### Ito Process

*• The stochastic process X = { X*^{t}*, t* *≥ 0 } that solves*
*X*_{t}*= X*_{0} +

∫ *t*
0

*a(X*_{s}*, s) ds +*

∫ *t*
0

*b(X*_{s}*, s) dW*_{s}*, t* *≥ 0*
is called an Ito process.

**– X**_{0} is a scalar starting point.

**–** *{ a(X*^{t}*, t) : t* *≥ 0 } and { b(X*^{t}*, t) : t* *≥ 0 } are*
stochastic processes satisfying certain regularity
conditions.

*• The terms a(X*^{t}*, t) and b(X*_{t}*, t) are the drift and the*

### Ito Process (continued)

*• A shorthand*^{a} is the following stochastic diﬀerential
*equation for the Ito diﬀerential dX** _{t}*,

*dX*_{t}*= a(X*_{t}*, t) dt + b(X*_{t}*, t) dW*_{t}*.* (48)
**– Or simply**

*dX*_{t}*= a*_{t}*dt + b*_{t}*dW*_{t}*.*

**– This is Brownian motion with an instantaneous drift**
*a*_{t}*and an instantaneous variance b*^{2}* _{t}*.

*• X is a martingale if a** ^{t}* = 0 (Theorem 17 on p. 485).

aPaul Langevin (1904).

### Ito Process (concluded)

*• dW is normally distributed with mean zero and*
*variance dt.*

*• An equivalent form of Eq. (48) is*
*dX*_{t}*= a*_{t}*dt + b*_{t}*√*

*dt ξ,* (49)

*where ξ* *∼ N(0, 1).*

### Euler Approximation

*• The following approximation follows from Eq. (49),*
*X(t*b * _{n+1}*)

= b*X(t*_{n}*) + a( bX(t*_{n}*), t*_{n}*) ∆t + b( bX(t*_{n}*), t*_{n}*) ∆W (t*_{n}*),*

(50)
*where t*_{n}*≡ n∆t.*

*• It is called the Euler or Euler-Maruyama method.*

*• Recall that ∆W (t** ^{n}*) should be interpreted as

*W (t*

*)*

_{n+1}*− W (t*

^{n}*), not W (t*

*)*

_{n}*− W (t*

^{n}*).*

_{−1}*• Under mild conditions, bX(t*_{n}*) converges to X(t** _{n}*).

### More Discrete Approximations

*• Under fairly loose regularity conditions, Eq. (50) on*
p. 492 can be replaced by

*X(t*b * _{n+1}*)

= b*X(t*_{n}*) + a( bX(t*_{n}*), t*_{n}*) ∆t + b( bX(t*_{n}*), t** _{n}*)

*√*

*∆t Y (t*_{n}*).*

**– Y (t**_{0}*), Y (t*_{1}*), . . . are independent and identically*
distributed with zero mean and unit variance.

### More Discrete Approximations (concluded)

*• An even simpler discrete approximation scheme:*

*X(t*b * _{n+1}*)

= b*X(t*_{n}*) + a( bX(t*_{n}*), t*_{n}*) ∆t + b( bX(t*_{n}*), t** _{n}*)

*√*

*∆t ξ.*

**– Prob[ ξ = 1 ] = Prob[ ξ =***−1 ] = 1/2.*

**– Note that E[ ξ ] = 0 and Var[ ξ ] = 1.**

*• This is a binomial model.*

*• As ∆t goes to zero, bX converges to X.*

### Trading and the Ito Integral

**• Consider an Ito process dS**^{t}*= µ*_{t}*dt + σ*_{t}*dW** _{t}*.

**– S**

_{t}*is the vector of security prices at time t.*

**• Let ϕ***t* be a trading strategy denoting the quantity of
*each type of security held at time t.*

**– Hence the stochastic process ϕ**_{t}**S*** _{t}* is the value of the

**portfolio ϕ**

_{t}*at time t.*

**• ϕ***t* **dS**_{t}**≡ ϕ***t**(µ*_{t}*dt + σ*_{t}*dW** _{t}*) represents the change in the

*value from security price changes occurring at time t.*

### Trading and the Ito Integral (concluded)

*• The equivalent Ito integral,*
*G*_{T}**(ϕ)***≡*

∫ *T*
0

**ϕ**_{t}**dS*** _{t}* =

∫ *T*
0

**ϕ**_{t}*µ*_{t}*dt +*

∫ *T*
0

**ϕ**_{t}*σ*_{t}*dW*_{t}*,*
measures the gains realized by the trading strategy over
*the period [ 0, T ].*

### Ito’s Lemma

A smooth function of an Ito process is itself an Ito process.

**Theorem 18 Suppose f : R***→ R is twice continuously*
*diﬀerentiable and dX = a*_{t}*dt + b*_{t}*dW . Then f (X) is the*
*Ito process,*

*f (X** _{t}*)

= *f (X*_{0}) +

∫ *t*
0

*f*^{′}*(X*_{s}*) a*_{s}*ds +*

∫ *t*
0

*f*^{′}*(X*_{s}*) b*_{s}*dW*
+1

2

∫ *t*
0

*f*^{′′}*(X*_{s}*) b*^{2}_{s}*ds*

### Ito’s Lemma (continued)

*• In diﬀerential form, Ito’s lemma becomes*
*df (X) = f*^{′}*(X) a dt + f*^{′}*(X) b dW +* 1

2 *f*^{′′}*(X) b*^{2} *dt.*

(51)

*• Compared with calculus, the interesting part is the third*
term on the right-hand side.

*• A convenient formulation of Ito’s lemma is*
*df (X) = f*^{′}*(X) dX +* 1

2 *f*^{′′}*(X)(dX)*^{2}*.*

### Ito’s Lemma (continued)

*• We are supposed to multiply out*

*(dX)*^{2} *= (a dt + b dW )*^{2} symbolically according to

*×* *dW* *dt*

*dW* *dt* 0

*dt* 0 0

**– The (dW )**^{2} *= dt entry is justiﬁed by a known result.*

*• Hence (dX)*^{2} *= (a dt + b dW )*^{2} *= b*^{2} *dt.*

*• This form is easy to remember because of its similarity*

### Ito’s Lemma (continued)

**Theorem 19 (Higher-Dimensional Ito’s Lemma) Let***W*_{1}*, W*_{2}*, . . . , W*_{n}*be independent Wiener processes and*

*X* *≡ (X*^{1}*, X*_{2}*, . . . , X*_{m}*) be a vector process. Suppose*

*f : R*^{m}*→ R is twice continuously diﬀerentiable and X*^{i}*is*
*an Ito process with dX*_{i}*= a*_{i}*dt +* ∑*n*

*j=1* *b*_{ij}*dW*_{j}*. Then*
*df (X) is an Ito process with the diﬀerential,*

*df (X) =*

∑*m*
*i=1*

*f*_{i}*(X) dX** _{i}* + 1
2

∑*m*
*i=1*

∑*m*
*k=1*

*f*_{ik}*(X) dX*_{i}*dX*_{k}*,*
*where f*_{i}*≡ ∂f/∂X*^{i}*and f*_{ik}*≡ ∂*^{2}*f /∂X*_{i}*∂X*_{k}*.*

### Ito’s Lemma (continued)

*• The multiplication table for Theorem 19 is*

*×* *dW*_{i}*dt*

*dW*_{k}*δ*_{ik}*dt* 0

*dt* 0 0

in which

*δ** _{ik}* =

1 *if i = k,*
0 *otherwise.*

### Ito’s Lemma (continued)

*• In applying the higher-dimensional Ito’s lemma, usually*
*one of the variables, say X*_{1}*, is time t and dX*_{1} *= dt.*

*• In this case, b*^{1j}*= 0 for all j and a*_{1} = 1.

*• Assume dX*^{t}*= a*_{t}*dt + b*_{t}*dW** _{t}*.

*• Consider the process f(X*^{t}*, t).*

### Ito’s Lemma (continued)

*• Then*

*df* = *∂f*

*∂X*_{t}*dX** _{t}* +

*∂f*

*∂t* *dt +* 1
2

*∂*^{2}*f*

*∂X*_{t}^{2} *(dX** _{t}*)

^{2}

= *∂f*

*∂X*_{t}*(a*_{t}*dt + b*_{t}*dW** _{t}*) +

*∂f*

*∂t* *dt*
+1

2

*∂*^{2}*f*

*∂X*_{t}^{2} *(a*_{t}*dt + b*_{t}*dW** _{t}*)

^{2}

=

( *∂f*

*∂X*_{t}*a** _{t}* +

*∂f*

*∂t* + 1
2

*∂*^{2}*f*

*∂X*_{t}^{2} *b*^{2}* _{t}*
)

*dt*
+ *∂f*

*b* *dW* *.* (52)

### Ito’s Lemma (continued)

**Theorem 20 (Alternative Ito’s Lemma) Let***W*_{1}*, W*_{2}*, . . . , W*_{m}*be Wiener processes and*

*X* *≡ (X*^{1}*, X*_{2}*, . . . , X*_{m}*) be a vector process. Suppose*

*f : R*^{m}*→ R is twice continuously diﬀerentiable and X*^{i}*is*
*an Ito process with dX*_{i}*= a*_{i}*dt + b*_{i}*dW*_{i}*. Then df (X) is the*
*following Ito process,*

*df (X) =*

∑*m*
*i=1*

*f*_{i}*(X) dX** _{i}* + 1
2

∑*m*
*i=1*

∑*m*
*k=1*

*f*_{ik}*(X) dX*_{i}*dX*_{k}*.*

### Ito’s Lemma (concluded)

*• The multiplication table for Theorem 20 is*

*×* *dW*_{i}*dt*

*dW*_{k}*ρ*_{ik}*dt* 0

*dt* 0 0

*• Above, ρ*^{ik}*denotes the correlation between dW** _{i}* and

*dW*

*.*

_{k}### Geometric Brownian Motion

*• Consider geometric Brownian motion Y (t) ≡ e*^{X(t)}**– X(t) is a (µ, σ) Brownian motion.**

**– Hence dX = µ dt + σ dW by Eq. (46) on p. 467.**

*• As ∂Y/∂X = Y and ∂*^{2}*Y /∂X*^{2} *= Y , Ito’s formula (51)*
on p. 498 implies

*dY* = *Y dX + (1/2) Y (dX)*^{2}

= *Y (µ dt + σ dW ) + (1/2) Y (µ dt + σ dW )*^{2}

= *Y (µ dt + σ dW ) + (1/2) Y σ*^{2} *dt.*

### Geometric Brownian Motion (concluded)

*• Hence*

*dY*

*Y* = (

*µ + σ*^{2}*/2*)

*dt + σ dW.* (53)

*• The annualized instantaneous rate of return is µ + σ*^{2}*/2*
*not µ.*

### Product of Geometric Brownian Motion Processes

*• Let*

*dY /Y* = *a dt + b dW*_{Y}*,*
*dZ/Z* = *f dt + g dW*_{Z}*.*

*• Consider the Ito process U ≡ Y Z.*

*• Apply Ito’s lemma (Theorem 20 on p. 504):*

*dU* = *Z dY + Y dZ + dY dZ*

= *ZY (a dt + b dW*_{Y}*) + Y Z(f dt + g dW** _{Z}*)

*+Y Z(a dt + b dW*

*Y*

*)(f dt + g dW*

*Z*)

= *U (a + f + bgρ) dt + U b dW*_{Y}*+ U g dW*_{Z}*.*

### Product of Geometric Brownian Motion Processes (continued)

*• The product of two (or more) correlated geometric*
Brownian motion processes thus remains geometric
Brownian motion.

*• Note that*

*Y* = exp[(

*a* *− b*^{2}*/2*)

*dt + b dW** _{Y}* ]

*,*

*Z*= exp[(

*f* *− g*^{2}*/2*)

*dt + g dW** _{Z}*]

*,*

*U*= exp[ (

*a + f* *−* (

*b*^{2} *+ g*^{2})

*/2*)

*dt + b dW*_{Y}*+ g dW** _{Z}* ]

*.*

### Product of Geometric Brownian Motion Processes (concluded)

*• ln U is Brownian motion with a mean equal to the sum*
*of the means of ln Y and ln Z.*

*• This holds even if Y and Z are correlated.*

*• Finally, ln Y and ln Z have correlation ρ.*

### Quotients of Geometric Brownian Motion Processes

*• Suppose Y and Z are drawn from p. 508.*

*• Let U ≡ Y/Z.*

*• We now show that*^{a}
*dU*

*U* *= (a* *− f + g*^{2} *− bgρ) dt + b dW*^{Y}*− g dW*^{Z}*.*

(54)

*• Keep in mind that dW*^{Y}*and dW*_{Z}*have correlation ρ.*

aExercise 14.3.6 of the textbook is erroneous.

### Quotients of Geometric Brownian Motion Processes (concluded)

*• The multidimensional Ito’s lemma (Theorem 20 on*
p. 504) can be employed to show that

*dU*

= *(1/Z) dY* *− (Y/Z*^{2}*) dZ* *− (1/Z*^{2}*) dY dZ + (Y /Z*^{3}*) (dZ)*^{2}

= *(1/Z)(aY dt + bY dW**Y* ) *− (Y/Z*^{2}*)(f Z dt + gZ dW**Z*)

*−(1/Z*^{2}*)(bgY Zρ dt) + (Y /Z*^{3}*)(g*^{2}*Z*^{2} *dt)*

= *U (a dt + b dW**Y*) *− U(f dt + g dW** ^{Z}*)

*−U(bgρ dt) + U(g*^{2} *dt)*

= *U (a* *− f + g*^{2} *− bgρ) dt + Ub dW*^{Y}*− Ug dW*^{Z}*.*

### Forward Price

*• Suppose S follows*
*dS*

*S* *= µ dt + σ dW.*

*• Consider F (S, t) ≡ Se*^{y(T}* ^{−t)}*.

*• Observe that*

*∂F*

*∂S* = *e*^{y(T}^{−t)}*,*

*∂*^{2}*F*

*∂S*^{2} = *0,*

*∂F* _{−t)}

### Forward Prices (concluded)

*• Then*

*dF* = *e*^{y(T}^{−t)}*dS* *− ySe*^{y(T}^{−t)}*dt*

= *Se*^{y(T}^{−t)}*(µ dt + σ dW )* *− ySe*^{y(T}^{−t)}*dt*

= *F (µ* *− y) dt + F σ dW*
by Eq. (52) on p. 503.

*• Thus F follows*
*dF*

*F* *= (µ* *− y) dt + σ dW.*

*• This result has applications in forward and futures*
contracts.^{a}

### Ornstein-Uhlenbeck Process

*• The Ornstein-Uhlenbeck process:*

*dX =* *−κX dt + σ dW,*
*where κ, σ* *≥ 0.*

*• It is known that*

*E[ X(t) ]* = *e*^{−κ(t−t0)}*E[ x*_{0} *],*
*Var[ X(t) ]* = *σ*^{2}

*2κ*
(

1 *− e** ^{−2κ(t−t0)}*)

*+ e*^{−2κ(t−t0)}*Var[ x*_{0}*],*

*Cov[ X(s), X(t) ]* = *σ*^{2}

*2κ* *e** ^{−κ(t−s)}*
[

1 *− e** ^{−2κ(s−t0)}* ]

*+e*

*−κ(t+s−2t0)*

*Var[ x*

_{0}

*],*

### Ornstein-Uhlenbeck Process (continued)

*• X(t) is normally distributed if x*^{0} is a constant or
normally distributed.

*• X is said to be a normal process.*

*• E[ x*^{0} *] = x*_{0} *and Var[ x*_{0} *] = 0 if x*_{0} is a constant.

*• The Ornstein-Uhlenbeck process has the following mean*
reversion property.

**– When X > 0, X is pulled toward zero.**

**– When X < 0, it is pulled toward zero again.**

### Ornstein-Uhlenbeck Process (continued)

*• A generalized version:*

*dX = κ(µ* *− X) dt + σ dW,*
*where κ, σ* *≥ 0.*

*• Given X(t*^{0}*) = x*_{0}, a constant, it is known that

*E[ X(t) ]* = *µ + (x*_{0} *− µ) e*^{−κ(t−t}^{0}^{)}*,* (55)
*Var[ X(t) ]* = *σ*^{2}

*2κ*
[

1 *− e*^{−2κ(t−t}^{0}^{)} ]
*,*
*for t* *≤ t.*

### Ornstein-Uhlenbeck Process (concluded)

*• The mean and standard deviation are roughly µ and*
*σ/√*

*2κ , respectively.*

*• For large t, the probability of X < 0 is extremely*

*unlikely in any ﬁnite time interval when µ > 0 is large*
*relative to σ/√*

*2κ .*

*• The process is mean-reverting.*

**– X tends to move toward µ.**

**– Useful for modeling term structure, stock price**
volatility, and stock price return.

### Square-Root Process

*• Suppose X is an Ornstein-Uhlenbeck process.*

*• Ito’s lemma says V ≡ X*^{2} has the diﬀerential,
*dV* = *2X dX + (dX)*^{2}

= 2*√*

*V (−κ√*

*V dt + σ dW ) + σ*^{2} *dt*

= (

*−2κV + σ*^{2})

*dt + 2σ√*

*V dW,*
a square-root process.

### Square-Root Process (continued)

*• In general, the square-root process has the stochastic*
diﬀerential equation,

*dX = κ(µ* *− X) dt + σ√*

*X dW,*

*where κ, σ* *≥ 0 and X(0) is a nonnegative constant.*

*• Like the Ornstein-Uhlenbeck process, it possesses mean*
*reversion: X tends to move toward µ, but the volatility*
is proportional to *√*

*X instead of a constant.*

### Square-Root Process (continued)

*• When X hits zero and µ ≥ 0, the probability is one*
that it will not move below zero.

**– Zero is a reﬂecting boundary.**

*• Hence, the square-root process is a good candidate for*
modeling interest rates.^{a}

*• The Ornstein-Uhlenbeck process, in contrast, allows*
negative interest rates.

*• The two processes are related (see p. 519).*

### Square-Root Process (concluded)

*• The random variable 2cX(t) follows the noncentral*
chi-square distribution,^{a}

*χ*

(*4κµ*

*σ*^{2} *, 2cX(0) e** ^{−κt}*
)

*,*
*where c* *≡ (2κ/σ*^{2})(1 *− e** ^{−κt}*)

*.*

^{−1}*• Given X(0) = x*^{0}, a constant,
*E[ X(t) ]* = *x*_{0}*e*^{−κt}*+ µ*(

1 *− e** ^{−κt}*)

*,*

*Var[ X(t) ]*=

*x*

_{0}

*σ*

^{2}

*κ*

(*e*^{−κt}*− e** ^{−2κt}*)

*+ µ* *σ*^{2}
*2κ*

(1 *− e** ^{−κt}*)2

*,*
*for t* *≥ 0.*

### Modeling Stock Prices

*• The most popular stochastic model for stock prices has*
been the geometric Brownian motion,

*dS*

*S* *= µ dt + σ dW.*

*• The continuously compounded rate of return X ≡ ln S*
follows

*dX = (µ* *− σ*^{2}*/2) dt + σ dW*
by Ito’s lemma.^{a}

aCompare it with Eq. (53) on p. 507.

### Modeling Stock Prices (concluded)

*• The more general deterministic volatility model posits*
*dS*

*S* *= µ dt + σ(S, t) dW,*

*where σ(S, t) is called the local volatility function.*^{a}

*• The trees for the deterministic volatility model are*
called implied trees.^{b}

*• Their construction requires option prices at all strike*
prices and maturities.

*• How to construct an eﬃcient implied tree without*
invalid probabilities remains open.

aDerman and Kani(1994).

*Continuous-Time Derivatives Pricing*

I have hardly met a mathematician who was capable of reasoning.

— Plato (428 B.C.–347 B.C.) Fischer [Black] is the only real genius I’ve ever met in ﬁnance. Other people, like Robert Merton or Stephen Ross, are just very smart and quick, but they think like me.

Fischer came from someplace else entirely.

— John C. Cox, quoted in Mehrling (2005)

### Toward the Black-Scholes Diﬀerential Equation

*• The price of any derivative on a non-dividend-paying*
stock must satisfy a partial diﬀerential equation (PDE).

*• The key step is recognizing that the same random*
process drives both securities.

*• As their prices are perfectly correlated, we ﬁgure out the*
amount of stock such that the gain from it oﬀsets

exactly the loss from the derivative.

*• The removal of uncertainty forces the portfolio’s return*
to be the riskless rate.

### Assumptions

*• The stock price follows dS = µS dt + σS dW .*

*• There are no dividends.*

*• Trading is continuous, and short selling is allowed.*

*• There are no transactions costs or taxes.*

*• All securities are inﬁnitely divisible.*

*• The term structure of riskless rates is ﬂat at r.*

*• There is unlimited riskless borrowing and lending.*

*• t is the current time, T is the expiration time, and*
*τ* *≡ T − t.*

### Black-Scholes Diﬀerential Equation

*• Let C be the price of a derivative on S.*

*• From Ito’s lemma (p. 500),*
*dC =*

(

*µS* *∂C*

*∂S* + *∂C*

*∂t* + 1

2 *σ*^{2}*S*^{2} *∂*^{2}*C*

*∂S*^{2}
)

*dt + σS* *∂C*

*∂S* *dW.*

**– The same W drives both C and S.**

*• Short one derivative and long ∂C/∂S shares of stock*
(call it Π).

*• By construction,*

### Black-Scholes Diﬀerential Equation (continued)

*• The change in the value of the portfolio at time dt is*^{a}
*dΠ =* *−dC +* *∂C*

*∂S* *dS.*

*• Substitute the formulas for dC and dS into the partial*
diﬀerential equation to yield

*dΠ =*
(

*−∂C*

*∂t* *−* 1

2 *σ*^{2}*S*^{2} *∂*^{2}*C*

*∂S*^{2}
)

*dt.*

*• As this equation does not involve dW , the portfolio is*
*riskless during dt time: dΠ = rΠ dt.*

aMathematically speaking, it is not quite right (Bergman, 1982).

### Black-Scholes Diﬀerential Equation (concluded)

*• So* (

*∂C*

*∂t* + 1

2 *σ*^{2}*S*^{2} *∂*^{2}*C*

*∂S*^{2}
)

*dt = r*
(

*C* *− S* *∂C*

*∂S*
)

*dt.*

*• Equate the terms to ﬁnally obtain*

*∂C*

*∂t* *+ rS* *∂C*

*∂S* + 1

2 *σ*^{2}*S*^{2} *∂*^{2}*C*

*∂S*^{2} *= rC.*

*• When there is a dividend yield q,*

*∂C* *+ (r* *− q) S* *∂C*

+ 1

*σ*^{2}*S*^{2} *∂*^{2}*C*

*= rC.*

### Rephrase

*• The Black-Scholes diﬀerential equation can be expressed*
in terms of sensitivity numbers,

*Θ + rS∆ +* 1

2 *σ*^{2}*S*^{2}*Γ = rC.* (56)

*• Identity (56) leads to an alternative way of computing*
Θ numerically from ∆ and Γ.

*• When a portfolio is delta-neutral,*
Θ + 1

2 *σ*^{2}*S*^{2}*Γ = rC.*

**– A deﬁnite relation thus exists between Γ and Θ.**

[Black] got the equation [in 1969] but then was unable to solve it. Had he been a better physicist he would have recognized it as a form of the familiar heat exchange equation, and applied the known solution. Had he been a better mathematician, he could have solved the equation from ﬁrst principles.

Certainly Merton would have known exactly what to do with the equation had he ever seen it.

### PDEs for Asian Options

*• Add the new variable A(t) ≡* ∫ *t*

0 *S(u) du.*

*• Then the value V of the Asian option satisﬁes this*
two-dimensional PDE:^{a}

*∂V*

*∂t* *+ rS* *∂V*

*∂S* + 1

2 *σ*^{2}*S*^{2} *∂*^{2}*V*

*∂S*^{2} *+ S* *∂V*

*∂A* *= rV.*

*• The terminal conditions are*
*V (T, S, A)* = max

(*A*

*T* *− X, 0*
)

*for call,*
*V (T, S, A)* = max

(

*X* *−* *A*
*T* *, 0*

)

*for put.*

### PDEs for Asian Options (continued)

*• The two-dimensional PDE produces algorithms similar*
to that on pp. 352ﬀ.

*• But one-dimensional PDEs are available for Asian*
options.^{a}

*• For example, Veˇceˇr (2001) derives the following PDE for*
Asian calls:

*∂u*

*∂t* *+ r*
(

1 *−* *t*

*T* *− z*

) *∂u*

*∂z* +

(1 *−* _{T}^{t}*− z*)2

*σ*^{2}
2

*∂*^{2}*u*

*∂z*^{2} = 0
*with the terminal condition u(T, z) = max(z, 0).*

### PDEs for Asian Options (concluded)

*• For Asian puts:*

*∂u*

*∂t* *+ r*
( *t*

*T* *− 1 − z*

) *∂u*

*∂z* +
( _{t}

*T* *− 1 − z*)2

*σ*^{2}
2

*∂*^{2}*u*

*∂z*^{2} = 0
with the same terminal condition.

*• One-dimensional PDEs lead to highly eﬃcient numerical*
methods.

### Heston’s Stochastic-Volatility Model

^{a}

*• Heston assumes the stock price follows*
*dS*

*S* = *(µ* *− q) dt +* *√*

*V dW*_{1}*,* (57)
*dV* = *κ(θ* *− V ) dt + σ√*

*V dW*_{2}*.* (58)
**– V is the instantaneous variance, which follows a**

square-root process.

**– dW**_{1} *and dW*_{2} *have correlation ρ.*

**– The riskless rate r is constant.**

*• It may be the most popular continuous-time*
stochastic-volatility model.

### Heston’s Stochastic-Volatility Model (continued)

*• Heston assumes the market price of risk is b*^{2}*√*
*V .*

*• So µ = r + b*^{2}*V .*

*• Deﬁne*

*dW*_{1}* ^{∗}* =

*dW*

_{1}

*+ b*

_{2}

*√*

*V dt,*
*dW*_{2}* ^{∗}* =

*dW*

_{2}

*+ ρb*

_{2}

*√*

*V dt,*
*κ** ^{∗}* =

*κ + ρb*

_{2}

*σ,*

*θ** ^{∗}* =

*θκ*

*κ + ρb*_{2}*σ.*

*• dW*1^{∗}*and dW*_{2}^{∗}*have correlation ρ.*

### Heston’s Stochastic-Volatility Model (continued)

*• Under the risk-neutral probability measure Q, both W*1^{∗}

*and W*_{2}* ^{∗}* are Wiener processes.

*• Heston’s model becomes, under probability measure Q,*
*dS*

*S* = *(r* *− q) dt +* *√*

*V dW*_{1}^{∗}*,*
*dV* = *κ*^{∗}*(θ*^{∗}*− V ) dt + σ√*

*V dW*_{2}^{∗}*.*

### Heston’s Stochastic-Volatility Model (continued)

*• Deﬁne*

*ϕ(u, τ )* = exp*{ ıu(ln S + (r − q) τ)*
*+θ*^{∗}*κ*^{∗}*σ*^{−2}

[

*(κ*^{∗}*− ρσuı − d) τ − 2 ln* 1 *− ge** ^{−dτ}*
1

*− g*

]

+ *vσ*^{−2}*(κ*^{∗}*− ρσuı − d)*(

1 *− e** ^{−dτ}*)
1

*− ge*

^{−dτ}}
*,*

*d* = √

*(ρσuı* *− κ** ^{∗}*)

^{2}

*− σ*

^{2}(

*−ıu − u*

^{2}

*) ,*

*g*=

*(κ*

^{∗}*− ρσuı − d)/(κ*

^{∗}*− ρσuı + d).*

### Heston’s Stochastic-Volatility Model (concluded)

The formulas are^{a}

*C* = *S*

[ 1

2 + 1
*π*

∫ _{∞}

0

Re

(*X*^{−ıu}*ϕ(u* *− ı, τ)*
*ıuSe*^{rτ}

)
*du*

]

*−Xe** ^{−rτ}*
[ 1

2 + 1
*π*

∫ _{∞}

0

Re

(*X*^{−ıu}*ϕ(u, τ )*
*ıu*

)
*du*

]
*,*
*P* = *Xe*^{−rτ}

[ 1

2 *−* 1
*π*

∫ _{∞}

0

Re

(*X*^{−ıu}*ϕ(u, τ )*
*ıu*

)
*du*

]
*,*

*−S*
[ 1

2 *−* 1
*π*

∫ _{∞}

0

Re

(*X*^{−ıu}*ϕ(u* *− ı, τ)*
*ıuSe*^{rτ}

)
*du*

]
*,*

*where ı =* *√*

*−1 and Re(x) denotes the real part of the*
*complex number x.*

### Stochastic-Volatility Models and Further Extensions

^{a}

*• How to explain the October 1987 crash?*

*• Stochastic-volatility models require an implausibly*
*high-volatility level prior to and after the crash.*

*• Merton (1976) proposed jump models.*

*• Discontinuous jump models in the asset price can*
alleviate the problem somewhat.

aEraker (2004).

### Stochastic-Volatility Models and Further Extensions (continued)

*• But if the jump intensity is a constant, it cannot explain*
the tendency of large movements to cluster over time.

*• This assumption also has no impacts on option prices.*

*• Jump-diﬀusion models combine both.*

**– E.g., add a jump process to Eq. (57) on p. 537.**

### Stochastic-Volatility Models and Further Extensions (concluded)

*• But they still do not adequately describe the systematic*
variations in option prices.^{a}

*• Jumps in volatility are alternatives.*^{b}

**– E.g., add correlated jump processes to Eqs. (57) and**
Eq. (58) on p. 537.

*• Such models allow high level of volatility caused by a*
jump to volatility.^{c}

aBates (2000) and Pan (2002).

bDuﬃe, Pan, and Singleton (2000).

cEraker, Johnnes, and Polson (2000).

### Complexities of Stochastic-Volatility Models

*• A few stochastic-volatility models suﬀer from*
subexponential tree size.

*• Examples include the Hull-White model (1987) and the*
Hilliard-Schwartz model (1996).^{a}

*• Future research may extend this negative result to more*
stochastic-volatility models.

**– We suspect many GARCH option pricing models**
entertain similar problems.^{b}

aChiu (R98723059) (2012).

b

*Hedging*

When Professors Scholes and Merton and I invested in warrants, Professor Merton lost the most money.

And I lost the least.

— Fischer Black (1938–1995)

### Delta Hedge

*• The delta (hedge ratio) of a derivative f is deﬁned as*

∆ *≡ ∂f/∂S.*

*• Thus ∆f ≈ ∆ × ∆S for relatively small changes in the*
*stock price, ∆S.*

*• A delta-neutral portfolio is hedged as it is immunized*
against small changes in the stock price.

*• A trading strategy that dynamically maintains a*
delta-neutral portfolio is called delta hedge.

### Delta Hedge (concluded)

*• Delta changes with the stock price.*

*• A delta hedge needs to be rebalanced periodically in*
order to maintain delta neutrality.

*• In the limit where the portfolio is adjusted continuously,*
perfect hedge is achieved and the strategy becomes

self-ﬁnancing.

### Implementing Delta Hedge

*• We want to hedge N short derivatives.*

*• Assume the stock pays no dividends.*

*• The delta-neutral portfolio maintains N × ∆ shares of*
*stock plus B borrowed dollars such that*

*−N × f + N × ∆ × S − B = 0.*

*• At next rebalancing point when the delta is ∆** ^{′}*, buy

*N* *× (∆*^{′}*− ∆) shares to maintain N × ∆** ^{′}* shares with a

*total borrowing of B*

^{′}*= N*

*× ∆*

^{′}*× S*

^{′}*− N × f*

*.*

^{′}*• Delta hedge is the discrete-time analog of the*

### Example

*• A hedger is short 10,000 European calls.*

*• σ = 30% and r = 6%.*

*• This call’s expiration is four weeks away, its strike price*
*is $50, and each call has a current value of f = 1.76791.*

*• As an option covers 100 shares of stock, N = 1,000,000.*

*• The trader adjusts the portfolio weekly.*

*• The calls are replicated well if the cumulative cost of*
*trading stock is close to the call premium’s FV.*^{a}

### Example (continued)

*• As ∆ = 0.538560, N × ∆ = 538, 560 shares are*

purchased for a total cost of 538,560 *× 50 = 26,928,000*
dollars to make the portfolio delta-neutral.

*• The trader ﬁnances the purchase by borrowing*
*B = N* *× ∆ × S − N × f = 25,160,090*
dollars net.^{a}

*• The portfolio has zero net value now.*

aThis takes the hedging viewpoint — an alternative. See an exercise in the text.

### Example (continued)

*• At 3 weeks to expiration, the stock price rises to $51.*

*• The new call value is f*^{′}*= 2.10580.*

*• So the portfolio is worth*

*−N × f** ^{′}* + 538,560

*× 51 − Be*

^{0.06/52}*= 171, 622*before rebalancing.

### Example (continued)

*• A delta hedge does not replicate the calls perfectly; it is*
not self-ﬁnancing as $171,622 can be withdrawn.

*• The magnitude of the tracking error—the variation in*
the net portfolio value—can be mitigated if adjustments
are made more frequently.

*• In fact, the tracking error over one rebalancing act is*
positive about 68% of the time, but its expected value is
essentially zero.^{a}

*• It is furthermore proportional to vega.*

aBoyle and Emanuel (1980).

### Example (continued)

*• In practice tracking errors will cease to decrease beyond*
a certain rebalancing frequency.

*• With a higher delta ∆*^{′}*= 0.640355, the trader buys*
*N* *× (∆*^{′}*− ∆) = 101, 795 shares for $5,191,545.*

*• The number of shares is increased to N × ∆*^{′}*= 640, 355.*

### Example (continued)

*• The cumulative cost is*

26,928,000 *× e*^{0.06/52}*+ 5,191,545 = 32,150,634.*

*• The portfolio is again delta-neutral.*

Option Change in No. shares Cost of Cumulative

value Delta delta bought shares cost

*τ* *S* *f* ∆ *N**×(5)* (1)*×(6)* FV(8’)+(7)

(1) (2) (3) (5) (6) (7) (8)

4 50 1.7679 0.53856 — 538,560 26,928,000 26,928,000
3 51 2.1058 0.64036 0.10180 101,795 5,191,545 32,150,634
2 53 3.3509 0.85578 0.21542 215,425 11,417,525 43,605,277
1 52 2.2427 0.83983 *−0.01595* *−15,955* *−829,660* 42,825,960
0 54 4.0000 1.00000 0.16017 160,175 8,649,450 51,524,853

The total number of shares is 1,000,000 at expiration (trading takes place at expiration, too).

### Example (concluded)

*• At expiration, the trader has 1,000,000 shares.*

*• They are exercised against by the in-the-money calls for*

$50,000,000.

*• The trader is left with an obligation of*

51,524,853 *− 50,000,000 = 1,524,853,*
which represents the replication cost.

*• Compared with the FV of the call premium,*
1,767,910 *× e*^{0.06}^{×4/52}*= 1,776,088,*

*− 1,524,853 = 251,235.*

### Tracking Error Revisited

*• Deﬁne the dollar gamma as S*^{2}Γ.

*• The change in value of a delta-hedged long option*
*position after a duration of ∆t is proportional to the*
dollar gamma.

*• It is about*

*(1/2)S*^{2}*Γ[ (∆S/S)*^{2} *− σ*^{2}*∆t ].*

**– (∆S/S)**^{2} is called the daily realized variance.

### Tracking Error Revisited (continued)

*• Let the rebalancing times be t*^{1}*, t*_{2}*, . . . , t** _{n}*.

*• Let ∆S*^{i}*= S*_{i+1}*− S** ^{i}*.

*• The total tracking error at expiration is about*

*n*∑*−1*
*i=0*

*e*^{r(T}^{−t}^{i}^{)}*S*_{i}^{2}Γ* _{i}*
2

[ (*∆S*_{i}*S*_{i}

)2

*− σ*^{2}*∆t*
]

*,*

*• The tracking error is path dependent.*

### Tracking Error Revisited (concluded)

^{a}

*• The tracking error ϵ*^{n}*over n rebalancing acts (such as*
251,235 on p. 558) has about the same probability of
being positive as being negative.

*• Subject to certain regularity conditions, the*
root-mean-square tracking error √

*E[ ϵ*^{2}_{n}*] is O(1/√*

*n ).*^{b}

*• The root-mean-square tracking error increases with σ at*
ﬁrst and then decreases.

aBertsimas, Kogan, and Lo (2000).

bSee also Grannan and Swindle (1996).

### Delta-Gamma Hedge

*• Delta hedge is based on the ﬁrst-order approximation to*
*changes in the derivative price, ∆f , due to changes in*
*the stock price, ∆S.*

*• When ∆S is not small, the second-order term, gamma*
Γ *≡ ∂*^{2}*f /∂S*^{2}, helps (theoretically).^{a}

*• A delta-gamma hedge is a delta hedge that maintains*
zero portfolio gamma, or gamma neutrality.

*• To meet this extra condition, one more security needs to*
be brought in.

aSee the numerical example on pp. 231–232 of the text.

### Delta-Gamma Hedge (concluded)

*• Suppose we want to hedge short calls as before.*

*• A hedging call f*^{2} is brought in.

*• To set up a delta-gamma hedge, we solve*

*−N × f + n*1 *× S + n*2 *× f*2 *− B = 0 (self-ﬁnancing),*

*−N × ∆ + n*1 *+ n*_{2} *× ∆*2 *− 0 = 0 (delta neutrality),*

*−N × Γ + 0 + n*2 *× Γ*2 *− 0 = 0 (gamma neutrality),*

*for n*_{1}*, n*_{2}*, and B.*

**– The gammas of the stock and bond are 0.**

### Other Hedges

*• If volatility changes, delta-gamma hedge may not work*
well.

*• An enhancement is the delta-gamma-vega hedge, which*
also maintains vega zero portfolio vega.

*• To accomplish this, one more security has to be brought*
into the process.

*• In practice, delta-vega hedge, which may not maintain*
gamma neutrality, performs better than delta hedge.