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.
• The terms a(Xt, t) and b(Xt, t) are the drift and the
Ito Process (continued)
• A shorthanda is the following stochastic differential equation for the Ito differential dXt,
dXt = a(Xt, t) dt + b(Xt, t) dWt. (48) – 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 (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 dXt = at dt + bt√
dt ξ, (49)
where ξ ∼ N(0, 1).
Euler Approximation
• The following approximation follows from Eq. (49), X(tb n+1)
= bX(tn) + a( bX(tn), tn) ∆t + b( bX(tn), tn) ∆W (tn),
(50) where tn ≡ n∆t.
• It is called the Euler or Euler-Maruyama method.
• Recall that ∆W (tn) should be interpreted as W (tn+1) − W (tn), not W (tn) − W (tn−1).
• Under mild conditions, bX(tn) converges to X(tn).
More Discrete Approximations
• Under fairly loose regularity conditions, Eq. (50) on p. 492 can be replaced by
X(tb n+1)
= bX(tn) + a( bX(tn), tn) ∆t + b( bX(tn), tn)√
∆t 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(tb n+1)
= bX(tn) + a( bX(tn), tn) ∆t + b( bX(tn), tn)√
∆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 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 ≡ 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) ≡
∫ 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
A smooth function of an Ito process is itself an Ito process.
Theorem 18 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
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.
(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 = b2 dt.
• This form is easy to remember because of its similarity
Ito’s Lemma (continued)
Theorem 19 (Higher-Dimensional Ito’s Lemma) Let W1, W2, . . . , Wn be independent Wiener processes and
X ≡ (X1, 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/∂Xi and fik ≡ ∂2f /∂Xi∂Xk.
Ito’s Lemma (continued)
• The multiplication table for Theorem 19 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.
• Assume 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
b dW . (52)
Ito’s Lemma (continued)
Theorem 20 (Alternative Ito’s Lemma) Let W1, W2, . . . , Wm be Wiener processes and
X ≡ (X1, 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 20 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) ≡ eX(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 ∂2Y /∂X2 = 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 σ2 dt.
Geometric Brownian Motion (concluded)
• Hence
dY
Y = (
µ + σ2/2)
dt + σ dW. (53)
• The annualized instantaneous rate of return is µ + σ2/2 not µ.
Product of Geometric Brownian Motion Processes
• Let
dY /Y = a dt + b dWY , dZ/Z = f dt + g dWZ.
• 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 dWY ) + Y Z(f dt + g dWZ) +Y Z(a dt + b dWY )(f dt + g dWZ)
= U (a + f + bgρ) dt + U b dWY + U g dWZ.
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 dWY ] , Z = exp[(
f − g2/2)
dt + g dWZ] , U = exp[ (
a + f − (
b2 + g2)
/2)
dt + b dWY + g dWZ ] .
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 thata dU
U = (a − f + g2 − bgρ) dt + b dWY − g dWZ.
(54)
• 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 20 on p. 504) 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)(f Z 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 F (S, t) ≡ Sey(T−t).
• Observe that
∂F
∂S = ey(T−t),
∂2F
∂S2 = 0,
∂F −t)
Forward Prices (concluded)
• Then
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 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[ 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.
• 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)
• 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), (55) Var[ X(t) ] = σ2
2κ [
1 − e−2κ(t−t0) ] , 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 ≡ X2 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 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
σ2 , 2cX(0) e−κt )
, where c ≡ (2κ/σ2)(1 − e−κt)−1.
• 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−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 − σ2/2) 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 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,
Black-Scholes Differential Equation (continued)
• The change in the value of the portfolio at time dt isa dΠ = −dC + ∂C
∂S dS.
• Substitute the formulas for dC and dS into the partial differential equation to yield
dΠ = (
−∂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 (
∂C
∂t + 1
2 σ2S2 ∂2C
∂S2 )
dt = r (
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 + (r − q) S ∂C
+ 1
σ2S2 ∂2C
= rC.
Rephrase
• The Black-Scholes differential equation can be expressed in terms of sensitivity numbers,
Θ + rS∆ + 1
2 σ2S2Γ = 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 σ2S2Γ = 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 σ2S2 ∂2V
∂S2 + 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 − Tt − z)2
σ2 2
∂2u
∂z2 = 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
σ2 2
∂2u
∂z2 = 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 dW1, (57) dV = κ(θ − V ) dt + σ√
V dW2. (58) – 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.
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.
Heston’s Stochastic-Volatility Model (continued)
• Under the risk-neutral probability measure Q, both W1∗
and W2∗ are Wiener processes.
• 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 = √
(ρσuı − κ∗)2 − σ2(−ıu − u2) , g = (κ∗ − ρσuı − d)/(κ∗ − ρσuı + d).
Heston’s Stochastic-Volatility Model (concluded)
The formulas area
C = S
[ 1
2 + 1 π
∫ ∞
0
Re
(X−ıuϕ(u − ı, τ) ıuSerτ
) du
]
−Xe−rτ [ 1
2 + 1 π
∫ ∞
0
Re
(X−ıuϕ(u, τ ) ıu
) du
] , P = Xe−rτ
[ 1
2 − 1 π
∫ ∞
0
Re
(X−ıuϕ(u, τ ) ıu
) du
] ,
−S [ 1
2 − 1 π
∫ ∞
0
Re
(X−ıuϕ(u − ı, τ) ıuSerτ
) 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 − Be0.06/52 = 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 × e0.06/52 + 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 × e0.06×4/52 = 1,776,088,
− 1,524,853 = 251,235.
Tracking Error Revisited
• Define the dollar gamma as S2Γ.
• 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)S2Γ[ (∆S/S)2 − σ2∆t ].
– (∆S/S)2 is called the daily realized variance.
Tracking Error Revisited (continued)
• Let the rebalancing times be t1, t2, . . . , tn.
• Let ∆Si = Si+1 − Si.
• The total tracking error at expiration is about
n∑−1 i=0
er(T−ti)Si2Γi 2
[ (∆Si Si
)2
− σ2∆t ]
,
• The tracking error is path dependent.
Tracking Error Revisited (concluded)
a• The tracking error ϵn 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[ ϵ2n ] 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 ≡ ∂2f /∂S2, 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 f2 is brought in.
• To set up a delta-gamma hedge, we solve
−N × f + n1 × S + n2 × f2 − B = 0 (self-financing),
−N × ∆ + n1 + n2 × ∆2 − 0 = 0 (delta neutrality),
−N × Γ + 0 + n2 × Γ2 − 0 = 0 (gamma neutrality),
for n1, n2, 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.
