Brownian Motion

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).

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). (46)

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

Brownian Motion as Limit of Random Walk

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. (24) on p. 242 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. (17) on p 143.

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 .

aKiyoshi Ito (1915–2008).

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 on 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) ], (47) 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. (47) on p. 471.

• 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. (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 b²_t.

• X is a martingale if the drift a_t is zero by Theorem 15 (p. 473).

aPaul Langevin (1904).

Ito Process (concluded)

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

• An equivalent form to 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(tb _n+1)

= bX(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.

• 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 (50) on p. 480 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.

• 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_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.

(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) = 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. (46) on p. 455.

• As ∂Y /∂X = Y and ∂²Y /∂X² = Y , Ito’s formula (51) on p. 486 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. 490):

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 dWY )(f dt + g dWZ)

= 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. 494.

• Let U ≡ Y /Z.

• We now show that^a dU

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

(52)

• 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 18 on p. 490) 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 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)}, (53) 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 .

• 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. 488), 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^a 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.

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

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. (54)

• Identity (54) 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.

PDEs for Asian Options (continued)

• The two-dimensional PDE produces algorithms similar to that on pp. 340ff.

• 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.

Heston’s Stochastic-Volatility Model

• Heston assumes the stock price follows dS

S = (µ − q) dt + √

V dW₁, (55) dV = κ(θ − V ) dt + σ√

V dW₂. (56) – V is the instantaneous variance, which follows a

square-root process.

– dW₁ and dW₂ have correlation ρ.

– The riskless rate r is constant.

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

aHeston (1993).

Heston’s Stochastic-Volatility Model (continued)

• Heston assumes the market price of risk is b₂√ V .

• So µ = r + b₂V .

• Define

dW₁^∗ = dW₁ + b₂√

V dt, dW₂^∗ = dW₂ + ρb₂√

V dt, κ^∗ = κ + ρb₂σ,

θ^∗ = θκ

κ + ρb₂σ.

• dW₁^∗ and dW₂^∗ have correlation ρ.

• Under the risk-neutral probability measure Q, both W₁^∗

Heston’s Stochastic-Volatility Model (continued)

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

S = (r − q) dt + √

V dW₁^∗, dV = κ^∗(θ^∗ − V ) dt + σ√

V dW₂^∗.

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 = p

(ρσuı − κ^∗)² − σ²(−ıu − u²) , g = (κ^∗ − ρσuı − d)/(κ^∗ − ρσuı + d).

Heston’s Stochastic-Volatility Model (concluded)

The formulas are^a

C = S

· 1

2 + 1 π

Z _∞

0

Re

µX^−ıuφ(u − ı, τ ) ıuSe^rτ

¶ du

¸

−Xe^−rτ

· 1

2 + 1 π

Z _∞

0

Re

µX^−ıuφ(u, τ ) ıu

¶ du

¸ , P = Xe^−rτ

· 1

2 − 1 π

Z _∞

0

Re

µX^−ıuφ(u, τ ) ıu

¶ du

¸ ,

−S

· 1

2 − 1 π

Z _∞

0

Re

µX^−ıuφ(u − ı, τ ) ıuSe^rτ

¶ du

¸ ,

where ı = √

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

aContributed by Mr. Chen, Jung-Ying (D95723006) on August 17,

Stochastic-Volatility Models and Further Extensions

• 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. (55) on p. 514.

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. (55) and Eq. (56) on p. 514.

• 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).

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.

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 ∆⁰, buy

N × (∆⁰ − ∆) shares to maintain N × ∆⁰ shares with a total borrowing of B⁰ = N × ∆⁰ × S⁰ − N × f⁰.

• Delta hedge is the discrete-time analog of the

continuous-time limit and will rarely be self-financing.

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^a well if the cumulative cost of trading stock is close to the call premium’s FV.

aThis example takes the replication viewpoint.

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 f⁰ = 2.10580.

• So the portfolio is worth

−N × f⁰ + 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 = 0.640355, the trader buys N × (∆⁰ − ∆) = 101, 795 shares for $5,191,545.

• The number of shares is increased to N × ∆⁰ = 640, 355.

Example (continued)

• The cumulative cost is

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

• The total borrowed amount is

B⁰ = 640,355 × 51 − N × f⁰ = 30,552,305.

• The portfolio is again delta-neutral with zero value.

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, the net gain is 1,776,088 − 1,524,853 = 251,235.