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NAG Toolbox: nag_specfun_opt_lookback_fls_price (s30ba)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_opt_lookback_fls_price (s30ba) computes the price of a floating-strike lookback option.

Syntax

[p, ifail] = s30ba(calput, sm, s, t, sigma, r, q, 'm', m, 'n', n)
[p, ifail] = nag_specfun_opt_lookback_fls_price(calput, sm, s, t, sigma, r, q, 'm', m, 'n', n)

Description

nag_specfun_opt_lookback_fls_price (s30ba) computes the price of a floating-strike lookback call or put option. 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. When r=q, the option price is given by
Pcall = S e-qT Φ a1 - Smin e-rT Φ a2 + S e-rT σT ϕ a1 + a1 Φ a1 -1 .  
The corresponding put price is (for b0),
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 .  
When r=q,
Pput = Smax e-rT Φ -a2 - S e-qT Φ -a1 + S e-rT σT ϕa1 + a1 Φa1 .  
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.
The option price Pij=PX=Xi,T=Tj is computed for each minimum or maximum observed price in a set Smin i  or Smax i , i=1,2,,m, and for each expiry time in a set Tj, j=1,2,,n.

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

Parameters

Compulsory Input Parameters

1:     calput – string (length ≥ 1)
Determines whether the option is a call or a put.
calput='C'
A call; the holder has a right to buy.
calput='P'
A put; the holder has a right to sell.
Constraint: calput='C' or 'P'.
2:     smm – double array
smi must contain Smin i , the ith minimum observed price of the underlying asset when calput='C', or Smax i , the maximum observed price when calput='P', for i=1,2,,m.
Constraints:
  • smiz ​ and ​ smi 1 / z , where z = x02am , the safe range parameter, for i=1,2,,m;
  • if calput='C', smiS, for i=1,2,,m;
  • if calput='P', smiS, for i=1,2,,m.
3:     s – double scalar
S, the price of the underlying asset.
Constraint: sz ​ and ​s1.0/z, where z=x02am, the safe range parameter.
4:     tn – double array
ti must contain Ti, the ith time, in years, to expiry, for i=1,2,,n.
Constraint: tiz, where z = x02am , the safe range parameter, for i=1,2,,n.
5:     sigma – double scalar
σ, the volatility of the underlying asset. Note that a rate of 15% should be entered as 0.15.
Constraint: sigma>0.0.
6:     r – double scalar
r, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.
Constraint: r0.0.
7:     q – double scalar
q, the annual continuous yield rate. Note that a rate of 8% should be entered as 0.08.
Constraint: q0.0.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the dimension of the array sm.
The number of minimum or maximum prices to be used.
Constraint: m1.
2:     n int64int32nag_int scalar
Default: the dimension of the array t.
The number of times to expiry to be used.
Constraint: n1.

Output Parameters

1:     pldpn – double array
ldp=m.
pij contains Pij, the option price evaluated for the minimum or maximum observed price Smin i or Smax i at expiry tj for i=1,2,,m and j=1,2,,n.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
On entry, calput=_ was an illegal value.
   ifail=2
Constraint: m1.
   ifail=3
Constraint: n1.
   ifail=4
Constraint: _smi_ for all i.
   ifail=5
Constraint: s_ and s_.
   ifail=6
Constraint: ti_ for all i.
   ifail=7
Constraint: sigma>0.0.
   ifail=8
Constraint: r0.0.
   ifail=9
Constraint: q0.0.
   ifail=11
Constraint: ldpm.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

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_specfun_cdf_normal (s15ab) and nag_specfun_erfc_real (s15ad)). An accuracy close to machine precision can generally be expected.

Further Comments

None.

Example

This example computes the price of a floating-strike lookback call with a time to expiry of 6 months and a stock price of 120. The minimum price observed so far is 100. The risk-free interest rate is 10% per year and the volatility is 30% per year with an annual dividend return of 6%.
function s30ba_example


fprintf('s30ba example results\n\n');

put = 'c';
s = 120;
sigma = 0.3;
r = 0.1;
q = 0.06;
sm = [100.0];
t = [0.5];

[p, ifail] = s30ba( ...
                    put, sm , s, t, sigma, r, q);

fprintf('\nFloating-strike Lookback\n European Call :\n');
fprintf('  Spot       =   %9.4f\n', s);
fprintf('  Volatility =   %9.4f\n', sigma);
fprintf('  Rate       =   %9.4f\n', r);
fprintf('  Dividend   =   %9.4f\n\n', q);

fprintf('   Strike    Expiry   Option Price\n');
for i=1:1
  for j=1:1
    fprintf('%9.4f %9.4f %9.4f\n', sm(i), t(j), p(i,j));
  end
end


s30ba example results


Floating-strike Lookback
 European Call :
  Spot       =    120.0000
  Volatility =      0.3000
  Rate       =      0.1000
  Dividend   =      0.0600

   Strike    Expiry   Option Price
 100.0000    0.5000   25.3534

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