python and monte carlo

Numerical methods in Financial Engineering Projects

Yannick ARMENTI

University of Paris-Saclay, Evry, LaMME - M2IF

March 28, 2018

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Outline

1 Partial Differential Equations

2 BSDE Monte-Carlo

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Partial Differential Equations FVA

{ v(T,S) = 0 on (0,∞) ∂tv + AbsS v + λ

( ubs −v −αf σS|∆bs −∂Sv|

)+ − rv = 0 on [0,T ) × (0,∞) (1)

with ubs the Black-Scholes call price and ∆bs = ∂subs .

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Partial Differential Equations KVA

{ w(T,S) = 0 on (0,∞) ∂tw + AbsS w + h max(αf σS|∆bs|,w) − (r + h)w = 0 on [0,T ) × (0,∞)

(2) with ubs the Black-Scholes call price and ∆bs = ∂subs .

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Outline

1 Partial Differential Equations

2 BSDE Monte-Carlo

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BSDE Monte-Carlo FVA

FVAt (%) = Et

[∫ T t

e−r(s−t)λs ( u(s,Ss ) −αf σSs

∣∣∂Su(s,Ss )∣∣)+ds ]

= v(t,St ) = ubs (t,St ) −u(t,St ) (3)

with ubs (t,s) the time-t Black-Scholes call price with spot s.

1 Replace u by ubs −v in the conditional expectation 2 Identify the generator f of the above BSDE (noting that

v(t,S) := FVAt )

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BSDE Monte-Carlo KVA

KVAt (%) = hEt

[∫ T t

e−(r+h)(s−t) max(ECs, KVAs )ds

] (4)

with

ECs = αf σSs |∆bs (s,Ss )| ∆bs (t,s) = ∂subs (t,s)

ubs (t,s) the time-t Black-Scholes call price with spot s

Hint: Identify the generator f of the above BSDE

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