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. 510):
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. 514.
• Let U ≡ Y/Z.
• We now show thata dU
U = (a − f + g2 − bgρ) dt + b dWY − g dWZ.
(57)
• 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. 510) 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 = −ySey(T−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. (55) on p. 509.
• 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],
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)
• 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), (58) 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.
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. 525).
aCox, Ingersoll, and Ross (1985).
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
aSee also Eq. (56) on p. 513. Also consistent with Lemma 9 (p. 259).
Modeling Stock Prices (continued)
• The more general deterministic volatility model posits dS
S = (rt − qt) 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.
aDerman and Kani (1994).
bDerman and Kani (1994); Dupire (1994); Rubinstein (1994).
Modeling Stock Prices (continued)
• How to construct an implied tree with efficiency, valid probabilities and stability remains 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, etc.
• Numerically, the inversion problem is an ill-posed problem.
• It is partially solved recently.b
aDerman and Kani (1994); Derman, Kani, and Chriss (1996); Cole- man, Kim, Li, and Verma (2000); Ayache, Henrotte, Nassar, and Wang (2004); Kamp (2009).
bFebruary 12, 2013.
Modeling Stock Prices (continued)
• Theoretically,a
σ(X, T )2 = 2
∂C
∂T + (rT − qT)X ∂X∂C + qTC X2 ∂∂X2C2
, (59) where C is the call price at time t with strike price X and maturity date T .
• In practice, like implied trees and for similar reasons, σ(K, T )2 may have spikes, vary wildly, or even be negative.
• Worse, the appearance of ∂2C/∂X2 in the denominator results in numerical instability.
Modeling Stock Prices (concluded)
• The relation between the implied volatility surface Σ(X, T ) and the local volatility surface σ(X, T ) isa
σ(X, T )2 = Σ2 + 2Στ [∂Σ
∂T + (rT − qT)X∂X∂Σ ] (1 − XyΣ ∂X∂Σ)2
+ XΣτ
[∂Σ
∂X − XΣτ4 (∂Σ
∂X
)2
+ X ∂X∂2Σ2 ] ,
where τ ≡ T − t and y ≡ ln(X/St) + ∫ T
t (qs − rs) ds.
• Although this version may be more stable than Eq. (59) on p. 532, it is expected to suffer from similar problems.
aAndreasen (1996); Andersen and Brotherton-Ratcliffe (1998);
Gatheral (2003); Wilmott (2006); Kamp (2009).
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.
• PDEs allow many numerical methods to be applicable.
Assumptions
a• 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.
aDerman and Taleb (2005) summarizes criticisms on these assump- tions and the replication argument.
Black-Scholes Differential Equation
• Let C be the price of a derivative on S.
• From Ito’s lemma (p. 506), 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,
Π = −C + S(∂C/∂S).
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
∂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. (60)
• Identity (60) 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.
— Perry Mehrling (2005)
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.
aKemna and Vorst (1990).
PDEs for Asian Options (continued)
• The two-dimensional PDE produces algorithms similar to that on pp. 357ff.
• 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).
aRogers and Shi (1995); Veˇceˇr (2001); Dubois and Leli`evre (2005).
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.
The Hull-White Model
• Hull and White (1987) postulate the following model, dS
S = r dt + √
V dW1, dV = µvV dt + bV dW2.
• Above, V is the instantaneous variance.
• They assume µv depends on V and t (but not S).
The SABR Model
• Hagan, Kumar, Lesniewski, and Woodward (2002) postulate the following model,
dS
S = r dt + SθV dW1, dV = bV dW2,
for 0 ≤ θ ≤ 1.
• A nice feature of this model is that the implied volatility surface has a compact approximate closed form.
The Hilliard-Schwartz Model
• Hilliard and Schwartz (1996) postulate the following general model,
dS
S = r dt + f (S)V a dW1, dV = µ(V ) dt + bV dW2,
for some well-behaved function f (S) and constant a.
The Blacher Model
• Blacher (2002) postulates the following model, dS
S = r dt + σ [
1 + α(S − S0) + β(S − S0)2 ]
dW1, dσ = κ(θ − σ) dt + ϵσ dW2.
• So the volatility σ follows a mean-reverting process to level θ.
Heston’s Stochastic-Volatility Model
a• Heston assumes the stock price follows dS
S = (µ − q) dt + √
V dW1, (61) dV = κ(θ − V ) dt + σ√
V dW2. (62) – 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.
aContributed by Mr. Chen, Chun-Ying (D95723006) on August 17,
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. (61) on p. 550.
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. (61) and Eq. (62) on p. 550.
• 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 (c√n) 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).
bChen (R95723051) (2008); Chen (R95723051), Lyuu, and Wen (D94922003) (2011).
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
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 well if the cumulative cost of trading stock is close to the call premium’s FV.a
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 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
• The tracking error at maturity is proportional to vega.b
aBoyle and Emanuel (1980).
bKamal and Derman (1999).
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, the net gain is 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. 571) 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.
Trees
I love a tree more than a man.
— Ludwig van Beethoven (1770–1827) And though the holes were rather small, they had to count them all.
— The Beatles, A Day in the Life (1967)
The Combinatorial Method
• The combinatorial method can often cut the running time by an order of magnitude.
• The basic paradigm is to count the number of admissible paths that lead from the root to any terminal node.
• We first used this method in the linear-time algorithm for standard European option pricing on p. 248.
• We will now apply it to price barrier options.
The Reflection Principle
a• Imagine a particle at position (0, −a) on the integral lattice that is to reach (n,−b).
• Without loss of generality, assume a > 0 and b ≥ 0.
• This particle’s movement:
(i, j) *(i + 1, j + 1) up move S → Su j(i + 1, j − 1) down move S → Sd
• How many paths touch the x axis?
aAndr´e (1887).
(0, a) (n, b) (0, a)
J
The Reflection Principle (continued)
• For a path from (0, −a) to (n, −b) that touches the x axis, let J denote the first point this happens.
• Reflect the portion of the path from (0, −a) to J.
• A path from (0, a) to (n, −b) is constructed.
• It also hits the x axis at J for the first time.
• The one-to-one mapping shows the number of paths from (0,−a) to (n, −b) that touch the x axis equals the number of paths from (0, a) to (n,−b).
The Reflection Principle (concluded)
• A path of this kind has (n + b + a)/2 down moves and (n − b − a)/2 up moves.
• Hence there are
( n
n+a+b 2
)
=
( n
n−a−b 2
)
(63) such paths for even n + a + b.
– Convention: (n
k
) = 0 for k < 0 or k > n.
Pricing Barrier Options (Lyuu, 1998)
• Focus on the down-and-in call with barrier H < X.
• Assume H < S without loss of generality.
• Define
a ≡
⌈ln (X/ (Sdn)) ln(u/d)
⌉
=
⌈ln(X/S) 2σ√
∆t + n 2
⌉ ,
h ≡
⌊ln (H/ (Sdn)) ln(u/d)
⌋
=
⌊ln(H/S) 2σ√
∆t + n 2
⌋ .
– h is such that ˜H ≡ Suhdn−h is the terminal price that is closest to, but does not exceed H.
– a is such that ˜X ≡ Suadn−a is the terminal price that is closest to, but is not exceeded by X.
Pricing Barrier Options (continued)
• The true barrier is replaced by the effective barrier ˜H in the binomial model.
• A process with n moves hence ends up in the money if and only if the number of up moves is at least a.
• The price Sukdn−k is at a distance of 2k from the lowest possible price Sdn on the binomial tree.
–
Sukdn−k = Sd−kdn−k = Sdn−2k. (64)
0 0
2 h n
2 a
S
0 0 0 0
2 j
X~ Su da n a Su dj n j
H~ Su dh n h
Pricing Barrier Options (continued)
• The number of paths from S to the terminal price Sujdn−j is (n
j
), each with probability pj(1 − p)n−j.
• With reference to p. 587, the reflection principle can be applied with a = n − 2h and b = 2j − 2h in Eq. (63) on p. 584 by treating the ˜H line as the x axis.
• Therefore,
( n
n+(n−2h)+(2j−2h) 2
)
=
( n
n − 2h + j )
paths hit ˜H in the process for h ≤ n/2.
Pricing Barrier Options (concluded)
• The terminal price Sujdn−j is reached by a path that hits the effective barrier with probability
( n
n − 2h + j )
pj(1 − p)n−j.
• The option value equals
∑2h
j=a
( n
n−2h+j
)pj(1 − p)n−j (
Sujdn−j − X)
Rn . (65)
– R ≡ erτ /n is the riskless return per period.
• It implies a linear-time algorithm.a
aLyuu (1998).
Convergence of BOPM
• Equation (65) results in the sawtooth-like convergence shown on p. 337.
• The reasons are not hard to see.
• The true barrier most likely does not equal the effective barrier.
• The same holds between the strike price and the effective strike price.
• The issue of the strike price is less critical.
• But the issue of the barrier is not negligible.
Convergence of BOPM (continued)
• Convergence is actually good if we limit n to certain values—191, for example.
• These values make the true barrier coincide with or just above one of the stock price levels, that is,
H ≈ Sdj = Se−jσ
√τ /n
for some integer j.
• The preferred n’s are thus n =
⌊
τ
(ln(S/H)/(jσ))2
⌋
, j = 1, 2, 3, . . .
Convergence of BOPM (continued)
• There is only one minor technicality left.
• We picked the effective barrier to be one of the n + 1 possible terminal stock prices.
• However, the effective barrier above, Sdj, corresponds to a terminal stock price only when n − j is even.a
• To close this gap, we decrement n by one, if necessary, to make n − j an even number.
aThis is because j = n − 2k for some k by Eq. (64) on p. 586. Of course we could have adopted the form Sdj (−n ≤ j ≤ n) for the effective barrier. It makes a good exercise.
Convergence of BOPM (concluded)
• The preferred n’s are now
n =
ℓ if ℓ − j is even ℓ − 1 otherwise , j = 1, 2, 3, . . . , where
ℓ ≡
⌊
τ
(ln(S/H)/(jσ))2
⌋ .
• Evaluate pricing formula (65) on p. 589 only with the n’s above.
0 500 1000 1500 2000 2500 3000 3500
#Periods 5.5
5.55 5.6 5.65 5.7
Down-and-in call value
Practical Implications
• Now that barrier options can be efficiently priced, we can afford to pick very large n’s (p. 596).
• This has profound consequences (to explain later).
n Combinatorial method Value Time (milliseconds)
21 5.507548 0.30
84 5.597597 0.90
191 5.635415 2.00
342 5.655812 3.60
533 5.652253 5.60
768 5.654609 8.00
1047 5.658622 11.10
1368 5.659711 15.00
1731 5.659416 19.40
2138 5.660511 24.70
2587 5.660592 30.20
3078 5.660099 36.70
3613 5.660498 43.70
4190 5.660388 44.10
4809 5.659955 51.60
5472 5.660122 68.70
6177 5.659981 76.70
6926 5.660263 86.90
Practical Implications (concluded)
• Pricing is prohibitively time consuming when S ≈ H because n ∼ 1/ ln2(S/H).
– This is called the barrier-too-close problem.
• This observation is indeed true of standard quadratic-time binomial tree algorithms.
• But it no longer applies to linear-time algorithms (see p. 598).
• This binomial model is O(1/√
n) convergent in general but O(1/n) convergent when the barrier is matched.a
aLin (R95221010) (2008) and Lin (R95221010) and Palmer (2010).
Barrier at 95.0 Barrier at 99.5 Barrier at 99.9
n Value Time n Value Time n Value Time
..
. 795 7.47761 8 19979 8.11304 253
2743 2.56095 31.1 3184 7.47626 38 79920 8.11297 1013 3040 2.56065 35.5 7163 7.47682 88 179819 8.11300 2200 3351 2.56098 40.1 12736 7.47661 166 319680 8.11299 4100 3678 2.56055 43.8 19899 7.47676 253 499499 8.11299 6300 4021 2.56152 48.1 28656 7.47667 368 719280 8.11299 8500
True 2.5615 7.4767 8.1130
(All times in milliseconds.)
