Ito’s Lemma

Trading and the Ito Integral

• Consider an Ito process dS_t = μ_t dt + σ_t dW_t. – S_t is the vector of security prices at time t.

• Let φ_t be a trading strategy denoting the quantity of each type of security held at time t.

– Hence the stochastic process φ_tSt is the value of the portfolio φ_t at time t.

• φ_t dSt ≡ φ_t(μ_t dt + σ_t dW_t) represents the change in the value from security price changes occurring at time t.

Trading and the Ito Integral (concluded)

• The equivalent Ito integral, G_T(φ) ≡

 _T

0 φ_t dS_t =

 _T

0 φ_tμ_t dt +

 _T

0 φ_tσ_t dW_t, 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 19 Suppose f : R → R is twice continuously differentiable and dX = a_t dt + b_t dW . Then f (X) is the Ito process,

f (X_t)

= f (X₀) +

 _t

0

f^(X_s) a_s ds +

 _t

0

f^(X_s) b_s dW +1

2

 _t

0

f^(X_s) b²_s ds for t ≥ 0.

aIto (1944).

Ito’s Lemma (continued)

• In differential form, Ito’s lemma becomes df (X) = f^(X) a dt + f^(X) b dW + 1

2 f^(X) b² dt.

(72)

• 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 to the Taylor expansion.

Ito’s Lemma (continued)

Theorem 20 (Higher-Dimensional Ito’s Lemma) Let W₁, W₂, . . . , W_n be independent Wiener processes and

X ≡ (X₁, X₂, . . . , X_m) be a vector process. Suppose

f : R^m → R is twice continuously differentiable and Xi is an Ito process with dX_i = a_i dt + _n

j=1 b_ij dW_j. Then df (X) is an Ito process with the differential,

df (X) =

m i=1

f_i(X) dX_i + 1 2

m i=1

m k=1

f_ik(X) dX_i dX_k, where f_i ≡ ∂f/∂X_i and f_ik ≡ ∂²f /∂X_i∂X_k.

Ito’s Lemma (continued)

• The multiplication table for Theorem 20 is

× dW_i dt

dW_k δ_ik dt 0

dt 0 0

in which

δ_ik =

⎧⎨

⎩

1 if i = k, 0 otherwise.

Ito’s Lemma (continued)

• In applying the higher-dimensional Ito’s lemma, usually one of the variables, say X₁, is time t and dX₁ = dt.

• In this case, b1j = 0 for all j and a₁ = 1.

• As an example, let

dX_t = a_t dt + b_t dW_t.

• Consider the process f(Xt, t).

Ito’s Lemma (continued)

• Then

df = ∂f

∂X_t dX_t + ∂f

∂t dt + 1 2

∂²f

∂X_t² (dX_t)²

= ∂f

∂X_t (a_t dt + b_t dW_t) + ∂f

∂t dt +1

2

∂²f

∂X_t² (a_t dt + b_t dW_t)²

=

 ∂f

∂X_t a_t + ∂f

∂t + 1 2

∂²f

∂X_t² b²_t

dt + ∂f

∂X_t b_t dW_t. (73)

Ito’s Lemma (continued)

Theorem 21 (Alternative Ito’s Lemma) Let W₁, W₂, . . . , W_m be Wiener processes and

X ≡ (X1, X₂, . . . , X_m) be a vector process. Suppose

f : R^m → R is twice continuously differentiable and Xi is an Ito process with dX_i = a_i dt + b_i dW_i. Then df (X) is the following Ito process,

df (X) =

m i=1

f_i(X) dX_i + 1 2

m i=1

m k=1

f_ik(X) dX_i dX_k.

Ito’s Lemma (concluded)

• The multiplication table for Theorem 21 is

× dW_i dt

dW_k ρ_ik dt 0

dt 0 0

• Above, ρ_ik denotes the correlation between dW_i and dW_k.

Geometric Brownian Motion

• Consider geometric Brownian motion Y (t) ≡ e^X(t) – X(t) is a (μ, σ) Brownian motion.

– Hence dX = μ dt + σ dW by Eq. (67) on p. 525.

• Note that

∂Y

∂X = Y,

∂²Y

∂X² = Y.

Geometric Brownian Motion (concluded)

• Ito’s formula (72) on p. 556 implies dY = Y dX + (1/2) Y (dX)²

= Y (μ dt + σ dW ) + (1/2) Y (μ dt + σ dW )²

= Y (μ dt + σ dW ) + (1/2) Y σ² dt.

• Hence

dY

Y =

μ + σ²/2

dt + σ dW. (74)

• The annualized instantaneous rate of return is μ + σ²/2 (not μ).^a

aConsistent with Lemma 10 (p. 282).

Product of Geometric Brownian Motion Processes

• Let

dY /Y = a dt + b dW_Y , dZ/Z = f dt + g dW_Z.

• Assume dWY and dW_Z have correlation ρ.

• Consider the Ito process U ≡ Y Z.

Product of Geometric Brownian Motion Processes (continued)

• Apply Ito’s lemma (Theorem 21 on p. 562):

dU = Z dY + Y dZ + dY dZ

= ZY (a dt + b dWY ) + Y Z(f dt + g dWZ) +Y Z(a dt + b dWY )(f dt + g dWZ)

= U(a + f + bgρ) dt + Ub dWY + Ug dWZ.

• The product of two (or more) correlated geometric Brownian motion processes thus remains geometric Brownian motion.

Product of Geometric Brownian Motion Processes (continued)

• Note that

Y = exp

a − b²/2

dt + b dW_Y , Z = exp

f − g²/2

dt + g dW_Z , U = exp

a + f −

b² + g²

/2

dt + b dW_Y + g dW_Z . – There is no bgρ term in U!

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. 566.

• Let U ≡ Y/Z.

• We now show that^a dU

U = (a − f + g² − bgρ) dt + b dW_Y − g dW_Z.

(75)

• Keep in mind that dWY and dW_Z have correlation ρ.

aExercise 14.3.6 of the textbook is erroneous.

Quotients of Geometric Brownian Motion Processes (concluded)

• The multidimensional Ito’s lemma (Theorem 21 on p. 562) can be employed to show that

dU

= (1/Z) dY − (Y/Z²) dZ − (1/Z²) dY dZ + (Y/Z³) (dZ)²

= (1/Z)(aY dt + bY dWY ) − (Y/Z²)(fZ dt + gZ dWZ)

−(1/Z²)(bgY Zρ dt) + (Y/Z³)(g²Z² dt)

= U(a dt + b dWY ) − U(f dt + g dWZ)

−U(bgρ dt) + U(g² dt)

= U(a − f + g² − bgρ) dt + Ub dWY − Ug dWZ.

Forward Price

• Suppose S follows dS

S = μ dt + σ dW.

• Consider F (S, t) ≡ Se^{y(T −t)}.

• Observe that

∂F

∂S = e^{y(T −t)},

∂²F

∂S² = 0,

∂F

∂t = −ySe^{y(T −t)}.

Forward Prices (concluded)

• Then

dF = e^{y(T −t)} dS − ySe^{y(T −t)} dt

= Se^{y(T −t)} (μ dt + σ dW ) − ySe^{y(T −t)} dt

= F (μ − y) dt + F σ dW.

– One can also prove it by Eq. (73) on p. 561.

• Thus F follows dF

F = (μ − y) dt + σ dW.

• This result has applications in forward and futures contracts.^a

aIt is also consistent with p. 515.

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)}, (76) 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.^b

• The two processes are related (see p. 578).

aCox, Ingersoll, and Ross (1985).

bBut some rates have gone negative in Europe in 2015!

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^−κt₂ , for t ≥ 0.

aWilliam Feller (1906–1970) in 1951.

Modeling Stock Prices

• The most popular stochastic model for stock prices has been the geometric Brownian motion,

dS

S = μ dt + σ dW.

• The continuously compounded rate of return X ≡ ln S follows

dX = (μ − σ²/2) dt + σ dW by Ito’s lemma.^a

aSee also Eq. (74) on p. 565. Consistent with Lemma 10 (p. 282).

Local-Volatility Models

• The more general deterministic volatility model posits dS

S = (r_t − qt) 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

aDerman and 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

. (77)

• 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 and 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 S_T = 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.

• Now, denote the implied volatility surface by Σ(X, T ) and the local volatility surface by σ(S, t).

aDerman and Kani (1997).

Local-Volatility Models (continued)

• 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,

y ≡ ln(X/St) +

 _T

t

(qs − rs) ds.

• Although this version may be more stable than Eq. (77) on p. 584, it is expected to suffer from similar problems.

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

Gatheral (2003); Wilmott (2006); Kamp (2009).

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)

• Let x ≡ ln(X/S0) − rT .

• For X large enough,^a

Σ(X, T )² < 2 | x | T .

• For X small enough,^b

Σ(X, T )² < β | x |

T for any β > 2.

aLee (2004).

bLee (2004).

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 and Hobson (2007).

bDumas, Fleming, and Whaley (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.

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.

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

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, etc.

• Numerically, inversion is an ill-posed problem.^b

aRubinstein (1994); Derman and Kani (1994); Derman, Kani, and Chriss (1996); Jackwerth and Rubinstein (1996); Jackwerth (1997); Cole- man, Kim, Li, and Verma (2000); Li (2000/2001); Moriggia, Muzzioli, and Torricelli (2009).

bAyache, Henrotte, Nassar, and Wang (2004).

Implied Trees (concluded)

• It is 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 complete solution is close.^c

aLok (D99922028) and Lyuu (2015, 2016).

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

cLok (D99922028) and Lyuu (2016).

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, (78) dV = κ(θ − V ) dt + σ√

V dW₂. (79) – 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, and 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.

aLeisen (2010); Beliaeva and 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. (78) on p. 599.

– 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. (78) and Eq. (79) on p. 599.

• 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); Lin (2007); Zhu and Lian (2012).

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).

bChen (R95723051) (2008); Chen (R95723051), Lyuu, and 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, and 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.

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. 558), 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.

aMathematically speaking, it is not quite right (Bergman, 1982).

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. (80)

• The local-volatility model (77) on p. 584 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 and Kani (1997).

Rephrase

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

Θ + rSΔ + 1

2 σ²S²Γ = rC. (81)

• Identity (81) 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.

• The option value becomes U(V, t) ≡ C(e^V , t).

• Furthermore,

∂C

∂t = ∂U

∂t ,

∂C

∂S = 1

S

∂U

∂V ,

∂²C

∂²S = 1 S²

∂²U

∂V ² − 1 S²

∂U

∂V . (82)

• Equation (82) is an alternative way to calculate gamma.^a

aSee also Eq. (43) on p. 341.

Black-Scholes Differential Equation: An Alternative (concluded)

• The Black-Scholes differential equation (80) on p. 617 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) ≡  _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. 399ff.^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).

aSee also Barraquand and Pudet (1996).

bRogers 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 − z₂ σ² 2

∂²u

∂z² = 0 with the same terminal condition.

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