Equivalence of Put Pricing FormulasPricing options under restricted domainNuméraire — couldn't understand the wiki explanationGil-Palaez Inversion Formula in Black Scholes worldA clarification on the Heston option pricing formulaNumerical Methods for Merton ModelCharacteristic functions for options on futuresODE Solution in Carr's Randomized American PutTrouble understanding jump part in Kou double exponential jump diffusion modelFair value of a binary cash-or-nothing option with a barrier
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Equivalence of Put Pricing Formulas
Pricing options under restricted domainNuméraire — couldn't understand the wiki explanationGil-Palaez Inversion Formula in Black Scholes worldA clarification on the Heston option pricing formulaNumerical Methods for Merton ModelCharacteristic functions for options on futuresODE Solution in Carr's Randomized American PutTrouble understanding jump part in Kou double exponential jump diffusion modelFair value of a binary cash-or-nothing option with a barrier
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;
$begingroup$
I have to show that:
beginequation
P_t,T(K)=e^-r(T-t) int_0^inftyleft(K-Sright)^+ q_T^S(S)dS
endequation
is equivalent to:
beginequation
P_t,T(K)=e^-r(T-t)int_-infty^Kleft(int_-infty^y q_T^S(z)dzright)dy
endequation
Breeden and Litzenberger have shown that using Leibniz integration rule and differentiating the first equation twice leads to:
beginequation
q_T^S(K)=e^rf(T-t)fracpartial^2P_t,T(K)partial K^2vert_K=S_T
endequation
However, I have difficulties to directly go from the first to the second equation in an elegant way. Does anyone have an idea how this can be achieved?
Many thanks for the help!
options option-pricing risk-neutral-measure pricing
asked Jun 25 at 13:26
William BurknechtWilliam Burknecht
505 bronze badges
$endgroup$
add a comment |
$begingroup$
I have to show that:
beginequation
P_t,T(K)=e^-r(T-t) int_0^inftyleft(K-Sright)^+ q_T^S(S)dS
endequation
is equivalent to:
beginequation
P_t,T(K)=e^-r(T-t)int_-infty^Kleft(int_-infty^y q_T^S(z)dzright)dy
endequation
Breeden and Litzenberger have shown that using Leibniz integration rule and differentiating the first equation twice leads to:
beginequation
q_T^S(K)=e^rf(T-t)fracpartial^2P_t,T(K)partial K^2vert_K=S_T
endequation
However, I have difficulties to directly go from the first to the second equation in an elegant way. Does anyone have an idea how this can be achieved?
Many thanks for the help!
options option-pricing risk-neutral-measure pricing
asked Jun 25 at 13:26
William BurknechtWilliam Burknecht
505 bronze badges
$endgroup$
add a comment |
$begingroup$
I have to show that:
beginequation
P_t,T(K)=e^-r(T-t) int_0^inftyleft(K-Sright)^+ q_T^S(S)dS
endequation
is equivalent to:
beginequation
P_t,T(K)=e^-r(T-t)int_-infty^Kleft(int_-infty^y q_T^S(z)dzright)dy
endequation
Breeden and Litzenberger have shown that using Leibniz integration rule and differentiating the first equation twice leads to:
beginequation
q_T^S(K)=e^rf(T-t)fracpartial^2P_t,T(K)partial K^2vert_K=S_T
endequation
However, I have difficulties to directly go from the first to the second equation in an elegant way. Does anyone have an idea how this can be achieved?
Many thanks for the help!
options option-pricing risk-neutral-measure pricing
asked Jun 25 at 13:26
William BurknechtWilliam Burknecht
505 bronze badges
$endgroup$
I have to show that:
beginequation
P_t,T(K)=e^-r(T-t) int_0^inftyleft(K-Sright)^+ q_T^S(S)dS
endequation
is equivalent to:
beginequation
P_t,T(K)=e^-r(T-t)int_-infty^Kleft(int_-infty^y q_T^S(z)dzright)dy
endequation
Breeden and Litzenberger have shown that using Leibniz integration rule and differentiating the first equation twice leads to:
beginequation
q_T^S(K)=e^rf(T-t)fracpartial^2P_t,T(K)partial K^2vert_K=S_T
endequation
However, I have difficulties to directly go from the first to the second equation in an elegant way. Does anyone have an idea how this can be achieved?
Many thanks for the help!
options option-pricing risk-neutral-measure pricing
options option-pricing risk-neutral-measure pricing
asked Jun 25 at 13:26
William BurknechtWilliam Burknecht
505 bronze badges
asked Jun 25 at 13:26
William BurknechtWilliam Burknecht
505 bronze badges
asked Jun 25 at 13:26
William BurknechtWilliam Burknecht
505 bronze badges
asked Jun 25 at 13:26
William BurknechtWilliam Burknecht
505 bronze badges
asked Jun 25 at 13:26
William BurknechtWilliam Burknecht
505 bronze badges
505 bronze badges
add a comment |
add a comment |
1 Answer
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$begingroup$
The first equation expresses the option price as a discounted expected value of the payoff contingent on an asset price $S geqslant 0$. Without loss of generality, we assume that the probability density function has support in $[0,infty)$, and rewrite as
$$beginalign P_t,T(K) &=e^-r(T-t) int_-infty^inftyleft(K-Sright)^+ q_T^S(S),dS \ &= e^-r(T-t) int_-infty^Kleft(K-Sright) q_T^S(S),dS endalign $$
Integrating by parts with $u = K-S$ and $dv = q_T^S(S),dS $, we have $du = -dS $ and
$$v = int_-infty^S q_T^S(z) , dz,$$
which with vanishing boundaries terms yields the result
$$P_t,T(K) = e^-r(T-t) int_-infty^K left(int_-infty^Sq_T^S(z) , dz right) , dS$$
answered Jun 25 at 18:31
RRLRRL
2,3007 silver badges13 bronze badges
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
The first equation expresses the option price as a discounted expected value of the payoff contingent on an asset price $S geqslant 0$. Without loss of generality, we assume that the probability density function has support in $[0,infty)$, and rewrite as
$$beginalign P_t,T(K) &=e^-r(T-t) int_-infty^inftyleft(K-Sright)^+ q_T^S(S),dS \ &= e^-r(T-t) int_-infty^Kleft(K-Sright) q_T^S(S),dS endalign $$
Integrating by parts with $u = K-S$ and $dv = q_T^S(S),dS $, we have $du = -dS $ and
$$v = int_-infty^S q_T^S(z) , dz,$$
which with vanishing boundaries terms yields the result
$$P_t,T(K) = e^-r(T-t) int_-infty^K left(int_-infty^Sq_T^S(z) , dz right) , dS$$
answered Jun 25 at 18:31
RRLRRL
2,3007 silver badges13 bronze badges
$endgroup$
add a comment |
$begingroup$
The first equation expresses the option price as a discounted expected value of the payoff contingent on an asset price $S geqslant 0$. Without loss of generality, we assume that the probability density function has support in $[0,infty)$, and rewrite as
$$beginalign P_t,T(K) &=e^-r(T-t) int_-infty^inftyleft(K-Sright)^+ q_T^S(S),dS \ &= e^-r(T-t) int_-infty^Kleft(K-Sright) q_T^S(S),dS endalign $$
Integrating by parts with $u = K-S$ and $dv = q_T^S(S),dS $, we have $du = -dS $ and
$$v = int_-infty^S q_T^S(z) , dz,$$
which with vanishing boundaries terms yields the result
$$P_t,T(K) = e^-r(T-t) int_-infty^K left(int_-infty^Sq_T^S(z) , dz right) , dS$$
answered Jun 25 at 18:31
RRLRRL
2,3007 silver badges13 bronze badges
$endgroup$
add a comment |
$begingroup$
The first equation expresses the option price as a discounted expected value of the payoff contingent on an asset price $S geqslant 0$. Without loss of generality, we assume that the probability density function has support in $[0,infty)$, and rewrite as
$$beginalign P_t,T(K) &=e^-r(T-t) int_-infty^inftyleft(K-Sright)^+ q_T^S(S),dS \ &= e^-r(T-t) int_-infty^Kleft(K-Sright) q_T^S(S),dS endalign $$
Integrating by parts with $u = K-S$ and $dv = q_T^S(S),dS $, we have $du = -dS $ and
$$v = int_-infty^S q_T^S(z) , dz,$$
which with vanishing boundaries terms yields the result
$$P_t,T(K) = e^-r(T-t) int_-infty^K left(int_-infty^Sq_T^S(z) , dz right) , dS$$
answered Jun 25 at 18:31
RRLRRL
2,3007 silver badges13 bronze badges
$endgroup$
The first equation expresses the option price as a discounted expected value of the payoff contingent on an asset price $S geqslant 0$. Without loss of generality, we assume that the probability density function has support in $[0,infty)$, and rewrite as
$$beginalign P_t,T(K) &=e^-r(T-t) int_-infty^inftyleft(K-Sright)^+ q_T^S(S),dS \ &= e^-r(T-t) int_-infty^Kleft(K-Sright) q_T^S(S),dS endalign $$
Integrating by parts with $u = K-S$ and $dv = q_T^S(S),dS $, we have $du = -dS $ and
$$v = int_-infty^S q_T^S(z) , dz,$$
which with vanishing boundaries terms yields the result
$$P_t,T(K) = e^-r(T-t) int_-infty^K left(int_-infty^Sq_T^S(z) , dz right) , dS$$
answered Jun 25 at 18:31
RRLRRL
2,3007 silver badges13 bronze badges
answered Jun 25 at 18:31
RRLRRL
2,3007 silver badges13 bronze badges
answered Jun 25 at 18:31
RRLRRL
2,3007 silver badges13 bronze badges
answered Jun 25 at 18:31
RRLRRL
2,3007 silver badges13 bronze badges
2,3007 silver badges13 bronze badges
add a comment |
add a comment |
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