Monte Carlo HJM Model List Group SpA 22 July 2015 Swaption pricing through Monte Carlo HJM Model where from Monte Carlo stochastic process swaption payoff level ID: 795242
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Slide1
Swaption
pricing
through
Monte Carlo
HJM
Model
Slide2List Group S.p.A. 22 July 2015
Swaption pricing through Monte Carlo HJM Model
where
from Monte Carlo stochastic process
swaption
payoff
level
floater present value
Pricing formula
Slide3List Group S.p.A. 22 July 2015
Xf0
=
X0
+
idt
* (
Y0
+
Y1
- X0
*lambda1)
+ sqrt(idt)*sqrt(V)*(sigma1*Z[1]);Xf1 = X1 +
idt
* (Y1+Y2 - X1*lambda2) + sqrt(idt)*sqrt
(V)*(
sigma2*Z[2]); Yf0 = Y0 + idt * (V*sigma1*sigma1-2*lambda1
*Y0
);Yf1 = Y1 + idt * (V*sigma1*sigma2*rho12 - Y1*(lambda1+lambda2));Yf2 = Y2
+ idt
* (V*
sigma2*
sigma2-2*lambda2*Y2);den = (theta + (V - theta)*exp(-
kappa*
idt));Gamma = sqrt(log(1 + (0.5 * (epsilon*epsilon) * V
* (1 - exp(-2*kappa*idt
))/kappa)/(den
*den)));Vf = (theta + (V - theta) * exp(-
kappa*idt)) * exp(-0.5*Gamma*Gamma + Gamma*Z[0]);
Markov (time) stepRandom variableModel parameter (constant)Dummy variablePrevious Markov process statusNext Markov process status
M(i+1)
=
F
(
M(
i
)
,
step
, param, random )
Markov process
Swaption pricing through Monte Carlo HJM Model
Slide4List Group S.p.A. 22 July 2015
stoch_disc
[t]
=
disc
[t
]
*
exp
( - G0_1
[t] * X0
[E] - G0_2[t] * X1[E] - 0.5 * ( G0_1[t] * ( Y0[E]
* G0_1
[t] + Y1[E] * G0_2[t] ) + G0_2[t] *
( Y1
[E] * G0_1[t] + Y2[E] * G0_2[t] ) ) );stoch_fwd[t]
= - shift
+ ( shift + fwd[t] ) * exp( G_1[t] * X0[E] + G_2[t] * X1[E] + G_1[t] * (
Y0
[E] * ( G0_1
[t] -
0.5*G_1[t] ) + Y1[E] * ( G0_2[t] - 0.5*G_2[t]
)
) + G_2[t] * ( Y1[E]
* ( G0_1[t]
- 0.5*G_1[t]
) + Y2[E] * ( G0_2[t] - 0.5*G_2[t]
) ) );evaluation timeHJM Markov process statusprecomputed quantities
(dependent on HJM parameters and swaption tenor)swaption leg datesdeterministic curvesstochastic curves
Reconstruction
Swaption pricing through Monte Carlo HJM Model
Slide5List Group S.p.A. 22 July 2015
for
(
path
;;)
//MC path
get_stochastic_curves
(
E
,
path,
fwd, stoch_disc);
for (t;;) //floater leg flt_pv += fwd[t] * stoch_disc[t] * tau; for (t;;)
//fix leg
level += stoch_disc[t] * tau; payoff+= stoch_disc[0] * max( flt_pv
- level * strike , 0.0 );
payoff/=npath;
level
floater present value
swaption
payoff
Pricing
Swaption pricing through Monte Carlo HJM Model
Slide6List Group S.p.A. 22 July 2015
Setup time grid scenario
(based on time extension
of basket’s instruments)
Generate MC scenario
of Markov statuses (X,Y,V)
(paths time steps)
Loop over basket instruments
Loop over MC paths
average(payoff
)
get_stochastic_curves
get_flt_pv
get_level
payoff
Workflow overview
Swaption pricing through Monte Carlo HJM Model
once for all
once for all
for each instrument
for each path
do price
Slide7List Group S.p.A. 22 July 2015
double
Markov time lattice
MC path
Markov process status (X,Y,V
)
Scenario gran total:
Pre-build tenors
Capital G
precomputed quantities:
basket
8 bytes
~300 steps
~10k paths6 doubles140MB1M (720 dates) 3M (240), 6M (120), 1Y (60) ~ 1k dates 4 doubles < 50KB ~2k instruments
Data size
Swaption pricing through Monte Carlo HJM Model