金融时间序列 opt

Option Pricing
Introduction
An option is afinancial instrument which gives its holder
the right to buy or sell an asset within a specified period of time
It is termed European if its holder is allowed only to
exercise the right to the claim on its expiration date T;
otherwise it is called American
A call option gives the right to buy an asset while a put
option gives the right to sell one
Specifically,a European call option with strike price K
gives its holder the right to buy the underlying asset at a price of K on date T
If S
T is the asset price on expiration date T,the call option payoff is(S T−K)+:=max{0,S T−K}
The Black-Scholes Model
In order to make the option pricing problem tractable,
Black and Scholes(1973)made the following
assumptions on the market and for the option:
The short-term interest rate is a known constant over
time
The asset pays no dividends or other distributions
There are no transaction costs in buying or selling
the asset or the option
It is possible to borrow any fraction of the price of a
security to buy it or to hold it,at the short-term
interest rate
There are no penalties to short selling
The asset price follows a geometric Brownian motion
with constant volatility
Specifically,it is assumed that there exist an asset and
a zero-coupon bond maturing at T,of constant return r The randomness of the asset price{S
t,t≥0}is attributable to a fully observable standard Brownian motion{W t,t≥0},which is a continuous-time stochastic process with independent increments that are normally distributed
If s≤t,W t−W s is independent of W u for all u≤s
[Markov property]
If s≤t,W t−W s is normally distributed with mean
zero and variance t−s[written W t−W s∼N(0,t−s)] As a consequence,W t−W0∼N(0,t)[usually it is also assumed that W0=0]
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The asset price process is governed by the stochastic differential equation(SDE):
dS t=µS t dt+σS t dW t(S0>0)(1) for constantsµandσ,respectively the instantaneous mean and standard deviation of the asset
return
The price process(1)is an example of an Itôprocess:
dX t=a(t,X t)dt+b(t,X t)dW t
or equivalently
X t=X0+ t0a(s,X s)ds+ t0b(s,X s)dW s
Itô’s
Lemma
Itô’s lemma is the basis of stochastic
calculus
If g(t,x)is a twice continuously differentiable function on [0,∞)×R,then Y t=g(t,X t)is again an Itôprocess:
dY t=c(t,X t)dt+d(t,X t)dW t
with
c(t,x)=∂g
∂t
(t,x)+a(t,x)
∂g
∂x
(t,x)+
b(t,x)2
2
∂2g
∂x2
(t,x)
d(t,x)=b(t,x)
∂g
∂x
(t,x)
An application to price process(1)shows that log S t is a Brownian motion(cf.Tsay§6.3.3)
Thus,log S
t is normally distributed with mean log S0+(µ−σ2/2)t and varianceσ2t Equivalently,S
t has a lognormal distribution with
E(S t)=S0eµt
Var(S t)=S20e2µt[eσ2t−1]
Arbitrage-Free Pricing
Consider a portfolioΠwhich,at time t,comprises1unit
of a European call option and−∆units of the asset
Assume that the value C(t,S)of the call option is twice
continuously differentiable on[0,∞)×R
Apply Itô’s lemma toΠ
t=C(t,S t)−∆S t and observe that the portfolioΠbecomes risk-free by choosing
∆=∂C/∂S(the“delta”of the call option)
Since there are no arbitrage opportunities,Πmust have
the same return as the risk-free asset:
dΠt=rΠt dt(i.e.,Πt=Π0e rt)
This argument yields the Black-Scholes partial
differential equation(PDE)whose solution is C(t,S)
Delta Hedging
Solving the Black-Scholes PDE subject to the terminal
condition C(T,S)=(S−K)+yields the Black-Scholes formula for the call option value:
C(t,S)=SΦ(d1(t,S))−Ke−r(T−t)Φ(d2(t,S)) where d2(t,S)=d1(t,S)−σ
√T−t and
d1(t,S)=log(S/K)+(r+σ2/2)(T−t)
σ
√T−t
The delta of the call option is given by
∆(t,S)=∂C
∂S
(t,S)=Φ(d1(t,S))
Put-Call
Parity
The value P (t,S )of a European put option satisfies the same Black-Scholes PDE (with C replaced by P )as the call option
value
Rather than solving this PDE,we can make use of the following put-call parity relation to deduce P (t,S ):
C (t,S )−P (t,S )=S −Ke
−r (T −t )
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Risk-Neutral Valuation
Central to option pricing theory is the observation that
no arbitrage opportunities can arise in equilibrium
Intuitively,an arbitrage comprises a strategy which
costs nothing to set up but guarantees the owner a
nonnegative payoff in the future
Formally,the market is said to be arbitrage free(and
complete)if there exists a(unique)equivalent
martingale measure
For such a market,the price at time t of any European
security that generates afinite variance terminal payoff can be obtained as the expected terminal payoff
discounted to present value,with the expectation taken w.r.t.the“risk-neutral”measure
For the Black-Scholes model,the asset price process (1)has the following representation under the equivalent martingale measure:
dS t=rS t dt+σS t dB t(S0>0)(2)
where{B t,t≥0}is another standard Brownian
motion
The value C(t,S)of the European call option is given by
C(t,S)=e−r(T−t)E[(S T−K)+|S t=S]
=e−r(T−t) ∞log(K/S)(Se x−K)f(x)dx
where f(x)is the N((r−σ2/2)(T−t),σ2(T−t))density function
Dividend-Paying Assets
The Black-Scholes theory can be readily extended to
the case in which the underlying asset pays dividends continuously(at the rate q,say)
Risk-neutral price process:dS
t=(r−q)S t dt+σS t dB t
B-S PDE:−rC+∂C
∂t +(r−q)S
∂C
∂S
+
1
2
σ2S2
∂2C
∂S2
=0
B-S formula:
C(t,S)=S e−q(T−t)Φ(d1(t,S))−Ke−r(T−t)Φ(d2(t,S)) where
d1(t,S)=[log(S/K)+(r−q+σ2/2)(T−t)]/σ
√T−t Put-call parity:C(t,S)−P(t,S)=S e−q(T−t)−Ke−r(T−t)
Implied Volatility
A call[put]option is said to be in the money(ITM),at
the money(A TM),or out of the money(OTM),according as S t>[<]K,S t=K,or S t<[>]K
Equating the Black-Scholes formula to the actual price
of an option yields a nonlinear equation forσ,whose
solution(denoted byσimp)is called the implied volatility of the underlying asset
T raders often quote option prices in terms of their
implied volatilities
How consistent are market option prices with the
Black-Scholes formula?—The Black-Scholes theory
assumes constant volatility for all options
Figure shows the implied volatilities of S&P500options
on May5,1993:(a)fixed expiration of44days for OTM calls and puts,(b)average of A TM call and put implied volatilities
is collectively referred
The asymmetry observed inσ
to as the volatility“skew”or“smile”
Though the exact shape and magnitude vary from day
to day,the asymmetry persists and belies the
Black-Scholes theory
Thus,it is probably incorrect to calculate option prices
using the Black-Scholes formula
Piecewise Polynomials and Splines
Piecewise Polynomials
Piecewise Constant
Piecewise Linear
Continuous Piecewise Linear
Piecewise-linear Basis Function
Assume that X is one-dimensional A piecewise polynomial function f (X )is
obtained by dividing the domain of X into contiguous intervals,and representing f by a separate polynomial in each interval
Figure 1shows two sim-ple piecewise polynomi-als
Basis Functions
The piecewise constantfit uses three basis functions:
h1(X)=I(X<ξ1)
h2(X)=I(ξ1≤X<ξ2)
h3(X)=I(ξ2≤X)
The LS estimate of the model f(X)= 3
βm h m(X)
amounts toˆβm=¯Y m,the mean of Y in the m th region The piecewise linearfit requires three additional basis
functions:h m+3=Xh m(X),m=1,2,3[and uses the model f(X)= 6m=1βm h m(X)]
Except in special cases,we would typically impose
continuity restrictions at the two knots(third panel)
These continuity restrictions lead to linear constraints
on the parameters:
f(ξ−1)=f(ξ+1)⇒β1+β4ξ1=β2+β5ξ1
f(ξ−2)=f(ξ+2)⇒β2+β5ξ2=β3+β6ξ2 Since there are two restrictions,we get back two
parameters,leaving four free parameters
Use a basis that directly incorporates the constraints:
h1(X)=1h3(X)=(X−ξ1)+[Figure1]
h2(X)=X h4(X)=(X−ξ2)+
where t+denotes the positive part
Piecewise Cubic Polynomials
Discontinuous
Continuous
Continuous First Derivative
Continuous Second Derivative
Piecewise Cubic Polynomials
Figure 2:Increasing orders of continuity at the knots
Fourth panel displays a cubic spline :
Continuous and has continuous first and second derivatives at the knots
Six basis functions:
h 1,2,3,4(X )=X 0,1,2,3
h 5,6(X )=(X
−ξ1,2)3
+
Order-M Splines
More generally,an order-M spline with knotsξ
j, j=1,...,K is a piecewise-polynomial of order M,and has continuous derivatives up to order M−2
The piecewise-constant function is an order-1spline
The continuous piecewise-linear function is an
order-2spline
A cubic spline has M=4
The general form for the truncated-power basis set is
h j(X)=X j−1,j=1,...,M
h M+ℓ(X)=(X−ξℓ)M−1
,ℓ=1,...,K
+
In practice the most widely used orders are M=1,2,4
Computational Considerations Thesefixed-knot splines are also known as regression
splines
One needs to select the order of the spline,the
number of knots and their placement
Can simply parameterize a family of splines by the
number of basis functions(or df),and have the
observations x i determine the positions of the knots T o generate a basis for polynomial splines in R,load
package splines and use the function bs(x,df= NULL,knots=NULL,degree=3,intercept =FALSE,Boundary.knots=range(x))
While the truncated power basis is conceptually simple,
it is not too attractive numerically;the B-spline basis
allows for efficient computations even when K is large
MARS:Multivariate Adaptive Regression Splines
MARS
MARS is an adaptive procedure for regression,and is well suited for problems with a large number of
inputs It can be viewed as a generalization of stepwise linear
regression
MARS uses expansions in piecewise linear basis functions of the form
(x −ξ)+=
x −ξif x >ξ
0otherwise and (ξ−x )+=
ξ−x if x <ξ
0otherwise
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The idea is to form reflected pairs for each input X j with knots at each observed value x ij of that
input Therefore,the collection of 2Np basis functions is
C ={(X j −ξ)+,(ξ−X j )+}
ξ∈{x 1j ,x 2j ,...,x Nj }j =1,2,...,p
Model
Building
The model-building strategy is like a forward stepwise
linear
regression
The model has the form
f(X)=β0+
M
m=1βm h m(X)
where each h m(X)is a function in C,or a product of two
or more such
functions
Given a choice for the h m,the coefficientsβm are
estimated by standard linear
regression
We start with only the constant function h0(X)=1in our model and all functions in C are candidate functions
y y
x
c
F IGURE 4:
Schematic of
the MARS forward model-building procedure.
At each stage we consider all products of a function h ℓ∈M (the model set)with one of the reflected pairs in C
We add to the model M the term of the form
ˆβ
M +1h ℓ(X )·(X j −ξ)++ˆβ
M +2h ℓ(X )·(ξ−X j )+that produces the largest decrease in training error All coefficients are LS
The winning products are added to the model and the
process is continued until the model set M contains some preset maximum number of terms
Example:At thefirst stage consider adding a pair of
functions(X j−ξ)+and(ξ−X j)+,ξ∈{x ij}(after multiplication by the constant function)
Add the best choice(X2−x72)+and(x72−X2)+
At the next stage consider including a pair of
products h m(X)·(X j−ξ)+and h m(X)·(ξ−X j)+,ξ∈{x ij},where for h m we have the choices
h0(X)=1,h1(X)=(X2−x72)+,h2(X)=(x72−X2)+ The third choice produces functions such as
(X1−x51)+·(x72−X2)+and(x51−X1)+·(x72−X2)+
Cross
Validation
We now have a large model which typically overfits the
data so a backward deletion procedure is
applied
The term whose removal causes the smallest increase in residual squared error is deleted from the model at each stage,producing an estimated best modelˆfλof
each size(number of terms)
λ
The optimal value ofλis estimated using GCV(rather than CV,for computational savings):
GCV(λ)= N i=1(y i−ˆfλ(x i))2
N(1−M(λ)/N)2
where M(λ)is the effective number of parameters in the model
We choose the model along the backward sequence
that minimizes GCV(λ)
M(λ)accounts both for the number of terms in the
model and the number of parameters used in selecting the optimal positions of the knots
If there are r linearly independent basis functions in the
model,and K knots were selected in the forward process,the formula is M(λ)=r+3K
One pays a price of three parameters for selecting a
knot in a piecewise linear regression
For an additive model(in which two-or higher-way
products are not allowed),M(λ)=r+2K
Model Strategy
The forward modeling strategy in MARS is hierarchical
with the restriction that each input can appear at most once in a product
Because linear splines
are zero over part of their
range,their products are
nonzero only over the
small part of the feature
space where all product
components are nonzero
The regression surface
is built up parsimoniously
using nonzero compo-
nents locally
Nonparametric Pricing
of Options
Self-Financing Trading
Strategy Another try at the Black-Scholes
formula Recall that the asset price process is given by (1)and the bond price process βsatisfies dβt =rβt
dt Denote the European call option price process by C
with C T =(S T −K )
+A self-financing trading strategy (a,b )in the asset and
bond generate no dividends and is such that
a t S t +
b t βt =a 0S 0+b 0β0+
t
0a u dS u + t 0
b u dβu for all
t Suppose there exists such a trading strategy with a T S T +b T βT =C T
Replication
If there is no arbitrage,then C
0=a0S0+b0β0
Indeed,if C
0>a0S0+b0β0,then consider the trading
strategy(−1,a,b)in the call option,asset and bond By assumption,−C T+a T S T+b TβT=0
The initial profit C0−a0S0−b0β0is riskless!
Conversely,if C
0<a0S0+b0β0,then consider the trading strategy(1,−a,−b)in the call option,asset and bond
The same arguments applied to each date t imply that
C t=a t S t+b tβt,i.e.,(a,b)is a“replicating”portfolio
In the absence of arbitrage,it follows that
dC t=a t dS t+b t dβt=(µa t S t+rb tβt)dt+σa t S t dB t
Alternatively,an SDE for C
t is given by Itô’s lemma
By the unique decomposition property of Itôprocesses,
we can“match coefficients”in both dt and dB t dB t:a t=∆(t,S t)where∆(t,S)=∂C(t,S)/∂S
Replication:b t=β−1
[C(t,S t)−S t∆(t,S t)]
dt:Black-Scholes PDE
Discrete-time replication:For t=iδ,i=1,...,n(=T/δ),
a u=a t−δand
b u=b t−δfor t−δ≤u<t
Discrete-time rebalancing of asset:a t=∆(t,S t)
T o satisfy the self-financing condition,we use
b t=β−1t[e rδb t−δβt−δ−S t(a t−a t−δ)]
Replication Error
For definiteness,consider the position(−1,a,b)in the
call option,asset and bond
If trading can take place continuously,perfect
replication is possible in the Black-Scholes model
In the case of discrete-time replication,we have
a0=∆(0,S0)(number of units of asset)and
Y0:=b0β0=C(0,S0)−S0∆(0,S0)(dollar investment
in bond)
For t=δ,...,nδ(=T),a t=∆(t,S t)and
Y t:=b tβt=e rδY t−δ−S t(a t−a t−δ)
Replication(or hedging)error is defined by
err=S T I(S T>K)+Y T−(S T−K)+=Y T+KI(S T>K)
Nonparametric Pricing
Assume:
r andσarefixed throughout the training sample
The statistical distribution of the underlying asset’s
return is independent of the level of the asset price Thus,the option pricing formulaˆC(·)is homogeneous
of degree one in both S t and K,and we need only
estimateˆC(T−t,S t/K)
Illustration(using data on H1’91settlement prices of
S&P500futures and futures options)
“Inputs”:time to expiration and“moneyness”
“Output”:option price divided by strike
Can relaxfirst assumption by using more inputs inˆC
C /K
h a
t(
C
)Lack of differentiability:ˆC
(T −t,S/K )is not differentiable w.r.t.S at the selected knots
In practice,need to “smooth the kink”。