Brownian Motion
a• 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 < · · · < tn, the random variables X(tk) − X(tk−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 = 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)
• The existence and uniqueness of such a process is guaranteed by Wiener’s theorem.a
• This process will be called a (μ, σ) Brownian motion with drift μ and variance σ2.
• Although Brownian motion is a continuous function of t with probability one, it is almost nowhere differentiable.
• The (0, 1) Brownian motion is called the Wiener process.
• If condition 3 is replaced by “ X(t) − X(s) depends only on t − s,” we have the more general Levy process.b
aNorbert Wiener (1894–1964). He received his Ph.D. from Harvard in 1912.
bPaul Levy (1886–1971).
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). (78)
• 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 Δx to the right with probability p after Δt time.
• It moves Δx to the left with probability 1 − p.
• Define
Xi =Δ
⎧⎨
⎩
+1 if the ith move is to the right,
−1 if the ith move is to the left.
– Xi are independent with
Prob[ Xi = 1 ] = p = 1 − Prob[ Xi = −1 ].
Brownian Motion as Limit of Random Walk (continued)
• Recall
E[ Xi ] = 2p − 1,
Var[ Xi ] = 1 − (2p − 1)2.
• Assume n = t/Δt is an integer.Δ
• Its position at time t is
Y (t) = Δx (XΔ 1 + X2 + · · · + Xn) .
Brownian Motion as Limit of Random Walk (continued)
• Therefore,
E[ Y (t) ] = n(Δx)(2p − 1), Var[ Y (t) ] = n(Δx)2
1 − (2p − 1)2 .
• With Δx = σΔ √
Δt and p = [ 1 + (μ/σ)Δ √
Δt ]/2,a E[ Y (t) ] = nσ√
Δt (μ/σ)√
Δt = μt, Var[ Y (t) ] = nσ2Δt
1 − (μ/σ)2Δt
→ σ2t, as Δt → 0.
aIdentical to Eq. (42) on p. 296!
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.
• Similarity to the the BOPM: The p is identical to the probability in Eq. (42) on p. 296 and Δx = ln u.
• Note that
Var[ Y (t + Δt) − Y (t) ]
=Var[ Δx Xn+1 ] = (Δx)2 × Var[ Xn+1 ] → σ2Δt.
Geometric Brownian Motion
• Let X =Δ { X(t), t ≥ 0 } be a Brownian motion process.
• The process
{ Y (t) = eΔ X(t), t ≥ 0 }, is called geometric Brownian motion.
• Suppose further that X is a (μ, σ) Brownian motion.
• By assumption, Y (0) = e0 = 1.
Geometric Brownian Motion (concluded)
• X(t) ∼ N(μt, σ2t) with moment generating function E
esX(t)
= E [ Y (t)s ] = eμts+(σ2ts2/2) from Eq. (27) on p 171.
• In particular,a
E[ Y (t) ] = eμt+(σ2t/2), Var[ Y (t) ] = E
Y (t)2
− E[ Y (t) ]2
= e2μt+σ2t
eσ2t − 1 .
aRecall Eqs. (29) on p. 180.
0.2 0.4 0.6 0.8 1 Time (t) -1
1 2 3 4 5 6 Y(t)
An Argument for Long-Term Investment
a• Suppose the stock follows the geometric Brownian motion
S(t) = S(0) eN(μt,σ2t) = S(0) etN(μ,σ2/t ), t ≥ 0, where μ > 0.
• The annual rate of return has a normal distribution:
N
μ, σ2 t
.
• The larger the t, the likelier the return is positive.
• The smaller the t, the likelier the return is negative.
aContributed by Dr. King, Gow-Hsing on April 9, 2015. See http://www.cb.idv.tw/phpbb3/viewtopic.php?f=7&t=1025
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 =Δ { W (t), t ≥ 0 } to denote the Wiener process.
• The goal is to develop integrals of X from a class of stochastic processes,a
It(X) =Δ t
0
X dW, t ≥ 0.
• It(X) is a random variable called the stochastic integral of X with respect to W .
• The stochastic process { It(X), t ≥ 0 } will be denoted by
X dW .
aKiyoshi Ito (1915–2008).
Stochastic Integrals (concluded)
• Typical requirements for X in financial applications are:
– Prob[ t
0 X2(s) ds < ∞ ] = 1 for all t ≥ 0 or the stronger 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.
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(tk−1) for t ∈ [ tk−1, tk), k = 1, 2, . . . for any realization (see figure on next page).a
aIt is right-continuous.
J J J J! J" J# : J
= B
J
Ito Integral (continued)
• The Ito integral of a simple process is defined as
It(X) =Δ
n−1
k=0
X(tk)[ W (tk+1) − W (tk) ], (79) where tn = t.
– The integrand X is evaluated at tk, not tk+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 It(X), unique almost certainly, such that It(Xn) converges in probability to It(X) for each sequence of simple
stochastic processes X1, X2, . . . such that Xn converges in probability to X.
• If X is continuous with probability one, then It(Xn) converges in probability to It(X) as
δn = maxΔ
1≤k≤n(tk − tk−1) goes to zero.
Ito Integral (concluded)
• It is a fundamental fact that
X dW is continuous almost surely.
• The following theorem says the Ito integral is a martingale.a
Theorem 18 The Ito integral
X dW is a martingale.
• A corollary is the mean value formula
E
b
a
X dW
= 0.
aExercise 14.1.1 covers simple stochastic processes.
Discrete Approximation and Nonanticipation
• Recall Eq. (79) on p. 592.
• The following simple stochastic process { X(t)} can be used in place of X to approximate t
0 X dW ,
X(s) = X(tΔ k−1) for s ∈ [ tk−1, tk), k = 1, 2, . . . , n.
• Note the nonanticipating feature of X.
– The information up to time s,
{ X(t), W (t), 0 ≤ t ≤ s },
cannot determine the future evolution of X or W .
Discrete Approximation and Nonanticipation (concluded)
• Suppose, unlike Eq. (79) on p. 592, we defined the stochastic integral from
n−1
k=0
X(tk+1)[ W (tk+1) − W (tk) ].
• Then we would be using the following different simple stochastic process in the approximation,
Y (s) = X(tΔ k) for s ∈ [ tk−1, tk), k = 1, 2, . . . , n.
• This clearly anticipates the future evolution of X.a
aSee Exercise 14.1.2 for an example where it matters.
:
J
:
J
;
= >
:
Ito Process
• The stochastic process X = { Xt, t ≥ 0 } that solves Xt = X0 +
t
0
a(Xs, s) ds + t
0
b(Xs, s) dWs, t ≥ 0 is called an Ito process.
– X0 is a scalar starting point.
– { a(Xt, t) : t ≥ 0 } and { b(Xt, t) : t ≥ 0 } are stochastic processes satisfying certain regularity conditions.
– a(Xt, t): the drift.
– b(Xt, t): the diffusion.
Ito Process (continued)
• Typical regularity conditions are:a
– For all T > 0, x ∈ Rn, and 0 ≤ t ≤ T ,
| a(x, t) | + | b(x, t) | ≤ C(1 + | x |) for some constant C.b
– (Lipschitz continuity) For all T > 0, x ∈ Rn, and 0 ≤ t ≤ T ,
| a(x, t) − a(y, t) | + | b(x, t) − b(y, t) | ≤ D | x − y | for some constant D.
aØksendal (2007).
bThis condition is not needed in time-homogeneous cases, where a and b do not depend on t.
Ito Process (continued)
• A shorthanda is the following stochastic differential equationb (SDE) for the Ito differential dXt,
dXt = a(Xt, t) dt + b(Xt, t) dWt. (80) – Or simply
dXt = at dt + bt dWt.
– This is Brownian motion with an instantaneous drift at and an instantaneous variance b2t.
• X is a martingale if at = 0.c
aPaul Langevin (1872–1946) in 1904.
bLike any equation, an SDE contains an unknown, the process Xt.
cRecall Theorem 18 (p. 594).
Ito Process (concluded)
• From calculus, we would expect t
0 W dW = W (t)2/2.
• But W (t)2/2 is not a martingale, hence wrong!
• The correct answer is [ W (t)2 − t ]/2.
• A popular representation of Eq. (80) is dXt = at dt + bt√
dt ξ, (81)
where ξ ∼ N(0, 1).
Euler Approximation
• Define tn Δ
= nΔt.
• The following approximation follows from Eq. (81),
X(t n+1)
= X(tn) +a( X(tn), tn) Δt + b( X(tn), tn) ΔW (tn). (82)
• It is called the Euler or Euler-Maruyama method.
• Recall that ΔW (tn) should be interpreted as W (tn+1) − W (tn),
not W (tn) − W (tn−1)!a
aRecall Eq. (79) on p. 592.
Euler Approximation (concluded)
• With the Euler method, one can obtain a sample path X(t 1), X(t2), X(t3), . . .
from a sample path
W (t0), W (t1), W (t2), . . . .
• Under mild conditions, X(tn) converges to X(tn).
More Discrete Approximations
• Under fairly loose regularity conditions, Eq. (82) on p. 602 can be replaced by
X(t n+1)
= X(tn) + a( X(tn), tn) Δt + b( X(tn), tn)√
Δt Y (tn).
– 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(t n+1)
= X(tn) + a( X(tn), tn) Δt + b( X(tn), tn)√
Δt ξ.
– Prob[ ξ = 1 ] = Prob[ ξ = −1 ] = 1/2.
– Note that E[ ξ ] = 0 and Var[ ξ ] = 1.
• This is a binomial model.
• As Δt goes to zero, X converges to X.a
aHe (1990).
Trading and the Ito Integral
• Consider an Ito process
dSt = μt dt + σt dWt.
– St is the vector of security prices at time t.
• Let φt be a trading strategy denoting the quantity of each type of security held at time t.
– Hence the stochastic process φtSt is the value of the portfolio φt at time t.
• φt dSt Δ
= φt(μt dt + σt dWt) represents the change in the value from security price changes occurring at time t.
Trading and the Ito Integral (concluded)
• The equivalent Ito integral, GT(φ) =Δ
T
0
φt dSt =
T
0
φtμt dt +
T
0
φtσt dWt, measures the gains realized by the trading strategy over the period [ 0, T ].
Ito’s Lemma
aA smooth function of an Ito process is itself an Ito process.
Theorem 19 Suppose f : R → R is twice continuously differentiable and dX = at dt + bt dW . Then f (X) is the Ito process,
f (Xt)
= f (X0) + t
0
f(Xs) as ds + t
0
f(Xs) bs dW +1
2 t
0
f(Xs) b2s ds for t ≥ 0.
aIto (1944).
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) b2 dt (83)
=
f(X) a + 12 f(X) b2
dt + f(X) b dW.
• Compared with calculus, the interesting part is the third term on the right-hand side of Eq. (83).
• A convenient formulation of Ito’s lemma is df (X) = f(X) dX + 1
2 f(X)(dX)2. (84)
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 = b2 dt in Eq. (84).
• This form is easy to remember because of its similarity to the Taylor expansion.
Ito’s Lemma (continued)
Theorem 20 (Higher-Dimensional Ito’s Lemma) Let W1, W2, . . . , Wn be independent Wiener processes and
X = (XΔ 1, X2, . . . , Xm) be a vector process. Suppose
f : Rm → R is twice continuously differentiable and Xi is an Ito process with dXi = ai dt + n
j=1 bij dWj. Then df (X) is an Ito process with the differential,
df (X) =
m i=1
fi(X) dXi + 1 2
m i=1
m k=1
fik(X) dXi dXk,
where fi = ∂f /∂XΔ i and fik = ∂Δ 2f /∂Xi∂Xk.
Ito’s Lemma (continued)
• The multiplication table for Theorem 20 is
× dWi dt
dWk δik dt 0
dt 0 0
in which
δik =
⎧⎨
⎩
1, if i = k, 0, otherwise.
Ito’s Lemma (continued)
• In applying the higher-dimensional Ito’s lemma, usually one of the variables, say X1, is time t and dX1 = dt.
• In this case, b1j = 0 for all j and a1 = 1.
• As an example, let
dXt = at dt + bt dWt.
• Consider the process f(Xt, t).
Ito’s Lemma (continued)
• Then df
= ∂f
∂Xt dXt + ∂f
∂t dt + 1 2
∂2f
∂Xt2 (dXt)2
= ∂f
∂Xt (at dt + bt dWt) + ∂f
∂t dt +1
2
∂2f
∂Xt2 (at dt + bt dWt)2
=
∂f
∂Xt at + ∂f
∂t + 1 2
∂2f
∂Xt2 b2t
dt + ∂f
∂Xt bt dWt. (85)
Ito’s Lemma (continued)
Theorem 21 (Alternative Ito’s Lemma) Let W1, W2, . . . , Wm be Wiener processes and
X = (XΔ 1, X2, . . . , Xm) be a vector process. Suppose
f : Rm → R is twice continuously differentiable and Xi is an Ito process with dXi = ai dt + bi dWi. Then df (X) is the following Ito process,
df (X) =
m i=1
fi(X) dXi + 1 2
m i=1
m k=1
fik(X) dXi dXk.
Ito’s Lemma (concluded)
• The multiplication table for Theorem 21 is
× dWi dt
dWk ρik dt 0
dt 0 0
• Above, ρik denotes the correlation between dWi and dWk.
Geometric Brownian Motion
• Consider geometric Brownian motion Y (t) = eΔ X(t). – X(t) is a (μ, σ) Brownian motion.
– By Eq. (78) on p. 577,
dX = μ dt + σ dW.
• Note that
∂Y
∂X = Y,
∂2Y
∂X2 = Y.
Geometric Brownian Motion (continued)
• Ito’s formula (83) on p. 609 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.
• Hence
dY
Y =
μ + σ2/2
dt + σ dW. (86)
• The annualized instantaneous rate of return is μ + σ2/2 (not μ).a
aConsistent with Lemma 10 (p. 301).
Geometric Brownian Motion (continued)
• Alternatively, from Eq. (78) on p. 577, Xt = X0 + μt + σ Wt, admits an explicit (strong) solution.
• Hence
Yt = Y0 eμt+σ Wt, (87) a strong solution to the SDE (86) where Y0 = eX0.
Geometric Brownian Motion (concluded)
• On the other hand, suppose dY
Y = μ dt + σ dW.
• Then X(t) = ln Y (t) followsΔ dX =
μ − σ2/2
dt + σ dW.
Exponential Martingale
• The Ito process
dXt = btXt dWt is a martingale.a
• It is called an exponential martingale.
• By Ito’s formula (83) on p. 609, X(t) = X(0) exp
−1 2
t
0
b2s ds + t
0
bs dWs
.
aRecall Theorem 18 (p. 594).
Product of Geometric Brownian Motion Processes
• Let
dY
Y = a dt + b dWY , dZ
Z = f dt + g dWZ.
• Assume dWY and dWZ have correlation ρ.
• Consider the Ito process
U = Y Z.Δ
Product of Geometric Brownian Motion Processes (continued)
• Apply Ito’s lemma (Theorem 21 on p. 615):
dU = Z dY + Y dZ + dY dZ
= ZY (a dt + b dWY ) + Y Z(f dt + g dWZ) +Y Z(a dt + b dWY )(f dt + g dWZ)
= U(a + f + bgρ) dt + Ub dWY + Ug dWZ.
• The product of correlated geometric Brownian motion processes thus remains geometric Brownian motion.
Product of Geometric Brownian Motion Processes (continued)
• Note that
Y = exp
a − b2/2
dt + b dWY , Z = exp
f − g2/2
dt + g dWZ , U = exp
a + f −
b2 + g2
/2
dt + b dWY + g dWZ .
• The strong solutions are:
Y (t) = exp
a − b2/2
t + b WY (t) , Z(t) = exp
f − g2/2
t + g WZ(t) , U (t) = exp
a + f −
b2 + g2 /2
t + b dWY + g WZ(t) .
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. 622.
• Let
U = Y /Z.Δ
• We now show thata dU
U = (a − f + g2 − bgρ) dt + b dWY − g dWZ.
(88)
• Keep in mind that dWY and dWZ 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 21 on p. 615) 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 dWY ) − (Y/Z2)(fZ dt + gZ dWZ)
−(1/Z2)(bgY Zρ dt) + (Y/Z3)(g2Z2 dt)
= U(a dt + b dWY ) − U(f dt + g dWZ)
−U(bgρ dt) + U(g2 dt)
= U(a − f + g2 − bgρ) dt + Ub dWY − Ug dWZ.
Forward Price
• Suppose S follows dS
S = μ dt + σ dW.
• Consider functional F (S, t) = SeΔ y(T −t) for constants y and T .
• As F is a function of two variables, we need the various partial derivatives of F (S, t) with respect to S and t.
• Note that in partial differentiation with respect to one variable, other variables are held constant.a
aContributed by Mr. Sun, Ao (R05922147) on April 26, 2017.
Forward Prices (continued)
• Now,
∂F
∂S = ey(T −t),
∂2F
∂S2 = 0,
∂F
∂t = −ySey(T −t).
• Thena
dF = ey(T −t) dS − ySey(T −t) dt
= Sey(T −t) (μ dt + σ dW ) − ySey(T −t) dt
= F (μ − y) dt + F σ dW.
aOne can also prove it by Eq. (85) on p. 614.
Forward Prices (concluded)
• Thus F follows dF
F = (μ − y) dt + σ dW.
• This result has applications in forward and futures contracts.
• In Eq. (60) on p. 490, μ = r = y.
• So dF
F = σ dW, a martingale.a
aIt is consistent with p. 566. Furthermore, it explains why Black’s formulas (68)–(69) on p. 518 use the same volatility σ as the stock’s.
Ornstein-Uhlenbeck (OU) Process
• The OU process:
dX = −κX dt + σ dW, where κ, σ ≥ 0.
• For t0 ≤ s ≤ t and X(t0) = x0, it is known that
E[ X(t) ] = e−κ(t−t0)E[ x0 ], Var[X(t) ] = σ2
2κ
1 − e−2κ(t−t0)
+ e−2κ(t−t0) Var[x0 ], Cov[X(s), X(t) ] = σ2
2κ e−κ(t−s)
1 − e−2κ(s−t0) +e−κ(t+s−2t0)Var[x0 ].
Ornstein-Uhlenbeck Process (continued)
• X(t) is normally distributed if x0 is a constant or normally distributed.
– E[ x0 ] = x0 and Var[ x0 ] = 0 if x0 is a constant.
• X is said to be a normal process.
• The OU process has the following mean-reverting property if κ > 0.
– 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(t0) = x0, a constant, it is known that
E[ X(t) ] = μ + (x0 − μ) e−κ(t−t0), (89) Var[ X(t) ] = σ2
2κ
1 − e−2κ(t−t0) , 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.a
aSee Knutson, Wimmer, Kuhnen, & Winkielman (2008) for the bio- logical basis for mean reversion in financial decision making.
Square-Root Process
• Suppose X is an OU process.
• Consider
V = XΔ 2.
• Ito’s lemma says V has the differential, dV = 2X dX + (dX)2
= 2√
V (−κ√
V dt + σ dW ) + σ2 dt
=
−2κV + σ2
dt + 2σ√
V dW, a square-root process.
Square-Root Process (continued)
• In general, the square-root process has the SDE, dX = κ(μ − X) dt + σ√
X dW,
where κ, σ > 0, μ ≥ 0, and X(0) ≥ 0 is a constant.
• Like the OU 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 OU process, in contrast, allows negative interest rates.b
• The two processes are related.c
aCox, Ingersoll, & Ross (1985).
bSome rates did go negative in Europe in 2015.
cRecall p. 635.
Square-Root Process (concluded)
• The random variable 2cX(t) follows the noncentral chi-square distribution,a
χ
4κμ
σ2 , 2cX(0) e−κt
, where c = (2κ/σΔ 2)(1 − e−κt)−1 and μ > 0.
• Given X(0) = x0, a constant, E[ X(t) ] = x0e−κt + μ
1 − e−κt , Var[ X(t) ] = x0 σ2
κ
e−κt − e−2κt
+ μ σ2 2κ
1 − e−κt2 , for t ≥ 0.
aWilliam Feller (1906–1970) in 1951.
Modeling Stock Prices
• The most popular stochastic model for stock prices has been the geometric Brownian motion,
dS
S = μ dt + σ dW.
• The logarithmic price X = ln S followsΔ dX =
μ − σ2 2
dt + σ dW by Eq. (86) on p. 618.
Local-Volatility Models
• The deterministic-volatility model for “smile” posits dS
S = (rt − qt) dt + σ(S, t) dW,
where instantaneous volatility σ(S, t) is called the local-volatility function.a
– “The most popular model after Black-Scholes is a local volatility model as it is the only completely consistent volatility model.”
• A (weak) solution exists if Sσ(S, t) is continuous and grows at most linearly in S and t.b
aDerman & Kani (1994); Dupire (1994).
bSkorokhod (1961); Achdou & Pironneau (2005).
Local-Volatility Models (continued)
• One needs to recover the local volatility surface σ(S, t) from the implied volatility surface.
• Theoretically,a
σ(X, T )2 = 2
∂C∂T + (rT − qT)X ∂X∂C + qTC X2 ∂∂X2C2 .
(90) – C is the call price at time t = 0 (today) with strike
price X and time to maturity T .
– σ(X, T ) is the local volatility that will prevail at future time T and stock price ST = X.
aDupire (1994); Andersen & Brotherton-Ratcliffe (1998).
Local-Volatility Models (continued)
• For more general models, this equation gives the
expectation as seen from today, under the risk-neural probability, of the instantaneous variance at time T given that ST = X.a
• In practice, the σ(S, t)2 derived by Dupire’s formula (90) may have spikes, vary wildly, or even be negative.
• The term ∂2C/∂X2 in the denominator often results in numerical instability.
aDerman & Kani (1997); R. W. Lee (2001); Derman & M. B. Miller (2016).
Local-Volatility Models (continued)
• Denote the implied volatility surface by Σ(X, T ) and the local volatility surface by σ(S, t).
• The relation between Σ(X, T ) and σ(X, T ) isa
σ(X, T )2 = Σ2 + 2Στ ∂Σ
∂T + (rT − qT)X ∂X∂Σ
1 − XyΣ ∂X∂Σ 2
+ XΣτ
∂X∂Σ − XΣτ4 ∂Σ
∂X
2
+ X ∂X∂2Σ2
,
τ = T − t,Δ
y = ln(X/SΔ t) +
T
t (qs − rs) ds.
aAndreasen (1996); Andersen & Brotherton-Ratcliffe (1998);
Gatheral (2003); Wilmott (2006); Kamp (2009).
Local-Volatility Models (continued)
• Although this version may be more stable than Eq. (90) on p. 641, it is expected to suffer from the same
problems.
• Small changes to the implied volatility surface may produce big changes to the local volatility surface.
Implied and Local Volatility Surfaces
a0 0.5
1 1.5
2 2.5
3
0 0.2 0.4 0.6 0.8 1 20 30 40 50 60 70 80 90 100 110
Strike ($)
Implied Vol Surface
Time to Maturity (yr)
Implied Vol (%)
0 0.5
1 1.5
2 2.5
3
0 0.2 0.4 0.6 0.8 1 20 30 40 50 60 70 80 90 100 110
Stock ($)
Local Vol Surface
Time (yr)
Local Vol (%)
aContributed by Mr. Lok, U Hou (D99922028) on April 5, 2014.
Local-Volatility Models (continued)
• In reality, option prices only exist for a finite set of maturities and strike prices.
• Hence interpolation and extrapolation may be needed to construct the volatility surface.a
• But then some implied volatility surfaces generate option prices that allow arbitrage opportunities.b
aDoing it to the option prices produces worse results (Li, 2000/2001).
bSee Rebonato (2004) for an example.
Local-Volatility Models (concluded)
• There exist conditions for a set of option prices to be arbitrage-free.a
• Some adopt parameterized implied volatility surfaces that guarantee freedom from certain arbitrages.b
• For some vanilla equity options, the Black-Scholes model seems better than the local-volatility model in predictive power.c
• The exact opposite is concluded for hedging in equity index markets!d
aKahal´e (2004); Davis & Hobson (2007).
bGatheral & Jacquier (2014).
cDumas, Fleming, & Whaley (1998).
dCr´epey (2004); Derman & M. B. Miller (2016).
Local-Volatility Models: Popularity
• Hirsa and Neftci (2014), “most traders and firms actively utilize this [local-volatility] model.”
• Bennett (2014), “Of all the four volatility regimes,
[sticky local volatility] is arguably the most realistic and fairly prices skew.”
• Derman & M. B. Miller (2016), “Right or wrong, local volatility models have become popular and ubiquitousin modeling the smile.”
Implied Trees
• The trees for the local-volatility model are called implied trees.a
• Their construction requires option prices at all strike prices and maturities.
– That is, an implied volatility surface.
• The local volatility model does not imply that the implied tree must combine.
• Exponential-sized implied trees exist.b
aDerman & Kani (1994); Dupire (1994); Rubinstein (1994).
bCharalambousa, Christofidesb, & Martzoukosa (2007); Gong & Xu (2019).
Implied Trees (continued)
• How to construct a valid implied tree with efficiency has been open for a long time.a
– Reasons may include: noise and nonsynchrony in data, arbitrage opportunities in the smoothed and interpolated/extrapolated implied volatility surface, wrong model, wrong algorithms, nonlinearity,
instability, etc.
• Inversion is an ill-posed numerical problem.b
aRubinstein (1994); Derman & Kani (1994); Derman, Kani, & Chriss (1996); Jackwerth & Rubinstein (1996); Jackwerth (1997); Coleman, Kim, Li, & Verma (2000); Li (2000/2001); Rebonato (2004); Moriggia, Muzzioli, & Torricelli (2009).
bAyache, Henrotte, Nassar, & X. Wang (2004).
Implied Trees (continued)
• It is finally solved for separable local volatilities.a
– The local-volatility function σ(S, t) is separableb if σ(S, t) = σ1(S) σ2(t).
• A solution is also available for any upper- and lower-bounded σ.c
aLok (D99922028) & Lyuu (2015, 2016, 2017).
bBrace, G¸atarek, & Musiela (1997); Rebonato (2004).
cLok (D99922028) & Lyuu (2016, 2017, 2020).
Implied Trees
a(concluded)
10 103 20
102
2 30
101 1 1.5
40
0.5 100 0
50
Root
aPlot supplied by Prof. Lok, U Hou (D99922028) on May 4, 2019.
Delta Hedge under the Local-Volatility Model
• Delta by the implied tree differs from delta by the Black-Scholes model’s implied volatility.
– The latter is by formula (46) or (47) (p. 343) after calculating the implied volatility from the same option price by the implied tree.
• Hence the profits and losses of their delta hedges will differ.
• The next plot shows the best 100 out of 100,000 random paths where the implied tree delta outperforms the
Black-Scholes delta.a
aIn terms of profits and losses. Plot supplied by Mr. Chiu, Tzu-Hsuan (R08723061) on November 20, 2021. We are hedging a long call.
Delta Hedge under the Local-Volatility Model (concluded)
• The next plot shows the best 100 out of 100,000 random paths where the Black-Scholes delta outperforms the
implied tree delta.a
aPlot supplied by Mr. Chiu, Tzu-Hsuan (R08723061) on November 20, 2021. We are again hedging a long call.