Implied Vol (%)Local Vol (%)

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^Δ follows

dX =



μ − σ² 2



dt + σ dW by Ito’s lemma.^a

aSee also Eq. (76) on p. 571. Consistent with Lemma 11 (p. 283).

Local-Volatility Models

• The more general deterministic volatility model posits dS

S = (r_t − q_t) dt + σ(S, t) dW,

where instantaneous volatility σ(S, t) is called the local volatility function.^a

• A (weak) solution exists if Sσ(S, t) is continuous and grows at most linearly in S and t.^b

• One needs to recover the local volatility surface σ(S, t) from the implied volatility surface.

aDerman & Kani (1994); Dupire (1994).

bSkorokhod (1961).

Local-Volatility Models (continued)

• Theoretically,^a

σ(X, T )² = 2

∂C∂T + (r_T − q_T)X _∂X^∂C + q_TC

X^{2 ∂}_∂X^2C2 . (79)

• 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 S_T = 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, σ(S, t)² may have spikes, vary wildly, or even be negative.

• The term ∂²C/∂X² in the denominator often results in numerical instability.

aDerman & Kani (1997).

Local-Volatility Models (continued)

• Now, 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 ) is^a

σ(X, T )² = Σ² + 2Στ _∂Σ

∂T + (rT − qT)X _∂X^∂Σ

1 − ^Xy_{Σ ∂X}^∂Σ ₂

+ XΣτ 

∂Σ

∂X − ^XΣτ₄  _∂Σ

∂X

₂

+ X _∂X^∂2Σ2 , τ = T − t,^Δ

y = ln(X/S^Δ t) +

 _T

t

(qs − rs) ds.

• Although this version may be more stable than Eq. (79) on p. 592, it is expected to suffer from similar problems.

aAndreasen (1996); Andersen & Brotherton-Ratcliffe (1998);

Implied and Local Volatility Surfaces

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

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)

• Small changes to the implied volatility surface may produce big changes to the local volatility surface.

• 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 some implied volatility surfaces generate option prices that allow arbitrage profits.

aDoing it to the option prices produces worse results (Li, 2000/2001).

Local-Volatility Models (continued)

• For example, consider the following implied volatility surface:^a

Σ(X, T )² = a_ATM(T ) + b(X − S₀)², b > 0.

• It generates higher prices for out-of-the-money options than in-the-money options for T large enough.^b

aATM: at-the-money.

bRebonato (2004).

Local-Volatility Models (continued)

• Below are bounds for implied volatilities of forward options.^a

• Let x = ln(X/F^Δ ₀) − rT .

• For X large enough,

Σ(X, T )² < 2 | x | T .

• For X small enough,

Σ(X, T )² < β | x |

T for any β > 2.

a

Local-Volatility Models (concluded)

• There exist conditions for a set of option prices to be arbitrage-free.^a

• For some vanilla equity options, the Black-Scholes model seems “better than” the local-volatility model.^b

aDavis & Hobson (2007).

bDumas, Fleming, & Whaley (1998).

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 require that the implied tree combine.

• An exponential-sized implied tree exists.^b

aDerman & Kani (1994); Dupire (1994); Rubinstein (1994).

bCharalambousa, Christofidesb, & Martzoukosa (2007).

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); Moriggia, Muzzioli, & Torri- celli (2009); Rebonato (2004).

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

Implied Trees (concluded)

• It is finally solved for separable local volatilities σ.^a – The local-volatility function σ(S, V ) is separable^b if

σ(S, t) = σ₁(S) σ₂(t).

• A very general solution is recently obtained.^c

aLok (D99922028) & Lyuu (2015, 2016, 2017).

bRebonato (2004); Brace, G¸atarek, & Musiela (1997).

cLok (D99922028) & Lyuu (2016, 2017).

The Hull-White Model

• Hull and White (1987) postulate the following model, dS

S = r dt + √

V dW₁, dV = μ_vV 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 + S^θV 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 + σ dW2.

• So the volatility σ follows a mean-reverting process to level θ.

Heston’s Stochastic-Volatility Model

• Heston (1993) assumes the stock price follows dS

S = (μ − q) dt + √

V dW₁, (80)

dV = κ(θ − V ) dt + σ√

V dW₂. (81) – 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.^a

aChristoffersen, Heston, & Jacobs (2009).

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 (continued)

The formulas are^a

C = S

1

2 + 1 π

 _∞

0 Re

X^−ıuφ(u − ı, τ) ıuSe^rτ

 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) 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.

Heston’s Stochastic-Volatility Model (concluded)

• For American options, we will need a tree for Heston’s model.^a

• They are all O(n³)-sized.

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

Stochastic-Volatility Models and Further Extensions

• How to explain the October 1987 crash?

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

• Merton (1976) proposed jump models.

• Discontinuous jump models in the asset price can alleviate the problem somewhat.

aEraker (2004).

Stochastic-Volatility Models and Further Extensions (continued)

• But if the jump intensity is a constant, it cannot explain the tendency of large movements to cluster over time.

• This assumption also has no impacts on option prices.

• Jump-diffusion models combine both.

– E.g., add a jump process to Eq. (80) on p. 607.

– 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. (80) and Eq. (81) on p. 607.

• 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 pair of stock price and a volatility.

aRecall p. 280.

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

aChiu (R98723059) (2012).

bY. Chen (R95723051) (2008); Y. Chen (R95723051), Lyuu, & Wen (D94922003) (2011).

Complexities of Stochastic-Volatility Models (concluded)

• Calibration can be computationally hard.

– Few have tried it on exotic options.^a

• There are usually several local minima for the calibration error.^b

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

– But which one captures the smile dynamics?

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

bAyache (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

• 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 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. 564), 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 is^a 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.

aBergman (1982) argues it is not quite right. But see Macdonald (1997).

Black-Scholes Differential Equation (continued)

• 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.

• This is a backward equation, which describes the

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

Black-Scholes Differential Equation (concluded)

• When there is a dividend yield q,

∂C

∂t + (r − q) S ∂C

∂S + 1

2 σ²S² ∂²C

∂S² = rC. (82)

• The local-volatility model (79) on p. 592 is simply the dual of this equation:^a

∂C

∂T + (r_T − q_T)X ∂C

∂X − 1

2 σ(X, T )²X² ∂²C

∂X² = −q_TC.

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

aDerman & Kani (1997).

Rephrase

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

Θ + rSΔ + 1

2 σ²S²Γ = rC. (83)

• Identity (83) leads to an alternative way of computing Θ numerically from Δ and Γ.

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

2 σ²S²Γ = 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 ,

∂²C

∂²S = 1 S²

∂²U

∂V ² − 1 S²

∂U

∂V . (84)

• Equation (84) is an alternative way to calculate gamma.^a

aSee also Eq. (45) on p. 345.

Black-Scholes Differential Equation: An Alternative (concluded)

• The Black-Scholes differential equation (82) on p. 629 becomes

1

2 σ² ∂²U

∂V ² +



r − q − σ² 2

 ∂U

∂V − rU + ∂U

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

max(X − e^V , 0).

[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 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. 403ff.^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 − z₂ σ² 2

∂²u

∂z² = 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 − z₂ σ² 2

∂²u

∂z² = 0 with the same terminal condition.

• One-dimensional PDEs lead to highly efficient numerical methods.

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 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.

• 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 × 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 and will rarely be self-financing.

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 example takes the replication viewpoint.

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 — an alternative. 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 the portfolio is worth

−N × f^ + 538,560 × 51 − Be^0.06/52 = 171, 622 before rebalancing.

Example (continued)

• A delta hedge does not replicate the calls perfectly; it is not self-financing as $171,622 can be withdrawn.

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

• In fact, the tracking error over one rebalancing act is positive about 68% of the time, but its expected value is essentially zero.^a

• The tracking error at maturity is proportional to vega.^b

aBoyle & Emanuel (1980).

bKamal & Derman (1999).

Example (continued)

• 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.

Example (continued)

• The cumulative cost is

26,928,000 × e^0.06/52 + 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 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 (concluded)

• At expiration, the trader has 1,000,000 shares.

• They are exercised against by the in-the-money calls for

$50,000,000.

• The trader is left with an obligation of

51,524,853 − 50,000,000 = 1,524,853, which represents the replication cost.

• Compared with the FV of the call premium, 1,767,910 × e^0.06×4/52 = 1,776,088, the net gain is 1,776,088 − 1,524,853 = 251,235.

Tracking Error Revisited

• Define the dollar gamma as S²Γ.

• 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)S²Γ[ (ΔS/S)² − σ²Δt ].

– (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, Δ = 1/52.

• The estimated tracking error is

−(1/2)×50²×0.0957074×

(1/50)² − (0.09/52)

= 159, 205.

• It is very close to our earlier number of 171,622.

• Delta hedge is also called gamma scalping.^a

aBennett (2014).

Tracking Error Revisited (continued)

• Let the rebalancing times be t₁, t₂, . . . , t_n.

• Let ΔSi = Si+1 − Si.

• The total tracking error at expiration is about

n−1

i=0

e^{r(T −ti)}S_i²Γ_i 2

ΔS_i S_i

₂

− σ²Δt 

.

• The tracking error is path dependent.

• It is also known that^a

n−1

i=0

ΔS_i S_i

₂

→ σ²T.

Tracking Error Revisited (concluded)

• The tracking error _n over n rebalancing acts (such as 251,235 on p. 649) 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 ).^b

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

aBertsimas, Kogan, & Lo (2000).

bGrannan & Swindle (1996).

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 /∂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 × Δ + n₁ + n₂ × Δ₂ − 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.