Financial Derivatives and Stochastic Analysis

Financial Derivatives and StochasticAnalysis

Carl LindbergChalmers University of Technology

[email protected]

December 3, 2010

Outline

Introduction

General Probability Theory

Information and Conditioning

Brownian Motion

Stochastic Calculus

Connections with Partial Differential Equations

Risk-Neutral Pricing

Options

I A derivative is a security for which the value is derivedfrom other securities

I This course deals primarily with the two most importantcases, call and put options for stocks

I The buyer of the call option has the right, but not theobligation to buy the underlying stock S(t) from the sellerof the option at time T (the expiration date) for the priceK (the strike price)

I At time T the call option is worth

max(0, S(T )−K)

I Fundamental question: What’s a fair price of a call option,denoted c(t, S(t)), at time t < T?

Options

I A put option, p(t, S(t)), have payoff max(0,K − S(T )) attime T

I Note that

S(T )−K = max(0, S(T )−K)−max(0,K − S(T ))

I Since the value of one stock at time T minus K is equal tothe value of one call option minus one put option at timeT , these values must agree at all previous times

Options

I Hence, if the risk-free interest rate is r, we have that

S(t)− exp(−r(T − t))K = c(t, S(t))− p(t, S(t))

I This relationship is called the put-call parityI The put-call parity holds regardless of which model we use,

and it always holds in practiceI This is as far as we come down this track.I To calculate actual prices of options we need more explicit

dynamics for how stock prices behave

Options

I Call and put options are traded on exchanges and OTCI Liquidity is often not great unless you are big enough to

call a market maker, but it’s growing fastlyI Largest liquidity in equity indexes such as S&P500 and

Eurostoxx50I There are options on everything that is traded

I GoldI WheatI Crude OilI Lean HogsI ...I ..

Options

I The option price is NOT just the expected valueI Compare price of car insurance and optionI The difference is that there is a tradeable underlyingI Hence options can be hedgedI The concept of hedging is ABSOLUTELY fundamental

to option pricing theoryI Understand hedging graphically!

Futures

I A future is the right and the obligation to buy an asset ata certain strike price K at a certain time T

I A future has the value S(T )−K at time TI If the risk-free interest rate is r, the value of the future at

time t < T is S(t)−Ke−r(T−t)

I In practice, futures are designed to they have value zero atinception, so the strike price K0(t) of such a future isK0(t) = S(t)er(T−t)

Futures

I Futures are traded extensively on ”everything”I Futures are traded on exchange, otherwise they are called

forwardsI The futures markets are often very liquid, more so than the

options markets

Arbitrage

I Both the put-call parity and the derivation of the fair strikeprice for a future involved finding the value of S(T )−Ktoday

I It is intuitive that the value of S(T )−K at time t < T isS(t)−Ke−r(T−t), but is it necessary?

I The answer is affirmative! By buying the stock today, andborrow the amount Ke−r(T−t) at the risk free moneyaccount an investor can guarantee that she has theamount S(T )−K at time T

Arbitrage

I Suppose for example that someone pays herS(t)−Ke−r(T−t) + δ for δ > 0 to deliver S(T )−K at timeT

I She will then buy the stock, lend the amount Ke−r(T−t),and keep δ as a profit

I By doing so, she will have made a sure profit withouttaking any risk, a so called arbitrage

I The financial markets are efficient in the sense that such”easy money” does not exist

Mathematics

I Newton teaches us (the chain rule) that

df(x(t), t) = f ′x(x(t), t)dx+ f ′t(x(t), t)dt

I However, if dx is ”appropriately” stochastic, we will seethat

df(x(t), t) = f ′x(x(t), t)dx+ f ′t(x(t), t)dt+12f ′′xx(x(t), t)dt

I This stochastic calculus is the main mathematical tool inFinancial Mathematics

Probability

I We want to model financial asset prices as stochasticprocesses in continuous time

I Undergraduate probability theory insufficient to handlefinancial problems

I To construct probability in continuous time we needmodern mathematics based on measure theory

Why measure theory?

I We need to assign probabilities to uncountably infiniteevents

I These events are complicated, it is not possible to just sumprobabilities as in undergraduate probability

I We must also restrict ourselves to ”reasonably”complicated events – Banach-Tarski!

σ-algebras

Let Ω be a nonempty set, and let F be a collection of subsets ofΩ. We say that F is a σ-algebra if:

I ∅ ∈ FI if A ∈ F , then Ac ∈ FI if the sets A1, A2, . . . ∈ F , then

⋃∞i=1Ai ∈ F

σ-algebras

I An element ω of Ω is an outcome, chosen by Tyche,goddess of Chance

I An element A of F is an event that includes certainoutcomes ω

Probability measures

A probability measure is a function that, to every set A ∈ F ,assigns a number in [0, 1], called the probability of A andwritten P(A). We require:

I P(Ω) = 1I if A1, A2, . . . ∈ F is a sequence of disjoint sets in F , then

P

( ∞⋃i=1

Ai

)=∞∑i=1

P(Ai)

I The triple (Ω,F ,P) is called a probability space

Example

I The collection∅,Ω, A,Ac

is a σ-algebraI The function defined by P(A) = 0.4 and P(Ω) = 1 is a

probability measure on F

Example

I The smallest σ-algebra that contains all closed sets on R iscalled the Borel σ-algebra

I Note that this the Borel σ-algebra by definition contains allopen sets, too

Random variable

I A random variable is a real-valued function X defined on Ωwith the property that for every Borel subset B of R, thesubset of Ω given by

X ∈ B = ω ∈ Ω;X(ω) ∈ B

is in FI If X ∈ B ∈ F , X is measurableI The events X ∈ B generate a σ-algebra, denoted σ(X)I We have that σ(X) ⊂ F , and σ(X) can be interpreted as

the information necessary to determine the value of XI If P(A) = 1 for an event A ∈ F , we say that A occurs

almost surely

Random variable

I A random variable is a function on elements of Ω thattakes values in R

I A probability measure is a function on elements of F thattakes values in [0, 1]

Riemann vs Lebesque integral

I Riemann: ever thinner tall rectangles, re-do everything ateach new partition

I Lebesque: Approximates the function from below, usesmaller and smaller LEGO blocks

Expectation

Expectation is an integral:

E[X] =∫

ΩX(ω)dP(ω)

Expectation

Note that:I Comparison: If X ≤ Y almost surely, then

E[X] ≤ E[Y ]

I Linearity: If α, β are constants,

E[αX + βY ] = αE[X] + βE[Y ]

I Jensen’s inequality: If ϕ is a convex, real-valuedfunction defined on R, then

ϕ(E[X]) ≤ E[ϕ(X)]

Almost sure convergence

The random variables X1, X2, . . . are said to converge almostsurely to X, and we write

limn→∞

Xn = X almost surely

if the set of outcomes ω ∈ Ω for which limn→∞Xn(ω) 6= X(ω)has probability zero

σ-algebras revisited

I The σ-algebra generated by X, denoted σ(X), is thecollection of all subsets of Ω of the form X ∈ B, where Branges over the Borel subsets of R

I The intuition is that σ(X) is the information necessary todetermine the value of X

I Let G be a σ-algebra. A random variable X isG-measurable iff the information in G is sufficient todetermine the value of X, i.e. if σ(X) ⊂ G

Filtration

I By filtration, we mean a family of σ-algrebras (Ft)0≤t≤∞that is increasing, i.e., Fs ⊂ Ft if s ≤ t

I The intuition is that Ft denotes the information availableto us at time t

Filtration

I A stochastic process is a collection of random variablesindexed by t ∈ [0, T ]

I The stochastic process is called adapted if, for each t, therandom variable Xt is Ft-measurable

Independence

I Let G,H be sub-σ-algebras of F . These two σ-algebras areindependent if

P(A ∩B) = P(A)× P(B), ∀ A ∈ G, B ∈ H

I Two random variables X and Y are independent if σ(X)and σ(Y ) are independent

I If X and Y are independent,

E[XY ] = E[X]× E[Y ]

I If X and Y are independent random variables, then so aref(X) and g(Y ), for f, g real-valued functions

Conditional expectation

The conditional expectation of X given G, denoted E[X|G], isany random variable that satisfies

I E[X|G] is G-measurableI∫A E[X|G](ω)dP(ω) =

∫AX(ω)dP(ω),∀A ∈ G

Conditional expectation

Let X and Y be integrable random variablesI E[αX + βY |G] = αE[X|G] + βE[Y |G]I If X is G-measurable, then E[XY |G] = XE[Y |G]I If H is a sub-σ-algebra of G (H contains less information

than G ), then E[E[X|G]|H] = E[X|H]I If X is independent of G, then E[X|G] = E[X]

Martingales

Let M(t) denote an adapted stochastic process, 0 ≤ t ≤ T . Wesay that M is a martingale if

E[M(t)|Fs] = M(s), ∀ 0 ≤ s ≤ t ≤ T

I A martingale has no drift

Markov processes

I Let M(t) denote an adapted stochastic process, 0 ≤ t ≤ T .We say M is Markov wrt the filtration Ft iff

E[M(t)|Fs] = E[M(t)|Ms] =: EM(s)[M(t)],

almost surely ∀0 ≤ s ≤ t ≤ TI An equivalent definition is that a process is Markov iff

there are functions f, g such that

E[f(M(t))|Fs] = g(M(s))

Markov processes

I A Markov process is independent of its pastI Intuitively, the Markov property essentially says that all

there is to know about the past of a process, for predictivepurposes is where it is now

I Everything we do in this course is Markov, and virtuallyeverything in reality, too

Brownian motion

A Brownian motion (BM) W is an adapted stochastic processthat satisfies:

I W (0) = 0I W (t) is almost surely a continuous functionI The increments W (t)−W (s), s ≤ t, have distributionN(0, t− s) and are independent of Fs

Intuitively, a Brownian motion is a symmetric random walkviewed from far away.

We assume that it exists since we are not mathematicallyequipped to prove it.

Brownian motion

Figure: A symmetric random walk

Brownian motion


Brownian motion


Martingale property of Brownian motion

Brownian motion is a martingale: Let 0 ≤ s ≤ t be given. Then

E[W (t)|Fs] = [±1]= E[(W (t)−W (s)) +W (s)|Fs]= [Linearity of expectation]= E[W (t)−W (s)|Fs] + E[W (s)|Fs]= [Independent increments and measurability]= E[W (t)−W (s)] +W (s)= [Expectation of Brownian motion]= W (s)

Quadratic Variation

I The functions encountered in ordinary calculus aretypically (at least piecewise) smooth

I Brownian motion is not!I The non-smoothness is the reason that Stochastic Calculus

is different from ordinary CalculusI For example, we will see that

d(W (t)2) = 2W (t)dW (t) + dt

I If we replace W (t) by the deterministic variable x, we haveNewton’s familiar formula

d(x2) = 2xdx

Quadratic Variation

I We focus on a non-smoothness property of Brownianmotion called Quadratic Variation (QV)

I For a function f defined for 0 ≤ t ≤ T , the QV of f up totime T is

[f, f ](T ) = lim‖Π‖→0

n−1∑j=0

[f(tj+1)− f(tj)]2,

where Π = t0, t1, . . . , tn, ‖Π‖ = maxj=0,...,n−1(tj+1 − tj),and 0 < t0 < t1 < . . . < tn = T

I It is easy to show that functions with continuousderivatives have QV zero

Quadratic Variation

We have that [W,W ](T ) = T for all T ≥ 0 in mean squared:Denote the sampled QV by

QΠ =n−1∑j=0

(W (tj+1)−W (tj))2.

Quadratic Variation

We have that [W,W ](T ) = T for all T ≥ 0 in mean squared:

E[QΠ] = [Expected value is linear]

=n−1∑j=0

E[(W (tj+1)−W (tj))2]

= [W (t)−W (s) ∈ N(0, t− s), for t ≥ s]

=n−1∑j=0

(tj+1 − tj)

= [definition of Π]= T

Quadratic Variation

We have that [W,W ](T ) = T for all T ≥ 0 in mean squared:

V ar[QΠ] = [BM has independent increments]

=n−1∑j=0

V ar[(W (tj+1)−W (tj))2]

= [may be assumed to be known]

=n−1∑j=0

2(tj+1 − tj)2

≤ [definition of Π]

≤n−1∑j=0

2‖Π‖(tj+1 − tj)

= 2‖Π‖T → 0, if ‖Π‖ → 0

Markov property of Brownian motion

Brownian motion has the Markov property: Let 0 ≤ s ≤ t begiven. Then

E[W (t)|W (s)] = [±1]= E[(W (t)−W (s)) +W (s)|W (s)]= [Linearity of expectation]= E[W (t)−W (s)|W (s)] + E[W (s)|W (s)]= [Independent increments and measurability]= E[W (t)−W (s)] +W (s)= [Expectation of BM ]= W (s)= [BM is a martingale]= E[W (t)|Fs]

Exponential martingale

I We define the exponential martingale, or geometric BM(GBM) as

Z(t) = exp(σW (t)− 12σ2t)

I The GBM will be used later as a model for stock prices

Exponential martingale

The GBM is a martingale: For 0 ≤ s ≤ t, we have

E[Z(t)|Fs] = [±1]

= E[exp(σ(W (t)−W (s)))× exp(σW (s)− 12σ2t)|Fs]

= [measurability]

= exp(σW (s)− 12σ2t)× E[exp(σ(W (t)−W (s)))|Fs]

= [independence]

= exp(σW (s)− 12σ2t)× E[exp(σ(W (t)−W (s)))]

= [W (t)−W (s) ∈ N(0, t− s)]

= exp(σW (s)− 12σ2t)× exp(

12σ2(t− s))]

= Z(s)

Reflection principleReflection equality:

I Define the first passage time to level m

τm = mint ≥ 0 : W (t) = m.

I Assume that W (t) = w < m, but that max0≥s≥tW (s) ≥ mso that τm ≤ t.

I When we stand at W (τm) = m, we consider the BM −Wstarting at m at time τm. This process switches sign onevery up and down move of W from τm onward, but hasthe same distribution as W . Hence it reflects the path ofW in m. It follows that if the level m is hit by W beforetime t, so that τm ≤ t, then the probability ofW (t) ≥ 2m−w is the same as the probability of W (t) ≤ w:

P[τm ≤ t,W (t) ≤ w] = P[W (t) ≥ 2m− w]

I UNDERSTAND GRAPHICALLY!

First passage time distribution

For all m 6= 0,

P[τm ≤ t] =2√2π

∫ ∞|m|√t

e−y2

2 dy, t ≥ 0

andfτm(t) =

d

dtP[τm ≤ t] =

|m|t√

2πte−

m2

2t , t ≥ 0


Proof: Take m > 0, and set m = w in the reflection equality toobtain

P[τm ≤ t,W (t) ≤ m] = P[W (t) ≥ m].

Further, if W (t) ≥ m, we must have that τm ≤ t, so

P[τm ≤ t,W (t) ≥ m] = P[W (t) ≥ m].

Hence

P[τm ≤ t] = P[τm ≤ t,W (t) ≤ m] + P[τm ≤ t,W (t) ≥ m]

= 2P[W (t) ≥ m] =2√2πt

∫ ∞m

e−x2

2t dx

The change of variable y = x√t

gives the answer for positive m


Proof: For m < 0, τm and τ|m| have the same distribution sinceBM is symmetric. We have already derived the distribution oftha latter. The density fτm is obtained by differentiatingP[τm ≤ t]

Maximum of BM

We define the maximum to date for BM to be

M(t) = max0≤s≤t

W (s)

Note thatM(t) ≥ m if and only if τm ≤ t

Hence the reflection equality can be written as

P[M(t) ≥ m,W (t) ≤ w] = P[W (t) ≥ 2m− w], w ≤ m, m > 0

Joint density

For t > 0, the joint density of (M(t),W (t)) is

fM(t),W (t)(m,w) =2(2m− w)t√

2πte−

(2m−w)2

2t , w ≤ m, m > 0

Proof: The reflection principle gives∫ ∞m

∫ w

−∞fM(t),W (t)(x, y)dydx = P[M(t) ≥ m,W (t) ≤ w]

= [Reflection principle] = P[W (t) ≥ 2m− w]

=1√2πt

∫ ∞2m−w

e−z2

2t dz

Differentiating gives the result

The Ito integral

I We wish to make sense of∫ T

0∆(t)dW (t)

I The integrand ∆(t) will be Ft-measurable, meaning thatthe information available at time t is sufficient to evaluate∆(t) at that time

I In our applications to financial mathematics, ∆ will berelated to positions in an asset

I The problem with defining an integral is that BM has noderivative. Hence∫ T

0∆dg(t) =

∫ T

0∆(t)g′(t)dt

will not work!

The Ito integral

I Let Π = t0, t1, t2, ...tn denote a partition of [0, T ], where

0 = t0 ≤ t1 ≤ . . . ≤ tn = T

I A process ∆ which is constant in t on each subinterval[tj , tj+1) is called a simple process

I We assume that ∆ is bounded a.s.

I This is not necessary but with this assumption we canmake some proofs rigorous

I We define the stochastic integral for simple functions as

I(t) =k−1∑j=0

∆(tj)[W (tj+1)−W (tj)] + ∆(tk)[W (t)−W (tk)]

where tk ≤ t ≤ tk+1

The Ito integral

I Note that this is actually∫ T

0∆(u)dW (u)

for simple functionsI Mathematically, we obtain the integral for anyFt-measurable integrand ∆ by letting the partition Π befiner, i.e. by letting n→∞

I We will return to this issue later

The Ito integral

I Stochastic calculus is Exceptionally technical - we haveto be sketchy in ”going to the limit”!

I However, the ideas in the proof sketches are the properones - you will get the correct intuition!

I I will indicate where we are not rigorous

Martingale property of Ito integral

The Ito integral is a martingale: We assume that t and s belongto different subintervals of the partition Π. We have

E[I(t)|Fs] = [±1] = E[I(t)− I(s) + I(s)|Fs]= [measurability] = E[I(t)− I(s)|Fs] + I(s)= [independent increments] = E[I(t)− I(s)] + I(s)= [expected value of BM ] = I(s)

Ito isometry

The Ito integral satisfies

E[I2(t)] = E∫ t

0∆2(u)du

Proof: We recall that ∆ is bounded. To simplify the notation,we set Dj = W (tj+1)−W (tj) for j = 0, . . . , k − 1 andDk = W (t)−W (tk) so that the Ito integral may be written as

I(t) =k∑j=0

∆(tj)Dj

We have that

I2(t) =k∑j=0

∆2(tj)D2j + 2

∑0≤i<j≤k

∆(ti)∆(tj)DiDj .

Ito isometry

Proof continued: Now, all cross-terms have zero expectationsince, for i < j

E[∆(ti)∆(tj)DiDj ]= [independence σ(Dj) and Ftj ]= E[∆(ti)∆(tj)Di]× E[Dj ]= [expectation of BM ]= E[∆(ti)∆(tj)Di]× 0 = 0

Ito isometry

Proof continued: We have further that

E[∆2(tj)D2j ]

= [independence σ(Dj) and Ftj ]= E[∆2(tj)]× E[D2

j ]

= [variance of BM ]

= E[∆2(tj)]× (tj+1 − tj)= [∆(t) constant in [tj , tj+1)]

= E[∫ tj+1

tj

∆2(u)du]

for j = 0, . . . , k − 1, and analogously for j = k. Summing overeach j gives the result

Quadratic variation of Ito integral

The quadratic variation of the Ito integral is

[I, I](t) =∫ t

0∆2(u)du

Proof: Straightforward, try yourself with the ideas from theproof of the quadratic variation of BM

Recall that ∫ t+δ

tdW (s)2 =

∫ t+δ

tds = δ

regardless of how small we choose δ > 0

Ito integral with deterministic integrand

The Ito integral I(t) with deterministic simple integrand ∆ isnormally distributed with zero mean and variance∫ t

0∆2(s)ds.

Proof: This follows immediately from the martingale propertyof I(t) and the Ito isometry, since I(t) is a (deterministic) sumof normal variables

Ito integral with general integrand

I So far, we have done everything for piecewise constantfunctions ∆n(t)

I By choosing the intervals on which ∆n(t) are constantincreasingly smaller, ∆n(t) will look increasingly like aright-continuous (continuous from the right, not necessarilyfrom the left) process ∆(t)

I We assume now that ∆(t, ω) is continuous for each ω, andhence also bounded


I We make now this argument mathematically rigourous,and all concepts and formulas we have treated for our”old” stochastic integral

In(t) =k−1∑j=0

∆n(tj)[W (tj+1)−W (tj)]+∆n(tk)[W (t)−W (tk)]

still hold for the ”new” stochastic integral defined by

I(t) :=∫ t

0∆(u)dW (u) = lim

n→∞

∫ t

0∆n(u)dW (u) = lim

n→∞In(t)

where ∆n(t) is an increasingly good approximation of ∆(t)in the sense that

E[∫ t

0(∆n(s)−∆(s))2ds

]→ 0 (1)


I We take the simple function ∆n(t) =∑

j ∆(tj)1[tj ,tj+1)

I Since it is bounded and continuous, we have∫ T

0(∆(t)−∆n(t))2dt→ 0

as n→∞ for each ω. Hence

E[∫ T

0(∆(t)−∆n(t))2dt]→ 0

by the Bounded Convergence Theorem


I The idea behind the construction is that, for each t, we canapply the Ito isometry to obtain

E[(In(t)− Im(t))2] = E[∫ t

0(∆n(u)−∆m(u))2du].

Here, the RHS goes to zero as m,n→∞ due toequation(1) above

I Hence E[(In(t)− Im(t))2]→ 0 (which is the definition of aCauchy sequence) in L2(Ω,F ,P), which is complete

I Completeness of L2(Ω,F ,P) means that every Cauchysequence converges to an elements of that space.

I We now define the Ito integral to be that limit

Ito’s formula

I Ito’s formula is THE fundamental result of stochasticcalculus

I Nonetheless, we are forced to cheat with the derivation of itI The following statements are all true, and the

intuition is the correct oneI Let f(t, x) be a twice continuously differentiable function,

then

df(t,W (t)) = ft(t,W (t))dt+fx(t,W (t))dW (t)+12fxx(t,W (t))dt

I Note that f(t)dt and∫f(t)dt are equivalent notations

Ito’s formula

”Proof”: Taylor’s theorem gives that

df(t,W (t)) := f(t+ dt,W (t+ dt))− f(t,W (t))= ft(t,W (t))dt+ fx(t,W (t))(W (t+ dt)−W (t))

+12fxx(t,W (t))(W (t+ dt)−W (t))2

+ ftx(t,W (t))dt(W (t+ dt)−W (t))

+12ftt(t,W (t))dt2

+ higher order terms

Ito’s formula

”Proof” continued:

= ft(t,W (t))dt+ fx(t,W (t))dW (t) +12fxx(t,W (t))dW (t)2

+ ftx(t,W (t))dtdW (t) +12ftt(t,W (t))dt2

+ higher order terms

Ito’s formula

”Proof” continued:I We know by the quadratic variation of the Ito integral that

12fxx(t,W (t))dW (t)2 =

12fxx(t,W (t))dt

Ito’s formula

”Proof” continued: We have also that

|∫ T

0ftx(t,W (t))dW (t)dt| ≤ maxdt×|

∫ T

0ftx(t,W (t))dW (t)| = 0

and

|∫ T

0ftt(t,W (t))dtdt| ≤ maxdt × |

∫ T

0

12ftt(t,W (t))dt| = 0

This gives the result, since the higher order terms vanish

Differential rules of calculation

We have now three important rules of calculation:I dW (t)dW (t) = dt (QV of BM)I dW (t)dt = 0I dtdt = 0

In addition, the same argument as we used to prove that[W (t),W (t)] = t gives that dW1(t)dW2(t) = 0 for independentBrownian motions W1,W2

Ito processes

I An Ito process is a stochastic process of the form

X(t) = X(0) +∫ t

0∆(u)dW (u) +

∫ t

0Θ(u)du

where ∆(u) and Θ(u) are adapted stochastic processesI Note: An Ito process is an Ito integral + drift

Quadratic variation of Ito process

I The quadratic variation of the Ito process is

[X,X](t) =∫ t

0dX(t)dX(t) =

∫ t

0∆2(u)du

I Note: The differentiable drift term does not contribute tothe QV, not ”spikey” enough!

Quadratic variation of Ito process

Proof: By using the definition dX(t) = ∆(t)dW (t) + Θ(t)dt, weget that

[X,X](t) =∫ t

0dX(t)dX(t)

=∫ t

0∆2(t)dW (t)dW (t)

+∫ t

02∆ΘdW (t)dt+

∫ t

0Θ2(t)dtdt

= [calculation rules]

=∫ t

0∆2(t)dt

Ito’s formula for Ito processes

Let X(t) be an Ito process, and let f(t, x) be a twicecontinuously differentiable function, then

df(t,X(t)) = ft(t,X(t))dt+ fx(t,X(t))dX(t)

+12fxx(t,X(t))dX(t)dX(t)

= ft(t,X(t))dt+ fx(t,X(t))dX(t)

+12fxx(t,X(t))∆2(t)dt

The stock price model

I Throughout the course, we will use the geometric BM(GBM) as the stock price model

I The GBM is the process

S(t) = S(0) exp(

(α− σ2

2)t+ σW (t)

)where the drift α and the volatility σ are constants

I By applying Ito’s formula, we get that the dynamics forGBM are

dS(t) = αS(t)dt+ σS(t)dW (t)

The money market account

I Throughout the course, we will assume constant interestrate for the money market account

I The money market account follows the process

B(t) = B(0) exp (rt)

where r is the constant interest rateI Since the money market account is deterministic, we see by

ordinary calculus that its dynamics are

dB(t) = rB(t)dt

Options revisited

I A European call option pays the amount maxS(T )−K, 0at time T

I An American call option allows the holder of the derivativeto exercise the contract at any time t ≤ T to obtainmaxS(t)−K, 0

I The terms European and American are used for anycontract to indicate whether or not the possibility of earlyexercise is available to the holder of the option

I We treat only European contracts in this course!I (You’re not missing out on much)

The idea

I The fundamental idea in option pricing theory is this: Ifwe can construct an actively managed self-financingportfolio consisting of the stock S(t) and the moneyaccount B(t) such that, at all times, this portfolio has thesame value as the option, then that portfolio and theoption have ... well... the same value

I (Self-financing means that no money is added or withdrawnfrom the portfolio)

I Take some time to understand this (graphically)!

The idea

I Intuition: If we can do this, we have ”produced” anoption from trading the stock and the money account, andits price is its production cost

I A producer of options can then, in principle, sell optionswithout taking risk!

I This idea is called ∆-hedging and it is THE major reasonfor the tremendous success of option pricing theory

I In practice, all risk cannot be eliminated, but almostI How do market makers do their business?

∆-hedging and the Black-Scholes pde

We construct a portfolio consisting of ∆(t) stocks S(t) and theremaining capital, positive or negative (which will be the case),is in a cash position in the money account. The portfolio valueX(t) has the dynamics

dX(t) = ∆(t)dS(t) + r(X(t)−∆(t)S(t))dt= ∆(t)(αS(t)dt+ σS(t)dW (t)) + r(X(t)−∆(t)S(t))dt= rX(t)dt+ ∆(t)(α− r)S(t)dt+ ∆(t)σS(t)dW (t)


We assume that the option price is twice continuouslydifferentiable, so that Ito’s formula applies. The call option chas dynamics

dc(t) = [Ito formula]

= ct(t, S(t))dt+ cx(t, S(t)dS(t) +12cxx(t, S(t))dS(t)dS(t)

= [calculation rules, dynamics of S(t)]= ct(t, S(t))dt+ αS(t)cx(t, S(t))dt+ σS(t)cx(t, S(t))dW (t))

+12cxx(t, S(t))σ2S2(t)dt


Now, X(t) = c(t) for all t if and only if X(0) = c(0, S(0)) (thestarting values are the same) and dX(t) = dc(t) (the valueschange identically) :

dc(t) = ct(t, S(t))dt+ αS(t)cx(t, S(t))dt+ σS(t)cx(t, S(t))dW (t))

+12cxx(t, S(t))σ2S2(t)dt

= rX(t)dt+ ∆(t)(α− r)S(t)dt+ ∆(t)σS(t)dW (t) = dX(t)

For the equality to hold, we need that

∆(t) = cx(t, S(t)) ∀t ∈ [0, T ).


I If we insert X(t) = c(t, S(t)) and ∆(t) = cx(t, S(t)) for allt ∈ [0, T ) in the equation above, we get that the drift termα cancels out, and that the call option price c(t, S(t)) mustsatisfy the partial differential equation

ct(t, x) + rxcx(t, x) +σ2x2

2cxx(t, x) = rc(t, x)

∀t ∈ [0, T ), x ≥ 0, with terminal condition

c(T, x) = max(0, x−K)

I The terminal condition must hold because the optionc(t, S(t)) has the value max(0, S(T )−K) at the time ofexpire T .

I This equation is known as the Black-Scholes equation


So what does this tell us?I An investor can sell an option at time 0 for the pricec(0, S(0))

I If she holds the hedge ∆(t) = cx(t, S(t)) stocks for allt ∈ [0, T ), finance by lending from the money account, thenX(t) = c(t, S(t)) for all t ∈ [0, T ) and at the terminal dateT she will have

X(T ) = c(T, S(T )) = max(0, S(T )−K)

to give to the option buyer, which is exactly what she owesI This holds regardless of the path the stock price

has followed, so c(t,S(t)) must be the only correctprice!

I ONCE AGAIN, UNDERSTAND GRAPHICALLY!

Arbitrage

So what if our model is correct, and a call option trades at aprice c(t, S(t)) + δ that is higher than c(t, S(t))?

I An investor can then sell an option at time t for the pricec(t, S(t)) + δ

I She then sets up the correct hedging portfolio, which isguaranteed to have the same value as the call option atall times

I At expiry T , she will have max(S(T )−K, 0) to give to theperson she sold the option to

I She is then left with the positive amount δ, a Sure Profit!I We can reason analogously if the price of the option is too

low

Arbitrage

I A trade that has strictly positive probability to make aprofit, but with no risk, i.e. zero probability of loss, iscalled an arbitrage

I Arbitrage cannot be possible, or at least not easilyaccessible in a reasonably efficient market

The solution to the Black-Scholes pde

I The solution to the Black-Scholes pdf is given by

c(t, s) = sΦ(d+(T − t, x))−Ke−r(T−t)Φ(d−(T − t), s),

for 0 ≤ t < T , and s > 0, where

d±(τ, s) =1

σ√τ

[log

s

K+(r ± σ2

2

)τ

]and Φ is the cdf for the standard normal distribution

I We will return to the derivation of this solution later

Feynman-Kac formula

Feynman-Kac: Consider the stochastic differential equation

dX(t) = β(t,X(t))dt+ γ(t,X(t))dW (t) (2)

Let X(t) be a solution to (2) with initial conditions given attime 0, h(y) is a function, r, T > 0, and let t ∈ [0, T ]. Define thefunction

f(t, x) = Et,x[e−r(T−t)h(X(T ))].

Then

ft(t, x) + β(t, x)fx(t, x) +12γ2(t, x)fxx(t, x) = rf(t, x) (3)

and the terminal condition

f(T, x) = h(x) ∀x ∈ R

Feynman-Kac formula

Note:I Eq (3) is actually on the same form as the Black-Scholes

equationI Hence this theorem will give us the price of an option with

payoff h(x) at time T as the solution to this equationI Solutions to stochastic differential equations of the form of

Eq (3) are Markov processesI This fact is intuitive but hard to prove

Feynman-Kac formula

Idea of proof:I We first prove that a function g(t,X(t)), is a martingale.

The function g will be related to fI We then take the stochastic differential of g(t,X(t)) and

set the drift term dt to zero. This must hold for g(t,X(t))to be a martingale, which we know it is

I Setting the drift term dt to zero yields Eq (3)

Feynman-Kac formula

A martingale lemma: Let X(u) be a solution to (2) with initialconditions given at time 0, h(y) is a function, fix T > 0, and let

g(t, x) = Et,x[h(X(T ))].

Then the stochastic process

g(t,X(t)), 0 ≤ t ≤ T

is a martingale

Feynman-Kac formula

Proof martingale lemma: Let 0 ≤ s ≤ t ≤ T be given. TheMarkov property of X implies that

E[h(X(T ))|Ft] = EX(t)[h(X(T ))] = g(t,X(t))

We have then

E[g(t,X(t))|Fs] = [Markov property]= E[E[h(X(T ))|Ft]|Fs]= [iterated conditioning]= E[h(X(T ))|Fs]= [Markov property]= g(s,X(s))

Feynman-Kac formula

Proof Feynman-Kac formula: Let X(t) solve Eq (2) starting attime 0. Then

e−rtf(t,X(t)) = E[e−rTh(X(T ))|Ft].

is on the form required by the martingale lemma, hencee−rtf(t,X(t) is a martingale. We have that the differential hasdynamics

d(e−rtf(t,X(t))) = e−rt[−rfdt+ ftdt+ fxdX +

12fxxdXdX

]= e−rt

[−rf + ft + βfx +

12fxx

]dt+ e−rtγfxdW

Hence, for e−rtf(t,X(t) to be a martingale, which we know itis, the dt term needs to be zero. This yields Eq (3)

Price of call option

I For h(x) = max(x− k, 0), we have now the price of a calloption given by

c(t, S(t)) = Et,S(t)[e−r(T−t) max(S(T )−K, 0)]

I HOWEVER, looking at Eq (2), the stock S(t) hasdifferent dynamics than what we assumed initially, namely

dS(t) = rS(t)dt+ σS(t)dW (t)

I Hence IT ALWAYS HOLDS that the price of an optionwith payoff h(x) is the expectation of h(S(T )) if wepretend that the drift of the stock is the interestrate r!

I The actual drift α is irrelevant for pricing purposes!


I The fact that the price of an option does not depend on thedrift α is THE major reason for its success

I The drift α is ”impossible” to estimate accurately:


Fix [0, T ] and choose n, set h = T/n, ti = ih, i = 1, . . . , n, and

Xi = logS(ti)S(ti−1)

= αh+ σ√hGi

for i = 1, . . . , n where Gi ∈ N(0, 1) are independentThe estimator

α =1T

n∑i=1

Xi

satisfies E[α] = α, and V ar(α) = σ2

T , which is independent of n


I Hence a 95% confidence interval with width 0.01 of αbecomes α± 1.96σ/

√T

I If σ = 0.3, then T = 3457 years!I After observing a stationary market for three thousand

years we still wouldn’t have accurate option prices!I On the contrary, the volatility can, in principle, be

estimated arbitrarily well by sampling more often


We calculate now the price of the call option c(t, S(t))

c(t, S(t)) = Et,S(t)[e−r(T−t) max(S(T )−K, 0)]

= [(W (T )−W (t)) =d

√T − tX, X ∈ N(0, 1)]

= Et,S(t)[e−r(T−t) max(S(t)e(r−σ

2

2)(T−t)+σ

√T−tX −K, 0)]

= [S(T ) ≥ K =⇒ −X ≤ d−(T − t, S(t))]

=[−X ∈ N(0, 1), d±(τ, s) =

1σ√τ

[log

s

K+(r ± σ2

2

)τ

]]=∫ d−(T−t,S(t))

−∞e−

σ2

2(T−t)−σ

√T−tx−x

2

2 dx

+ e−r(T−t)∫ d−(T−t,S(t))

−∞Ke−

x2

2 dx


= [change of variable : y = x+ σ√T − t]

=∫ d+(T−t,S(t))

−∞e−

y2

2 dy

+ e−r(T−t)∫ d−(T−t,S(t))

−∞Ke−

x2

2 dx

= S(t)Φ(d+(T − t, x))−Ke−r(T−t)Φ(d−(T − t), s)

I By the put-call parity, or by calculations analogous toabove, we can get the price of the put option (with payoffmax(K − S(t), 0)):

p(t, S(t)) = Ke−r(T−t)Φ(−d−(T−t), s)−S(t)Φ(−d+(T−t, x))

More exotic options

I Recall that the framework above allows us to price anycontract with a payoff that can be described by a functionh(x)

I However, it can also be modified to handle contracts thatare more advanced, such as path-dependent options, andoptions on several underlying stocks

I The intuition and ideas are the same, and we will not coverthese contracts in detail in this course

I The vast majority of options are calls or putsI You can trade virtually any contract you want

over-the-counter (at horrible bid-ask spreads!)

Risk-neutral pricing

I We have now solved the problem of pricing options


I We recall that options are priced as if the stock S(t) haddrift r

I If we discount the stock process used in the pricing,henceforth denoted S(t), we get that

d(e−rtS(t)) = σe−rtS(t)[dW (t)],

which is a martingaleI If we discount the actual stock S(t), we get that

d(e−rtS(t)) = (α− r)e−rtS(t)dt+ σe−rtS(t)dW (t)= σe−rtS(t)[Θdt+ dW (t)]

for the market price of risk

Θ =α− rσ


I Analogously, we get that the differential of the discountedactual portfolio process X(t) and the discounted pricedportfolio process X(t) become

d(e−rtX(t)) = ∆(t)σe−rtS(t)[Θdt+ dW (t)]

andd(e−rtX(t)) = ∆(t)σe−rtS(t)dW (t),

where the latter is a martingaleI We make the interpretation that the market prices

according to a different probability measure than the actualprobability measure P, such that stock prices and portfolioprocesses are martingales

Change of Measure

Let the random variable Z be an almost surely nonnegativerandom variable with E[Z] = 1. For A ∈ F , define

P(A) =∫AZ(ω)dP(ω).

Then P is a probability measure. Furthermore, if X is a randomvariable with E[|X|] <∞, then

E[X] = E[XZ]

Equivalent measures

Two probability measures P and P on (Ω,F) are said to beequivalent if they agree which sets in F have probability zero

Radon-Nikodym

Let P and P be two probability measures defined on (Ω,F).Then there exists an almost surely positive random variable Zsuch that E[Z] = 1 and

P(A) =∫AZ(ω)dP(ω) for every A ∈ F

Hence E[X] = E[ZX]. The variable Z is called theRadon-Nikodym derivative of P w r t P, and we write

Z =dPdP

Further,Z(t) = E[Z|Ft],

for 0 ≤ t ≤ T is the Radon-Nikodym process.

Three small Radon-Nikodym lemmas

Lemma 1: The Radon-Nikodym process Z(t) is a martingale.

Proof:

E[Z(t)|Fs] = [definition]= E[E[Z|Ft]|Fs] = [tower property]= E[Z|Fs] = [definition]= Z(s)

Three small Radon-Nikodym lemmas

Lemma 2 (Lemma 5.2.1. in the textbook): Let Y beFt-measurable. Then

E[Y ] = E[Y Z(t)].

Proof:

E[Y ] = [definition]= E[Y Z] = [tower property]= E[E[Y Z|Ft]] = [measurability]= E[Y E[Z|Ft]] = [definition]= E[Y Z(t)]

Three small Radon-Nikodym lemmasLemma 3 (Lemma 5.2.2. in the textbook): Let Y beFt-measurable. Then

E[Y |Fs] =1

Z(s)E[Y Z(t)|Fs].

Proof: We want to verify that the definition of conditionalexpectation holds for 1

Z(s)E[Y Z(t)|Fs]. For A ∈ Fs,∫A

1Z(s)

E[Y Z(t)|Fs]dP = E[1A1

Z(s)E[Y Z(t)|Fs]] = [Lemma 5.2.1.]

= E[1AE[Y Z(t)|Fs]] = [1A ∈ Fs]= E[E[1AY Z(t)|Fs]] = [tower property]= E[1AY Z(t)]

=∫AY dP

Levy’s theorem:

The following result helps us to recognize a Brownian motion:

Levy’s theorem: Let M(t) be a continuous martingale w.r.t. Ft,with M(0) = 0. If [M,M ](t) = t for all t ≥ 0, then M(t) is aBrownian motion.

Girsanov

Girsanov’s theorem: Let Θ be an adapted process. Define

Z(t) = exp(−ΘW (t)− 1

2Θ2t

),

W (t) = W (t) + Θt,

and assume that

E[∫ T

0Θ(u)2Z(u)2du] <∞.

Set Z = Z(T ). Then E[Z] = 1 and W (t) is a BM under theprobability measure P

Girsanov

Proof Girsanov’s theorem: We want to use Levy’s theorem toprove that W is a Brownian motion. The process W starts atzero and is continuous almost surely by definition, hencecontinuity holds for both P and P. Further, we may assumethat we know that [W,W ](t) = t holds almost surely, not just inmean square which we proved. It follows that for both P and P,

dW (t)dW (t) = [definition]

= (dW (t) + Θ(t)dt)2 = [calculation rules]= dW (t)dW (t) = t

Girsanov:

Proof Girsanov’s theorem, continued: It remains to show thatW is a martingale under P. We have that Z(t) is a martingalesince

dZ(t) = [Ito] = −ΘZ(t)dW (t).

Since Z(t) is a martingale and Z = Z(T ), we have

Z(t) = [Z(t) martingale]= E[Z(T )|Ft] = [Z = Z(T )]= E[Z|Ft]

Hence, Z(t) is a Radon-Nikodym process, and Lemmas 5.2.1.and 5.2.2. apply.

Girsanov:

Proof Girsanov’s theorem, continued: We have that

d(W (t)Z(t)) = [Ito product rule]

= (−W (t)Θ(t) + 1)Z(t)dW (t),

so (W (t)Z(t) is a martingale under P. Now,

E[W (t)|Fs] = [Lemma 5.2.2.]

=1

Z(s)E[W (t)Z(t)|Fs] = [W (t)Z(t) a martingale]

=1

Z(s)W (s)Z(s)

= W (s)

Girsanov:

I Note that P and P are equivalent since P[Z > 0] = 1I The probability measure P is often referred to as the

risk-neutral (martingale) probability measureI In our model, the risk-neutral measure is unique, since no

other Θ gives zero drift of W (t) under the probabilitymeasure P

I In general, we call a probability measure risk-neutral if it isequivalent to the actual measure P and renders anydiscounted portfolio price process e−rtX(t) (includinge−rtS(t)) into a martingale

Why measure change?

I We know that the price c(t, S(t)) of a call option must be

c(t, S(t)) = e−rτ E[max(S(t)e(r−σ2/2)(T−t)+σ(W (T )−W (t))−K, 0)],

where we take expectation w.r.t. the measure P for whichS(t) has drift r, and W is a Brownian motion

I But S(t) can still be viewed as having drift α

S(t) = S(0)e(r−σ2/2)t+σW (t)

= [W (t) =α− rσ

t+W (t)]

= S(0)e(α−σ2/2)t+σW (t).

I But W is not a Brownian motion under P!

Why measure change?

I We need to change measure to P to evaluate the price

e−rτ E[max(S(t)e(α−σ2/2)(T−t)+σ(W (T )−W (t)) −K, 0)].

I But then we get

e−rτE[Z(t) max(S(t)e(α−σ2/2)(T−t)+σ(W (T )−W (t)) −K, 0)].

I In other words, the pricing formula becomes neater underP, AND it is then also independent of the unobservabledrift α

General pricing formula

I We can now express the Feynman-Kac price, for which wepretended that the drift of the stock was r, as theexpected value of the payoff h(S(T )) with respect to therisk-neutral measure P:

f(t, S(t)) =1e−rt

E[e−rTh(S(T ))|Ft]

I Since the price f , by ∆-hedging, is always equal to thehedging portfolio process X(t) (which is a martingale), thediscounted price e−rtf(t, S(t)) is also a martingale:

e−rtf(t, S(t)) = E[e−rT f(T, S(T ))|Ft]

I We see by these formulas that the price and the payoffcoincides at T (which we already knew by ∆-hedging!)

General pricing formula

I Take now any FT -measurable payoff V (T )I It can be shown, although we will not do so, that the pricev of a derivative with payoff V (T ) is

v(t) =1e−rt

E[e−rTV (T )|Ft]

I Hence we can price anything by calculating itsexpected value under the risk-neutral probabilitymeasure!!

Monte Carlo

Not every contract can be priced explicitly....I The Monte Carlo technique is the most applied pricing

method, alongside solving the partial differential equations(pde) of the option prices numerically

I Monte Carlo can, in principle, handle all contracts - pdecan not!

I Further, it will always give back the correct price! - pdedoes not!

I Unfortunately, it is rather slow - pde is not!I We will not treat the pde method in this course...

Monte Carlo

I The idea of Monte Carlo is to use the Strong Law of LargeNumbers, which tells us that

v(t) =1n

n∑i=1

Vi(T )→ 1e−rt

E[e−rTV (T )|Ft] = v(t),

almost surely as n→∞, where Vi(T ) are i.i.d. outcomes ofV (T ) and v(t) is the price of the derivative

I The i.i.d. outcomes Vi(T ) will in practice be simulated on acomputer

I Since the Vi(T ) are i.i.d., the Central Limit Theorem givesus confidence intervals for the pricing error:

v(t)± 1.96σV (T )/√n

Monte Carlo and hedging

I We can also hedge using the Monte Carlo methodI The idea here is to use numerical differentiationI Assume that we have a contract with payoff h(S(T )),

which has value

f(t, x) = e−r(T−t)Et, x[h(S(T ))]

by the Feynman-Kac formulaI We know from the derivation of Black-Scholes equation

that the correct hedge ratio is

∂f(t, x)∂x

≈ f(t, x+ h)− f(t, x− h)2h

for a small h

Monte Carlo and hedging

I Note that we use the centred approximation of thederivative, as it is more accurate

I The right hand side of the equation can be evaluatednumerically

I Use the same simulated outcomes for both f(t, x+ h) andf(t, x− h) - this saves time and reduces the error

I For general models, the correct hedging ratio for a pricev(t, x) is Not necessarily v′x(t, x) - Remember this!

Arbitrage revisited

I An arbitrage is a portfolio process X(t), X(0) = 0,satisfying for some time T

P[X(T ) ≥ 0] = 1, P[X(T ) > 0] > 0

I It is a reasonable sanity check for a mathematical modelthat it does not allow arbitrage, since arbitrage cannotexist consistently in reality

I The Black-Scholes model is arbitrage free, as we will nowsee

Fundamental theorem of asset pricing

Fundamental theorem of asset pricing: If a market model has arisk-neutral probability measure, then it does not admitarbitrage

I The Black-Scholes model has a risk-neutral probabilitymeasure

I Hence there is no way to make a sure and risk free profitwithin this model

I This theorem is essentially a model sanity check

Fundamental theorem of asset pricing:

Proof of fundamental theorem of asset pricing:I We assume that a model has a risk-neutral probability

measure PI Let X(t) be a portfolio process with X(0) = 0I Since any discounted portfolio process is a martingale

under P, we have that

E[e−rTX(T )] = X(0) = 0 (4)

Fundamental theorem of asset pricing:

Proof of fundamental theorem of asset pricing:I Suppose now that P[X(T ) < 0] = 0I Since P and P are equivalent, P[X(T ) < 0] = 0I This implies P[X(T ) > 0] = 0, since Eq (4) holds ande−rt > 0

I Once again, since P and P are equivalent, P[X(T ) > 0] = 0I Hence X(t) is not an arbitrage!

Implied volatility

I The measure P ”emerges” from our calculations as amathematical construction, and has no real meaning

I Nonetheless P is often given the interpretation that itreflects the views of the market

Implied volatility

I However, the volatility σ is the same for the stock S(t)under both probability measures

I The volatility σ that makes our theoretical price c(t, S(t))agree with the market price is called the volatility impliedby the market, or the implied volatility

I If the Black-Scholes model would have given a perfectdescription of real option markets the implied volatilitywould be more or less independent of the time to expiry(T − t) and the strike price K

I This is Not the case!I The implied volatility is the wrong number in the

wrong model to yield the right price!

Implying the risk-neutral distribution:

We know that the call option has the price

c(0, T, S(0),K) = E[e−rT max(S(T )−K, 0)]

If we denote the risk-neutral density of S(T ) when S(0) = x byp(0, T, x, y), then

c(0, T, x,K) = e−rT∫ ∞K

(y −K)p(0, T, x, y)dy (5)

Implying the risk-neutral distribution:

Now,

cK(0, T, x,K) = −e−rT∫ ∞K

p(0, T, x, y)dy = −e−rT P[S(T ) > K]

andcKK(0, T, x,K) = e−rT p(0, T, x, y)

I This argument holds regardless of the dynamics ofS(t)!

I Hence, if we have sufficiently many strike prices K for anoption, we can find the distribution of the underlyingstock, as it is implied by the market

I The distribution p(0, T, x, y) is what you compare yourviews to when you trade

Local volatility, implying the volatilitysurface:

I We know from Feynman-Kac that we can find the price ofany option for which the dynamics of the stock S(t) underP satisfies

dS(t) = β(t, S(t))dt+ γ(t, S(t))dW (t)

I Recall that the change of measure from P to P changes thedrift αS(t) to rS(t) (or more generally, to β(t, S(t)))

I The volatility σ(t, S(t))S(t) (or more generally, γ(t, S(t)))is unchanged by the measure change

I We want to find a non-constant and non-random functionσ(t, x) such that the model prices give a perfect fit to agiven set of option prices observed in the market


I We assume as given that the risk-neutral densityp(t, T, x, y) satisfies the Kolmogorov forward equation

∂

∂Tp(t, T, x, y) = − ∂

∂y(ryp(t, T, x, y))

+12∂2

∂y2(σ2(T, y)y2p(t, T, x, y))

I We have from Equation (5) that

cT (0, T, x,K) = −rc(0, T, x,K) (6)

+ e−rT∫ ∞K

(y −K)pT (0, T, x, y)dy


We want to use the Kolmogorov forward equation, so weintegrate by parts to show that

−∫ ∞K

(y −K)∂

∂y(ryp(0, T, x, y))dy =

∫ ∞K

ryp(0, T, x, y)dy,

(7)

where we assume that

limy→∞

(y −K)ryp(0, T, x, y) = 0


Further, we integrate by parts and integrate again to show that

12

∫ ∞K

(y −K)∂2

∂y2(σ2(T, y)y2p(0, T, x, y))dy (8)

=12σ2(T,K)K2p(0, T, x,K),

if

limy→∞

(y −K)∂

∂y(σ2(T, y)y2p(0, T, x, y)) = 0

limy→∞

σ2(T, y)y2p(0, T, x, y) = 0


I Using Equations (6), (7), and (8) in the Kolmogorovforward equation, and the fact that

p(0, T, x,K) = erT cKK(0, T, x,K)

we get that

cT (0, T, x,K) = −rKcK(0, T, x,K) (9)

+12σ2(T,K)K2cKK(0, T, x,K)

I Since cT , cK , and cKK can be approximated from themarket prices, we can find the function σ(T,K)

I This approach is called Local VolatilityI It is very much used in the financial industry

Financial Derivatives and Stochastic Analysis

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