stochastic calculus, application of real analysis in finance
Post on 11-Feb-2022
3 Views
Preview:
TRANSCRIPT
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus, Application of RealAnalysis in Finance
Workshop for Young Mathematicians in Korea
Seungkyu Lee
Pohang University of Science and Technology
August 4th, 2010
Stochastic Calculus, Application of Real Analysis in Finance
Contents
1 BINOMIAL ASSET PRICING MODEL
2 Stochastic Calculus
Stochastic Calculus, Application of Real Analysis in Finance
Contents
1 BINOMIAL ASSET PRICING MODEL
2 Stochastic Calculus
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Contents
1 BINOMIAL ASSET PRICING MODEL
2 Stochastic Calculus
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Binomial Asset Pricing Model
Assumptions:
Model stock prices in discrete time
at each step, the stock price will change to one of twopossible values
begin with an initial positive stock price S0
there are two positive numbers, d and u, with 0 < d < 1 < usuch that at the next period, the stock price will be either dS0or uS0, d = 1/u.
⇒ Of course, real stock price movements are much morecomplicated
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Binomial Asset Pricing Model
Assumptions:
Model stock prices in discrete time
at each step, the stock price will change to one of twopossible values
begin with an initial positive stock price S0
there are two positive numbers, d and u, with 0 < d < 1 < usuch that at the next period, the stock price will be either dS0or uS0, d = 1/u.
⇒ Of course, real stock price movements are much morecomplicated
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Binomial Asset Pricing Model
Objectives:
the definition of probability space and
the concept of Arbitrage Pricing and its relation toRisk-Neutral Pricing is clearly illuminated
Flip CoinsToss a coin and when we get a ”Head,” the stock price moves up,but when we get a ”Tail,” the price moves down.⇒ the price at time 1 by S1(H) = uS0 and S1(T) = dS1⇒ After the second toss,
S2(HH) = uS1(H) = u2S0, S2(HT) = dS1(H) = duS0
S2(TH) = uS1(H) = duS0, S2(TT) = dS1(T) = d2S0
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Binomial Asset Pricing Model
Objectives:
the definition of probability space and
the concept of Arbitrage Pricing and its relation toRisk-Neutral Pricing is clearly illuminated
Flip CoinsToss a coin and when we get a ”Head,” the stock price moves up,but when we get a ”Tail,” the price moves down.
⇒ the price at time 1 by S1(H) = uS0 and S1(T) = dS1⇒ After the second toss,
S2(HH) = uS1(H) = u2S0, S2(HT) = dS1(H) = duS0
S2(TH) = uS1(H) = duS0, S2(TT) = dS1(T) = d2S0
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Binomial Asset Pricing Model
Objectives:
the definition of probability space and
the concept of Arbitrage Pricing and its relation toRisk-Neutral Pricing is clearly illuminated
Flip CoinsToss a coin and when we get a ”Head,” the stock price moves up,but when we get a ”Tail,” the price moves down.⇒ the price at time 1 by S1(H) = uS0 and S1(T) = dS1
⇒ After the second toss,
S2(HH) = uS1(H) = u2S0, S2(HT) = dS1(H) = duS0
S2(TH) = uS1(H) = duS0, S2(TT) = dS1(T) = d2S0
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Binomial Asset Pricing Model
Objectives:
the definition of probability space and
the concept of Arbitrage Pricing and its relation toRisk-Neutral Pricing is clearly illuminated
Flip CoinsToss a coin and when we get a ”Head,” the stock price moves up,but when we get a ”Tail,” the price moves down.⇒ the price at time 1 by S1(H) = uS0 and S1(T) = dS1⇒ After the second toss,
S2(HH) = uS1(H) = u2S0, S2(HT) = dS1(H) = duS0
S2(TH) = uS1(H) = duS0, S2(TT) = dS1(T) = d2S0
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Binomial Asset Pricing Model
For the moment, let us assume that the third toss is the last oneand denote by
Ω = HHH,HHT ,HTH,HTT , THH, THT , TTH, TTT
the set of all possible outcomes of the three tosses.
The set Ω of all possible outcomes of a random experiment iscalled the sample space for the experiment, and the element ω ofΩ are called sample points.Denote the k-th component of ω by ωk.⇒ S3 depends on all of ω.⇒ S2 depends on only the first two components of ω, ω1 and ω2.
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Binomial Asset Pricing Model
For the moment, let us assume that the third toss is the last oneand denote by
Ω = HHH,HHT ,HTH,HTT , THH, THT , TTH, TTT
the set of all possible outcomes of the three tosses.The set Ω of all possible outcomes of a random experiment iscalled the sample space for the experiment, and the element ω ofΩ are called sample points.
Denote the k-th component of ω by ωk.⇒ S3 depends on all of ω.⇒ S2 depends on only the first two components of ω, ω1 and ω2.
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Binomial Asset Pricing Model
For the moment, let us assume that the third toss is the last oneand denote by
Ω = HHH,HHT ,HTH,HTT , THH, THT , TTH, TTT
the set of all possible outcomes of the three tosses.The set Ω of all possible outcomes of a random experiment iscalled the sample space for the experiment, and the element ω ofΩ are called sample points.Denote the k-th component of ω by ωk.⇒ S3 depends on all of ω.⇒ S2 depends on only the first two components of ω, ω1 and ω2.
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Basic Probability Theory
Probability space (Ω,F,P) like (R,B,µ), Lebesgue Measure
Ω: the set of all possible realizations of the stochasticeconomy
Ω = HHH,HHT ,HTH,HTT , THH, THT , TTH, TTT
ω: a sample path, ω ∈ ΩFt: the sigma field of distinguishable events at time t
F2 = ∅, HHH,HHT , HTH,HTT , THH, THT , TTH, TTT , . . . ,Ω
P: a probability measure defined on the elements of Ft
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Probability Spaces
DefinitionIf Ω is a given set, then a σ-algebra F on Ω is a family F ofsubsets of Ω with the following properties:
∅ ∈ F
F ∈ F ⇒ FC ∈ F, where FC = Ω \ F is the complement of F inΩ
A1,A2, . . . ∈ F ⇒ A :=⋃∞i=1Ai ∈ F
The pair (Ω,F) is called a measurable space.
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Probability Spaces
DefinitionA Probability measure P on a measurable space (Ω,F) is afunction P : F → [0, 1] such that
P(∅) = 0, P(Ω) = 1
if A1,A2, . . . ∈ F and Ai∞i=1 is disjoint then
P
( ∞⋃i=1
Ai
)=
∞∑i=1
P(Ai)
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Random Variable
DefinitionIf (Ω,F,P) is a given probability space, then a functionY : Ω→ Rn is called F-measurable if
Y−1(U) := ω ∈ Ω; Y(ω) ∈ U ∈ F
for all open sets U ∈ Rn(or, equivalently, for all Borel setsU ⊂ Rn).In the following we let (Ω,F,P) denote a given completeprobability space.A random variable X is an F-measurable function X : Ω→ Rn.
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Arbitrage Price of Call Options
A money market with interest rate r
$1 invested in the money market ⇒ $(1+ r) in the next period
d < (1 + r) < u
European call option with strike price K > 0 and expiration time 1
this option confers the right to buy the stock at time 1 for Kdollars
so is worth max[S1 − K, 0]
denote V1(ω) = (S1(ω) − K)+ , max[S1(ω) − K] thevalue(payoff) at expiration.
⇒ Compute the arbitrage price of the call option at time zero, V0.
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Arbitrage Price of Call Options
A money market with interest rate r
$1 invested in the money market ⇒ $(1+ r) in the next period
d < (1 + r) < u
European call option with strike price K > 0 and expiration time 1
this option confers the right to buy the stock at time 1 for Kdollars
so is worth max[S1 − K, 0]
denote V1(ω) = (S1(ω) − K)+ , max[S1(ω) − K] thevalue(payoff) at expiration.
⇒ Compute the arbitrage price of the call option at time zero, V0.
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Arbitrage Price of Call Options
Suppose at time zero you sell the call option for V0 dollars.
ω1 = H ⇒ you should pay off (uS0 − K)+
ω1 = T ⇒ you should pay off (dS0 − K)+
At time 0, we don’t know the value of ω1.
Making replicating portfolio:
V0 − ∆0S0 dollars in the money market.
∆0 shares of stock
⇒ the value of the portfolio is V0 at time 0.⇒ the value should be (S1 − K)
+ at time 1.
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Arbitrage Price of Call Options
Suppose at time zero you sell the call option for V0 dollars.
ω1 = H ⇒ you should pay off (uS0 − K)+
ω1 = T ⇒ you should pay off (dS0 − K)+
At time 0, we don’t know the value of ω1.Making replicating portfolio:
V0 − ∆0S0 dollars in the money market.
∆0 shares of stock
⇒ the value of the portfolio is V0 at time 0.⇒ the value should be (S1 − K)
+ at time 1.
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Arbitrage Price of Call Options
Thus,V1(H) = ∆0S1(H) + (1 + r)(V0 − ∆0S0) (1)
V1(T) = ∆0S1(T) + (1 + r)(V0 − ∆0S0) (2)
Subtracting (2) from (1), we obtain
V1(H) − V1(T) = ∆0(S1(H) − S1(T)), (3)
so that
∆0 =V1(H) − V1(T)
S1(H) − S1(T)(4)
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Arbitrage Price of Call Options
Thus,V1(H) = ∆0S1(H) + (1 + r)(V0 − ∆0S0) (1)
V1(T) = ∆0S1(T) + (1 + r)(V0 − ∆0S0) (2)
Subtracting (2) from (1), we obtain
V1(H) − V1(T) = ∆0(S1(H) − S1(T)), (3)
so that
∆0 =V1(H) − V1(T)
S1(H) − S1(T)(4)
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Arbitrage Price of Call Options
Substitute (4) into either (1) or (2) and solve for V0;
V0 =1
1 + r
[1 + r− d
u− dV1(H) +
u− (1 + r)
u− dV1(T)
](5)
Simply,
V0 =1
1 + r[pV1(H) + qV1(T)] (6)
where
p =1 + r− d
u− d, q =
u− (1 + r)
u− d= 1 − p
⇒ We can regard them as probabilities of H and T, respectively.⇒ They are the risk-neutral probabilities.
Stochastic Calculus, Application of Real Analysis in Finance
BINOMIAL ASSET PRICING MODEL
Arbitrage Price of Call Options
Substitute (4) into either (1) or (2) and solve for V0;
V0 =1
1 + r
[1 + r− d
u− dV1(H) +
u− (1 + r)
u− dV1(T)
](5)
Simply,
V0 =1
1 + r[pV1(H) + qV1(T)] (6)
where
p =1 + r− d
u− d, q =
u− (1 + r)
u− d= 1 − p
⇒ We can regard them as probabilities of H and T, respectively.⇒ They are the risk-neutral probabilities.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Contents
1 BINOMIAL ASSET PRICING MODEL
2 Stochastic Calculus
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Problem - The Movement of Stock Price
A risky investment(e.g. a stock), where the price X(t) per unit attime t satisfies a stochastic differential equation:
dX
dt= b(t,Xt) + σ(t,Xt) · ”noise”, (7)
where b and σ are some given functions.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
dXdt
= b(t,Xt) + σ(t,Xt)Wt
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
dXdt
= b(t,Xt) + σ(t,Xt)Wt
Based on many situations, one is led to assume that the ”noise”,Wt, has these properties:
t1 6= t2 ⇒ Wt1 and Wt2 are independent.
Wt is stationary, i.e. the (joint)distribution ofWt1+t, . . . ,Wtk+t does not depend on t.
E[Wt] = 0 for all t.
Let 0 = t0 < t1 < · · · < tm = t and consider a discrete version of(7):
Xk+1 − Xk = b(tk,Xk)∆tk + σ(tk,Xk)Wk∆tk, (8)
where Xj = X(tj), Wk =Wtk , ∆tk = tk+1 − tk.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
dXdt
= b(t,Xt) + σ(t,Xt)Wt
Based on many situations, one is led to assume that the ”noise”,Wt, has these properties:
t1 6= t2 ⇒ Wt1 and Wt2 are independent.
Wt is stationary, i.e. the (joint)distribution ofWt1+t, . . . ,Wtk+t does not depend on t.
E[Wt] = 0 for all t.
Let 0 = t0 < t1 < · · · < tm = t and consider a discrete version of(7):
Xk+1 − Xk = b(tk,Xk)∆tk + σ(tk,Xk)Wk∆tk, (8)
where Xj = X(tj), Wk =Wtk , ∆tk = tk+1 − tk.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
dXdt
= b(t,Xt) + σ(t,Xt)Wt
Replace Wk∆k by ∆Vk = Vtk+1 − Vtk , where Vtt>0 is somesuitable stochastic process.
It can be easily proved that Vtt>0 satisfies the four propertieswhich define the standard Brownian Motion, Btt>0:
B0 = 0.
The increments of Bt are independent; i.e. for any finite set oftimes 0 6 t1 < t2 < · · · < tn < T the random variablesBt2 − Bt1 ,Bt3 − Bt2 , . . . ,Btn − Btn−1 are independent.
For any 0 6 s 6 t < T the increment Bt − Bs has the Gaussiandistribution with mean 0 and variance t− s.
For all w in a set of probability one, Bt(w) is a continuousfunction of t.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
dXdt
= b(t,Xt) + σ(t,Xt)Wt
Replace Wk∆k by ∆Vk = Vtk+1 − Vtk , where Vtt>0 is somesuitable stochastic process.It can be easily proved that Vtt>0 satisfies the four propertieswhich define the standard Brownian Motion, Btt>0:
B0 = 0.
The increments of Bt are independent; i.e. for any finite set oftimes 0 6 t1 < t2 < · · · < tn < T the random variablesBt2 − Bt1 ,Bt3 − Bt2 , . . . ,Btn − Btn−1 are independent.
For any 0 6 s 6 t < T the increment Bt − Bs has the Gaussiandistribution with mean 0 and variance t− s.
For all w in a set of probability one, Bt(w) is a continuousfunction of t.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
dXdt
= b(t,Xt) + σ(t,Xt)Wt
Replace Wk∆k by ∆Vk = Vtk+1 − Vtk , where Vtt>0 is somesuitable stochastic process.It can be easily proved that Vtt>0 satisfies the four propertieswhich define the standard Brownian Motion, Btt>0:
B0 = 0.
The increments of Bt are independent; i.e. for any finite set oftimes 0 6 t1 < t2 < · · · < tn < T the random variablesBt2 − Bt1 ,Bt3 − Bt2 , . . . ,Btn − Btn−1 are independent.
For any 0 6 s 6 t < T the increment Bt − Bs has the Gaussiandistribution with mean 0 and variance t− s.
For all w in a set of probability one, Bt(w) is a continuousfunction of t.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
dXdt
= b(t,Xt) + σ(t,Xt)Wt
Thus we put Vt = Bt and obtain from (8):
Xk = X0 +
k−1∑j=0
b(tj,Xj)∆tj +k−1∑j=0
σ(tj,Xj)∆Bj. (9)
When ∆tj → 0, by applying the usual integration notation, weshould obtain
Xt = X0 +
∫t0b(s,Xs)ds+ ”
∫t0σ(s,Xs)dBs” (10)
Now, in the remainder of this chapter we will prove the existence,in a certain sense, of
”
∫t0f(s,ω)dBs(ω)”
where Bt(ω) is 1-dim’l Brownian motion.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
dXdt
= b(t,Xt) + σ(t,Xt)Wt
Thus we put Vt = Bt and obtain from (8):
Xk = X0 +
k−1∑j=0
b(tj,Xj)∆tj +k−1∑j=0
σ(tj,Xj)∆Bj. (9)
When ∆tj → 0, by applying the usual integration notation, weshould obtain
Xt = X0 +
∫t0b(s,Xs)ds+ ”
∫t0σ(s,Xs)dBs” (10)
Now, in the remainder of this chapter we will prove the existence,in a certain sense, of
”
∫t0f(s,ω)dBs(ω)”
where Bt(ω) is 1-dim’l Brownian motion.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
dXdt
= b(t,Xt) + σ(t,Xt)Wt
Thus we put Vt = Bt and obtain from (8):
Xk = X0 +
k−1∑j=0
b(tj,Xj)∆tj +k−1∑j=0
σ(tj,Xj)∆Bj. (9)
When ∆tj → 0, by applying the usual integration notation, weshould obtain
Xt = X0 +
∫t0b(s,Xs)ds+ ”
∫t0σ(s,Xs)dBs” (10)
Now, in the remainder of this chapter we will prove the existence,in a certain sense, of
”
∫t0f(s,ω)dBs(ω)”
where Bt(ω) is 1-dim’l Brownian motion.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Construction of the Ito Integral
It is reasonable to start with a definition for a simple class offunctions f and then extend by some approximation procedure.Thus, let us first assume that f has the form
φ(t,ω) =∑j>0
ej(ω) · X[j·2−n,(j+1)2−n)(t), (11)
where X denotes the characteristic (indicator) function.
For such functions it is reasonable to define∫t0φ(t,ω)dBt(ω) =
∑j>0
ej(ω)[Btj+1 − Btj ](ω). (12)
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Construction of the Ito Integral
It is reasonable to start with a definition for a simple class offunctions f and then extend by some approximation procedure.Thus, let us first assume that f has the form
φ(t,ω) =∑j>0
ej(ω) · X[j·2−n,(j+1)2−n)(t), (11)
where X denotes the characteristic (indicator) function.For such functions it is reasonable to define∫t
0φ(t,ω)dBt(ω) =
∑j>0
ej(ω)[Btj+1 − Btj ](ω). (12)
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Construction of the Ito Integral
In general, as we did in Real Analysis, it is natural to approximatea given function f(t,ω) by∑
j
f(t∗j ,ω) · X[tj,tj+1)(t)
where the points t∗j belong to the intervals [tj, tj+1], specifically in
Ito Integral, t∗j = tj, and then define∫t0 f(s,ω)dBs(ω).
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Construction of the Ito Integral
DefinitionThe Ito integral of f is defined by∫t
0f(s,ω)dBs(ω) = lim
n→∞∫t0φn(s,ω)dBs(ω) (limit in L2(P))
(13)where φn is a sequence of elementary functions such that
E
[∫t0(f(s,ω) − φn(s,ω))2dt
]→ 0 as n→∞. (14)
Note that such a sequence φn satisfying (14) exists.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Example
∫t0BsdBs =
1
2B2t −
1
2t.
(proof)
Put φn(s,ω) =∑Bj(ω) · X[tj,tj+1)(s), where Bj = Btj . Then
E
[∫t0(φn − Bs)
2ds
]= E
∑j
∫tj+1
tj
(Bj − Bs)2ds
=
∑j
∫tj+1
tj
(s− tj)ds
=∑ 1
2(tj+1 − tj)
2 → 0 as ∆tj → 0
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Example
∫t0BsdBs =
1
2B2t −
1
2t.
(proof)Put φn(s,ω) =
∑Bj(ω) · X[tj,tj+1)(s), where Bj = Btj . Then
E
[∫t0(φn − Bs)
2ds
]= E
∑j
∫tj+1
tj
(Bj − Bs)2ds
=
∑j
∫tj+1
tj
(s− tj)ds
=∑ 1
2(tj+1 − tj)
2 → 0 as ∆tj → 0
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Example
∫t0BsdBs =
1
2B2t −
1
2t.
(proof)Put φn(s,ω) =
∑Bj(ω) · X[tj,tj+1)(s), where Bj = Btj . Then
E
[∫t0(φn − Bs)
2ds
]= E
∑j
∫tj+1
tj
(Bj − Bs)2ds
=
∑j
∫tj+1
tj
(s− tj)ds
=∑ 1
2(tj+1 − tj)
2 → 0 as ∆tj → 0
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Example
∫t0BsdBs = lim
∆tj→0
∫t0φndBs = lim
∆tj→0
∑j
Bj∆Bj.
Now
∆(B2j ) = B2j+1 − B
2j = (Bj+1 − Bj)
2 + 2Bj(Bj+1 − Bj)
= (∆Bj)2 + 2Bj∆Bj,
and therefore,
B2t =∑j
∆(B2j ) =∑j
(∆Bj)2 + 2
∑j
Bj∆Bj
or ∑j
Bj∆Bj =1
2B2t −
1
2
∑j
(∆Bj)2.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Example
∫t0BsdBs = lim
∆tj→0
∫t0φndBs = lim
∆tj→0
∑j
Bj∆Bj.
Now
∆(B2j ) = B2j+1 − B
2j = (Bj+1 − Bj)
2 + 2Bj(Bj+1 − Bj)
= (∆Bj)2 + 2Bj∆Bj,
and therefore,
B2t =∑j
∆(B2j ) =∑j
(∆Bj)2 + 2
∑j
Bj∆Bj
or ∑j
Bj∆Bj =1
2B2t −
1
2
∑j
(∆Bj)2.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Example
∫t0BsdBs = lim
∆tj→0
∫t0φndBs = lim
∆tj→0
∑j
Bj∆Bj.
Now
∆(B2j ) = B2j+1 − B
2j = (Bj+1 − Bj)
2 + 2Bj(Bj+1 − Bj)
= (∆Bj)2 + 2Bj∆Bj,
and therefore,
B2t =∑j
∆(B2j ) =∑j
(∆Bj)2 + 2
∑j
Bj∆Bj
or ∑j
Bj∆Bj =1
2B2t −
1
2
∑j
(∆Bj)2.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Example
∫t0BsdBs = lim
∆tj→0
∫t0φndBs = lim
∆tj→0
∑j
Bj∆Bj.
Now
∆(B2j ) = B2j+1 − B
2j = (Bj+1 − Bj)
2 + 2Bj(Bj+1 − Bj)
= (∆Bj)2 + 2Bj∆Bj,
and therefore,
B2t =∑j
∆(B2j ) =∑j
(∆Bj)2 + 2
∑j
Bj∆Bj
or ∑j
Bj∆Bj =1
2B2t −
1
2
∑j
(∆Bj)2.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
The Ito Formula
The previous example illustrates that the basic definition of Itointegrals is not very useful when we try to evaluate a given integralas ordinary Riemann integrals without the fundamental theorem ofcalculus plus the chain rule in the explicit calculations.It turns out that it is possible to establish an Ito integral version ofthe chain rule, called the Ito formula.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
The Ito Formula
TheoremLet Xt be an Ito process given by
dXt = udt+ vdBt.
Let g(t, x) ∈ C2([0,∞)× R). Then Yt = g(t,Xt) is again an Itoprocess, and
dYt =∂g
∂t(t,Xt)dt+
∂g
∂x(t,Xt)dXt +
1
2
∂2g
∂x2(t,Xt) · (dXt)2, (15)
where (dXt)2 = (dXt) · (dXt) is computed according to the rules
dt · dt = dt · dBt = dBt · dt = 0, dBt · dBt = dt.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Example, Again
Calculate the integral
I =
∫t0BsdBs
Choose Xt = Bt and g(t, x) = 12x
2. Then, Yt =12B
2t.
By Ito’s formula,
d
(1
2B2t
)= BtdBt +
1
2dt.
In other words,1
2B2t =
∫t0BsdBs +
1
2t.
Stochastic Calculus, Application of Real Analysis in Finance
Stochastic Calculus
Example, Again
Calculate the integral
I =
∫t0BsdBs
Choose Xt = Bt and g(t, x) = 12x
2. Then, Yt =12B
2t.
By Ito’s formula,
d
(1
2B2t
)= BtdBt +
1
2dt.
In other words,1
2B2t =
∫t0BsdBs +
1
2t.
top related