### The Hull-White Model

*• Hull and White (1987) postulate the following*
*stochastic-volatility model,*

*dS*

*S* = *r dt +* *√*

*V dW*_{1}*,*
*dV* = *μ*_{v}*V dt + bV dW*_{2}*.*

*• Above, V is the instantaneous variance.*

*• They assume μ*v *depends on V and t (but not S).*

### The Barone-Adesi–Rasmussen–Ravanelli Model

*• Barone-Adesi, Rasmussen, and Ravanelli (2005)*
postulate the following model,

*dS*

*S* = *μ dt +* *√*

*V dW*_{1}*,*

*dV* = *κ(θ − V ) dt + bV dW*2*.*

*• Above, W*1 *and W*2 are correlated.

### The Stein-Stein Model

*• E. Stein and J. Stein (1991) postulate the following*
model,

*dS*

*S* = *r dt + V dW*_{1}*,*

*dV* = *κ(μ − V ) dt + σ dW.*

*• Closed-form formulas exist for European calls and puts.*^{a}

aSch¨obel & Zhu (1999).

### The SABR Model

*• Hagan, Kumar, Lesniewski, and Woodward (2002)*
postulate the following model,

*dS*

*S* = *r dt + S*^{θ}*V dW*1*,*
*dV* = *bV dW*2*,*

for 0 *≤ θ ≤ 1.*

*• A nice feature of this model is that the implied volatility*
surface has a compact approximate closed form.

### The Blacher Model

*• Blacher (2001) postulates the following model,*
*dS*

*S* = *r dt + σ*

*1 + α(S − S*_{0}*) + β(S − S*_{0})^{2}

*dW*_{1}*,*
*dσ = κ(θ − σ) dt + σ dW*2*.*

*• The volatility σ follows a mean-reverting process to level*
*θ.*

### The Hilliard-Schwartz Model

*• Hilliard and Schwartz (1996) postulate the following*
very general model,

*dS*

*S* = *r dt + f (S)V* ^{a}*dW*1*,*
*dV* = *μ(V ) dt + bV dW*2*,*

*for some well-behaved function f (S) and constant a.*

*• It includes all previously mentioned stochastic-volatility*
models as special cases.^{a}

aH. Chiu (R98723059) (2012).

### Heston’s Stochastic-Volatility Model

*• Heston (1993) assumes the stock price follows*
*dS*

*S* = *(μ − q) dt +* *√*

*V dW*_{1}*,* (91)
*dV* = *κ(θ − V ) dt + σ√*

*V dW*_{2}*.* (92)
**– V is the instantaneous variance, which follows a**

square-root process.

**– dW**_{1} *and dW*_{2} *have correlation ρ.*

**– The riskless rate r is constant.**

*• It may be the most popular continuous-time*
stochastic-volatility model.^{a}

aChristoﬀersen, Heston, & Jacobs (2009).

### Heston’s Stochastic-Volatility Model (continued)

*• Heston assumes the market price of risk is b*2*√*
*V .*

*• So μ = r + b*_{2}*V .*

*• Deﬁne*

*dW*_{1}* ^{∗}* =

*dW*

_{1}

*+ b*

_{2}

*√*

*V dt,*
*dW*_{2}* ^{∗}* =

*dW*2

*+ ρb*2

*√V dt,*
*κ** ^{∗}* =

*κ + ρb*2

*σ,*

*θ** ^{∗}* =

*θκ*

*κ + ρb*_{2}*σ.*

*• dW*_{1}^{∗}*and dW*_{2}^{∗}*have correlation ρ.*

### Heston’s Stochastic-Volatility Model (continued)

*• Under the risk-neutral probability measure Q, both W*_{1}^{∗}*and W*_{2}* ^{∗}* are Wiener processes.

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

*S* = *(r − q) dt +* *√*

*V dW*_{1}^{∗}*,*
*dV* = *κ*^{∗}*(θ*^{∗}*− V ) dt + σ√*

*V dW*_{2}^{∗}*.*

### 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}

*− σ*

^{2}

*(−ıu − u*

^{2}

*) ,*

*g*=

*(κ*

^{∗}*− ρσuı − d)/(κ*

^{∗}*− ρσuı + d).*

### Heston’s Stochastic-Volatility Model (continued)

The formulas for European calls and puts are^{a}

*C* = *S*

1

2 + 1
*π*

_{∞}

0

Re

*X*^{−ıu}*φ(u − ı, τ )*
*ıuSe*^{rτ}

*du*

*−Xe*^{−rτ}

1

2 + 1
*π*

_{∞}

0 Re

*X*^{−ıu}*φ(u, τ )*
*ıu*

*du*

*,*
*P* = *Xe*^{−rτ}

1

2 *−* 1
*π*

_{∞}

0

Re

*X*^{−ıu}*φ(u, τ )*
*ıu*

*du*

*,*

*−S*

1

2 *−* 1
*π*

_{∞}

0

Re

*X*^{−ıu}*φ(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, Chun-Ying (D95723006) on August 17, 2008 and Mr. Liou, Yan-Fu (R92723060) on August 26, 2008. See Lord &

Kahl (2009) and Cui, Rollin, & Germano (2017) for alternative formulas.

### Heston’s Stochastic-Volatility Model (concluded)

*• For American options, trees are needed.*

*• They are all O(n*^{3})-sized and do not match all
moments.^{a}

*• An O(n*^{2.5}*)-sized 9-jump tree that matches all means*
and variances with valid probabilities is available.^{b}

*• The size reduces to O(n*^{2}) for knock-out double-barrier
options.^{c}

aNelson & Ramaswamy (1990); Nawalkha & Beliaeva (2007); Leisen (2010); Beliaeva & Nawalkha (2010); M. Chou (R02723073) (2015); M.

Chou (R02723073) & Lyuu (2016).

bZ. Lu (D00922011) & Lyuu (2018).

cZ. Lu (D00922011) & Lyuu (2018).

### Stochastic-Volatility Models and Further Extensions

^{a}

*• How to explain the October 1987 crash?*

**– The Dow Jones Industrial Average fell 22.61% on**
October 19, 1987 (called the Black Monday).

**– The CBOE S&P 100 Volatility Index (VXO) shot up**
to 150%, the highest VXO ever recorded.^{b}

*• Stochastic-volatility models require an implausibly*
*high-volatility level prior to and after the crash.*

**– Because the processes are continuous.**

*• Discontinuous jump models in the asset price can*
alleviate the problem somewhat.^{c}

aEraker (2004).

bCaprio (2012).

cMerton (1976).

### 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. (91) on p. 663.**

**– Closed-form formulas exist for GARCH-jump option**
pricing models.^{a}

aLiou (R92723060) (2005).

### 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. (91) and*
Eq. (92) on p. 663.

*• Such models allow high level of volatility caused by a*
jump to volatility.^{c}

aBates (2000); Pan (2002).

bDuﬃe, Pan, & Singleton (2000).

cEraker, Johnnes, & Polson (2000); Y. Lin (2007); Zhu & Lian (2012).

### Why Are Trees for Stochastic-Volatility Models Diﬃcult?

*• The CRR tree is 2-dimensional.*^{a}

*• The constant volatility makes the span from any node*
ﬁxed.

*• But a tree for a stochastic-volatility model must be*
3-dimensional.

**– Every node is associated with a combination of stock**
*price and volatility.*

aRecall p. 298.

### Why Are Trees for Stochastic-Volatility Models

### Diﬃcult (Binomial Case)?

### Why Are Trees for Stochastic-Volatility Models

### Diﬃcult (Trinomial Case)?

### Why Are Trees for Stochastic-Volatility Models Diﬃcult? (concluded)

*• Locally, the tree looks ﬁne for one time step.*

*• But the volatility regulates the spans of the nodes on*
the stock-price plane.

*• Unfortunately, those spans diﬀer from node to node*
because the volatility varies.

*• So two time steps from now, the branches will not*
combine!

*• Smart ideas are thus needed.*

### Complexities of Stochastic-Volatility Models

*• A few stochastic-volatility models suﬀer from*
*subexponential (c*^{√}* ^{n}*) tree size.

*• Examples include the Hull-White (1987),*

Hilliard-Schwartz (1996), and SABR (2002) models.^{a}

*• Future research may extend this negative result to more*
stochastic-volatility models.

**– We suspect many GARCH option pricing models**
entertain similar problems.^{b}

aH. Chiu (R98723059) (2012).

bY. C. Chen (R95723051) (2008); Y. C. Chen (R95723051), Lyuu, &

Wen (D94922003) (2011).

### Complexities of Stochastic-Volatility Models (concluded)

*• Flexible placement of nodes and removal of*
low-probability nodes may make the models
*O(n** ^{2.5}*)-sized!

^{a}

*• Calibration can be computationally hard.*

**– Few have tried it on exotic options.**^{b}

*• There are usually several local minima.*^{c}

**– They will give diﬀerent prices to options not used in**
the calibration.

**– But which set capture the smile dynamics?**

aZ. Lu (D00922011) & Lyuu (2018).

bAyache, Henrotte, Nassar, & X. Wang (2004).

cAyache (2004).

*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.

**– Their prices are perfectly correlated.**

*• We then 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}

### and Notations

*• 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.*^{Δ}

aDerman & Taleb (2005) summarizes criticisms on these assumptions and the replication argument.

### Black-Scholes Diﬀerential Equation

*• Let C be the price of a simple derivative*^{a} *on S.*

*• From Ito’s lemma (p. 611),*
*dC =*

*μS* *∂C*

*∂S* + *∂C*

*∂t* + 1

2 *σ*^{2}*S*^{2} *∂*^{2}*C*

*∂S*^{2}

*dt + σS* *∂C*

*∂S* *dW.*

**– The same W drives both C and S.**

**– Unlike dS/S, the diﬀusion of dC/C is stochastic!**

*• Short one derivative and long ∂C/∂S shares of stock*
(call it Π).

*• By construction,*

Π = *−C + S(∂C/∂S).*

a

### Black-Scholes Diﬀerential Equation (continued)

*• The change in the value of the portfolio at time dt is*^{a}
*dΠ = −dC +* *∂C*

*∂S* *dS.* (93)

*• Substitute the formulas for dC and dS into the above*
to yield

*dΠ =*

*−∂C*

*∂t* *−* 1

2 *σ*^{2}*S*^{2} *∂*^{2}*C*

*∂S*^{2}

*dt.*

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

aBergman (1982) and Bartels (1995) argue this is not quite right. But see Macdonald (1997). Mathematically, it is wrong (Bingham & Kiesel, 2004).

### Black-Scholes Diﬀerential Equation (continued)

*• So*

*∂C*

*∂t* + 1

2 *σ*^{2}*S*^{2} *∂*^{2}*C*

*∂S*^{2}

*dt = r*

*C − S* *∂C*

*∂S*

*dt.*

*• Equate the terms to ﬁnally obtain*^{a}

*∂C*

*∂t* *+ rS* *∂C*

*∂S* + 1

2 *σ*^{2}*S*^{2} *∂*^{2}*C*

*∂S*^{2} *= rC.*

*• This is a backward equation, which describes the*

*dynamics of a derivative’s price forward in physical time.*

aKnown as the Feynman-Kac stochastic representation formula.

### Black-Scholes Diﬀerential Equation (concluded)

*• When there is a dividend yield q,*

*∂C*

*∂t* *+ (r − q) S* *∂C*

*∂S* + 1

2 *σ*^{2}*S*^{2} *∂*^{2}*C*

*∂S*^{2} *= rC.* (94)

*• Dupire’s formula (90) for the local-volatility model*^{a} is
simply the dual of this equation:^{b}

*∂C*

*∂T* *+ (r**T* *− q**T**)X* *∂C*

*∂X* *−* 1

2 *σ(X, T )*^{2}*X*^{2} *∂*^{2}*C*

*∂X*^{2} = *−q**T**C.*

*• This is a forward equation, which describes the dynamics*
*of a derivative’s price backward in maturity time.*

aSee p. 641.

bDerman & Kani (1997).

### Rephrase

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

*Θ + rSΔ +* 1

2 *σ*^{2}*S*^{2}*Γ = rC.* (95)

*• Identity (95) leads to an alternative way of computing*
Θ numerically from Δ and Γ.

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

2 *σ*^{2}*S*^{2}*Γ = rC.*

**– A deﬁnite relation thus exists between Γ and Θ.**

### Black-Scholes Diﬀerential Equation: An Alternative

*• Perform the change of variable V* *= ln S.*^{Δ}

*• The option value becomes U(V, t)* *= C(e*^{Δ} ^{V}*, t).*

*• Furthermore,*

*∂C*

*∂t* = *∂U*

*∂t* *,*

*∂C*

*∂S* = 1

*S*

*∂U*

*∂V* *,* (96)

*∂*^{2}*C*

*∂*^{2}*S* = 1
*S*^{2}

*∂*^{2}*U*

*∂V* ^{2} *−* 1
*S*^{2}

*∂U*

*∂V* *.* (97)

### Black-Scholes Diﬀerential Equation: An Alternative (concluded)

*• Equations (96) and (97) are alternative ways to*
calculate delta and gamma.^{a}

*• They are particularly useful for a tree of logarithmic*
prices.

*• The Black-Scholes diﬀerential equation (94) on p. 685*
becomes

1

2 *σ*^{2} *∂*^{2}*U*

*∂V* ^{2} +

*r − q −* *σ*^{2}
2

*∂U*

*∂V* *− rU +* *∂U*

*∂t* = 0
*subject to U (V, T ) being the payoﬀ such as*

*max(X − e*^{V}*, 0).*

aSee Eqs. (52) on p. 365 and (53) on p. 367.

[ 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)* =^{Δ} _{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 *σ*^{2}*S*^{2} *∂*^{2}*V*

*∂S*^{2} *+ 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 & Vorst (1990).

### PDEs for Asian Options (continued)

*• The two-dimensional PDE produces algorithms similar*
to that on pp. 447ﬀ.^{a}

*• But one-dimensional PDEs are available for Asian*
options.^{b}

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

*∂u*

*∂t* *+ r*

1 *−* *t*

*T* *− z*

*∂u*

*∂z* +

1 *−* _{T}^{t}*− z*_{2}
*σ*^{2}
2

*∂*^{2}*u*

*∂z*^{2} = 0
*with the terminal condition u(T, z) = max(z, 0).*

aBarraquand & Pudet (1996).

bRogers & Shi (1995); Veˇceˇr (2001); Dubois & 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}
*σ*^{2}
2

*∂*^{2}*u*

*∂z*^{2} = 0
with the same terminal condition.

*• One-dimensional PDEs result in highly eﬃcient*
numerical algorithms.

*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

*• Recall the delta (hedge ratio) of a derivative f:*

Δ =^{Δ} *∂f*

*∂S.*

*• Thus*

*Δf ≈ Δ × ΔS*

*for relatively small changes in the stock price, ΔS.*

*• A delta-neutral portfolio is hedged as it is immunized*
against small changes in the stock price.

### Delta Hedge (concluded)

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

**– Trading strategies can also be static (or constant).**^{a}

*• 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.”

aRecall p. 494 for one in hedging the short forward contract with the underlying asset and loans.

### Implementing Delta Hedge

*• We want to hedge N short derivatives.*

*• Assume the stock pays no dividends.*

*• The delta-neutral portfolio maintains N × Δ shares of*
*stock plus B borrowed dollars such that*

*−N × f + N × Δ × S − B = 0.*

*• At next rebalancing point when the delta is Δ** ^{}*, buy

*N × (Δ*

^{}*− Δ) shares to maintain N × Δ*

*shares.*

^{}*• Delta hedge is the discrete-time analog of the*
continuous-time limit.

*• It will rarely be self-ﬁnancing however small Δt is.*

### Example

*• A hedger is short 10,000 European calls.*

*• S = 50, 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 takes the replication viewpoint: One starts with zero dollar.

### Example (continued)

*• As Δ = 0.538560*

*N × Δ = 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 × Δ × S − N × f = 25,160,090*
dollars net.^{a}

aThis takes the hedging viewpoint: One starts with the option pre- mium. See Exercise 16.3.2 of the text.

### Example (continued)

*• At 3 weeks to expiration, the stock price rises to $51.*

*• The new call value is f*^{}*= 2.10580.*

*• So before rebalancing, the portfolio is worth*

*− N × f*^{}*+ 538,560 × 51 − Be*^{0.06/52}*= 171, 622.* (98)

*• The delta hedge is not self-ﬁnancing as $171,622 can be*
withdrawn.

**– It does not replicate the calls perfectly.**

### Example (continued)

*• The magnitude of the tracking error—the variation in*
the net portfolio value—can be mitigated if adjustments
are made more frequently.

*• The tracking error over one rebalancing act is positive*
about 68% of the time.

*• But its expected value is ∼ 0 under the risk-neutral*
probability measure.^{a}

**– Of course, the stock price should be sampled under**
*the real-world probability measure.*^{b}

aBoyle & Emanuel (1980).

bRecall Eq. (93) on p. 683 or see p. 711.

### Example (continued)

*• The tracking error at maturity is proportional to vega.*^{a}

*• In practice tracking errors will cease to decrease beyond*
a certain rebalancing frequency.

*• With a higher delta Δ*^{}*= 0.640355, the trader buys*
*N × (Δ*^{}*− Δ) = 101, 795*

shares for $5,191,545.

*• The number of shares is increased to N × Δ*^{}*= 640, 355.*

aKamal & Derman (1999).

### Example (continued)

*• The cumulative cost is*^{a}

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

*• The portfolio is again delta-neutral.*

aWe take the replication viewpoint. Under the BOPM, the replicating strategy is self-ﬁnancing and matches the payoﬀ perfectly.

Option Change in No. shares Cost of Cumulative

value Delta delta bought shares cost

*τ* *S* *f* Δ *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 (continued)

*• 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.

*• So if we had started with the PV of $1,524,853, we*

*would have replicated 10,000 such calls in this scenario.*

### Example (concluded)

*• The FV of the call premium equals*

1,767,910 *× e*^{0.06×4/52}*= 1,776,088.*

*• That means the net gain in this scenario is*
1,776,088 *− 1,524,853 = 251,235*
if we are hedging 10,000 European calls.

### Tracking Error Revisited

*• Deﬁne the dollar gamma as S*^{2}Γ.

*• The change in value of a delta-hedged long option*
*position after a duration of Δt is proportional to the*
dollar gamma.

*• It is about*

*(1/2)S*^{2}*Γ[ (ΔS/S)*^{2} *− σ*^{2}*Δt ].*

**– (ΔS/S)**^{2} is called the daily realized variance.

### Tracking Error Revisited (continued)

*• In our particular case,*

*S = 50, Γ = 0.0957074, ΔS = 1, σ = 0.3, Δt = 1/52.*

*• The estimated tracking error is*

*−(1/2)×50*^{2}*×0.0957074×*

*(1/50)*^{2} *− (0.09/52)*

*= 159, 205.*

*• It is very close to our earlier number of 171,622.*^{a}

*• Delta hedge is also called gamma scalping.*^{b}

aRecall Eq. (98) on p. 700.

bBennett (2014).

### Tracking Error Revisited (continued)

*• Let the rebalancing times be t*1*, t*_{2}*, . . . , t** _{n}*.

*• Let ΔS**i* *= S**i+1* *− S**i*.

*• The total tracking error at expiration is about*

*n−1*

*i=0*

*e*^{r(T −t}^{i}^{)}*S*_{i}^{2}Γ* _{i}*
2

*ΔS**i*

*S**i*

_{2}

*− σ*^{2}*Δt*

*.*

*• The tracking error is clearly path dependent.*

*• Mathematically,*^{a}

*n−1*

*i=0*

*ΔS**i*

*S*_{i}

_{2}

*→ σ*^{2}*T.*

aProtter (2005).

### Tracking Error Revisited (concluded)

^{a}

*• The tracking error*^{b} _{n}*over n rebalancing acts has*
about the same probability of being positive as being
negative.

*• Subject to certain regularity conditions, the*
root-mean-square tracking error

*E[ *^{2}_{n}*] is O(1/√*

*n ).*^{c}

*• The root-mean-square tracking error increases with σ at*
ﬁrst and then decreases.

aBertsimas, Kogan, & Lo (2000).

bSuch as 251,235 on p. 706.

cGrannan & Swindle (1996).

### Which Probability Measure?

^{a}

*• Should the proﬁt and loss (i.e., tracking error) of a*
*hedging strategy be calculated under the real-world or*
risk-neutral probability measure?

*• It is the former.*

*• But the deltas and option prices should be calculated*
under the risk-neutral probability measure.

*• If whenever we sample the next stock price, backward*
induction is performed for the delta, it will take a long
time to obtain the distribution of the proﬁt and loss.

*• How to do it eﬃciently?*^{b}

aContributed by Mr. Chiu, Tzu-Hsuan (R08723061) on April 9, 2021.

bContributed by Mr. Lu, Zheng-Liang (D00922011) in August, 2021.

### Delta-Gamma Hedge

*• Delta hedge is based on the ﬁrst-order approximation to*
*changes in the derivative price, Δf , due to changes in*
*the stock price, ΔS.*

*• When ΔS is not small, the second-order term, gamma*
Γ *= ∂*^{Δ} ^{2}*f /∂S*^{2}, helps.

*• A delta-gamma hedge is a delta hedge that maintains*
zero portfolio gamma; it is gamma neutral.

*• To meet this extra condition, one more security needs to*
be brought in.

### Delta-Gamma Hedge (concluded)

*• Suppose we want to hedge short calls as before.*

*• A hedging call f*_{2} is brought in.

*• To set up a delta-gamma hedge, we solve*

*−N × f + n*1 *× S + n*2 *× f*2 *− B = 0 (self-ﬁnancing),*

*−N × Δ + n*1 *+ n*2 *× Δ*2 *− 0 = 0 (delta neutrality),*

*−N × Γ + 0 + n*_{2} *× Γ*_{2} *− 0 = 0 (gamma neutrality),*

*for n*_{1}*, n*_{2}*, and B.*

**– The gammas of the stock and bond are 0.**

*• See the numerical example on pp. 231–232 of the text.*

### 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, still 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) All those holes and pebbles.

Who could count them?

*— James Joyce, Ulysses (1922)*
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. 286.

*• 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.**^{a}

*• Hence there are*

*n*

* n+a+b*
2

=

*n*

* n−a−b*
2

(99)
**such paths for even n + a + b.**

**– Convention:** _{n}

*k*

*= 0 for k < 0 or k > n.*

aVerify it!

### Pricing Barrier Options (Lyuu, 1998)

*• Focus on the down-and-in call with barrier H < X.*

*• So H < S.*

*• Deﬁne*

*a* =^{Δ}

*ln (X/ (Sd** ^{n}*))

*ln(u/d)*

=

*ln(X/S)*
*2σ**√*

*Δt* + *n*
2

*,*
*h* =^{Δ}

*ln (H/ (Sd** ^{n}*))

*ln(u/d)*

=

*ln(H/S)*
*2σ**√*

*Δt* + *n*
2

*.*

**– a is such that ˜**X*= Su*^{Δ} ^{a}*d*^{n−a}*is the terminal price*
*that is closest to X from above.*

**– h is such that ˜**H*= Su*^{Δ} ^{h}*d*^{n−h}*is the terminal price*
*that is closest to H from below.*^{a}

aSo we underestimate the price.

### 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 Su*^{k}*d*^{n−k}*is at a distance of 2k from the*
*lowest possible price Sd** ^{n}* on the binomial tree.

**–**

*Su*^{k}*d*^{n−k}*= Sd*^{−k}*d*^{n−k}*= Sd*^{n−2k}*.*

(100)

0 0

n 2 h 2 a

S

0 0 0 0

2 j

X Su d~ ^{a n a}
Su d^{j n j}

H Su d~ ^{h n h}

### Pricing Barrier Options (continued)

*• A path from S to the terminal price Su*^{j}*d** ^{n−j}* has

*probability p*

*(1*

^{j}*− p)*

*of being taken.*

^{n−j}*• With reference to p. 724, the reﬂection principle (p. 719)*
can be applied with

**a = n − 2h,****b = 2j − 2h,**

in Eq. (99) on p. 721 by treating the ˜*H line as the x*
axis.

### Pricing Barrier Options (continued)

*• Therefore,*

*n*

*n+(n−2h)+(2j−2h)*
2

=

*n*

*n − 2h + j*

paths hit ˜*H in the process for h ≤ n/2.*

*• The terminal price Su*^{j}*d** ^{n−j}* is reached by a path that
hits the eﬀective barrier with probability

*n*

*n − 2h + j*

*p** ^{j}*(1

*− p)*

^{n−j}*,*

*j ≤ 2h.*

### Pricing Barrier Options (concluded)

*• The option value equals*

_{2h}

*j=a*

_{n}

*n−2h+j*

*p*^{j}*(1 − p)*^{n−j}

*Su*^{j}*d*^{n−j}*− X*

*R*^{n}*.*

(101)

**– R***= e*^{Δ} * ^{rτ/n}* is the riskless return per period.

*• It yields a linear-time algorithm.*^{a}

aLyuu (1998).

### Convergence of BOPM

*• Equation (101) results in the same sawtooth-like*

convergence shown on p. 408 (repeated on next page).

*• The reasons are not hard to see.*

*• The eﬀective barrier ˜H rarely equals the true barrier H.*

### Convergence of BOPM (continued)

### 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 ≈ Sd*^{j}*= 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, Sd** ^{j}*, 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.*

a*This is because j = n − 2k for some k by Eq. (100) on p. 723. Of*
*course we could have adopted the more general form Sd** ^{j}* (

*−n ≤ j ≤ n)*for the eﬀective barrier. It makes a good exercise.

### Convergence of BOPM (concluded)

*• The preferred n’s are now*

*n =*

⎧⎨

⎩

*,* *if − j is even,*
* − 1, otherwise,*

*j = 1, 2, 3, . . . , where*
=^{Δ}

*τ*

*(ln(S/H)/(jσ))*^{2}

*.*

*• Evaluate pricing formula (101) on p. 727 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

^{a}

*• This binomial model is O(1/√*

*n) convergent in general*
*but O(1/n) convergent when the barrier is matched.*^{b}

*• Now that barrier options can be eﬃciently priced, we*
*can aﬀord to pick very large n’s (p. 735).*

*• This has profound consequences.*^{c}

aLyuu (1998).

bJ. Lin (R95221010) (2008); ; J. Lin (R95221010) & Palmer (2013).

cSee pp. 749ﬀ.

*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

7717 5.660272 97.20

### Practical Implications (concluded)

*• Pricing is prohibitively time consuming when S ≈ H*
because

*n ∼ 1/ ln*^{2}*(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. 737).

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.)