Pricing American Options - Duality approach in Monte Carlo

Pricing American Options - Duality approach in Monte Carlo

Pricing American Options - Duality approach inMonte Carlo

Ilnaz Asadzadeh

The University of Calgary

April 16, 2013


Introduction

Pricing American Option

I Pricing an American option is essentially an equivalence to aproblem of solving an optimal stopping problem by definingthe optimal exercise rule.

I The value of an American option is thus calculated bycomputing the expected discounted payoff under this rule.

I valuations of American options mostly rely on numericalsimulations and the optimization.

I Because of curse of dimensional Monte Carlo methods arebeing used for this problem.

I Here we are going to use Duality approach developed byRogers and independently by Haugh and Kogan.


Introduction








Introduction








Introduction








Introduction








Problem Formulation

Assumptions

I we denote the payoff of an American option at any timet ∈ (0,T ] as h(t) = h(St , t).

I Given a class of admissible stopping times Tt the Americanoption pricing problem can be formulated asV0 = supτ∈T0E [e−

∫ τ0 r(u)duh(Sτ , τ)].

I The value of American put option at t = 0 is then given byV0 = supτ∈T0E [e−rτ (K − Sτ )+]

I we denote Si = S(t = ti ) i ∈ (0, 1, 2, ...,M) as the state ofasset price. By Euler-Maruyama scheme we can approximatethe asset price process:

I Si = Si−1 + (r − δ)Si−1∆t + σSi−1∆Wi i = 1, 2, ...,M


Problem Formulation

Assumptions








Problem Formulation

Assumptions








Problem Formulation

Assumptions








Problem Formulation

Assumptions








Asset Price Paths

Asset Price Movements

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

40

45


Asset Price Paths

Optimal Stopping Rule and Continuation Values

I Optimal stopping time isτ∗ = min{τi ∈ {t1, ..., tM} : hi (Si ) ≥ Vi (Si )}

I Continuation Value of an American option is the value ofholding rather than exercising the option

I The value function by recursive relation is

VM(s) = hM(s), SM = s;Vi−1(s) = max{hi−1(s),E [Vi (Si )|Si−1 = s]}, i = 1, 2, ...,M.

I The continuation value is defined

CM(s) = 0;Ci (s) = E [Vi+1(Si+1)|Si = s], i = 1, 2, ...,M.


Asset Price Paths









Asset Price Paths









Asset Price Paths









Approximating Continuation Values by Least Square Regression

The continuation value is approximated by linear combination ofknown functions (basis functions) of the current state of assetprice s. By least square based regression. We want to find anexpression of the form

Ck(s) = E [Vk+1(Sk+1)|Sk = s] =K∑

k=1

βkiψi (s) = βTk ψ(s) (1)



If there exists a relationship between the continuation value Ci (s)and current state of asset price s in the form (1), the least-squaresregression will give the coefficients βi in the form

βi = (E [ψ(Si )ψ(Si )T ])−1E [ψ(Si )Vi+1(Si+1)] ≡ B−1ψ BψV (2)

where Bψ = E [ψ(Si )ψ(Si )T ] is a K × K matrix and

BψV = E [ψ(Si )Vi+1(Si+1)] is a vector of length K.



Simulating Coefficients βi by Monte Carlo

I First we need N different simulated paths of asset price

(S(n)1 ,S

(n)2 , ...,S

(n)M )

I Second we assume that the value functions Vi+1(S(n)i+1) are

known.

I Third the least square estimation of βi is given byβi = B−1ψ,i BψV ,i

I where

Bψ,i = 1/NN∑

n=1

ψ(S(n)i )ψ(S

(n)i )T (3)

BψV ,i = 1/NN∑

n=1

ψ(S(n)i )Vi+1(S

(n)i+1) (4)





(S(n)1 ,S

(n)2 , ...,S

(n)M )


known.


I where

Bψ,i = 1/NN∑

n=1

ψ(S(n)i )ψ(S

(n)i )T (3)

BψV ,i = 1/NN∑

n=1

ψ(S(n)i )Vi+1(S

(n)i+1) (4)





(S(n)1 ,S

(n)2 , ...,S

(n)M )


known.


I where

Bψ,i = 1/NN∑

n=1

ψ(S(n)i )ψ(S

(n)i )T (3)

BψV ,i = 1/NN∑

n=1

ψ(S(n)i )Vi+1(S

(n)i+1) (4)





(S(n)1 ,S

(n)2 , ...,S

(n)M )


known.


I where

Bψ,i = 1/NN∑

n=1

ψ(S(n)i )ψ(S

(n)i )T (3)

BψV ,i = 1/NN∑

n=1

ψ(S(n)i )Vi+1(S

(n)i+1) (4)



In practice Vi+1 need to be replaced by the estimated values

Vi+1(Si+1) = max{hi+1(Si+1), Ci+1(Si+1)}

Then the continuation value can be estimated by

Ci (Si ) = βTi ψ(Si ) (5)


Duality Approach in Monte Carlo

Using Duality for computing an upper bound for theOption Price

I The duality minimizes over a class of supermartingales ormartingales and provides upper bounds on prices

I Base on recursive relation we have

Vi (Si ) ≥ E [Vi (Si+1)|Si ], i = 0, ...,M − 1

which is the definition of supermartingale.

I We are using the result from Hugh and Kogan withspecializations to martingales.

I Let M = {Mi , i = 0, ...,M} be a martingale with M0 = 0.






Vi (Si ) ≥ E [Vi (Si+1)|Si ], i = 0, ...,M − 1









Vi (Si ) ≥ E [Vi (Si+1)|Si ], i = 0, ...,M − 1






Theorem

For any martingale M = {Mi , i = 0, ...,M} satisfying M0 = 0,the price of American option V0(S0) satisfies the inequality

V0(S0) ≤ infME [maxi=1,...,M{hi (Si )−Mi}] (6)

The equality holds with the optimal martingale M∗defined by

M∗0 = 0;

M∗i =∑i

j=1 ∆j , fori = 1, 2, ...,M.

where ∆i is given by ∆i = Vi (Si )− E [Vi (Si )|Si−1]



Duality Estimator

I we can estimate the value of an American option by

V0(S0) = E [maxi=1,...,M{hi (Si )− Mi}] (7)

which is our duality estimator

I For estimating the duality we should construct a martingaleM that is close to optimal martingale M∗.

I define M as

M0 = 0

Mi =∑i

k=1 ∆k , fori = 1, ...,M(8)

Where∆k = Vi (Si )− E [Vi (Si )|Si−1] (9)



Duality Estimator


V0(S0) = E [maxi=1,...,M{hi (Si )− Mi}] (7)



I define M as

M0 = 0

Mi =∑i

k=1 ∆k , fori = 1, ...,M(8)




Duality Estimator


V0(S0) = E [maxi=1,...,M{hi (Si )− Mi}] (7)



I define M as

M0 = 0

Mi =∑i

k=1 ∆k , fori = 1, ...,M(8)




Duality Estimator

for constructing a martingale we should use a nested simulation,

I at each step after simulating the main paths

{S (n)i : n = 1, ...,N}, we simulate m sub-paths

I estimate the conditional expectation E [Vi (Si )|Si−1] by

E [Vi (Si )|Si−1] = 1/mm∑

k=1

Vi (S(k)i ) (10)

where S(k)i is the simulation of sub paths.



Duality Estimator

for constructing a martingale we should use a nested simulation,

I at each step after simulating the main paths

{S (n)i : n = 1, ...,N}, we simulate m sub-paths

I estimate the conditional expectation E [Vi (Si )|Si−1] by

E [Vi (Si )|Si−1] = 1/mm∑

k=1

Vi (S(k)i ) (10)

where S(k)i is the simulation of sub paths.



Numerical Result

We implement the method with the following parameters S0 = 30,K = 30, σ = 0.4, r = 0.05, T = 1, the European Put option pricewith the mentioned parameters with Black-Scholes formula is3.944.

Table : American Option Upper Bound Price

Path Sub Path Time Step Duality Value Variance Duality

10000 200 32 4.3762 5.5478e-0410000 400 32 4.3610 5.3513e-0410000 600 32 4.3299 5.2989e-0410000 800 32 4.3043 5.2645e-04


References

References

I Martin B. Haugh, Leonid Kogan (2003). Pricing AmericanOptions: A duality Approach, Operations Research.

I Francis A. Longstaff, Eduardo S.Schwartz (2001), ValuingAmerican Options by Simulation: A simple Least-SquaresApproach, Anderson Graduate School of Management, UCLos Angeles.

I Nikolay Aleksandrov Aleksandrov (2006). Pricing AmericanOptions - Monte Carlo approach, Christ Church CollegeUniversity of Oxford

I P. Glasserman. Monte Carlo Methods in FinancialEngineering. Springer, 2004.

I Wikipedia

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Pricing American Options - Duality approach in Monte Carlo