range parameter.
7:     t[n]const doubleInput
On entry: t[i-1] must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: t[i-1]z, where z = nag_real_safe_small_number , the safe range parameter, for i=1,2,,n.
8:     sigmadoubleInput
On entry: σ, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: sigma>0.0.
9:     rdoubleInput
On entry: the annual risk-free interest rate, r, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0 and absr-q>10×eps×maxabsr,1, where eps=nag_machine_precision, the machine precision.
10:   qdoubleInput
On entry: the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: q0.0 and absr-q>10×eps×maxabsr,1, where eps=nag_machine_precision, the machine precision.
11:   p[m×n]doubleOutput
Note: the i,jth element of the matrix P is stored in
  • p[j-1×m+i-1] when order=Nag_ColMajor;
  • p[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array p contains the computed option prices.
12:   delta[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • delta[j-1×m+i-1] when order=Nag_ColMajor;
  • delta[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array delta contains the sensitivity, PS, of the option price to change in the price of the underlying asset.
13:   gamma[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • gamma[j-1×m+i-1] when order=Nag_ColMajor;
  • gamma[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array gamma contains the sensitivity, 2PS2, of delta to change in the price of the underlying asset.
14:   vega[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • vega[j-1×m+i-1] when order=Nag_ColMajor;
  • vega[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array vega contains the sensitivity, Pσ, of the option price to change in the volatility of the underlying asset.
15:   theta[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • theta[j-1×m+i-1] when order=Nag_ColMajor;
  • theta[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array theta contains the sensitivity, -PT, of the option price to change in the time to expiry of the option.
16:   rho[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • rho[j-1×m+i-1] when order=Nag_ColMajor;
  • rho[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array rho contains the sensitivity, Pr, of the option price to change in the annual risk-free interest rate.
17:   crho[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • crho[j-1×m+i-1] when order=Nag_ColMajor;
  • crho[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array crho containing the sensitivity, Pb, of the option price to change in the annual cost of carry rate, b, where b=r-q.
18:   vanna[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • vanna[j-1×m+i-1] when order=Nag_ColMajor;
  • vanna[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array vanna contains the sensitivity, 2PSσ, of vega to change in the price of the underlying asset or, equivalently, the sensitivity of delta to change in the volatility of the asset price.
19:   charm[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • charm[j-1×m+i-1] when order=Nag_ColMajor;
  • charm[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array charm contains the sensitivity, -2PS T, of delta to change in the time to expiry of the option.
20:   speed[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • speed[j-1×m+i-1] when order=Nag_ColMajor;
  • speed[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array speed contains the sensitivity, 3PS3, of gamma to change in the price of the underlying asset.
21:   colour[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • colour[j-1×m+i-1] when order=Nag_ColMajor;
  • colour[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array colour contains the sensitivity, -3PS2 T, of gamma to change in the time to expiry of the option.
22:   zomma[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • zomma[j-1×m+i-1] when order=Nag_ColMajor;
  • zomma[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array zomma contains the sensitivity, 3PS2σ, of gamma to change in the volatility of the underlying asset.
23:   vomma[m×n]doubleOutput
Note: the i,jth element of the matrix is stored in
  • vomma[j-1×m+i-1] when order=Nag_ColMajor;
  • vomma[i-1×n+j-1] when order=Nag_RowMajor.
On exit: the m×n array vomma contains the sensitivity, 2Pσ2, of vega to change in the volatility of the underlying asset.
24:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, q=value.
Constraint: q0.0.
On entry, r=value.
Constraint: r0.0.
On entry, s=value.
Constraint: svalue and svalue.
On entry, sigma=value.
Constraint: sigma>0.0.
NE_REAL_2
On entry, r=value and q=value.
Constraint: absr-q>10×eps×maxabsr,1, where eps is the machine precision.
NE_REAL_ARRAY
On entry, sm[value]=value.
Constraint: valuesm[i]value for all i.
On entry, t[value]=value.
Constraint: t[i]value for all i.
On entry with a call option, sm[value]=value.
Constraint: for call options, sm[i]value for all i.
On entry with a put option, sm[value]=value.
Constraint: for put options, sm[i]value for all i.

7  Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function, Φ. This is evaluated using a rational Chebyshev expansion, chosen so that the maximum relative error in the expansion is of the order of the machine precision (see nag_cumul_normal (s15abc) and nag_erfc (s15adc)). An accuracy close to machine precision can generally be expected.

8  Further Comments

None.

9  Example

This example computes the price of a floating-strike lookback put with a time to expiry of 6 months and a stock price of 87. The maximum price observed so far is 100. The risk-free interest rate is 6% per year and the volatility is 30% per year with an annual dividend return of 4%.

9.1  Program Text

Program Text (s30bbce.c)

9.2  Program Data

Program Data (s30bbce.d)

9.3  Program Results

Program Results (s30bbce.r)


nag_lookback_fls_greeks (s30bbc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012