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package Math::Business::BlackScholes::Binaries::Greeks::Vega; |
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use strict; |
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use warnings; |
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our $VERSION = '0.06'; ## VERSION |
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=head1 NAME |
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Math::Business::BlackScholes::Binaries::Greeks::Vega |
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=head1 DESCRIPTION |
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Gets the Vega for different options, Vanilla and Foreign for all our bet types |
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=cut |
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=head1 SUBROUTINES |
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See L |
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=cut |
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use List::Util qw( max ); |
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use Math::CDF qw( pnorm ); |
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use Math::Trig; |
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use Math::Business::BlackScholesMerton::Binaries; |
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use Math::Business::BlackScholes::Binaries::Greeks::Math qw( dgauss ); |
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sub vanilla_call { |
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my ($S, $K, $t, $r_q, $mu, $vol) = @_; |
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my $d1 = (log($S / $K) + ($mu + $vol * $vol / 2.0) * $t) / ($vol * sqrt($t)); |
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my $vega = $S * sqrt($t) * exp(($mu - $r_q) * $t) * dgauss($d1); |
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return $vega; |
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} |
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sub vanilla_put { |
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my ($S, $K, $t, $r_q, $mu, $vol) = @_; |
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# Same as vega of vanilla call |
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return vanilla_call($S, $K, $t, $r_q, $mu, $vol); |
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} |
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sub call { |
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my ($S, $U, $t, $r_q, $mu, $vol) = @_; |
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my $d1 = (log($S / $U) + ($mu + $vol * $vol / 2.0) * $t) / ($vol * sqrt($t)); |
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my $d2 = $d1 - $vol * sqrt($t); |
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my $vega = -exp(-$r_q * $t) * dgauss($d2) * $d1 / $vol; |
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return $vega; |
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} |
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sub put { |
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my ($S, $D, $t, $r_q, $mu, $vol) = @_; |
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my $d1 = (log($S / $D) + ($mu + $vol * $vol / 2.0) * $t) / ($vol * sqrt($t)); |
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my $d2 = $d1 - $vol * sqrt($t); |
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my $vega = exp(-$r_q * $t) * dgauss($d2) * $d1 / $vol; |
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return $vega; |
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} |
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sub expirymiss { |
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my ($S, $U, $D, $t, $r_q, $mu, $vol) = @_; |
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return call($S, $U, $t, $r_q, $mu, $vol) + put($S, $D, $t, $r_q, $mu, $vol); |
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} |
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sub expiryrange { |
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my ($S, $U, $D, $t, $r_q, $mu, $vol) = @_; |
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return -1 * expirymiss($S, $U, $D, $t, $r_q, $mu, $vol); |
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} |
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sub onetouch { |
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my ($S, $U, $t, $r_q, $mu, $vol, $w) = @_; |
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if (not defined $w) { |
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$w = 0; |
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} |
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my $sqrt_t = sqrt($t); |
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my $theta = ($mu / $vol) + (0.5 * $vol); |
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my $theta_ = ($mu / $vol) - (0.5 * $vol); |
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# Floor v_ squared at just above zero in case negative interest rates push it negative. |
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my $v_ = sqrt(max($Math::Business::BlackScholesMerton::Binaries::SMALL_VALUE_MU, ($theta_ * $theta_) + (2 * (1 - $w) * $r_q))); |
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my $e = (log($S / $U) - ($vol * $v_ * $t)) / ($vol * $sqrt_t); |
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my $e_ = (-log($S / $U) - ($vol * $v_ * $t)) / ($vol * $sqrt_t); |
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my $eta = ($S > $U) ? 1 : -1; |
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my $pa_e = (log($U / $S) / ($vol * $vol * $sqrt_t)) + (($theta_ * $theta) / ($vol * $v_) * $sqrt_t); |
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my $pa_e_ = (-log($U / $S) / ($vol * $vol * $sqrt_t)) + (($theta_ * $theta) / ($vol * $v_) * $sqrt_t); |
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my $A = -($theta + $theta_ + ($theta_ * $theta / $v_) + $v_) / ($vol * $vol); |
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my $A_ = -($theta + $theta_ - ($theta_ * $theta / $v_) - $v_) / ($vol * $vol); |
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my $part1 = |
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pnorm(-$eta * $e) * $A * log($U / $S) - $eta * dgauss($e) * $pa_e; |
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my $part2 = |
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pnorm($eta * $e_) * $A_ * log($U / $S) + $eta * dgauss($e_) * $pa_e_; |
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my $vega = (($U / $S)**(($theta_ + $v_) / $vol)) * $part1 + (($U / $S)**(($theta_ - $v_) / $vol)) * $part2; |
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return $vega * exp(-$w * $r_q * $t); |
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} |
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sub notouch { |
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my ($S, $U, $t, $r_q, $mu, $vol, $w) = @_; |
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# No touch bet always pay out at end |
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$w = 1; |
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return -1 * onetouch($S, $U, $t, $r_q, $mu, $vol, $w); |
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} |
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sub upordown { |
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my ($S, $U, $D, $t, $r_q, $mu, $vol, $w) = @_; |
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# $w = 0, paid at hit |
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# $w = 1, paid at end |
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if (not defined $w) { $w = 0; } |
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return ot_up_ko_down_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $vol, $w) + ot_down_ko_up_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $vol, $w); |
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} |
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sub w_common_function_pelsser_1997 { |
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my ($S, $U, $D, $t, $r_q, $mu, $vol, $w, $eta) = @_; |
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my $pi = Math::Trig::pi; |
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134
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my $h = log($U / $D); |
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my $x = log($S / $D); |
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137
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# $eta = 1, onetouch up knockout down |
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# $eta = 0, onetouch down knockout up |
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# This variable used to check stability |
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if (not defined $eta) { |
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die |
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"$0: (w_common_function_pelsser_1997) Wrong usage of this function for S=$S, U=$U, D=$D, t=$t, r_q=$r_q, mu=$mu, vol=$vol, w=$w. eta not defined."; |
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} |
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if ($eta == 0) { $x = $h - $x; } |
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my $r_dash = $r_q * (1 - $w); |
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my $mu_new = $mu - (0.5 * $vol * $vol); |
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my $mu_dash = sqrt(max($Math::Business::BlackScholesMerton::Binaries::SMALL_VALUE_MU, ($mu_new * $mu_new) + (2 * $vol * $vol * $r_dash))); |
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my $omega = ($vol * $vol); |
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my $series_part = 0; |
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my $hyp_part = 0; |
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my $stability_constant = |
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Math::Business::BlackScholesMerton::Binaries::get_stability_constant_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $vol, $w, $eta, 1); |
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my $iterations_required = Math::Business::BlackScholesMerton::Binaries::get_min_iterations_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $vol, $w); |
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for (my $k = 1; $k < $iterations_required; $k++) { |
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1710
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my $lambda_k_dash = (0.5 * (($mu_dash * $mu_dash) / $omega + ($k * $k * $pi * $pi * $vol * $vol) / ($h * $h))); |
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# d{lambda_k}/dw |
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1710
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my $dlambdak_domega = 0.5 * (-($mu_new / $omega) - (($mu_new * $mu_new) / ($omega * $omega)) + (($k * $k * $pi * $pi) / ($h * $h))); |
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my $beta_k = exp(-$lambda_k_dash * $t) / $lambda_k_dash; |
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# d{beta_k}/d{lambda_k} |
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1710
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my $dbetak_dlambdak = -exp(-$lambda_k_dash * $t) * (($t * $lambda_k_dash + 1) / ($lambda_k_dash**2)); |
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# d{beta_k}/dw |
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1710
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my $dbetak_domega = $dlambdak_domega * $dbetak_dlambdak; |
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my $phi = |
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(1.0 / ($h * $h)) * ($omega * $dbetak_domega + $beta_k) * $k; |
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2929
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$series_part += $phi * $pi * sin($k * $pi * ($h - $x) / $h); |
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# |
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# For vega, the stability function is 2* $vol * $phi, for volga/vanna it is different, |
181
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# but we shall ignore for now. |
182
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# |
183
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1710
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50
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66
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4298
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if ($k == 1 |
184
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and (not(abs(2 * $vol * $phi) < $stability_constant))) |
185
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{ |
186
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0
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0
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die |
187
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"$0: PELSSER VEGA formula for S=$S, U=$U, D=$D, t=$t, r_q=$r_q, mu=$mu, vol=$vol, w=$w, eta=$eta cannot be evaluated because PELSSER VEGA stability conditions (2 * $vol * $phi less than $stability_constant) not met. This could be due to barriers too big, volatilities too low, interest/dividend rates too high, or machine accuracy too low."; |
188
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} |
189
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} |
190
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191
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78
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124
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my $alpha = $mu_dash / ($vol * $vol); |
192
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78
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161
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my $dalpha_domega = -(($mu_new * $omega) + (2 * $mu_new * $mu_new) + (2 * $r_dash * $omega)) / (2 * $alpha * $omega * $omega * $omega); |
193
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194
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# We have to handle the special case where the denominator approaches 0, see our documentation in |
195
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# quant/Documents/Breakout_bet.tex under the SVN "quant" module. |
196
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78
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50
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214
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if ((Math::Trig::sinh($alpha * $h)**2) == 0) { |
197
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0
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0
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$hyp_part = 0; |
198
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} else { |
199
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78
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962
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$hyp_part = |
200
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($dalpha_domega / (2 * (Math::Trig::sinh($alpha * $h)**2))) * |
201
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(($h + $x) * Math::Trig::sinh($alpha * ($h - $x)) - ($h - $x) * Math::Trig::sinh($alpha * ($h + $x))); |
202
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} |
203
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204
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78
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2112
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my $dc_domega = ($hyp_part - $series_part) * exp(-$r_q * $w * $t); |
205
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206
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78
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194
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return $dc_domega; |
207
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} |
208
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209
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sub ot_up_ko_down_pelsser_1997 { |
210
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13
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13
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0
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34
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my ($S, $U, $D, $t, $r_q, $mu, $vol, $w) = @_; |
211
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212
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13
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33
|
my $mu_new = $mu - (0.5 * $vol * $vol); |
213
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13
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38
|
my $h = log($U / $D); |
214
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13
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24
|
my $x = log($S / $D); |
215
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13
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25
|
my $omega = ($vol * $vol); |
216
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217
|
13
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|
55
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my $c = Math::Business::BlackScholesMerton::Binaries::common_function_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $vol, $w, 1); |
218
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13
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|
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|
|
3582
|
my $dc_domega = w_common_function_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $vol, $w, 1); |
219
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220
|
13
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49
|
my $dVu_domega = |
221
|
|
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|
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-((0.5 * $omega + $mu_new) * ($h - $x) / ($omega * $omega)) * $c; |
222
|
13
|
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21
|
$dVu_domega += $dc_domega; |
223
|
13
|
|
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|
|
30
|
$dVu_domega *= exp($mu_new * ($h - $x) / $omega); |
224
|
|
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|
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|
|
225
|
13
|
|
|
|
|
49
|
return $dVu_domega * (2 * $vol); |
226
|
|
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|
|
} |
227
|
|
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|
|
228
|
|
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|
|
|
|
sub ot_down_ko_up_pelsser_1997 { |
229
|
13
|
|
|
13
|
0
|
41
|
my ($S, $U, $D, $t, $r_q, $mu, $vol, $w) = @_; |
230
|
|
|
|
|
|
|
|
231
|
13
|
|
|
|
|
22
|
my $mu_new = $mu - (0.5 * $vol * $vol); |
232
|
13
|
|
|
|
|
28
|
my $x = log($S / $D); |
233
|
13
|
|
|
|
|
26
|
my $omega = ($vol * $vol); |
234
|
|
|
|
|
|
|
|
235
|
13
|
|
|
|
|
35
|
my $c = Math::Business::BlackScholesMerton::Binaries::common_function_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $vol, $w, 0); |
236
|
13
|
|
|
|
|
3690
|
my $dc_domega = w_common_function_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $vol, $w, 0); |
237
|
|
|
|
|
|
|
|
238
|
13
|
|
|
|
|
33
|
my $dVl_domega = |
239
|
|
|
|
|
|
|
((0.5 * $omega + $mu_new) * $x / ($omega * $omega)) * $c; |
240
|
13
|
|
|
|
|
21
|
$dVl_domega += $dc_domega; |
241
|
13
|
|
|
|
|
25
|
$dVl_domega *= exp(-$mu_new * $x / $omega); |
242
|
|
|
|
|
|
|
|
243
|
13
|
|
|
|
|
44
|
return $dVl_domega * (2 * $vol); |
244
|
|
|
|
|
|
|
} |
245
|
|
|
|
|
|
|
|
246
|
|
|
|
|
|
|
sub range { |
247
|
6
|
|
|
6
|
0
|
4281
|
my ($S, $U, $D, $t, $r_q, $mu, $vol, $w) = @_; |
248
|
|
|
|
|
|
|
|
249
|
|
|
|
|
|
|
# Range always pay out at end |
250
|
6
|
|
|
|
|
13
|
$w = 1; |
251
|
|
|
|
|
|
|
|
252
|
6
|
|
|
|
|
22
|
return -1 * upordown($S, $U, $D, $t, $r_q, $mu, $vol, $w); |
253
|
|
|
|
|
|
|
} |
254
|
|
|
|
|
|
|
|
255
|
|
|
|
|
|
|
1; |
256
|
|
|
|
|
|
|
|