mit15_450f10_lec02

5/23/2018 MIT15_450F10_lec02

1/75

Stochastic Integral Its Lemma Black-Scholes Model Multivariate It Processes SDEs SDEs and PDEs Risk-Neutral Probability Risk-Neutral Pricing

Stochastic Calculus and Option Pricing

Leonid Kogan

MIT, Sloan

15.450, Fall 2010

Leonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 1 / 74c
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

2/75


Outline

1 Stochastic Integral

2 Its Lemma

3 Black-Scholes Model

4 Multivariate It Processes

5 SDEs

6 SDEs and PDEs

7 Risk-Neutral Probability

8 Risk-Neutral Pricing

c Stochastic Calculus 2 / 74Leonid Kogan ( MIT, Sloan ) 15.450, Fall 2010
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

3/75


Outline


2 Its Lemma



5 SDEs

6 SDEs and PDEs



c Stochastic Calculus 3 / 74Leonid Kogan ( MIT, Sloan ) 15.450, Fall 2010
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

4/75


Brownian Motion

Consider a random walk xt+nt with equally likely increments of

t.

Let the time step of the random walk shrink to zero: t 0.

The limit is a continuous-time process called Brownian motion, which we

denote Zt , or Z (t).

We always set Z0 =0.Brownian motion is a basic building block of continuous-time models.

cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 4 / 74
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

5/75


Properties of Brownian Motion

Brownian motion has independent increments: if t

5/23/2018 MIT15_450F10_lec02

6/75


Ito Integral

Ito integral, also called the stochastic integral (with respect to the Brownian

motion) is an object tu dZu

0

where u is a stochastic process.

Important: u can depend on the past history of Zu, but it cannot depend onthe future. u is called adapted to the history of the Brownian motion.

Consider discrete-time approximations

N

t(i1)t (Zit Z(i1)t ), t =

Ni=1

and then take the limit of N (the limit must be taken in themean-squared-error sense).

The limit is well defined, and is called the Ito integral.

cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 6 / 74S h i I l I L Bl k S h l M d l M l i i I P SDE SDE d PDE Ri k N l P b bili Ri k N l P i i
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

7/75


Properties of It Integral

It integral is linear

t t t(au +c bu)dZu = au dZu +c bu dZu

0 0 0tXt = 0 u dZu, is continuous as a function of time.Increments of Xt have conditional mean of zero (under some technicalrestrictions on u):

Et [Xt Xt] =0, t >tT T Note: a sufficient condition for Et t u dZu =0 i s E0 0 2u du

5/23/2018 MIT15_450F10_lec02

8/75


It Processes

An It process is a continuous-time stochastic process Xt , or X(t), of the form

t tu du + u dZu

0 0

u is called the instantaneous drift of Xt at time u, and u is called theinstantaneous volatility, or the diffusion coefficient.

t dt captures the expected change of Xt between t and t +dt.t dZt captures the unexpected (stochastic) component of the change of Xtbetween t and t +dt.Conditional mean and variance:

Et (Xt+dt Xt ) =t dt +o(dt), Et (Xt+dt Xt )2 =t2 dt +o(dt)

cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 8 / 74Stochastic Integral Its Lemma Black Scholes Model Multivariate It Processes SDEs SDEs and PDEs Risk Neutral Probability Risk Neutral Pricing
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

9/75

Stochastic Integral It s Lemma Black-Scholes Model Multivariate It Processes SDEs SDEs and PDEs Risk-Neutral Probability Risk-Neutral Pricing

Outline

1

Stochastic Integral

2 Its Lemma



5 SDEs

6 SDEs and PDEs



c Stochastic Calculus 9 / 74Leonid Kogan ( MIT, Sloan ) 15.450, Fall 2010Stochastic Integral Its Lemma Black-Scholes Model Multivariate It Processes SDEs SDEs and PDEs Risk-Neutral Probability Risk-Neutral Pricing
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

10/75

Stochastic Integral It s Lemma Black Scholes Model Multivariate It Processes SDEs SDEs and PDEs Risk Neutral Probability Risk Neutral Pricing

Quadratic Variation

Consider a time discretization

0 =t1

5/23/2018 MIT15_450F10_lec02

11/75

Stochastic Integral It s Lemma Black Scholes Model Multivariate It Processes SDEs SDEs and PDEs Risk Neutral Probability Risk Neutral Pricing

Quadratic Variation

Quadratic variation of an It process

t tXt = u du + u dZu

0 0

is given by

T

[X]T = 2t

dt0

Heuristically, the quadratic variation formula states that

(dZt )2 =dt, dt dZt =o(dt), (dt)2 =o(dt)

Random walk intuition:

|dZt |=dt, |dt dZt |= (dt)3/2 =o(dt), dZt2 =dtConditional variance of the It process can be estimated by approximating its

quadratic variation with a discrete sum. This is the basis for variance

estimation using high-frequency data.

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

12/75

g y g

Its Lemma

Its Lemma states that if Xt is an It process,t tXt = u du + u dZu

0 0

then so is f (t,Xt ), where f is a sufficiently smooth function, and

df (t,Xt ) = f (t,Xt ) +f (t,Xt )t +1 2f (t,Xt )2t dt +f (t,Xt )t dZtt Xt 2 Xt2 XtIts Lemma is, heuristically, a second-order Taylor expansion in t and Xt ,

using the rule that

(dZt )2 =dt, dt dZt =o(dt), (dt)2 =o(dt)

cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 12 / 74Stochastic Integral Its Lemma Black-Scholes Model Multivariate It Processes SDEs SDEs and PDEs Risk-Neutral Probability Risk-Neutral Pricing

5/23/2018 MIT15_450F10_lec02

13/75

g y g

Its Lemma

Using the Taylor expansion,

df (t,Xt )f (

t,t

Xt )dt +f (

t

X

,Xt

t )dXt +

2

1 2f

(t

t

,2

Xt )dt2

+2

1 2f

(

X

t,t

2

Xt )(dXt )

2 +2

f

t

(

t,X

X

t

t )dt dXt

f (t,Xt ) f (t,Xt ) f (t,Xt )= dt + t dt + t dZt

t Xt Xt

+2

1 2f

(

X

t,t2

Xt )2t dt +o(dt)

Short-hand notation

df (t,Xt ) = f (

t,t

Xt )dt +f (

t

X

,Xt

t )dXt +

2

1 2f

(

X

t,t2

Xt )(dXt )

2


5/23/2018 MIT15_450F10_lec02

14/75

Its LemmaExample

Let Xt =exp(at +bZt ).We can write Xt =f (t,Zt ), where

f (t,Zt ) =exp(at +bZt )Using

f (t,Zt )=af (t,Zt ), f (t,Zt ) =bf (t,Zt ), 2f (t,Zt ) =b2f (t,Zt )

t Zt Zt2

Its Lemma implies

b2dXt = a + Xt dt +bXt dZt

2

Expected growth rate of Xt is a +b2/2.c Stochastic Calculus 14 / 74Leonid Kogan ( MIT, Sloan ) 15.450, Fall 2010

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

15/75

Outline

1

Stochastic Integral

2 Its Lemma



5 SDEs

6 SDEs and PDEs



http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

16/75

The Black-Scholes Model of the Market

Consider the market with a constant risk-free interest rate r and a single risky

asset, the stock.Assume the stock does not pay dividends and the price process of the stock

is given by St =S0 exp 1 2 t +Zt

2

Because Brownian motion is normally distributed, usingE0[exp(Zt )]=exp(t/2), find

E0[St] =S0 exp(t)Using Its Lemma (check)

dSt=dt +dZt

St

is the expected continuously compounded stock return, is the volatility ofstock returns.

Stock returns have constant volatility.

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

17/75

Dynamic Trading

Consider a trading strategy with continuous rebalancing.

At each point in time, hold t shares of stocks in the portfolio.

Let the portfolio value be Wt . Then Wt tSt dollars are invested in theshort-term risk-free bond.

Portfolio is self-financing: no exogenous incoming or outgoing cash flows.Portfolio value changes according to

dWt =t dSt + (Wt tSt )r dtDiscrete-time analogy

Bt+tWt+t Wt =t (St+t St ) + (Wt tSt ) 1

Bt

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

18/75

Option Replication

Consider a European option with the payoff H(ST ).

We will construct a self-financing portfolio replicating the payoff of the option.

Look for the portfolio such that

Wt =f (t,St )for some function f (t,St ).By Law of One Price, f (t,St )must be the price of the option at time t, beingthe cost of a trading strategy with an identical payoff.

Note that the self-financing condition is important for the above argument: wedo not want the portfolio to produce intermediate cash flows.


5/23/2018 MIT15_450F10_lec02

19/75

Option Replication

Apply Its Lemma to portfolio value, Wt =f (t,St ):f f 1 2fdWt =t dSt + (Wt tSt )r dt =t

dt +St

dSt +2 St

2(dSt )

2

where (dSt )2 =2St2 dt

The above equality holds at all times if

t = f (tS

,St

t ), f (t,tSt ) +

21 2f

(St,

t2St )2St2 r f (t,St ) Sft St =0

If we can find the solution f (t,S)to the PDEf (t,S) f (t,S) 1

2S22f (t,S)

r f (t,S) + +rS + =0t S 2 S

2

with the boundary condition f (T ,S) =H(S), then the portfolio withf (t,St )

W0 =f (0,S0), t =St

replicates the option!cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 19 / 74


5/23/2018 MIT15_450F10_lec02

20/75

Black-Scholes Option Price

We conclude that the option price can be computed as a solution of the

Black-Scholes PDEf f 1 2f

r f + +rS + 2S2 =0t S 2 S2

If the option is a European call with strike K , the PDE can be solved in closed

form, yielding the Black-Scholes formula:

C(t,St ) =StN(z1) exp(r(T t))KN(z2),where N(.)is the cumulative distribution function of the standard normaldistribution,

log SK

t + r +21 2 (T t)

z1 =

,

T tand

z2 =z1 T t.Note that does not enter the PDE or the B-S formula. This is intuitive fromthe perspective of risk-neutral pricing. Discuss later.


5/23/2018 MIT15_450F10_lec02

21/75

3

Black-Scholes Option Replication

The replicating strategy requires holding

f (t,St )t =

St

stock shares in the portfolio. t is called the options delta.

It is possible to replicate any option in the Black-Scholes setting because

Price of the stock St is driven by a Brownian motion;

Rebalancing of the replicating portfolio is continuous;

There is a single Brownian motion affecting the payoff of the option and the price

of the stock. More on this later, when we cover multivariate It processes.


5/23/2018 MIT15_450F10_lec02

22/75

Single-Factor Term Structure Model

Consider a model of the term structure of default-free bond yields.

Assume that the short-term interest rate follows

drt =(rt )dt +(rt )dZtLet P(t,)denote the time-t price of a discount bond with unit face valuematuring at time .

We want to construct an arbitrage-free model capturing, simultaneously, the

dynamics of bond prices of many different maturities.


5/23/2018 MIT15_450F10_lec02

23/75


Assume that bond prices can be expressed as a function of the short rate

only (single-factor structure)

P(t,) =f (t,rt ,)It formula implies that

dP(t,) = f +f (rt ) +1 2f 2(rt ) dt +f (rt )dZtt r 2 r2 r

t t

Consider a self-financing portfolio which invests P(t,)/t dollars in thebond maturing at , and P(t

,)/t

dollars in the bond maturing at

.Self-financing requires that the investment in the risk-free short-term bond is

Wt P(

t,)

+P(

t,

)

t t


5/23/2018 MIT15_450F10_lec02

24/75


Portfolio value evolves according to

dWt = Wt P(

t,

)+P(

t,

)

rt +

t

t

dt+

t t t t

t t dZt

t t

= Wt P(

t,

)+P(

t,

)

rt +

t

t

dt

t t t t

Portfolio value changes are instantaneously risk-free.

To avoid arbitrage, the portfolio value must grow at the risk-free rate:

Wt P(

t,

)+P(

t,

)

rt +

t

t

=Wtrtt t t t


5/23/2018 MIT15_450F10_lec02

25/75


We conclude that

t rtP(t,)

= t rtP(t,), for any and t t

Assume that, for some ,

t

rtP(t,) =(t,rt )

t

Then, for all bonds, must have

t rtP(t,) =(t,rt )t

Recall the definition of t, t to derive the pricing PDE on P(t,) =f (t,rt ,)

f f 1 2f f+ (r) + 2(r) rf =(t,r)(r) , f (,r,) =1

t r 2 r2 r

The solution, indeed, has the form f (t,r,).cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 25 / 74


5/23/2018 MIT15_450F10_lec02

26/75


What ift rtP(t,)

t=(t,rt )

for any function , i.e., the LHS depends on something other than rt and t?

Then the term structure will not have a single-factor form.We have seen that, for each choice of (t,rt ), absence of arbitrage impliesthat bond prices must satisfy the pricing PDE.

The reverse is true: if bond prices satisfy the pricing PDE (with well-behaved

(t,rt )), there is no arbitrage (show later, using risk-neutral pricing).The choice of (t,rt )determines the joint arbitrage-free dynamics of bondprices (yields).

(t,rt )is the price of interest rate risk.


5/23/2018 MIT15_450F10_lec02

27/75

Outline


2 Its Lemma



5 SDEs

6 SDEs and PDEs



http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

28/75

Multiple Brownian motions

Consider two independent Brownian motions Zt1 and Zt

2. Construct a third

process Xt =Zt1 + 1 2Zt2Xt is also a Brownian motion:

Xt has IID normal increments;

Xt is continuous.

Xt and Zt1

are correlated: E0 XtZt1 =tCorrelated Brownian motions can be constructed from uncorrelated ones,

just like with normal random variables.

Cross-variation

N1

[Zt1,Zt2]T = lim (Z 1(tn+1) Z 1(tn))(Z 2(tn+1) Z 2(tn))=0

0n=1

Short-hand rule

dZt1 dZt

2 =0 dZt1 dXt =dt



5/23/2018 MIT15_450F10_lec02

29/75

Multivariate It Processes

A multivariate It process is a vector process with each coordinate driven byan It process.

Consider a pair of processes

dXt =tX dt +Xt dZtX ,

dYt =tY

dt +

Y

t dZtY

,dZt

X dZtY =t dt

Its formula can be extended to multiple process as follows:

f f f

df (t,Xt ,Yt ) = dt + dXt + dYt +t Xt Yt

1 2f 1 2f 2f(dXt )

2 + (dYt )2 + dXt dYt

2 Xt2 2 Yt

2 Xt Yt


5/23/2018 MIT15_450F10_lec02

30/75

Example of Its formula

Consider two asset price processes, Xt and Yt , both given by It processes

dXt =Xt dt +Xt dZtXdYt =tY dt +Yt dZtY

dZtX dZt

Y =0Using Its formula, we can derive the process for the ratio ft =Xt /Yt (usef (X,Y ) =X/Y ):

2

dft dXt dYt dXt dYt dYt= +

ft Xt Yt Xt Yt Yt 2X Y Y X Y= t t + t dt + t dZtX t dZtYXt Yt Yt2 Xt Yt


f f

5/23/2018 MIT15_450F10_lec02

31/75

Example of Its formula

We find that the expected growth rate of the ratio Xt /Yt is X Y Y 2t t t +Xt Yt Yt

2

Assume that tX =tY . Then, dft Yt 2

Et = dtft Yt

2

Repeating the same calculation for the inverse ratio, ht =Yt /Xt , we find 2

Etdht

= tX dtht Xt

2

It is possible for both the ratio Xt /Yt and its inverse Yt /Xt to be expected to

grow at the same time. Application to FX.


O li

5/23/2018 MIT15_450F10_lec02

32/75

Outline


2 Its Lemma



5 SDEs

6

SDEs and PDEs




St h ti Diff ti l E ti

5/23/2018 MIT15_450F10_lec02

33/75

Stochastic Differential Equations

Example: Hestons stochastic volatility model

dSt =dt +vt dZSSt

t

dvt = (vt v)dt +vt dZtvdZt

S dZtv =dt

Conditional variance vt is described by a Stochastic Differential Equation.

Definition (SDE)

The It process Xt satisfies a stochastic differential equation

dXt

=(t,Xt)dt +(t,X

t)dZ

t

with an initial condition X0 if it satisfies

Xt =X0 +t

0

(s,Xs)ds +(s,Xs)dZs.cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 33 / 74


E i t f S l ti f SDE

5/23/2018 MIT15_450F10_lec02

34/75

Existence of Solutions of SDEs

Assume that for some C,D >0|(t,X)|+|(t,X)|C(1 +|X |)

and

|(t,X) (t,Y )|+|(t,X) (t,Y )|D|X Y |for any X and Y (Lipschitz property).

Then, the SDE

dXt =(t,Xt )dt +(t,Xt )dZt , X0 =x,has a unique continuous solution Xt .


Common SDEs

5/23/2018 MIT15_450F10_lec02

35/75

Common SDEsArithmetic Brownian Motion

The solution of the SDE

dXt =dt +dZtis given by

Xt =X0 +t +Zt .The process Xt is called an arithmetic Brownian motion, or Brownian motion

with a drift.

Guess and verify.

We typically reduce an SDE to a few common cases with explicit solutions.


Common SDEs

5/23/2018 MIT15_450F10_lec02

36/75

Common SDEsGeometric Brownian Motion

Consider the SDEdXt =Xt dt +Xt dZt

Define the process

Yt =ln(Xt ).By Its Lemma,

dYt =X

1

t

Xt dt +X

1

t

Xt dZt +12

(X

1

t2

)2Xt2 dt = (2/2)dt +dZt .

Yt is an arithmetic Brownian motion, given in the previous example, and

Yt =Y0 + (2/2)t +Zt .Then

Xt =eYt =X0 exp (2/2)t +ZtcLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 36 / 74


Common SDEs

5/23/2018 MIT15_450F10_lec02

37/75

Common SDEsOrnstein-Uhlenbeck process

The mean-reverting Ornstein-Uhlenbeck process is the solution Xt to the

stochastic differential equation

dXt = (X Xt )dt +dZtWe solve this equation using et as an integrating factor.

Setting Yt =et and using Its lemma for the function f (X ,Y ) =X Y , we findd(etXt ) =et (X dt +dZt ).

Integrating this between 0 and t, we find

t tetXt X0 = esXds + esdZs,0 0

i.e., tXt =etX0 + (1 et )X + estdZs.

0


Option Replication in the Heston Model

5/23/2018 MIT15_450F10_lec02

38/75

Option Replication in the Heston Model

Assume the Heston stochastic-volatility model for the stock.

Attempt to replicate the option payoff with the stock and the risk-free bond.Can we find a trading strategy that would guarantee perfect replication?

It is possible to replicate an option using a bond, a stock, and another option.


Recap of APT

5/23/2018 MIT15_450F10_lec02

39/75

Recap of APT

Recall the logic of APT.

Suppose we have N assets with two-factor structure in their returns:Rt

i =ai +bi1Ft1 +bi2Ft2Interest rate is r.

While not stated explicitly, all factor loadings may be stochastic.

Unlike the general version of the APT, we assume that returns have noidiosyncratic component.

At time t, consider any portfolio with fraction i in each asset i that has zero

exposure to both factors:

b111 +b122 +...+b1N N =0

b211 +b222 +...+b2N N =0

This portfolio must have zero expected excess return to avoid arbitrage

1 Et [Rt1] r +2 Et [Rt2] r +...+N Et [RtN] r =0


Recap of APT

5/23/2018 MIT15_450F10_lec02

40/75

Recap of APT

To avoid arbitrage, any portfolio satisfying

b111 +b122 +...+b1N N =0b2

11 +b222 +...+b2N N =0must satisfy

1 Et [Rt1

] r +2 Et [Rt2] r +...+N Et [RtN] r =0Restating this in vector form, any vector orthogonal to

(b11,b12,...,b1N )and (b21,b22,...,b2N )

must be orthogonal to (Et [Rt1] r,Et [Rt2] r,...,Et [RtN] r).Conclude that the third vector is spanned by the first two: there exist

constants (prices of risk) (1t ,2t )such thatEt [Rt

i] r =1tb1i +t2b2i , i =1,...,NcLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 40 / 74


Option Pricing in the Heston Model

5/23/2018 MIT15_450F10_lec02

41/75


Suppose there are N derivatives with prices given by

fi (t,St ,vt )where the first option is the stock itself: f 1(t,St ,vt ) =St .Using Itos lemma, their prices satisfy

df i (t,St ,vt ) =aitdt +fi (t,St ,vt )dSt +fi (t,St ,vt )dvtSt vtCompare the above to our APT argument

Conclude that there exist St and vt such that

Edf i (t,St ,vt ) rf i (t,St ,vt )dt= fi (t

,S

S

t

t ,vt )t

S dt +fi (t

,v

S

t

t ,vt )t

v dt

We work with price changes instead of returns, as we did in the APT,

because some of the derivatives may have zero price.



5/23/2018 MIT15_450F10_lec02

42/75


The APT pricing equation, applied to the stock, implies that

E [dSt rSt dt]= (r)St dt =tS dtvt is the price of volatility risk, which determines the risk premium on any

investment with exposure to dvt .

Writing out the pricing equation explicitly, with the Itos lemma providing an

expression for E df i (t,St ,vt ) ,fi fi fi

+ S + ()(v v)+t S v

1 2fi 1 2fi 2fi fi fivS2 + 2v + Sv rf i = (r)S + v

2 S2 2 v2 Sv S v t

As long as we assume that the price of volatility risk is of the form

v =v (t,St ,vt )tthe assumed functional form for option prices is justified and we obtain an

arbitrage-free option pricing model.


Numerical Solution of SDEs

5/23/2018 MIT15_450F10_lec02

43/75

Numerical Solution of SDEs

Except for a few special cases, SDEs do not have explicit solutions.

The most basic and common method of approximating solutions of SDEs

numerically is using the first-order Euler scheme.

Use the grid ti

=i.Xi+1 =Xi +(ti ,Xi )+(ti ,Xi )i ,

where

i are IID N(0,1)random variables.

Using a binomial distribution for i , with equal probabilities of 1, is also avalid procedure for approximating the distribution of Xt


Outline

5/23/2018 MIT15_450F10_lec02

44/75

Outline


2 Its Lemma



5 SDEs

6 SDEs and PDEs




Moments of Diffusion Processes

5/23/2018 MIT15_450F10_lec02

45/75

Moments of Diffusion Processes

Often need to compute conditional moments of diffusion processes:

Expected returns and variances of returns on financial assets over finite time

intervals;

Use the method of moments to estimate a diffusion process from discretelysampled data;

Compute prices of derivatives.

One approach is to reduce the problem to a PDE, which can sometimes be

solved analytically.


Kolmogorov Backward Equation

5/23/2018 MIT15_450F10_lec02

46/75

g q

Diffusion process Xt with coefficients (t,X)and (t,X).Objective: compute a conditional expectation

f (t,X) =E[g(XT )|Xt =X]Suppose f (t,X)is a smooth function of t and X . By the law of iteratedexpectations,

f (t,Xt) =E

t[f (t +dt,X

t+

dt)] E

t[df (t,X

t)]=0

Using Itos Lemma,

Et [df (t,Xt )]= f (t,X) +(t,X)f (t,X) +1 (t,X)2 2f (t,X) dt =0t X 2 X 2

Kolmogorov backward equationf (t,X)

+(t,X)f (t,X) +1 (t,X)2 2f (t,X) =0,t X 2 X 2

with boundary condition

f (T ,X) =g(X)cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 46 / 74


Example: Square-Root Diffusion

5/23/2018 MIT15_450F10_lec02

47/75

p q

Consider a popular diffusion process used to model interest rates and

stochastic volatility

dXt = (Xt X)dt + Xt dZtWe want to compute the conditional moments of this process, to be used as

a part of GMM estimation.

Compute the second non-central moment

f (t,X) =E(XT2|Xt =X)Using Kolmogorov backward equation,

f (t,X) (X X)f (t,X) +1 2X 2f (t,X) =0,t X 2 X 2

with boundary condition

f (T ,X) =X 2cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 47 / 74



5/23/2018 MIT15_450F10_lec02

48/75

p q

Look for the solution in the the form

f (t,X) =a0(t) +a1(t)X +a2(t)X 22Substitute f (t,X)into the PDE

a0(t) +a1(t)X +a2(t)X 2 (X X)(a1(t) +a2(t)X) +2 Xa2(t) =02 2

Collect terms with different powers of X , zero, one and two, to get

a0(t) +Xa1(t) =0

2a1(t) a1(t) + +X a2(t) =0

2a2(t) 2a2(t) =0

with initial conditions

a0(T ) =a1(T ) =0, a2(T ) =2cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 48 / 74



5/23/2018 MIT15_450F10_lec02

49/75

p q

We solve the system of equations starting from the third one and working up

to the first:

a0(t)=XA

1

2e2(tT ) e(tT ) +1

2

a1(t)=A e(tT ) e2(tT )

a2(t)=2e2(tT )A = 2 +2X

Compare the exact expression above to an approximate expression, obtained

by assuming that T t =t is small.



5/23/2018 MIT15_450F10_lec02

50/75

Assume T t =

t is small. Using Taylor expansion,

a0(t) =o(t), a1(t) =At +o(t)a2(t) =2(1 2t) +o(t)

Then Et (Xt2+t |Xt =X)X 2 +t (2 +2X)X 2X2Alternatively,

Xt+t Xt (Xt X)t +

Xt

tt , t N(0,1)Therefore

Et (Xt2+t |Xt =X)X 2 +t 2X 2X(X X)


Outline

5/23/2018 MIT15_450F10_lec02

51/75


2 Its Lemma



5 SDEs

6 SDEs and PDEs




Risk-Neutral Probability Measure

5/23/2018 MIT15_450F10_lec02

52/75

Under the risk-neutral probability measureQ

, expected conditional assetreturns must equal the risk-free rate.

Alternatively, using the discounted cash flow formula, the price Pt of an asset

with payoff HT at time T is given by

T Pt t exp rsds HT=EQ twhere rs is the instantaneous risk-free interest rate at time s.

The DCF formula under the risk-neutral probability can be used to compute

asset prices by Monte Carlo simulation (e.g., using an Euler scheme to

approximate the solutions of SDEs).

Alternatively, one can derive a PDE characterizing asset prices using the

connection between PDEs and SDEs.


Change of Measure

5/23/2018 MIT15_450F10_lec02

53/75

We often want to consider a probability measure Qdifferent from P, but theone that agrees with Pon which events have zero probability (equivalent toP) (e.g., Qcould be a risk-neutral measure).Different probability measures assign different relative likelihoods to the

trajectories of the Brownian motion.

It is easy to express a new probability measure Qusing its densitydQ

T =dP T

For any random variable XT ,

EQ0 [XT] =EP0 [T XT], EQt [XT] =EP T XT , t EPt [T]t tQis equivalent to Pif T is positive (with probability one).cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 53 / 74


Change of Measure

5/23/2018 MIT15_450F10_lec02

54/75

If we consider a probability measure Qdifferent from P, but the one that

agrees with Pon which events have zero probability, then the P-Brownianmotion Zt

P becomes an Ito process under Q:dZt

P =dZtQ t dtfor some t . ZtQ

is a Brownian motion under Q.

When we change probability measures this way, only the drift of the Brownian

motion changes, not the variance.

Intuition: a probability measure assigns relative likelihood to different

trajectories of the Brownian motion. Variance of the Ito process can be

recovered from the shape of a single trajectory (quadratic variation), so itdoes not depend on the relative likelihood of the trajectories, hence, does not

depend on the choice of the probability measure.



5/23/2018 MIT15_450F10_lec02

55/75

Under the risk-neutral probability measure, expected conditional asset

returns must equal the risk-free rate.

Start with the stock price process under P:

dSt =tSt dt +tSt dZtPUnder the risk-neutral measure Q,

dSt =rtSt dt +tSt dZtQThus, if dZt

P = t dt +dZtQ,t tt =rt

t is the price of risk.

The risk-neutral measure Qis such that the process ZtQ defined bydZt

Q = t rt dt +dZtPtis a Brownian motion under Q.



5/23/2018 MIT15_450F10_lec02

56/75

Under the risk-neutral probability measure, expected conditional asset

returns must equal the risk-free rate.

We conclude that for any asset (paying no dividends), the conditional risk

premium is given by

EP dSt rt dt =EP dSt EQ dStt t tSt St StThus, mathematically, the risk premium is the difference between expected

returns under theP

andQ

probabilities.



5/23/2018 MIT15_450F10_lec02

57/75

If we want to connect Qto Pexplicitly, how can we compute the density,dQ/dP?

In discrete time, the density was conditionally lognormal.The density T = (dQ/dP)T is given by t t

t =exp u dZuP 1 2u du , 0 t T20 0The state-price density is given by t

t =exp ru du t0

The reverse is true: if we define T as above, for any process t satisfying

certain regularity conditions (e.g.,t is bounded, or satisfies the Novikovs

condition as in Back 2005, Appendix B.1), then measure Qis equivalent to Pand t

ZtQ =ZtP + u du

0

is a Brownian motion under Q.


Risk-Neutral Probability and Arbitrage

5/23/2018 MIT15_450F10_lec02

58/75

If there exists a risk-neutral probability measure, then the model is

arbitrage-free.

If there exists a unique risk-neutral probability measure in a model, then all

options are redundant and can be replicated by trading in the underlying

assets and the risk-free bond.

A convenient way to build arbitrage-free models is to describe them directly

under the risk-neutral probability.

One does not need to describe the Pmeasure explicitly to specify anarbitrage-free model.

However, to estimate models using historical data, particularly, to estimaterisk premia, one must specify the price of risk, i.e., the link between Qand P.


Outline
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

59/75


2 Its Lemma



5 SDEs

6 SDEs and PDEs




Black-Scholes Model
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

5/23/2018 MIT15_450F10_lec02

60/75

Assume that the stock pays no dividends and the stock price follows

dSt =dt +dZtPStAssume that the interest rate is constant, r.

Under the risk-neutral probability Q, the stock price process is

dSt =r dt +dZtQStTerminal stock price ST is lognormally distributed:

ln ST =ln S0 + r 2 T +T Q, Q N(0,1)2

Price of any European option with payoff H(ST )can be computed as=EQ ePt t r(T t)H(ST )


Term Structure of Interest Rates

5/23/2018 MIT15_450F10_lec02

61/75

Consider the Vasicek model of bond prices.A single-factor arbitrage-free model.

To guarantee that the model is arbitrage-free, build it under the risk-neutral

probability measure.

Assume the short-term risk-free rate process under Qdrt = (rt r)dt +dZtQ

Price of a pure discount bond maturing at T is given by

T

P(t,T ) =E

Qt exp rs dst

Characterize P(t,T )as a solution of a PDE.



5/23/2018 MIT15_450F10_lec02

62/75

Look for P(t,T ) =f (t,rt ).Using Itos lemma,

EQt [df (t,rt )]= f (t,rt ) (rt r)f (t,rt ) +1 2 2f (t2,rt ) dtt rt 2 rtRisk-neutral pricing requires that

EQt [df (t,rt )]=rtf (t,rt )dtand therefore f (t,rt )must satisfy the PDE

f (t,r)(r r)f (t,r) +1 2

2f (t,r)=rf (t,r)t r 2 r2

with the boundary condition

f (T ,r) =1cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 62 / 74



5/23/2018 MIT15_450F10_lec02

63/75

Look for the solution in the form

f (t,rt ) =exp (a(T t) b(T t)rt )Derive a system of ODEs on a(t)and b(t)to find

a(T t) =r(T t) r 1 e(T t)

2 2(T t) e2(T t) +4e(T t) 3

43

1 (T t)b(T t) = 1 e



5/23/2018 MIT15_450F10_lec02

64/75

Assume a constant price of risk . What does this imply for the interest rate

process under the physical measure Pand for the bond risk premia?Use the relation

dZtP = dt +dZtQ

to derive

drt = (rt r)dt +dt +dZt

P = rt r + dt +dZt

P

Expected bond returns satisfy

EP dP(t,T ) = (rt +Pt )dtt P(t,T )where

Pt = 1 f (t,rt )= b(T t)f (t,rt ) rt|b(T t)|is the volatility of bond returns.cLeonid Kogan ( MIT, Sloan ) Stochastic Calculus 15.450, Fall 2010 64 / 74


Equity Options with Stochastic Volatility

5/23/2018 MIT15_450F10_lec02

65/75

Consider again the Hestons model. Assume that under the risk-neutral

probabilityQ

, stock price is given by

d ln St = (r 1vt )dt +vtdZtQ,S2dvt = (vt v)dt +vtdZtQ,S +

1 2vtdZtQ,v

dZtQ,S

dZtQ,v

=0

ZtQ,v models volatility shocks uncorrelated with stock returns.

Constant interest rate r.

The price of a European option with a payoff H(ST )can be computed asPt =EQt [exp (r(T t))H(ST )]

We havent said anything about the physical process for stochastic volatility.

In particular, how is volatility risk priced?



5/23/2018 MIT15_450F10_lec02

66/75

In the Hestons model, assume that the price of volatility risk is constant,v ,

and the price of stock price risk is constant, S.

Then, under P, stock returns follow

d ln St = r +Svt 2

1vt dt +vtdZtP,S

dvt =

(vt v) +vt

S +

1 2v

dt+

vtdZtP,S +1 2vtdZtP,vdZt

P,S dZtP,v =0

We have used

dZtP,S = S dt +dZtQ,S

dZtP,v = v dt +dZtQ,v

Conditional expected excess stock return is

EPdSt

r dt=EP d ln St +vt dt r dt = Svt dtt tSt 2



5/23/2018 MIT15_450F10_lec02

67/75

Our assumptions regarding the market prices of risk translate directly intoimplications for return predictability.

For stock returns, our assumption of constant price of risk predicts a

nonlinear pattern in excess returns: expected excess stock returns

proportional to conditional volatility.

Suppose we construct a position in options with the exposure t to stochasticvolatility shocks and no exposure to the stock price:

dWt = [...]dt +tdZtP,vThen the conditional expected gain on such a position is

(Wtr +tv )dt


Hestons Model of Stochastic Volatility

5/23/2018 MIT15_450F10_lec02

68/75

Assume that under the risk-neutral probability Q, stock price is given by

d ln St = (r 1vt )dt +vtdZtQ,S2dvt = (vt v)dt +vtdZtQ,S +1 2vtdZtQ,v

dZtQ,S dZt

Q,v =0ZQ,v models volatility shocks uncorrelated with stock returns.t

Constant interest rate r.

The price of a European option with a payoff H(ST )can be computed as=EQ[exp (r(T t))H(ST )]Pt t

Assume that the price of volatility risk is constant, v , and the price of stock

price risk is constant, S.


Variance Swap in Hestons Model

5/23/2018 MIT15_450F10_lec02

69/75

Consider a variance swap, paying

T (d ln Su)2 Kt2t

at time T . What should be the strike price of the swap, Kt , to make sure that

the market value of the swap at time t is zero?

Using the result on quadratic variation,

(d ln St )2 =vt dt

the strike price must be such that

T

e

r(T t)

E

Qt vu du

Kt

2

=0t

Need to compute T K 2 =Et Qt

t

vu du


Variance Swap in Hestons Model

5/23/2018 MIT15_450F10_lec02

70/75

Since

vu =vt u (vs v)ds + u vs dZsQ,S +1 2 u vs dZsQ,vt t t

we find that

u u

EQt

[vu] =v

tEQ

t(v

sv)ds =v

t (EQ

t[v

s]v)ds

t t

Solving the above equation for EQt [vu], we findEQt [vu] =v +e(ut)(vt v)

We obtain the strike priceT 1 Kt

2 =EQt vu du =v(T t) + (vt v) 1 e(T t)t


Expected Profit/Loss on a Variance Swap

5/23/2018 MIT15_450F10_lec02

71/75

To compute expected profit/loss on a variance swap, we need to evaluate

T EP (d ln Su)2 K 2t tt

Instantaneous expected gain on a long position in the swap is easy to

compute in closed form.

The market value of a swap starts at 0 at time t, and at s >t becomes s T

Ps EQ er(T s) vu du + vu du K 2s t

t s

s

=er(T s) vu du +Ks2 Kt2t

We conclude that the instantaneous gain on the long swap position at time s

equals

[...]ds +1er(T s) 1 e(T s) dvs


Expected Profit/Loss on a Variance Swap

5/23/2018 MIT15_450F10_lec02

72/75

We conclude that the instantaneous gain on the long swap position at time s

equals

[...]ds +1er(T s)

1 e(T s)

dvs

Given our assumed market prices of risk, S and v , the time-s expected

instantaneous gain on the swap opened at time t is

r(T s) (T s)Psr ds +vs 1e

1 e

S +

1 2v

ds


Summary

5/23/2018 MIT15_450F10_lec02

73/75

Risk-neutral pricing is a convenient framework for developing arbitrage-free

pricing models.

Connection to classical results: risk-neutral expectation can be characterized

by a PDE.

Risk premium on an asset is the difference between expected return under Pand under Qprobability measures.


Readings

5/23/2018 MIT15_450F10_lec02

74/75

Back 2005, Sections 2.1-2.6, 2.8-2.9, 2.11, 13.2, 13.3, Appendix B.1.


MIT OpenCourseWarehttp://ocw.mit.edu
http://-/?-http://-/?-http://ocw.mit.edu/http://ocw.mit.edu/

5/23/2018 MIT15_450F10_lec02

75/75

15.450 Analytics of FinanceFall 2010

For information about citing these materials or our Terms of Use, visit:http://ocw.mit.edu/terms.
http://ocw.mit.edu/http://ocw.mit.edu/http://ocw.mit.edu/

mit15_450f10_lec02

Documents