Download Optimal stopping time for American options.

MMA707-Analytical Finance I
Optimal stopping time for American options
19th October 2016
Authors
Angelos Kalaitzis
Tariq Ade
Yige Liu
Teacher
Jan Röman
Division of Applied Mathematics
School of Education, Culture and Communication
Mälardalen University
Box 883, SE-721 23 Västerås, Sweden
1
Abstract
The purpose of this report is to find the optimal stopping time for American Options
that are useful in the field of financial engineering and in stock markets. We determine
the importance and the usage of the aforementioned optimal stopping time and we the
help of python we implemented some programmes in order to find this time.
2
Table of Contents
Abstract………………………………………………………………………. 2
Introduction...................................................................................................... .4
History of option............................................................................................... 5
Option................................................................................................................ 5
Call option..........................................................................................................5
Put option...........................................................................................................5
European option.................................................................................................6
American option.................................................................................................6
Call option..........................................................................................................6
Put option...........................................................................................................6
Value..................................................................................................................6
Optiomal stopping time......................................................................................6
Optimal stopping problem………………………………………………….......7
The American Option………………………………………………………......7
Conclution..........................................................................................................9
Bibliography.....................................................................................................10
Appendix…………………………………………………………………. 11-17
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INTRODUCTION
Through this seminar project we will focus on american option and the optimal
stopping time.
Since American option can be exercised anytime before the maturity,the aim of this
seminar is to find the optimal stopping time during which the exercise of the option
will be benefical, in other terms, the optimal time when the payoff of the american
option is maximum
4
History of Options
Options
An option is a financial contract sold by the option writer to the option holder.The
contract offers the buyer the right but not the obligation to buy (Call) or sell(Put ) the
security or other financial asset at an agreed-upon price(the strike price) during a
certain period or on a specific day (maturity)
Call Option
Call Option is a financial contract between the buyer and the seller.The buyer of the
call option has the right but not the obligation to BUY the underlying assset at a
certain price (strike price) on a specific day (maturity)
The buyer pays a fee (called a premium) for this right.The seller has the obligation to
sell this financial instrument to the buyer if the buyer decides.
Put Option
Put option are essentially the opposite of call Option.
Put Option gives the owner of a put the right but not the obligation to SELL the
underlying asset at a specified price (the strike ) on a specific day (the maturity)
European Option
A european option is an option that can only be exercised at the maturity
It gives the owner the right but not the obligation to buy or sell the underlying
security at a specific price known as the strike price on the maturity.
A European call option gives the owner the right to buy the underlying asset while a
European put gives the holder the right to sell the underlying asset at a specific price.
The value of the European option (or the Payoff) if it is exercised is given by
Call Option Payoff=Max((S-K),0)
Put Option Payoff=Max((K-S),0)
S : stock Price
K :strike Price
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American Options
An American option is an option that can be exercised at any time during the life of
the option.
Call Option
The holder of American Call option, has the right to Buy the option at any point in
time until the maturity date
Put Option
The holder of American Put option has the right to sell the option at any point in time
until the maturity date.
Value
The Value of the American option is given by :
Call Option =max(0,S-K)
Put Option = max(0,K-S)
In American Option,the option can be exercised any time until their expiration time T.
If we model the price of the assets by a stochastic porocess Xt,the optimal choice of
the moment to exercise the option in order to maximize the expected payoff
corresponds to the optimal stopping problem.
Optimal Stopping Time
Optimal stopping time is concerned with the problem of choosing the time to take a
particular action in order to maximise an expected payoff or to minimize and expected
cost.
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Optimal stopping problem
The American option
Let’s consider a stopping problem
V(x) = max E[𝑒 −𝜇𝑡 f(x,t)]
X=X(t) :is a geometric Brownian motion
dx(t)= r 𝑋𝑡 dt + σ𝑋𝑡 d𝐵𝑡 (1)
The solution of (1) is unique and is
1
𝑋𝑡 = 𝑋𝑂 exp[( r- 2 𝜎 2 )t + σ𝐵𝑡 ] (2)
𝑩𝒕 : a standard Brownian Motion process started at zero
r : interest rate
K: strike price
σ : volatility
The arbitrage free price of the perpetual (infinite horizon) American put option is
V(x) = sup 𝑬𝒙 [𝒆−𝒓𝝉 ( 𝒌 − 𝑿𝑻 )+ (3)
To exercise the option and gain the maximal value we have to find the optimal
arbitrage free price and the optimal stopping time 𝜏∗
From equations (2) and (3) we assume that there exist a point b ε(0,Κ) such that the
stopping time 𝜏𝑏 = min{t≥0 : 𝑋𝑡≤𝑏 } is optimal in equation (3)
From strong Markov property we get the boundarie conditions
𝐿𝑋 V = rV, x ≥ b
V(x) = (𝐾 − 𝑥), x = b
𝑉 ′ (x) = - 1 , x = b
V(x) > (𝐾 − 𝑥)+ , x > 𝑏
V(x) = (𝐾 − 𝑥)+ , 0< 𝑥 < 𝑏
From equation (3) the process X is strong Markov (diffusion) and the infinitesimal
generator given by
Lx= rx
𝜃
𝜃𝜒
+
𝜎2
2
𝜒2
𝜃2
𝜃𝜒2
This is the equation to solve the free-boundary problem
D𝑥 2 V’’+rxV’-rV=0
We are setting D=σ2/2 in above equation to recognize the Cauchy-Euler equation.
The solution in the form is
V(x)=xp
Inserting V(x)=xp into D𝑥 2 V’’+rxV’-rV=0
we have
r
r
p2-(1- )p=0
D
D
The above equation called quadratic equation which has two roots, 𝑝1=1 and
𝑝2 =-r/D.
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So the general solution of D𝑥 2 V’’+rxV’-rV=0
can be written
V(x)=C1x+C2x-r/D
The C1 and C2 are undetermined constants. For the condition that V(x)≤K for all x>0,
we can see that C1 must be zero. So that V(x)=(K-x)+ for x=b and V’(x)=-1 for x=b
are the two algebraic equations in two unknowns C2 and b. Then we can get
D
K
（
）1+r/D
r
1  D/r
k
b=
1  D/r
C2 =
Inserting C2 into V(x)=C1x+C1x-r/D under the condition C1=0 we can conclude that
𝐷
V(x)=f(x) =
(
𝐾
{ 𝑟 1+𝐷/𝑟
)1+𝑟/𝐷 𝑥 −𝑟/𝐷 ), if x ∈ [b, ∞)
K − x, if x ∈ (0, b]
V is C2 on (0,b)∪(b, ∞) but only C1 at b, and also that V is convex on (0, ∞)
In conclusion
The arbitrage-free price V from V(x) = sup 𝑬𝒙 [𝒆−𝒓𝝉 ( 𝒌 − 𝑿𝑻 )+ is given explicitly
by
𝑫
𝑲
(
)𝟏+𝒓/𝑫 𝒙−𝒓/𝑫 ), 𝒊𝒇 𝒙 ∈ [𝒃, ∞)
V(x) = 𝒇(𝒙) = { 𝒓 𝟏+𝑫/𝒓
.
𝑲 − 𝒙, 𝒊𝒇 𝒙 ∈ (𝟎, 𝒃]
k
The stopping time τb from 𝝉𝒃 = min{t≥0 : 𝑿𝒕≤𝒃} with b given by b=
is
1  D/r
optimal in the problem V(x) = sup 𝐸𝑥 [𝑒 −𝑟𝜏 ( 𝑘 − 𝑋𝑇 )+
8
Conclusion
After finding the optimal stopping time problem of american options,we have listed
the boundaries conditions.
To exercise the option and gain the maximal value we had to find the optimal
arbitrage free price and the optimal stopping time τ_* under the boundaries
conditions.
In order to prove the efficiency of the optimal stopping time of american option,we
implemented the payoff of american option using monte carlo and compared it to the
payoff of european option.
We observed that the american option payoff at the optimal stopping time is superior
to the payoff of european option
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Bibliography
[1] Options : http://www.investopedia.com/terms/o/option.asp
[2] Call Option : https://en.wikipedia.org/wiki/Call_option
[3] Put Option : https://en.wikipedia.org/wiki/Put_option
[4] European Option : http://www.investopedia.com/terms/e/europeanoption.asp
[5] American Option : http://www.investopedia.com/terms/a/americanoption.asp
[6] Optimal stopping time
https://en.wikipedia.org/wiki/Optimal_stopping
[7]Goran Peskir & Albert Shiryaev,Optimal stopping and Free-Boundary
Problems(2006) ISBN :978-3-7643-2419-3
[8]Yves Hilpisch, Python for Finance-Analyze Big Financial Data(2014) ISBN :9781491945285
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Appendix
Python programmes
#!/usr/bin/env python2
# -*- coding: utf-8 -*"""
Created on Thu Oct 13 21:13:51 2016
@author: SNS_FSB
"""
def bsm_call_value(S0, K, T, r, sigma):
#Valuation of European call option in BSM model.
#Analytical formula.
#Parameters
#==========
#S0 : float
#initial stock/index level
#K : float
#strike price
#T : float
#maturity date (in year fractions)
#r : float
#constant risk-free short rate
#sigma : float
#volatility factor in diffusion term
#Returns
#=======
#value : float
#present value of the European call option
from math import log, sqrt, exp
from scipy import stats
S0 = float(S0)
d1 = (log(S0 / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
d2 = (log(S0 / K) + (r - 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
value = (S0 * stats.norm.cdf(d1, 0.0, 1.0) - K * exp(-r * T) * stats.norm.cdf(d2, 0.0,
1.0))
# stats.norm.cdf --> cumulative distribution function
# for normal distribution
return value
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#!/usr/bin/env python2
# -*- coding: utf-8 -*"""
Created on Fri Oct 14 10:43:13 2016
@author: SNS_FSB
"""
def gbm_mcs_amer(K, option='call'):
''' Valuation of American option in Black-Scholes-Merton
by Monte Carlo simulation by LSM algorithm
Parameters
==========
K : float
(positive) strike price of the option
option : string
type of the option to be valued ('call', 'put')
Returns
C0 : float
estimated present value of European call option
'''
dt = T / M
df = np.exp(-r * dt)
# simulation of index levels
S = np.zeros((M + 1, I))
S[0] = S0
sn = gen_sn(M, I)
for t in range(1, M + 1):
S[t] = S[t - 1] * np.exp((r - 0.5 * sigma ** 2) * dt + sigma * np.sqrt(dt) * sn[t])
# case-based calculation of payoff
if option == 'call':
h = np.maximum(S - K, 0)
else:
h = np.maximum(K - S, 0)
# LSM algorithm
V = np.copy(h)
for t in range(M - 1, 0, -1):
reg = np.polyfit(S[t], V[t + 1] * df, 7)
C = np.polyval(reg, S[t])
V[t] = np.where(C > h[t], V[t + 1] * df, h[t])
# MCS estimator
C0 = df * 1 / I * np.sum(V[1])
return C0
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#!/usr/bin/env python2
# -*- coding: utf-8 -*"""
Created on Fri Oct 14 11:39:50 2016
@author: SNS_FSB
"""
M = 50
def gbm_mcs_dyna(K, option='call'):
''' Valuation of European options in Black-Scholes-Merton
by Monte Carlo simulation (of index level paths)
Parameters
==========
K : float
(positive) strike price of the option
option : string
type of the option to be valued ('call', 'put')
Returns
=======
C0 : float
estimated present value of European call option
'''
dt = T / M
# simulation of index level paths
S = np.zeros((M + 1, I))
S[0] = S0
sn = gen_sn(M, I)
for t in range(1, M + 1):
S[t] = S[t - 1] * np.exp((r - 0.5 * sigma ** 2) * dt
+ sigma * np.sqrt(dt) * sn[t])
# case-based calculation of payoff
if option == 'call':
hT = np.maximum(S[-1] - K, 0)
else:
hT = np.maximum(K - S[-1], 0)
# calculation of MCS estimator
C0 = np.exp(-r * T) * 1 / I * np.sum(hT)
return C0
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#!/usr/bin/env python2
# -*- coding: utf-8 -*"""
Created on Fri Oct 14 11:07:26 2016
@author: SNS_FSB
"""
S0 = 100.
r = 0.05
sigma = 0.25
T = 1.0
I = 50000
def gbm_mcs_stat(K):
import numpy as np
import numpy.random as npr
import matplotlib.pyplot as plt
''' Valuation of European call option in Black-Scholes-Merton
by Monte Carlo simulation (of index level at maturity)
Parameters
==========
K : float
(positive) strike price of the option
Returns
=======
C0 : float
estimated present value of European call option
'''
sn = gen_sn(1, I)
# simulate index level at maturity
ST = S0 * np.exp((r - 0.5 * sigma ** 2) * T
+ sigma * np.sqrt(T) * sn[1])
# calculate payoff at maturity
hT = np.maximum(ST - K, 0)
# calculate MCS estimator
C0 = np.exp(-r * T) * 1 / I * np.sum(hT)
return C0
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#!/usr/bin/env python2
# -*- coding: utf-8 -*"""
Created on Fri Oct 14 11:05:44 2016
@author: SNS_FSB
"""
def gen_sn(M, I, anti_paths=True, mo_match=True):
''' Function to generate random numbers for simulation.
Parameters
==========
M : int
number of time intervals for discretization
I : int
number of paths to be simulated
anti_paths: Boolean
use of antithetic variates
mo_math : Boolean
use of moment matching
'''
import numpy as np
import numpy.random as npr
import matplotlib.pyplot as plt
if anti_paths is True:
sn = npr.standard_normal((M + 1, I / 2))
sn = np.concatenate((sn, -sn), axis=1)
else:
sn = npr.standard_normal((M + 1, I))
if mo_match is True:
sn = (sn - sn.mean()) / sn.std()
return sn
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#!/usr/bin/env python2
# -*- coding: utf-8 -*"""
Created on Fri Oct 14 11:23:12 2016
@author: SNS_FSB
"""
I = 10000
M = 50
dt = T / M
S = np.zeros((M + 1, I))
S[0] = S0
for t in range(1, M + 1):
S[t] = S[t - 1] * np.exp((r - 0.5 * sigma ** 2) * dt
+ sigma * np.sqrt(dt) * npr.standard_normal(I))
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#!/usr/bin/env python2
# -*- coding: utf-8 -*#"""
#Created on Thu Oct 13 22:06:57 2016
#@author: SNS_FSB
#"""
#
# Monte Carlo valuation of European call options with NumPy
# mcs_vector_numpy.py
#
import math
import numpy as np
from time import time
np.random.seed(20000)
t0 = time()
# Parameters
S0 = 100.; K = 105.; T = 1.0; r = 0.05; sigma = 0.2
M = 50; dt = T / M; I = 250000
# Simulating I paths with M time steps
S = np.zeros((M + 1, I))
S[0] = S0
for t in range(1, M + 1):
z = np.random.standard_normal(I)
# pseudorandom numbers
S[t] = S[t - 1] * np.exp((r - 0.5 * sigma ** 2) * dt + sigma * math.sqrt(dt) * z)
# vectorized operation per time step over all paths
# Calculating the Monte Carlo estimator
C0 = math.exp(-r * T) * np.sum(np.maximum(S[-1] - K, 0)) / I
# Results output
tnp1 = time() - t0
print "European Option Value %7.3f" % C0
print "Duration in Seconds %7.3f" % tnp1
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#!/usr/bin/env python2
# -*- coding: utf-8 -*"""
Created on Thu Oct 13 21:13:51 2016
@author: SNS_FSB
"""
def bsm_call_value(S0, K, T, r, sigma):
#Valuation of European call option in BSM model.
#Analytical formula.
#Parameters
#==========
#S0 : float
#initial stock/index level
#K : float
#strike price
#T : float
#maturity date (in year fractions)
#r : float
#constant risk-free short rate
#sigma : float
#volatility factor in diffusion term
#Returns
#=======
#value : float
#present value of the European call option
from math import log, sqrt, exp
from scipy import stats
S0 = float(S0)
d1 = (log(S0 / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
d2 = (log(S0 / K) + (r - 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
value = (S0 * stats.norm.cdf(d1, 0.0, 1.0) - K * exp(-r * T) * stats.norm.cdf(d2, 0.0,
1.0))
# stats.norm.cdf --> cumulative distribution function
# for normal distribution
return value
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#!/usr/bin/env python2
# -*- coding: utf-8 -*"""
Created on Fri Oct 14 10:34:28 2016
@author: SNS_FSB
"""
# Vega function
def bsm_vega(S0, K, T, r, sigma):
''' Vega of European option in BSM model.
Parameters
==========
S0 : float
initial stock/index level
K : float
strike price
T : float
maturity date (in year fractions)
r : float
constant risk-free short rate
sigma : float
volatility factor in diffusion term
Returns
=======
vega : float
partial derivative of BSM formula with respect
to sigma, i.e. Vega
'''
import numpy as np
import numpy.random as npr
import matplotlib.pyplot as plt
from math import log, sqrt
from scipy import stats
S0 = float(S0)
d1 = (log(S0 / K) + (r + 0.5 * sigma ** 2) * T / (sigma * sqrt(T))
vega = S0 * stats.norm.cdf(d1, 0.0, 1.0) * sqrt(T)
return vega
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