Brownian Motion

Example

• Take the binomial model with two assets.

• In a period, asset one’s price can go from S to S₁ or S₂.

• In a period, asset two’s price can go from P to P₁ or P₂.

• Assume

S₁

P₁ < S

P < S₂ P₂ to rule out arbitrage opportunities.

Example (continued)

• For any derivative security, let C₁ be its price at time one if asset one’s price moves to S₁.

• Let C₂ be its price at time one if asset one’s price moves to S₂.

• Replicate the derivative by solving

αS₁ + βP₁ = C₁, αS₂ + βP₂ = C₂,

using α units of asset one and β units of asset two.

Example (continued)

• This yields

α = P₂C₁ − P₁C₂

P₂S₁ − P₁S₂ and β = S₂C₁ − S₁C₂ S₂P₁ − S₁P₂ .

• The derivative costs C = αS + βP

= P₂S − P S₂

P₂S₁ − P₁S₂ C₁ + P S₁ − P₁S

P₂S₁ − P₁S₂ C₂.

Example (concluded)

• It is easy to verify that C

P = p C₁

P₁ + (1 − p) C₂ P₂ . – Above,

p ≡ (S/P ) − (S₂/P₂) (S₁/P₁) − (S₂/P₂).

• The derivative’s price using asset two as numeraire is thus a martingale under the risk-neutral probability p.

• The expected returns of the two assets are irrelevant.

Brownian Motion

• Brownian motion is a stochastic process { X(t), t ≥ 0 } with the following properties.

1. X(0) = 0, unless stated otherwise.

2. for any 0 ≤ t₀ < t₁ < · · · < t_n, the random variables X(t_k) − X(t_k−1)

for 1 ≤ k ≤ n are independent.^b

3. for 0 ≤ s < t, X(t) − X(s) is normally distributed with mean µ(t − s) and variance σ²(t − s), where µ and σ 6= 0 are real numbers.

aRobert Brown (1773–1858).

bSo X(t) − X(s) is independent of X(r) for r ≤ s < t.

Brownian Motion (concluded)

• Such a process will be called a (µ, σ) Brownian motion with drift µ and variance σ².

• The existence and uniqueness of such a process is guaranteed by Wiener’s theorem.^a

• Although Brownian motion is a continuous function of t with probability one, it is almost nowhere differentiable.

• The (0, 1) Brownian motion is also called the Wiener process.

aNorbert Wiener (1894–1964).

Example

• If { X(t), t ≥ 0 } is the Wiener process, then X(t) − X(s) ∼ N (0, t − s).

• A (µ, σ) Brownian motion Y = { Y (t), t ≥ 0 } can be expressed in terms of the Wiener process:

Y (t) = µt + σX(t). (45)

• Note that Y (t + s) − Y (t) ∼ N (µs, σ²s).

Brownian Motion Is a Random Walk in Continuous Time

Claim 1 A (µ, σ) Brownian motion is the limiting case of random walk.

• A particle moves ∆x to the left with probability 1 − p.

• It moves to the right with probability p after ∆t time.

• Assume n ≡ t/∆t is an integer.

• Its position at time t is

Y (t) ≡ ∆x (X₁ + X₂ + · · · + X_n) .

Brownian Motion as Limit of Random Walk (continued)

• (continued) – Here

X_i ≡





+1 if the ith move is to the right,

−1 if the ith move is to the left.

– X_i are independent with

Prob[ X_i = 1 ] = p = 1 − Prob[ X_i = −1 ].

• Recall E[ X_i ] = 2p − 1 and Var[ X_i ] = 1 − (2p − 1)².

Brownian Motion as Limit of Random Walk (continued)

• Therefore,

E[ Y (t) ] = n(∆x)(2p − 1), Var[ Y (t) ] = n(∆x)² £

1 − (2p − 1)² ¤ .

• With ∆x ≡ σ√

∆t and p ≡ [ 1 + (µ/σ)√

∆t ]/2, E[ Y (t) ] = nσ√

∆t (µ/σ)√

∆t = µt, Var[ Y (t) ] = nσ²∆t £

1 − (µ/σ)²∆t¤

→ σ²t, as ∆t → 0.

Brownian Motion as Limit of Random Walk (concluded)

• Thus, { Y (t), t ≥ 0 } converges to a (µ, σ) Brownian motion by the central limit theorem.

• Brownian motion with zero drift is the limiting case of symmetric random walk by choosing µ = 0.

• Note that

Var[ Y (t + ∆t) − Y (t) ]

=Var[ ∆x X_n+1 ] = (∆x)² × Var[ X_n+1 ] → σ²∆t.

• Similarity to the the BOPM: The p is identical to the probability in Eq. (23) on p. 239 and ∆x = ln u.

Geometric Brownian Motion

• Let X ≡ { X(t), t ≥ 0 } be a Brownian motion process.

• The process

{ Y (t) ≡ e^X(t), t ≥ 0 }, is called geometric Brownian motion.

• Suppose further that X is a (µ, σ) Brownian motion.

• X(t) ∼ N (µt, σ²t) with moment generating function E

h

e^sX(t) i

= E [ Y (t)^s ] = e^{µts+(σ2ts2/2)} from Eq. (16) on p 141.

Geometric Brownian Motion (continued)

• In particular,

E[ Y (t) ] = e^µt+(σ2t/2), Var[ Y (t) ] = E £

Y (t)² ¤

− E[ Y (t) ]²

= e^2µt+σ2t

³

e^σ2t − 1

´ .

0.2 0.4 0.6 0.8 1 Time (t) -1

1 2 3 4 5 6 Y(t)

Geometric Brownian Motion (continued)

• It is useful for situations in which percentage changes are independent and identically distributed.

• Let Y_n denote the stock price at time n and Y₀ = 1.

• Assume relative returns

X_i ≡ Y_i Y_i−1

are independent and identically distributed.

Geometric Brownian Motion (concluded)

• Then

ln Y_n =

Xn i=1

ln X_i

is a sum of independent, identically distributed random variables.

• Thus { ln Y_n, n ≥ 0 } is approximately Brownian motion.

– And { Y_n, n ≥ 0 } is approximately geometric Brownian motion.

Continuous-Time Financial Mathematics

A proof is that which convinces a reasonable man;

a rigorous proof is that which convinces an unreasonable man.

— Mark Kac (1914–1984) The pursuit of mathematics is a divine madness of the human spirit.

— Alfred North Whitehead (1861–1947), Science and the Modern World

Stochastic Integrals

• Use W ≡ { W (t), t ≥ 0 } to denote the Wiener process.

• The goal is to develop integrals of X from a class of stochastic processes,^a

I_t(X) ≡ Z _t

0

X dW, t ≥ 0.

• I_t(X) is a random variable called the stochastic integral of X with respect to W .

• The stochastic process { I_t(X), t ≥ 0 } will be denoted by R

X dW .

aIto (1915–).

Stochastic Integrals (concluded)

• Typical requirements for X in financial applications are:

– Prob[R_t

0 X²(s) ds < ∞ ] = 1 for all t ≥ 0 or the stronger R _t

0 E[ X²(s) ] ds < ∞.

– The information set at time t includes the history of X and W up to that point in time.

– But it contains nothing about the evolution of X or W after t (nonanticipating, so to speak).

– The future cannot influence the present.

• { X(s), 0 ≤ s ≤ t } is independent of { W (t + u) − W (t), u > 0 }.

Ito Integral

• A theory of stochastic integration.

• As with calculus, it starts with step functions.

• A stochastic process { X(t) } is simple if there exist 0 = t₀ < t₁ < · · · such that

X(t) = X(t_k−1) for t ∈ [ t_k−1, t_k), k = 1, 2, . . . for any realization (see figure next page).

t0 t

1 t

2 t

3 t

4 t

5

X t

a f

t

Ito Integral (continued)

• The Ito integral of a simple process is defined as I_t(X) ≡

n−1X

k=0

X(t_k)[ W (t_k+1) − W (t_k) ], (46) where t_n = t.

– The integrand X is evaluated at t_k, not t_k+1.

• Define the Ito integral of more general processes as a limiting random variable of the Ito integral of simple stochastic processes.

Ito Integral (continued)

• Let X = { X(t), t ≥ 0 } be a general stochastic process.

• Then there exists a random variable I_t(X), unique almost certainly, such that I_t(X_n) converges in probability to I_t(X) for each sequence of simple

stochastic processes X₁, X₂, . . . such that X_n converges in probability to X.

• If X is continuous with probability one, then I_t(X_n) converges in probability to I_t(X) as

δ_n ≡ max_1≤k≤n(t_k − t_k−1) goes to zero.

Ito Integral (concluded)

• It is a fundamental fact that R

X dW is continuous almost surely.

• The following theorem says the Ito integral is a martingale.

– A corollary is the mean value formula E

" Z _b

a

X dW

#

= 0.

Theorem 15 The Ito integral R

X dW is a martingale.

Discrete Approximation

• Recall Eq. (46) on p. 455.

• The following simple stochastic process { bX(t) } can be used in place of X to approximate the stochastic

integral R_t

0 X dW ,

X(s) ≡ X(tb _k−1) for s ∈ [ t_k−1, t_k), k = 1, 2, . . . , n.

• Note the nonanticipating feature of bX.

– The information up to time s,

{ bX(t), W (t), 0 ≤ t ≤ s },

cannot determine the future evolution of X or W .

Discrete Approximation (concluded)

• Suppose we defined the stochastic integral as

n−1X

k=0

X(t_k+1)[ W (t_k+1) − W (t_k) ].

• Then we would be using the following different simple stochastic process in the approximation,

Y (s) ≡ X(tb _k) for s ∈ [ t_k−1, t_k), k = 1, 2, . . . , n.

• This clearly anticipates the future evolution of X.

X

t

X

t

$ Y

(a) (b)

$X

Ito Process

• The stochastic process X = { X_t, t ≥ 0 } that solves X_t = X₀ +

Z _t

0

a(X_s, s) ds + Z _t

0

b(X_s, s) dW_s, t ≥ 0 is called an Ito process.

– X₀ 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 diffusion, respectively.

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. (47) – 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 b²_t.

• X is a martingale if the drift a_t is zero by Theorem 15 (p. 457).

aPaul Langevin (1904).

Ito Process (concluded)

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

• An equivalent form to Eq. (47) is dX_t = a_t dt + b_t√

dt ξ, (48)

where ξ ∼ N (0, 1).

• This formulation makes it easy to derive Monte Carlo simulation algorithms.

Euler Approximation

• The following approximation follows from Eq. (48), X(tb _n+1)

= bX(t_n) + a( bX(t_n), t_n) ∆t + b( bX(t_n), t_n) ∆W (t_n),

(49) where t_n ≡ n∆t.

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

• Under mild conditions, bX(t_n) converges to X(t_n).

• Recall that ∆W (t_n) should be interpreted as W (t_n+1) − W (t_n) instead of W (t_n) − W (t_n−1).

More Discrete Approximations

• Under fairly loose regularity conditions, approximation (49) on p. 464 can be replaced by

X(tb _n+1)

= bX(t_n) + a( bX(t_n), t_n) ∆t + b( bX(t_n), t_n)√

∆t Y (t_n).

– Y (t₀), Y (t₁), . . . are independent and identically distributed with zero mean and unit variance.

More Discrete Approximations (concluded)

• An even simpler discrete approximation scheme:

X(tb _n+1)

= bX(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 clearly defines a binomial model. 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_tS_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) ≡

Z _T

0

φ_t dS_t =

Z _T

0

φ_tµ_t dt +

Z _T

0

φ_tσ_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 16 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 (X₀) + Z _t

0

f⁰(X_s) a_s ds + Z _t

0

f⁰(X_s) b_s dW +1

2 Z _t

0

f⁰⁰(X_s) b²_s ds for t ≥ 0.

Ito’s Lemma (continued)

• In differential form, Ito’s lemma becomes df (X) = f⁰(X) a dt + f⁰(X) b dW + 1

2 f⁰⁰(X) b² dt.

(50)

• 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) = f⁰(X) dX + 1

2 f⁰⁰(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.

• This form is easy to remember because of its similarity to the Taylor expansion.

Ito’s Lemma (continued)

Theorem 17 (Higher-Dimensional Ito’s Lemma) Let W₁, W₂, . . . , W_n be independent Wiener processes and

X ≡ (X₁, X₂, . . . , 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 + P_n

j=1 b_ij dW_j. Then df (X) is an Ito process with the differential,

df (X) =

Xm i=1

f_i(X) dX_i + 1 2

Xm i=1

Xm k=1

f_ik(X) dX_i dX_k, where f_i ≡ ∂f /∂x_i and f_ik ≡ ∂²f /∂x_i∂x_k.

Ito’s Lemma (continued)

• The multiplication table for Theorem 17 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)

Theorem 18 (Alternative Ito’s Lemma) Let W₁, W₂, . . . , W_m be Wiener processes and

X ≡ (X₁, X₂, . . . , 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) =

Xm i=1

f_i(X) dX_i + 1 2

Xm i=1

Xm k=1

f_ik(X) dX_i dX_k.

Ito’s Lemma (concluded)

• The multiplication table for Theorem 18 is

× dW_i dt

dW_k ρ_ik dt 0

dt 0 0

• Here, ρ_ik denotes the correlation between dW_i and dW_k.

Geometric Brownian Motion

• Consider the geometric Brownian motion process Y (t) ≡ e^X(t)

– X(t) is a (µ, σ) Brownian motion.

– Hence dX = µ dt + σ dW by Eq. (45) on p. 439.

• As ∂Y /∂X = Y and ∂²Y /∂X² = Y , Ito’s formula (50) on p. 470 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 σ² dt.

Geometric Brownian Motion (concluded)

• Hence

dY

Y = ¡

µ + σ²/2¢

dt + σ dW.

• The annualized instantaneous rate of return is µ + σ²/2 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 18 on p. 474):

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 − b²/2¢

dt + b dW_Y ¤ , Z = exp£¡

f − g²/2¢

dt + g dW_Z¤ , U = exp£ ¡

a + f − ¡

b² + g²¢

/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. 478.

• Let U ≡ Y /Z.

• We now show that dU

U = (a − f + g² − bgρ) dt + b dW_Y − g dW_Z.

(51)

• Keep in mind that dW_Y and dW_Z have correlation ρ.

Quotients of Geometric Brownian Motion Processes (concluded)

• The multidimensional Ito’s lemma (Theorem 18 on p. 474) can be employed to show that

dU

= (1/Z) dY − (Y /Z²) dZ − (1/Z²) dY dZ + (Y /Z³) (dZ)²

= (1/Z)(aY dt + bY dW_Y ) − (Y /Z²)(f Z dt + gZ dW_Z)

−(1/Z²)(bgY Zρ dt) + (Y /Z³)(g²Z² dt)

= U (a dt + b dW_Y) − U (f dt + g dW_Z)

−U (bgρ dt) + U (g² dt)

= U (a − f + g² − bgρ) dt + U b dW_Y − U g dW_Z.

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[ x₀ ], Var[ X(t) ] = σ²

2κ

³

1 − e^{−2κ(t−t0)}

´

+ e^{−2κ(t−t0)} Var[ x0], Cov[ X(s), X(t) ] = σ²

2κ e^−κ(t−s) h

1 − e^{−2κ(s−t0)} i

+e−κ(t+s−2t0)Var[ x₀],

for t₀ ≤ s ≤ t and X(t₀) = x₀.

Ornstein-Uhlenbeck Process (continued)

• X(t) is normally distributed if x₀ is a constant or normally distributed.

• X is said to be a normal process.

• E[ x₀ ] = x₀ and Var[ x₀ ] = 0 if x₀ is a constant.

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

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

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

Ornstein-Uhlenbeck Process (continued)

• Another version:

dX = κ(µ − X) dt + σ dW, where σ ≥ 0.

• Given X(t₀) = x₀, a constant, it is known that

E[ X(t) ] = µ + (x₀ − µ) e^{−κ(t−t0)}, (52) Var[ X(t) ] = σ²

2κ h

1 − e^{−2κ(t−t0)} i

, for t₀ ≤ 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 (say µ > 4σ/√

2κ).

• The process is mean-reverting.

– X tends to move toward µ.

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

Continuous-Time Derivatives Pricing

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

— Plato (428 B.C.–347 B.C.)

Toward the Black-Scholes Differential Equation

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

• 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. 472), dC =

µ

µS ∂C

∂S + ∂C

∂t + 1

2 σ²S² ∂²C

∂S²

¶

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,

Π = −C + S(∂C/∂S).

Black-Scholes Differential Equation (continued)

• The change in the value of the portfolio at time dt is dΠ = −dC + ∂C

∂S dS.

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

dΠ = µ

−∂C

∂t − 1

2 σ²S² ∂²C

∂S²

¶ dt.

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

Black-Scholes Differential Equation (concluded)

• So µ

∂C

∂t + 1

2 σ²S² ∂²C

∂S²

¶

dt = r µ

C − S ∂C

∂S

¶ dt.

• Equate the terms to finally obtain

∂C

∂t + rS ∂C

∂S + 1

2 σ²S² ∂²C

∂S² = rC.

• When there is a dividend yield q,

∂C

∂t + (r − q) S ∂C

∂S + 1

2 σ²S² ∂²C

∂S² = rC.

Rephrase

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

Θ + rS∆ + 1

2 σ²S²Γ = rC. (53)

• Identity (53) leads to an alternative way of computing Θ numerically from ∆ and Γ.

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

2 σ²S²Γ = rC.

– A definite relation thus exists between Γ and Θ.

PDEs for Asian Options

• Add the new variable A(t) ≡ R _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 σ²S² ∂²V

∂S² + S ∂V

∂A = rV.

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

µA

T − X, 0

¶

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

µ

X − A T , 0

¶

for put.

aKemna and Vorst (1990).

PDEs for Asian Options (continued)

• The two-dimensional PDE produces algorithms similar to that on pp. 329ff.

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

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

∂u

∂t + r µ

1 − t

T − z

¶ ∂u

∂z +

¡1 − _T^t − z¢₂ σ² 2

∂²u

∂z² = 0 with the terminal condition u(T, z) = max(z, 0).

aRogers and Shi (1995); Veˇceˇr (2001); Dubois and Leli`evre (2005).

PDEs for Asian Options (concluded)

• For Asian puts:

∂u

∂t + r µ t

T − 1 − z

¶ ∂u

∂z +

¡ _t

T − 1 − z¢₂ σ² 2

∂²u

∂z² = 0 with the same terminal condition.

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

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 in the sense that it is immunized against small changes in the stock price.

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