Theory of Contingent Claims - EISTIet.perso.eisti.fr/pdfs/th-cont-claims2013-11-06.pdf ·...

IntroductionMono-Period Market, RECALL

Discrete Time MarketsContinuous Time Markets

Theory of Contingent Claims

Erik Taflin, EISTI

M2 QFRM and IFI ING3: Version 2013-11-06

Erik Taflin, EISTI Theory of Contingent Claims



Outlines I

IntroductionSome different types of derivativesMarket ModelsArbitrage PricingFrictionless and Ideal Market

Mono-Period Market, RECALLProbabilistic ModelArbitrage and Equivalent Martingale MeasurePricing of Derivative Products




Outlines IIDiscrete Time Markets

Probabilistic ModelBinomial Model I

PortfoliosArbitrage and Equivalent Martingale MeasurePricing of Derivative Products

Some examples for the Binomial Model

American Derivatives

Continuous Time MarketsErik Taflin, EISTI Theory of Contingent Claims



Outlines III

Original Black-Scholes Model and FormulaThe greeksGeneralized Black-Scholes modelArbitrage and e.m.m.Pricing derivativesExamples of Generalized Black-Scholes model

Non-deterministic Interest Rate1-dim. local volatility modelsSimple example of stochastic volatility model and volatility smile




Outlines IVThe Heston model




Some different types of derivativesMarket ModelsArbitrage PricingFrictionless and Ideal Market

1. Introduction

• Financial Asset: Firstly, a contract, which only generate flows of money is afinancial assets. Secondly, a contract, which only generate flows of otherfinancial assets is also a financial asset.• Financial Derivative: Financial asset, whose price only depends on the valueof other more basic underlying variables, such as Stock prices, Bond prices,Temperature, Snow depth, . . .• (Non-financial) Derivatives ∃ since thousands of years: Forward Contractson raw products, s.a. wheat• Synonyms in this course: Financial Derivative, Derivative, DerivativeSecurity, Contingent Claim, . . .• Fundamental Problem: Find a fair price of a derivative (evaluation). Black,Merton and Scholes (∼1973)





• Role of Derivatives:

- Hedge (cover) risks for some

- Portfolio management and speculation for others

- Arbitraging for a small number





1.1 Some different types of derivatives

Underlying Assets: Spot price of a stock, Future price of a stock, Exchangerate ($/e, . . . ), . . . . Underlying Assets also called Primary AssetsA) ForwardForward (contract): Contract that stipulates that its holder can and shall buythe underlying for a predetermined amount K (the delivery price) at a givenfuture time T (the time of maturity). The delivery price K is determined suchthat the value of the contract is zero at the contract dateForward Price: Let t0 be the contract date. Then, by definition, the ForwardPrice of the underlying at t0 for delivery at T is K .





B) Options• European Call Option: Contract that stipulates that its holder can buy theunderlying for a predetermined amount K (the strike) at a given future timeT (the time of maturity or exercise date).• American Call Option: As European except that all exercises dates t ≤ Tare allowed.• Put Options: Substitute sell in place of buy in def. of Call• Pay-Off at exercise time te and price of underlying Ste :

- Call: (Ste − K )+

- Put: (K − Ste )+

• Other Options: Asian, Barrier, Caps, Floors, Swaptions, Straddle,Bermudan, Russian, . . .





1.2 Market Models• Mono-Period Models: Trading dates T = 0,T,

| |

t = 0 t = T

• Discrete Time Models: Trading dates T = 0, 1, . . . ,T − 1,T,| | | | | |

0 1 2 T −1 T

• Continuous Time Models: Trading dates T = [0,T ] (Most simple andrealistic),

| | | | | |

0 1 2 T −1 T





1.3 Arbitrage Pricing

• An Arbitrage Portfolio in a financial market (with a risk-free asset) is aself-financing portfolio θ, whose value Vt(θ) at date t ∈ T satisfies:

i) V0(θ) = 0

ii) VT (θ) ≥ 0

iii) P(VT (θ) > 0) > 0

• Arbitrage Free Market: @ an arbitrage portfolio; AOA• Arbitrage Pricing of Derivatives in a arbitrage free market of underlyingassets: The price of a derivative is determined such that the extended marketof underlying assets and the derivative is Arbitrage Free





Efficient Market Hypothesis: Asset prices reflect all information and no onecan earn excess returns with certainty.AOA is a more general and precise mathematical formulation of the EfficientMarket Hypothesis.





1.4 Frictionless and Ideal MarketWe make several simplifying hypotheses concerning the financial market(frictionless market):

I The number of financial assets is constant in time

I Asset prices take real values. No dividends are payed

I One can sell and buy any real number multiple of an asset

I Buy and sell prices are equal (i.e. no transaction costs)

I Lend and borrow interest rates are equal

I The price of asset is independent of the amount one buy and sell(Price-taker market)

I All information is public

A more realistic market can be obtained by modifying the Ideal Market, withtransaction costs and other frictions




Probabilistic ModelArbitrage and Equivalent Martingale MeasurePricing of Derivative Products

2. Mono-Period Market2.1 Probabilistic Model

• Trading dates: T = 0,T• Probability space: (Ω,P,F), Ω set of elementary events, P a prioriprobability, F σ-algebra of possible events• Usually, but not always: Ω = ω1, . . . , ωK, where K is the number ofelementary events, pi = P(ωi ) > 0, F = subsets of Ω• N general Assets with prices S1, . . . ,SN (Quoted Spot Prices) and possiblyone more asset, a risk-free Asset with deterministic price S0.• Spot prices at t = 0 : S0 = (S1

0 , . . . ,SN0 ) ∈ RN if N assets;

S0 = (S00 ,S

10 , . . . ,S

N0 ) ∈ RN+1 and S0

0 > 0, if N + 1 assets. By defaultS0

0 = 1, if not specified differently.





• Spot prices at T : ST = (S1T , . . . ,S

NT ) is a random vector in RN if N assets;

ST = (S0T ,S

1T , . . . ,S

NT ) is a random vector in RN+1 if 1 + N assets, S0

T > 0,• Interest Rate r , when S0 ∃: S0

T = (1 + r)S00 , so 1 + r > 0

• Portfolio: θi is the number of units of the i-th asset held in the portfolio θ.θ = (θ1, . . . , θN) ∈ RN if N assets. θ = (θ0, . . . , θN) ∈ R1+N if 1 + N assets• Value V(θ) of a prtf θ:

- at t = 0 : V0(θ) =∑

i θiS i

0 = θ · S0 ∈ R- at t = T : VT (θ) =

∑i θ

iS iT = θ · ST is random variable in R

• Gains from date 0 to date T on the investment V0(θ) in the prtf θ:G(θ) = G(θ) = VT (θ)− V0(θ) = θ · (ST − S0) is random variable in R• Return on the investment in the prtf θ: R(θ) = VT (θ)/V0(θ) if V0(θ) 6= 0





• Discounted prices, gains and return, when risk-free asset with price S0 ∃:S it = S i

t/S0t , Vt(θ) = θ · S0

t , G(θ) = VT (θ)− V0(θ), R = VT (θ)/V0(θ)• Representation when Ω is a finite set:

V0(θ)

(VT (θ))(ωK)

(VT (θ))(ω1)

(VT (θ))(ω2)

...





• Matrix notation for calculations when Ω is a finite set:

- Let S and R be, when risk-free asset with price S0 ∃ the K × (1 + N)matrix and when risk-free asset with price S0 @ the K × N matrix, withelements Sij = S j

T (ωi ) and Rij = Sij/S j0 respectively. S is the price

matrix and R the matrix of returns.

- Given a prtf θ, with V0(θ)) 6= 0. Let Θ and ϑ be, when risk-free assetwith price S0 ∃ the (1 + N)× 1 matrix and when risk-free asset withprice S0 @ the N × 1 matrix, with elements Θi = θi andϑi = θiS i

0/V0(θ)) respectively. Θ is the portfolio matrix and ϑ theportfolio fractions matrix.

- Let V be the K × 1 matrix with elements Vi = (VT (θ))(ωi ).

- Then V = SΘ and (R(θ))(ωi ) = (Rϑ)i for 1 ≤ i ≤ K .





2.2 Arbitrage and Equivalent Martingale Measure

• An Arbitrage Portfolio (or an Arbitrage Opportunity) is a portfolio θ suchthat one of the following two statements A and B is true:

A: The following three statements are true

i) V0(θ) = 0ii) VT (θ) ≥ 0iii) P(VT (θ) > 0) > 0

B: The following two statements are true

i) V0(θ) < 0ii) VT (θ) ≥ 0

• Caution: In this course the above definition is specific for mono-period case• An Arbitrage Free Market is a market where @ an Arbitrage Portfolio; (Also:Arbitraged Market, AOA)





• State Price Vector: A vector β = (β1, . . . , βK ) ∈ RK satisfying ∀i

S i0 =

∑

1≤j≤KS iT (ωj)βj (1)

is called a State Price Vector. The number βi is called a state price (of ωi )• Arrow-Debreu assets with pay-off e1, . . . , eK : ei has pay-off 1 in the state ωi

and pay-off 0 in ωj if i 6= j , i.e. ei (ωj) = δij , ∀i , j ∈ 1, . . . , K.• Interpretation of βi : If, for a given i , ei is one of the primary assets, then(1) gives that βi is the price of ei at t = 0





• Price Π0(X ) at date t = 0 of a general pay-off X at date T . Intuitively,since Arrow-Debreu assets ei not always tradable: An asset with pay-offX (ωi )ei at date T has price X (ωi )βi at t = 0, so a price candidate is(justification in §3)

Π0(X ) =∑

1≤i≤KX (ωi )βi (2)

Caution: β not unique in general ⇒ Π0(X ) not unique in general

Theorem 2.1@ an arbitrage portfolio if and only if ∃ β such that βi > 0, ∀ 1 ≤ i ≤ K .Proof:





• Equivalent martingale Measure (E.M.M.) and Interest Rate:

- Suppose that ∃ state price vector β, with βi > 0 ∀i . Define r by

1/(1 + r) =∑

1≤i≤Kβi (3)

- r is the interest rate defined by a risk-free asset. In fact, Eq. (2) ⇒

Π0(1 + r) =∑

1≤i≤K(1 + r)βi = 1 (4)

- Define qi = (1 + r)βi and the measure Q by Q(ωi ) = qi . Then qi > 0and

∑1≤i≤K qi = 1, so Q is a probability measure and P ∼ Q.





- Expected value w.r.t. Q is: EQ [X ] =∑

1≤i≤K X (ωi )qi . Eq. (2) ⇒

Π0(X ) = EQ

[X

1 + r

](5)

- Ingredients : Arbitrage Price, Q ∼ P and Pay-Off discounted to t = 0





• Interpretation of Q, when risk-free asset with price S0 ∃:

- S , the discounted prices are defined by: S it = S i

t/S0t for t ∈ T. So

S00 = S0

T = 1

- Eq. (5) givesS i

0 = EQ

[S iT

], 0 ≤ i ≤ N (6)

- Eq. (6) ⇒ S is a martingale under Q

• Definition of e.m.m. (in market with or without S0):

Definition 2.2An Equivalent Martingale Measure (e.m.m.) Q is a probability measure on(Ω,F) equivalent to P, such that for some r > −1 and ∀i

S i0 = EQ

[S iT

1 + r

]. (7)





Remark 2.3

i) In general there does not exists a unique e.m.m., i.e. equation (7) for Qand r has not always a unique solution.ii) If Q and r is a solution of (7), then βi = qi/(1 + r), with qi = Q(ωi ),defines a state price vector β satisfying (3).

• First Fundamental Theorem of Asset Pricing: Theorem 2.1 gives

Theorem 2.4∃ an Equivalent Martingale Measure if and only if the market is arbitrage free





Corollary 2.5

In an arbitrage free market, we have for all interest rates r and e.m.m. Qsolution of (7) and all portfolios θ that

V0(θ) = EQ

[VT (θ)

1 + r

]. (8)

• Vt(θ) portfolio price discounted to date 0 if S0 ∃: Let Vt(θ) = Vt(θ)/S0t .

Corollary 2.5 ⇒V0(θ) = EQ

[VT (θ)

]. (9)

• So, V(θ) is a Q-martingale





2.3 Pricing of Derivative Products

We here introduce arbitrage pricing methods clarifying the validity andmeaning of pricing formulas (2) and (5).• Derivative Product: A derivative is defined by its pay-off X at date T ,where X is any F-measurable r.v. (Only mono-period case)• Hedging Portfolio: A derivative X is hedgeable (or attainable) if there ∃ aprtf. θ s.t.

VT (θ) = X . (10)

Such θ is called a Hedging portfolio of X .• M: In the sequel M denotes the spot market defined by S and the set ofpossible portfolios.





Example 2.6

We consider the mono-period market with interest rate r = 10% and twostocks S1 et S2 :

- Price at t = 0 : S10 = S2

0 = 100

- Price at t = T :

S1T (ω1) = 88, S1

T (ω2) = 110, S1T (ω3) = 132, (11)

S2T (ω1) = 132, S2

T (ω2) = 88, S2T (ω3) = 110. (12)

Find a hedging portfolio of a Put on S1 with strike 105.





Solution:The pay-off is X = (105− S1

T )+, so X (ω1) = 17, X (ω2) = 0 and X (ω3) = 0.The hedging portfolio θ shall satisfy VT (θ) = X .In matrix notation

X = SΘ, (13)

where

X =

1700

and S =

1110 88 1321110 110 881110 132 110

. (14)





S is invertible, which gives

Θ = S−1X =

17033−17

661766

. (15)

So the hedging portfolio is θ = ( 17033 ,−17

66 ,1766 ) and its price at t = 0 is

V0(θ) = 17033 .

Fin Example 2.6.





• The value of Hedging Portfolios at t = 0 of X is unique:

Proposition 2.7

Suppose that the market M is Arbitrage Free. Let X be a hedgeablederivative and let θ and η be hedging portfolios of X . Then V0(θ) = V0(η).Let Q and r satisfy Eq. (7). Then

V0(θ) = EQ

[X

1 + r

]. (16)

Proof: Follows from (8) of Corollary 2.5.• M′: Let X be a derivative and x ∈ R. In next theorem M′ denotes themarket with prices S0 and x at t = 0 and prices ST and X at t = T .





Theorem 2.8Let M be Arbitrage Free. Then the following three statements are equivalent:

i) The market M′ is arbitrage free

ii) ∃ an e.m.m. Q and a interest rate r , satisfying Eq. (7) and

x = EQ

[X

1 + r

]. (17)

iii) ∃ β, satisfying (1), s.t. βi > 0, ∀ 1 ≤ i ≤ K and s.t.

x =∑

1≤i≤KX (ωi )βi . (18)





Corollary 2.9

Let M be Arbitrage Free. If X is hedgeable, then the price x for whichstatement ii) of Theorem 2.8 holds true is unique.

• Thus, for a hedgeable derivative X , Theorem 2.8 and Corollary 2.9 justify tocall this unique price x , The Arbitrage Price of X . It is denoted Π0(X ) and

Π0(X ) = EQ

[X

1 + r

], (19)

for any e.m.m. Q and a interest rate r , satisfying Eq. (7).





Example 2.10

We consider the mono-period market with interest rate r = 5%, two stocksS1 and S2 and three states ω1, ω2 and ω3 :

S1T

S10

(ω1) =42

31,

S1T

S10

(ω2) =21

31,

S1T

S10

(ω3) =21

62

andS2T

S20

(ω1) =21

124,

S2T

S20

(ω2) =42

31,

S2T

S20

(ω3) =168

31.

The a priori probability of ω1, ω2 and ω3 are 1/8, 3/8 and 4/8 respectively.What is the price (at t = 0) of a Call on S2 with strike (150/31)S2

0 , obtainedby using an e.m.m. Q? Also, find a hedging portfolio of the Call. What is theprice of the hedging portfolio?





Solution:Let qk = Q(ωk). The eq. EQ

[ST

]= S0 gives EQ [ST ] = (1 + r)S0, which

then gives

q1S iT (ω1) + q2S i

T (ω2) + q3S iT (ω3) = (1 + r)S i

0 i = 0, 1, 2.

With matrix notation we obtain

(R)t

q1

q2

q3

= (1 + r)

111

. (20)

Here

R =

2120

4231

21124

2120

2131

4231

2120

2162

16831

. (21)





The unique solution of (20) is

q1 =3

5, q2 =

3

10, q3 =

1

10. (22)

The pay-off of the Call is X = (S2T − 150

31 S20 )+, so X (ω1) = 0, X (ω2) = 0 and

X (ω3) = 1831 S2

0 . Its price at t = 0 is

EQ

[X

1 + r

]= q3

X (ω3)

1 + r=

12

217S2

0 . (23)





A hedging portfolio θ of the Call shall satisfy VT (θ) = X . In matrix notation

X = SΘ = R

θ0S0

0

θ1S10

θ2S20

, where X =

00

1831 S2

0

. (24)

This gives θ0S0

0

θ1S10

θ2S20

= S2

0

−3600

8897124148

287

. (25)

We have V0(θ) = 12217 S2

0 , which, as it should be, is the same as the the pricegiven in (23). Fin Example 2.10.





• Complete Market: The market M is said to be Complete, when “all”derivatives are hedgeable.• Second Fundamental Theorem:

Theorem 2.11The following two statements are equivalent:

i) The market M is arbitrage free and complete

ii) ∃ a unique e.m.m. Q.

Corollary 2.12

In an arbitrage free and complete market, every derivative X has a uniquearbitrage price Π0(X ) at t = 0, given by (19).





• Pricing of a derivative X in an Incomplete Market:

I If X is hedgeable, then a unique arbitrage price Π0(X ) is given by (19)according to Corollary 2.9.

I If X is not hedgeable, then (as we shall see) the arbitrage price is notunique.

• Me : Let Me be the set of e.m.m. for the market M, which is supposedarbitrage free. So Me 6= ∅.• Possible arbitrage prices of a non-hedgeable derivative X : Theorem 2.8 andformula (17) gives that for every Q ∈Me and corresponding interest rate r , apossible arbitrage price is given by

EQ

[X

1 + r

].

This leads to an interval of possible arbitrage prices.





• Lower-Upper price spread: ]Π∗0(X ),Π∗0(X )[ , where

Π∗0(X ) = infQ∈Me

EQ

[X

1 + r

]and Π∗0(X ) = sup

Q∈Me

EQ

[X

1 + r

]. (26)

Remind that in general r in this formula depends on Q.• To sum up:

Proposition 2.13

Let M be Arbitrage Free and let the price of X at t = 0 be x . Then themarket M′ is arbitrage free iff x = Π0(X ) (see (19)) when X is hedgeable andx ∈ ]Π∗0(X ),Π∗0(X )[ when X is not hedgeable.

• Problem: How to choose the price in ]Π∗0(X ),Π∗0(X )[ ?




Probabilistic ModelPortfoliosArbitrage and Equivalent Martingale MeasurePricing of Derivative ProductsAmerican Derivatives

3 Discrete Time Markets, 3.1 Probabilistic Model

• Trading dates: t ∈ T = 0, 1, . . . ,T, where T ≥ 1 is an integer• Filtered Probability space (Complete): (Ω,P,F , Ftt∈T), where

- Ω is a set of elementary events, it can be a finite or infinite set

- P is an a priori probability measure

- F is a σ-algebra of possible events

- Ftt∈T is a filtration of sub-σ-algebras Fs ⊂ Ft ⊂ F for 0 ≤ s ≤ t;FT = F . If Ω is a finite set, then F0 = Ω, ∅. If Ω is an infinite set,then F0 is the σ-algebra generated by Ω and the sets of measure zero(null sets)





• 1 + N Assets (1 risk-free and N general assets):

- S it is the price of asset nr. i at date t, 0 ≤ i ≤ N and t ∈ T. It is a real

r.v. and it is Ft-measurable, i.e. the value of S it is known at t

- St = (S0t , S

1t , . . . ,S

Nt ) is the price vector (or just the price) at t

- S = (S0,S1, . . . ,ST ) is the price process. S is adapted to the filtrationFtt∈T

- S0 is the price (process) of a risk-free asset and S i , 1 ≤ i ≤ N are theprice (processes) of general assets. For a stock, S i

t > 0.

• We suppose that:

- S0 > 0, i.e. S0t > 0 ∀ t ∈ T.

- S0t+1 is Ft-measurable (justifying “risk-free”), cf. predictable

- S00 = 1, with some exceptions





• (Spot) Interest Rate r :

- S0t+1/S0

t is Ft-measurable, i.e. known at t

- The Spot Interest Rate rt for the period [t, t + 1[ is defined by

1 + rt =S0t+1

S0t

. (27)

rt is known at t in the beginning of the period

- S0t = S0

0 (1 + r0) · · · (1 + rt−1), so S0t is the amount of money at time t at

a bank account generated by an initial investment of S00 and interest

rates r0, . . . , rt−1.

- Warning: r does not have to be constant in time or deterministic. It canbe stochastic.

• Discounted prices: S it = S i

t/S0t is the discounted price at date t of asset i ,

0 ≤ i ≤ N. In particular S0t = 1, for t ∈ T. S is the discounted price process.





3.1.1 Binomial Model• Assets: One risk-free with price S0 and one risky asset with price S1, for ex.a stock• 2 possible evolutions from t to t + 1: Up denoted u and Down denoted d .The probabilities of u and d are p and 1− p respectively, where 0 < p < 1and p is independent of what has happened before t. Typically, if u (resp. d)then the price S1 evolves by a factor U (resp. D) where D ≤ U :

u

d

S0t

(1+ r)S0t

(1+ r)S0t

t t +1

u

d

S1t

US1t

DS1t

t t +1

Figure: Price binom. mod.





• The Random Source in Bin. Mod. is a Bernoulli process ν:

- ν = (ν1, . . . , νT ), where the νt are i.i.d., P(νt = 1) = p andP(νt = 0) = 1− p

- Ω = 0, 1T ; νt(ω) is the t:th coordinate of ω ∈ Ω, soω = (ν1(ω), . . . , νT (ω)). Convention: ωk has the coordinates given bythe binary representation of the integer k , 0 ≤ k ≤ 2T − 1

- If ω ∈ Ω corresponds to n Up’s (so T − n Down’s) thenP(ω) = pn(1− p)T−n

- Filtration Ftt∈T, where F0 = Ω, ∅ and Ft = σ(ν1, . . . , νt), for1 ≤ t ≤ T . So ν1, . . . , νt are known at t.

- νt+1(ω) = 0 and νt+1(ω) = 1 are identified with the evolution d and urespectively, from t to t + 1.





• Information tree: one-to-one correspondence between states and final leaves

t = 0 t = 1 t = 2 t = 3 t = T − 1 t = Tω0

ω1

ωn

ωn+1

ωK−2

ωK−1

Figure: Information tree; Number of states K = 2T





• Path lattice: one-to-one correspondence between states ω and paths

t = 0 t = 1 t = 2 t = 3 t = T − 1 t = T

Figure: Path lattice; Number of paths K = 2T





Example 3.1

t = 0 t = 1 t = 2 t = T

ω0 = (0,0,0)

ω1 = (0,0,1)ω2 = (0,1,0)

ω3 = (0,1,1)ω4 = (1,0,0)

ω5 = (1,0,1)ω6 = (1,1,0)

ω7 = (1,1,1)

Figure: Information tree, with T = 3, K = 8





2) Filtration:

- F0 = Ω, ∅.- F1 = σ(ν1). To find it, let A1(0) = ν−1

1 (0) and A1(1) = ν−11 (1),

where ν−11 (B) is the inverse image of the set B.

Then A1(0) = ω0, ω1, ω2, ω3,A1(1) = ω4, ω5, ω6, ω7

so F1 = σ(A1(0),A1(1)).

Since A1(0),A1(1) is a partition of Ω it follows that, at time t = 1 wecan distinguish between events in A1(0) and events in A1(1), but we cannot distinguish between events within A1(0) or events within A1(1).





- F2 = σ(ν1, ν2). Let A2(i , j) = ν−11 (i) ∩ ν−1

2 (j). Then

A2(0, 0) = ω0, ω1,A2(0, 1) = ω2, ω3,A2(1, 0) = ω4, ω5,A2(1, 1) = ω6, ω7,

(28)

so F2 = σ(A2(0, 0),A2(0, 1),A2(1, 0),A2(1, 1). Since the A2(i , j) definesa partition of Ω, at t = 2 one can distinguish between events which are intwo different such sets, but not within the same set.

- F3 = σ(ν1, ν2, ν3). Let A3(i , j , k) = ν−11 (i) ∩ ν−1

2 (i) ∩ ν−13 (k).

Then A3(i , j , k) = (i , j , k). Explicitly,

A3(0, 0, 0) = ω0, . . . ,A3(1, 1, 1) = ω7.

So F3 is the set of all subsets of Ω.





3) Let X be a r.v. (later on it will be a derivative product)

- X is F0-measurable, i.e known at t = 0, means exactly that X isconstant on Ω, i.e. X (ω) = X (ω′) ∀ω, ω′ ∈ Ω

- X is F1-m. ⇔ X is constant on A1(0) and constant on A1(1).

- X is F2-m. ⇔ X is constant on each one of the setsA2(0, 0),A2(0, 1),A2(1, 0),A2(1, 1)

- X is F3-m. ⇔ X is arbitrary





• Number of Up’s: Let Nt be the number of Up’s up to date t included:

N0 = 0, Nt = ν1 + . . .+ νt , if t > 0. (29)

• Binomial Distribution: If 0 ≤ n ≤ t, then

P(Nt = n) =

(t

n

)pn(1− p)t−n.

• Price Distribution:S1t = S1

0 UNt Dt−Nt

gives P(S1t = S1

0 UnDt−n) =

(t

n

)pn(1− p)t−n.

• Recall that:E [Nt ] = tp and var [Nt ] = tp(1− p).





• Markov Process sinceS1t+1

S1t

are independent of S10 , ..,S

1t

• To sum up, the binomial price model is given by the filtered probabilityspace (Ω,P,F ,A), the two dimensional price process S and the possibletrading times t ∈ T = 0, . . . ,T, where

I Ω is the set of elementary events

I P is the a priori probability measure

I F = FT is the σ-algebra of all events

I A = (Ft)t∈T is the filtration





Example 3.2

U = 2 , D = 12 , p = 3

4 , S10 = 1 , T = 3 Price S1 path lattice :

1

2

12

4

1

14

8

2

12

18

Figure: Price path lattice, ω1 = (0, 0, 1), ω2 = (0, 1, 0) and ω4 = (1, 0, 0).





P (ω4) = p(1− p)2 = 34

(14

)2= 3

64

P (ω2) = p(1− p)2 = 364

P (ω1) = p(1− p)2 = 34

(14

)2= 3

64

P (ω0) = (1− p)3 =(14

)3

ω7

ω6

ω5

ω4

ω3

ω2

ω1

ω0

P (S13 = 1

2) = P (ω1 , ω2 , ω4)

= 3 · 364

= 964

Figure: Information tree





3.2 Portfolios• Portfolio θ:

- θit is the number of units of the i-th asset held in the portfolio at time t.θit is a R-valued r.v. and it is Ft-measurable, i.e. is known at t.

- θt = (θ0t , θ

1t , . . . , θ

Nt ) is the instantaneous portfolio at t. It is a

RN+1-valued random vector. θ = (θ0, θ1, . . . , θT ) is the portfolio,also called the trading strategy or dynamical portfolio.

- Mathematically the portfolio θ is a RN+1-valued Ftt∈T-adaptedstochastic process

- θit ≥ 0 long position in i :th asset; θit ≤ 0 short position in i :th asset;

• Price process V (θ) of a portfolio θ:

Vt(θ) = θt · St , t ∈ T. (30)

• Discounted Price process V(θ) of a portfolio θ: (Discounted to date 0)

Vt(θ) = Vt(θ)/S0t . (31)





• (Spot) Market M: The filtered probability space (Ω,P,F , Ftt∈T), theprice process S and the set of all possible portfolios defines the UnderlyingSpot Market M.M said to be a finite market if Ω is finite.• Gains from t to t + 1, for a portfolio θ: Change due only to the variation ofthe asset prices:

θt · (St+1 − St).

• Gains process G(θ) of prtf. θ: Sum of the gains from date 0 up to date tincluded

Gt(θ) =∑

0≤s<t

θs · (Ss+1 − Ss), t ∈ T. (32)

• Discounted Gains process G(θ) of prtf. θ:

Gt(θ) =∑

0≤s<t

θs · (Ss+1 − Ss), t ∈ T. (33)

Warning: Gt(θ) is not always the discounted value of Gt(θ)Erik Taflin, EISTI Theory of Contingent Claims




• Self-financing prtf. θ is a prtf. where the changes in its price only comesfrom variations in the asset prices:

Definition 3.3θ is said to be self-financing if

Vt(θ) = V0(θ) + Gt(θ), ∀t ∈ T. (34)

Proposition 3.4

The following three statements are equivalent:

i) θ is self-financing

ii) Vt(θ) = V0(θ) + Gt(θ), ∀t ∈ T.iii) θt−1 · St = θt · St , ∀t ∈ 1, . . . ,T.





Proof: By definition Vt+1(θ) = θt+1 · St+1.iii) ⇒ i) : When iii) is true it follows that

Vt+1(θ) = θt · St+1 = θt · St + θt · (St+1 − St) = Vt(θ) + θt · (St+1 − St)

Repeated use of this equality then shows that θ is self-financed.i) ⇒ iii) : When θ is self-financed it follows that

Vt+1(θ) = Vt(θ) + θt · (St+1 − St) = θt · St + θt · (St+1 − St) = θt · St+1

Repeated use of this equality proves that statement iii) is true.Similarly its proved that iii) ⇔ ii).

Example 3.5

A buy-and-hold portfolio θ, i.e. θt = θ0, ∀t ∈ T is self-financing.





3.3 Arbitrage and Equivalent Martingale Measure (e.m.m.)

Question for motivation of the introduction of e.m.m

• Binomial model T = 3, U = 2, D = 12 , r = 0, p = 3

4 , S10 = 1

• European Call: Strike K = 1 (at the money), Maturity T

1

2

12

4

1

14

8

2

12

18

Pay Off = X = (S1T −K)+ = (S1

T − 1)+

7 = X(ω7)

1 = X(ω3) = X(ω5) = X(ω6)

0 = X(ω1) = X(ω2) = X(ω4)

0 = X(ω0)

• What is the price of the Call at t = 0?Erik Taflin, EISTI Theory of Contingent Claims




Answer:

• The price is 1327 . In fact

13

27= Q(S1

3 = 2) · 1 + Q(S13 = 8) · 7 = 3q2(1− q) + 7q3. (35)

Q is here an Equivalent Martingale Measure (e.m.m) given by

Q(S1t+1 = US1

t+1) ≡ q =1 + r − D

U − D=

1− 1/2

2− 1/2=

1

3. (36)


• Arbitrage Portfolio (or Arbitrage Opportunity):

Definition 3.6An Arbitrage Portfolio θ, is a self-financed portfolio such that:i) V0(θ) = 0ii) VT (θ) ≥ 0iii) EP [VT (θ)] > 0.

• An Arbitrage Free Market is a market where @ an Arbitrage Portfolio; (Also:Arbitraged Market)

Remark 3.7• Definition 3.6 is equivalent to:An Arbitrage Portfolio θ, is a self-financed portfolio such that:i) V0(θ) = 0ii) VT (θ) ≥ 0iii) EP

[VT (θ)

]> 0.




Proposition 3.8

Let H = X |X = GT (θ) for some prtf θ, Γ = X |X ≥ 0 and X FT −mand F = X ∈ Γ |E [X ] = 1. The following four statements are equivalent:i) The market M is Arbitrage Freeii) ∀θ GT (θ) ≥ 0 ⇒ GT (θ) = 0iii) H ∩ Γ = 0iv) H ∩ F = ∅.Proof: QED





Proposition 3.9

The market M is arbitrage free iff every monoperiod sub-model is arbitragefree.

Proof:• Suppose first that there exists a mono-period sub-market from t0 tot0 + 1 ≤ T with an OA, i.e. ∃ θt0 s.t.

θt0 · (St0+1 − St0) ≥ 0 and P(θt0 · (St0+1 − St0

) > 0) > 0. (37)

Define θt = 0 for t 6= t0 and θ = (θ0, . . . , θT ). ThenGT (θ) = θt0 · (St0+1 − St0

) ≥ 0. Due to (37), P(GT (θ) > 0) > 0. So, θ is anarbitage portfolio according to Proposition 3.8.





• Suppose then that there does not exist a mono-period sub-market with anOA. Let θ be a portfolio such that GT (θ) ≥ 0.

I Then GT−1(θ) ≥ 0, (or otherwise there exists an arbitrage portfolio fromT − 1 to T )

I So, by iteration GT (θ) ≥ 0, GT−1(θ) ≥ 0, . . . , G1(θ) ≥ 0.

I Since G1(θ) ≥ 0, we must have G1(θ) = 0 (if not ∃ OA from t = 0 tot = 1). Then by iteration

G1(θ) = 0 ⇒ G2(θ) = 0 ⇒ · · · GT−1(θ) = 0

This shows that GT (θ) = 0. So Proposition 3.8 ⇒ the market is arbitragefree.





• The market M is Arbitrage Free iff certain constraints are satisfied, seeDefinition 3.6. The introduction of an Equivalent Martingale Measure Qpermits to express them as linear relations between the prices St at differenttimes t. E.m.m is the main tool to study AOA, completeness of a financialmarket and to price derivatives.





Definition 3.10 (Equivalent Martingale Measure)

An Equivalent Martingale Measure (e.m.m.) Q is a probability measure on(Ω,F) equivalent to P and such that S is a Q-martingale, i.e.

EQ

[|S i

t |]<∞, ∀t ∈ T and 0 ≤ i ≤ N (38)

andSt = EQ

[ST | Ft

], ∀t ∈ T. (39)

Note that the last equation is equivalent to

St = EQ

[St+1|Ft

], for all t = 0, 1, ...,T − 1. (40)





Proposition 3.11

Suppose that Q is an e.m.m. If θ is such that EQ

[|GT (θ)|

]<∞, then G(θ)

is a Q-martingale. If moreover θ is self-financed, then V(θ) is a Q-martingale.Proof: QED

Theorem 3.12 (First Fundamental Theorem of Asset Pricing; Case Ωfinite)

∃ an Equivalent Martingale Measure if and only if the market is arbitrage free.Proof: QED





Example 3.13 (e.m.m. in the Binomial model)

For the Binomial financial market there exists an e.m.m. iff

D < 1 + r < U or D = 1 + r = U.

• In fact Q ∼ P and (40) ⇔ (with qt(ω) = EQ [νt+1|Ft ] (ω))

S1t (ω)

(1 + r)t= EQ

[St+1

(1 + r)t+1|Ft

](ω)

⇔ S1t (ω) = qt(ω)

S1t (ω)U

(1 + r)+ (1− qt(ω))

S1t (ω)D

(1 + r)

⇔ qt(ω) = q ≡ 1 + r − D

U − D, if U > D and 0 < qt(ω) < 1 if U = D.





• The Fundamental Theorem of Asset Pricing (FTAP), for Ω general is due toDalang-Morton-Willinger.

Theorem 3.14 (First FTAP; DMW)

∃ an Equivalent Martingale Measure Q if and only if the market is arbitragefree.Moreover, when Q exists, then it can be choosen such that

i) St ∈ L1(Ω,Q,FT )

ii) dQdP ∈ L∞(Ω,Q,FT )

Proof: QED

Example 3.15 (“Discrete time BS” Market’)





3.4 Pricing of Derivative Products

We here introduce arbitrage pricing methods. First we only consider EuropeanDerivatives (EU Derivatives).• An European Derivative is defined by its expiration date T , where0 ≤ T ≤ T and by its pay-off X at T . The pay-off X is a FT -measurable r.v.• T = T by default. More general cases than the above “definition” can beconsidered, s.a. T being a random time (stopping time). One can alsoconsider european derivatives paying dividends.• How to price an European Derivative X ?

- Let the market M be Arbitrage Free

- The price Πt(X ) at time t of X shall be s.t. the extended market M′ ofM by Π(X ) is also Arbitrage Free

We shall next realize this idea.


• Hedging: An European Derivative X is said to be hedgeable (also used:attainable, replicable) if there ∃ a self-financed prtf. θ s.t.

VT (θ) = X . (41)

Such θ is called a Hedging portfolio of X .

Proposition 3.16

Suppose that the market M is Arbitrage Free and let Q be an e.m.m. If X isa hedgeable derivative and if θ and η are hedging portfolios of X , thenV(θ) = V(η) (equality in the sens of stoch. proc.),

Vt(θ) = EQ

[X

S0T

| Ft

](42)

and V(θ) is a Q-martingale.

Proof: This follows from Proposition 3.11.




• Arbitrage Price: Let X be a hedgeable EU derivative. The Arbitrage Priceof X at date t is

Πt(X ) = S0t EQ

[X

S0T

| Ft

], (43)

where Q is any e.m.m. Thus, the discounted price is

Πt(X ) = EQ

[X

S0T

| Ft

]. (44)

So the discounted price process Π(X ) is a Q-martingale.• Justification of “Arbitrage Price”: For a hedgeable derivative X , the uniqueprice, which makes M′ an arbitrage free market, is given by (43).• Complete Market: The market M is said to be Complete, when “all” EUderivatives are hedgeable. (Positivity and integrability conditions are left outin this course).





Theorem 3.17 (Second Fundamental Theorem)

The following two statements are equivalent:

i) The market M is arbitrage free and complete

ii) ∃ a unique e.m.m. Q.

Corollary 3.18

In an arbitrage free and complete market, every derivative X has a uniquearbitrage price (process) Π(X ), given by (43).





The proof of Th 3.17 is based on:

Theorem 3.19If M is arbitrage-free and Q ∈Me , then X is hedgeable iff

∀ Q ′ ∈Me , EQ′

[X

S0T

]= EQ

[X

S0T

]. (45)





• Pricing of a derivative X in an Incomplete Market:

- Let X be a non-hedgeable EU derivative. What are the possible arbitrageprices at time t of X ? As we shall see, there is no unique arbitrage price.

- Let Me be the set of e.m.m. for the market M, which is supposedarbitrage free. So Me 6= ∅.

- Introduce, for a given e.m.m. Q ∈Me ,

πQt (X ) = S0t EQ

[X/S0

T | Ft

], ∀t ∈ T. (46)





• Possible arbitrage prices of X : Since there is an e.m.m. in the extendedmarket M′, according to formula (46), it follows from Theorem 3.12 that M′is arbitrage free. So, πQ(X ) is a possible arbitrage prices process of X . Moreprecisely,

Proposition 3.20

Let Q ∈Me and let πQ(X ) be given by formula (46). Then the market M isarbitrage free iff the market M′ is arbitrage free.


For a general X (hedgeable or not), we define the lower and upperarbetrage price bounds Π∗t(X ) and Π∗t (X ) by

Π∗t(X ) = infQ∈Me

πQt (X ) and Π∗t (X ) = supQ∈Me

πQt (X ). (47)

• Lower-Upper price spread at t ∈ T on the set A where Π∗t(X ) < Π∗t (X ) :

πQt (X )(ω) ∈ ]Π∗t(X )(ω),Π∗t (X )(ω)[ a.s. ω ∈ A

and on the set Ac where Π∗t(X ) = Π∗t (X ) :

πQt (X )(ω) = Π∗t(X )(ω) = Π∗t (X )(ω) a.s. ω ∈ Ac

• In particular for t = 0:

I If X is not hedgeable, then Π∗0(X ) < Π∗0(X ) and

πQ0 (X ) ∈ ]Π∗0(X ),Π∗0(X )[

I If X is hedgeable, then

πQt (X ) = Π∗t(X ) = Π∗t (X ) .




• One proves that

1. Π∗t (X ) = Π∗t (X )/S0t is the superhedging cost at t, i.e.

Π∗t (X ) = infVt(θ) | θ self-financing and VT (θ) ≥ X/S0T (48)

2. ∀ Q ∈Me , Π∗(X ) is a Q-supermartingale

3. and Π∗(X ) is the the smallest such process also satisfying

Π∗T (X ) ≥ X/S0T . (49)





3.4.1 Some examples for the Binomial Model

Example 3.21

Prices for all ω and t in (35)

1

2

12

4

1

14

8

2

12

18

Price of X Pay Off = X = (S1T −K)+ = (S1

T − 1)+

7 = X(ω7)

1 = X(ω3) = X(ω5) = X(ω6)

0 = X(ω1) = X(ω2) = X(ω4)

0 = X(ω0)

1327

119

19

3

13

0





Example 3.22 (Barrier)

Let T = 3, r = 110 , U = 5

4 , D = 45 , S1

0 = 400, p = 34 . Find the price at t = 0

of a Barrier Option of the type Down-And-Out Call with

Barrier H = 350, Strike K = 450.

N.B : The Pay Off at T in the state ω is given by

X (ω) =

0 if min0≤t≤T S1

t (ω) < H

(S1T (ω)− K )+ if min0≤t≤T S1

t (ω) ≥ H(50)

We also have, denoting by 1A the caracteristic function of a set A :

X = (S1T − K )+ 1min0≤t≤T S1

t ≥H. (51)





Solution: q = 1+r−DU−D =

1110− 4

554− 4

5

. So q = 23

400

500

320

635

400

256

31354

= 781, 24

500

320

10245

= 204, 8

t = 0 t = 1 t = 2 t = 3 = T

S1t

350 = H





400

500

320

635

400

400

256

31254

= 781, 25

500

500

500

320

320

320

10245

= 204, 8

Information tree with S1

PAY-OFF in ω

(31254

− 450)+ = 13254

(ω7)

(500− 450)+ = 50 (ω6)

50 (ω5)

0 (ω4)

0 (ω3)

0 (ω2)

0 (ω1)

0 (ω0)





• Price at t = 0 :

Π0(X ) = EQ

[X

S0T

]=

1

(1 + r)TEQ [X ]

1

(1 + r)TEQ [X ] =

(10

11

)3(Q(ω5)50 + Q(ω6)50 + Q(ω7)

1325

4

)

=

(10

11

)3(2 · 50q2(1− q) +

1325

4q3

)

=

(10

11

)3(

100 ·(

2

3

)2

(1

3) +

1325

4·(

2

3

)3)

=

(10

33

)3

(400 + 2 · 1325) ≈ 84.87.





Example 3.23 (Asian Call Option)

Let T = 4, r = 14 , D = 1, U = 2, S1

0 = 10, p = 34 .

Consider an option with pay-off X at maturity T :

X = (1

T

T∑

t=1

S1t − K )+, with strike K = 35. (52)

Find the price (ΠX (t))(ω), for all t, ω.





Before studying the price of the option, we note that the pay-off X is pathdependant.

10

20

10

40

20

10

80

40

20

10

160

80

40

20

10

⇒ 1T

∑Tt=1 S

1t = 220

4

⇒ 1T

∑Tt=1 S

1t = 150

4





S10 = 10

20

10

40

20

20

10

80

40

40

20

10

20

20

40

160

8080

4080

4040

20

10

2020

4020

4040

80

ω ∈ Ω ; Ω = ω0, ω1, ..., ω15 K = 35

Arbre des prix S1t (ω)





S14(ω)

14

∑4t=1 S

1t (ω) X = (1

4

∑4t=1 S

it −K)+

S14(ω15) = 160 300

4= 75 X(ω15) = 40

S14(ω14) = 80 220

4= 55 X(ω14) = 20

S14(ω13) = 80 180

4= 45 X(ω13) = 10

S14(ω12) = 40 140

4= 35 X(ω12) = 0

S14(ω11) = 80 160

4= 40 X(ω11) = 5

S14(ω10) = 40 120

4= 30 X(ω10) = 0

S14(ω9) = 40 100

4= 25 X(ω9) = 0

S14(ω8) = 20 80

4= 20 X(ω8) = 0

S14(ω7) = 80 150

4= 75

2X(ω7) =

52

S14(ω6) = 40 110

4= 55

2X(ω6) = 0

S14(ω5) = 40 90

4= 45

2X(ω5) = 0

S14(ω4) = 20 70

4= 35

2X(ω4) = 0

S14(ω3) = 40 80

4= 20 X(ω3) = 0

S14(ω2) = 20 60

4= 15 X(ω2) = 0

S14(ω1) = 20 50

4= 25

2X(ω1) = 0

S14(ω0) = 10 40

4= 10 X(ω0) = 0





61250

2925

150

265

15

0

110

20

2

1

0

0

0

0

12

q =54−1

2−1= 1

4; Πt(X) = 1

1+rEQ [Πt+1(X)|Ft] =

45

(14Πt+1(X)(up) + 3

4Πt+1(X)(down)

)

⇒ Πt(X) = 15Πt+1(X)(up) + 3

5Πt+1(X)(down)

X

X(ω15) = 40

X(ω14) = 20X(ω13) = 10

X(ω12) = 0X(ω11) = 5

X(ω10) = 0X(ω9) = 0

X(ω8) = 0

X(ω0) = 0

X(ω1) = 0X(ω2) = 0

X(ω3) = 0X(ω4) = 0

X(ω5) = 0X(ω6) = 0

X(ω7) =52


3.5 American Derivatives

• An American Derivative X , with expiry date T , is an adapted processX0, . . . ,XT of pay-offs, where Xt is the amount payed to the holder if heexercise at date t. The holder can chose to exercise at any datet ∈ 0, . . . ,T and he can exercise only once.• Suppose that the market M is arbitrage free and for simplicity in the sequelof this sub-section that M is also complete and let Q be the e.m.m.

Example 3.24

Let Y be the spotprice of a stock or more generally of an asset s.t. Y /S0 is aQ-martingale, 1) An American Call, with strike K and expiry date T , on Y isgiven by Xt = (Yt − K )+.2) An American Put on an asset with price process Y , with strike K andexpiry date T , is given by Xt = (K − Yt)+.

• How to price an AM derivative X ? The idea is still Arbitrage Pricing.

Suppose for the moment that it works and gives a unique price Π(a)t (X ) at t.




• Dynamical Programming approach to Π(a)(X )

- Price at T (if not exercised before): Π(a)T (X ) = XT

- Price at T − 1 (if not exercised before):

Π(a)T−1(X ) = max

(XT−1, S0

T−1EQ

[Π

(a)T (X )

S0T

| FT−1

])(53)

- Price at t < T (if not exercised before):

Π(a)t (X ) = max

(Xt , S0

t EQ

[Π

(a)t+1(X )

S0t+1

| Ft

])(54)

• Is the discounted price process Π(a)(X ) = Π(a)(X )/S0 a Q-martingale?

Π(a)t (X )

S0t

= max

(Xt

S0t

, EQ

[Π

(a)t+1(X )

S0t+1

| Ft

])≥ EQ

[Π

(a)t+1(X )

S0t+1

| Ft

]

If Xt is sufficiently big, then the inequality is strict.Erik Taflin, EISTI Theory of Contingent Claims




So Π(a)(X ) not always a Q-martingale! But always a Super Martingale, i.e.

Π(a)t (X ) ≥ EQ

[Π

(a)t+1(X ) | Ft

]∀ t ∈ 0, . . . ,T − 1

andEQ

[(Π

(a)t (X ))−

]<∞ ∀ t ∈ 0, . . . ,T

Moreover Π(a)t (X ) ≥ Xt/S0

t .

Proposition 3.25

Π(a)(X ) is the smallest supermartingale, s.t. Π(a)t (X ) ≥ Xt/S0

t . (One saysthat Π(a)(X ) is the Snell-envelope of X/S0).

Proof:





• Ts,t is the set of all stopping times τ, s.t. s ≤ τ ≤ t. We have

Proposition 3.26

The price Π(a)t (X ) at the time t ∈ 0, . . . ,T of an American derivative X is

given by

Π(a)t (X ) = S0

t maxτ∈Tt,T

EQ

[XτS0τ

| Ft

]. (55)

Proof:• τt : Let τt be a stopping time for which the max in (55) is attained. Thesequence τ0, . . . , τT (where τT = T ) is called an arbitrage free exercisetime strategy (also; rational exercise time strategy). Not always unique.





Example 3.27

Let X be s.t. Xt/S0t defines a Q-martingale. Then by optional stopping, for

all τ ∈ Tt,TEQ

[XτS0τ

| Ft

]=

Xt

S0t

.

It follows in this case that Π(a)t (X ) = Xt for all t and that any sequence of

stopping times τ0, . . . , τT, where τt ∈ Tt,T is an arbitrage free exercise timestrategy.





Example 3.28

We consider an American Put with strike K = 3/2 in the market defined bythe Binomial model with S1

0 = 1, T = 1, D = 1/2, U = 2 and D < 1 + r < U.For the different r , what are the arbitrage free exercise time strategies?





• American Call; If the spot interest rates are strictly positive, then theAmerican Call of Example 3.24 should be exercised at time T , as in the caseof European Call:

Proposition 3.29

Let the spot interest rate rt ≥ 0 for t ∈ 0, . . . ,T − 1. For the American Callof Example 3.24 the exercise time strategy given by τt = T is arbitrage freeand for all t ∈ 0, . . . ,T

Π(a)t (X )

S0t

= EQ

[(YT − K )+

S0T

| Ft

].

Moreover, if rt > 0 for t ∈ 0, . . . ,T − 1, then this τt = T is the uniquearbitrage free exercise time strategy.





Proof: The function R 3 x 7→ (x)+ is convex. Using first that S0T is positive

and then Jensen’s inequality it follows that:

S0T−1EQ

[(YT − K )+

S0T

| FT−1

]= S0

T−1EQ

[(

YT − K

S0T

)+ | FT−1

]

≥ S0T−1(EQ

[YT − K

S0T

| FT−1

])+ = S0

T−1(YT−1 − KB(T − 1,T )

S0T−1

)+

= (YT−1 − KB(T − 1,T ))+ ≥ (YT−1 − K )+.

Here the last inequality follows since B(T − 1,T ) ≤ 1, due to positive interestrate rT−1 from T − 1 to T . This shows that one possibility at T − 1 is not toexercise. Since the last inequality is strict if rT−1 > 0, this also shows thatone should not exercise at T − 1 if rT−1 > 0. Recursion from t = T − 1 tot = 0 now proves the statement.





4 Continuous Time Markets

• Aim: Introduce the model of Black, Merton and Scholes and itsgeneralizations• Why Continuous time models?

- Assets are (in many cases) quoted at high frequency and withoutinterruption; Ex: Foreign Exchange rates and Stock-indices⇒ Almost continuously quoted

- A robust discrete time model must give predictions almost independentof the time increment ∆, when ∆ is small:

| | | | | |

0 t t +Δ

⇒ Continuous limit ∃





- Quotes are not equidistant in reality:

| | | | | | | | | | || | | | | |

⇒ Quotes can be considered as a sample of a Continuous Time Process

- Technical reasons:Discrete mathematics complicatedTheory of continuous time processes ∃ and computational easier(Stochastic integration, Ito’s calculus, Girsanov’s transformation, . . . )





4.1 Original Black-Scholes ModelStock price model• Trading dates: T = [0,T ]• Random source: A one dimensional Brownian motion W is defined on a(complete) probability space (Ω,P,F), where P is the a priori probabilitymeasure.• Filtration: Ftt∈T generated by W (and the null-sets); F = FT

• Two assets: A Risk-free bank account, with price process

S0t = S0

0 exp(rt) (56)

and a Risky stock with price process, which is an “exponential Brownian”

S1t = S1

0 exp(σWt + (µ − 1

2σ2)t), (57)

where r , µ, σ ∈ R and σ 6= 0. S1t /S1

0 is log-normal; ln(S1t /S1

0 ) isN ((µ − 1

2σ2)t, σ2t)





• SDEs (Stochastic Differential Equation) by Ito’s lemma:

dS0t = rS0

t dt (58)

dS1t = S1

t µdt + S1t σdWt . (59)

Here r is the (continuous compounded) spot interest rate, µ the drift and σthe volatility.• Discounted price S : S i = S i/S0

dS0t = 0 (60)

dS1t = S1

t (µ − r)dt + S1t σdWt . (61)

Exercise 4.1Let γ = µ−r

σ and ξt = exp(−γWt − 12γ

2t). Establish that ξS is a

P-martingale. (N.b. γ is called the market price of risk).





Portfolio• The portfolio θ = (θ0, θ1) :

- θ0t is the number of units of the risk-free asset held at time t

- θ1t is the number of units of the risky asset held at time t

- θ0t and θ1

t are known at time t, i.e. they are Ft-measurable.

- θt = (θ0t , θ

1t ) is the instantaneous portfolio at t.

• Price process V (θ) of a portfolio θ:

Vt(θ) = θt · St , t ∈ T. (62)

• Discounted Price process V(θ) of a portfolio θ: (Discounted to date 0)

Vt(θ) = Vt(θ)/S0t . (63)

Obviously Vt(θ) = θt · St .





• Gains process G(θ) of prtf. θ: Sum of the gains from date 0 up to date t

Gt(θ) =

∫ t

0θs · dSs , t ∈ T. (64)

Warning: To be rigorous, one should here introduce a set of admissibleportfolios, such that the gains process is well-defined and such that the modelis arbitrage free (excluding for example doubling strategies). This is outsidethe scope of this course and we shall just suppose that the prtf. is sufficientlyintegrable or uniformly bounded from below, as to guarantee these properties.• Discounted Gains process G(θ) of prtf. θ:

Gt(θ) =

∫ t

0θs · dSs , t ∈ T. (65)

• Self-financing prtf. θ is a prtf. where the changes in its price only comesfrom variations in the asset prices, i.e.

Vt(θ) = V0(θ) + Gt(θ), ∀t ∈ T. (66)





Exercise 4.2Establish that the definition of a self-financing prtf. by formula (66) isequivalent to

Vt(θ) = V0(θ) + Gt(θ), ∀t ∈ T. (67)





Black-Scholes Equation• Simple EU derivative X , i.e.

X = f (S1T ), for some f

• Hedging prtf. θ of X :

VT (θ) = X , for some self-fin. prtf. θ

• To construct the hedging prtf. θ of X we try the ansatz: There is a functionF ∈ C 1,2([0,T [ × ]0,∞[ ) s.t.

Vt(θ) = F (t,S1t ), ∀t ∈ T (68)

andF (T , x) = f (x), ∀x > 0. (69)

Introduce: F1(t, x) = ∂F (t, x)/∂t, F2(t, x) = ∂F (t, x)/∂x andF22(t, x) = ∂2F (t, x)/∂x2.





Differentiation of the l.h.s. of (68) gives, since θ is self-fin.:

dVt(θ) = θt · dSt = (rS0t θ

0t + µS1

t θ1t )dt + σS1

t θ1t dWt

= (rVt(θ) + (µ − r)S1t θ

1t )dt + σS1

t θ1t dWt

(70)

(68) and (70) give,

dVt(θ) = (rF (t,S1t ) + (µ − r)S1

t θ1t )dt + σS1

t θ1t dWt (71)

Differentiation of the r.h.s. of (68) gives, by Ito’s lemma, ∀t ∈ [0,T [ :

dF (t,S1t ) = (F1(t, S1

t ) + µS1t F2(t, S1

t ) +1

2σ2(S1

t )2F22(t,S1t ))dt

+ σS1t F2(t,S1

t )dWt .(72)





Identification of (71) and (72) gives, first

θ1t = F2(t, S1

t ), (73)

and then

rF (t,S1t )+(µ−r)S1

t θ1t = F1(t, S1

t )+µS1t F2(t, S1

t )+1

2σ2(S1

t )2F22(t,S1t ). (74)

• Eqs. (74) and (69) give the Black-Scholes Equation: ∀ (t, x) ∈ T×]0,∞[,

∂∂t F (t, x) + rx ∂

∂x F (t, x) + 12σ

2x2 ∂2

∂x2 F (t, x) = rF (t, x),

F (T , x) = f (x), ∀x > 0.(75)

• The hedging prtf. θ is given by (73)

θ1t = F2(t,S1

t )

θ0t = 1

S0t

(F (t,S1

t )− F2(t, S1t )S1

t

).

(76)





A special case of the Feynman-Kac formula gives the solution of the B-S eq.:

Theorem 4.3(Under certain conditions on f ). If

F (t, x) = e−r(T−t)

∫ ∞

−∞f (xey )

1√2πσ2(T − t)

exp

(−(y − (T − t)(r − 1

2σ2))2

2σ2(T − t)

)dy ,

(77)

where (t, x) ∈ [0,T [ × ]0,∞[ , then F is the solution of the B-S Eq. (75) andθ defined by (76) is a hedging prtf. of X = f (S1

T ).

Proof:





Black-Scholes Formula for a Call• Let f be the pay-off of a EU Call with strike K > 0 :

f (x) = (x − K )+, x > 0 (78)

and let C be the solution of the B-S eq. given by formula (77). Then

C(t, x) = e−r(T−t)∫ ∞−∞

(xey − K)+1√

2πσ2(T − t)exp

(−

(y − (T − t)(r − 12σ2))2

2σ2(T − t)

)dy

= e−r(T−t)∫ ∞

ln(K/x)(xey − K)

1√2πσ2(T − t)

exp

(−

(y − (T − t)(r − 12σ2))2

2σ2(T − t)

)dy.

(79)

We make the substitution

z = −y − (T − t)(r − σ2/2)

σ√

T − t, so y = −zσ

√T − t + (T − t)(r − σ2/2).


Let

z0 =ln(x/K ) + (T − t)(r − σ2/2)

σ√

T − t.

Formula (79) now gives

C (t, x) = e−r(T−t)

∫ z0

−∞

(x exp(−zσ

√T − t + (T − t)(r − σ2/2))− K

) e−z2

2√2π

dz .

(80)Let

I1(t, x) = e−r(T−t)

∫ z0

−∞x exp(−zσ

√T − t + (T − t)(r − σ2/2))

e−z2

2√2π

dz

and

I2(t, x) = −Ke−r(T−t)

∫ z0

−∞

e−z2

2√2π

dz .

ThenC (t, x) = I1(t, x) + I2(t, x).

Let N be the normal (0, 1) pdf (probability distribution function).




It follows that

I1(t, x) = x e−r(T−t)

∫ z0

−∞exp(−zσ

√T − t + (T − t)(r − σ2/2))

e−z2

2√2π

dz

= x

∫ z0

−∞exp

(− (z + σ

√T − t)2

2

)dz√2π

= x N(z0 + σ√

T − t ).

For I2 it follows thatI2(t, x) = −Ke−r(T−t)N(z0).

To sum up, introduce τ = T − t, d1(τ) = z0 + σ√τ and d2(τ) = z0, i.e.

d1(τ) =1

σ√τ

(lnx

K+ τ(r +

σ2

2)) (81)

d2(τ) =1

σ√τ

(lnx

K+ τ(r − σ2

2)), (82)

then (80) gives the Black-Scholes Formula for the price of a EU Call

C (t, x) = xN(d1(τ))− Ke−rτN(d2(τ)). (83)





0.25 0.5 0.75 1 1.25 1.5 1.75 2x

0.25

0.5

0.75

1

1.25

1.5

1.75

2C

Figure: Price C (0, x). Here T = 1, K = 1, r = 1/10 and σ = 4/10





1 2 3 4 5 6 7 8x

1

2

3

4

5

6

7

8C

Figure: Price C (0, x). Here T = 1, K = 1, r = 1/10 and σ = 4/10





• Call-Put relation: Let P(T , x) = (K − x)+. Then

C (T ,S1T )− P(T , S1

T ) = S1T − K .

The arbitrage price at time t of each side then gives

C (t, S1t )− P(t,S1

t ) = S1t − Ke−r(T−t). (84)

Since this is true for all values of S1t , we obtain the Call-Put relation

C (t, x)− P(t, x) = x − Ke−r(T−t), (85)

for x > 0.• Price of a Put P(t, x) is obtained from formulas (83) and (85) and therelation 1− N(z) = N(−z)

P(t, x) = −xN(−d1(τ)) + Ke−rτN(−d2(τ)). (86)

for x > 0.Erik Taflin, EISTI Theory of Contingent Claims




Example 4.4

Consider a standard B-S market, with annual interest rate r = 3.5% andannual volatility σ = 30%. The stock price is 132e today. What is the pricetoday of a Call and a Put, with 9 months to maturity and strike 125?Let t be today. We have τ = 3/4Y and according to (81) and (82):d1(τ) = 1

3/10√

3/4(ln (132/125) + 3/4(35/1000 + (3/10)2/2)) ≈ 0.440665 and

d2(τ) = 1

3/10√

3/4(ln (132/125) + 3/4(35/1000− (3/10)2/2)) ≈ 0.180858.

N(d1(τ)) ≈ 0.670272 and N(d2(τ)) ≈ 0.57176. Formula (83) givesC (t, 132) ≈ 18, 8576. The Call-Put parity relation (85) then givesP(t, 132) ≈ 8.61903.





4.2 The greeks• B-S model is complete, so by definition every derivative is hedgeable• However, in practice a hedge is in general only approximate for severalreasons, as1) The portfolio can not be re-balanced continuously in time.2) The underlying price model, here the original B-S model, is only anapproximation of the real price dynamics.3) The model parameters are only determined with a certain accuracy.• Let F be the price function (here given by B-S eq. (75)) of a simple EUderivative X and let θ be an approximate hedging prtf., s.t. its value at time tis a function of the stock price S1

t . Then the price difference p at t ofderivative and hedge is:

p(t, S1t ) = F (t,S1

t )− Vt(θ). (87)





• p is also a function of the model parameters r , µ and σ, if this is the casefor θ. Sensitivities of derivative price F (t, x), w.r.t. variations in t, x , r and σneeded;• “Greeks” for a Call F (t, x) = C (t, x) :

Delta ∆ =∂C (t, x)

∂x= N(d1) > 0, for x > 0. (88)

Gamma Γ =∂2C (t, x)

∂x2=

1

xσ√τ

N ′(d1) > 0, for x > 0. (89)

(So, the Call-price is strictly concave.)

Theta Θ =∂C

∂t= − xσ

2√τ

N ′(d1)− Ke−rτ r N(d2) < 0, for x > 0, (90)

where τ = T − t. (So C is increasing in τ .)Erik Taflin, EISTI Theory of Contingent Claims




Vega V =∂C

∂σ= x√τ N ′(d1) > 0, for x > 0. (91)

Rho ρ =∂C

∂r= Kτe−rτ N(d2) > 0, for x > 0. (92)

• ∆, Γ and Θ satisfies, for a simple derivative with price F (see B-S eq. (75)) :

Θ + rx∆ +1

2σ2x2Γ− rF = 0. (93)





Example 4.5

In the situation of Example 4.4, what are today the hedging portfolios of theCall and the Put respectively?The hedging prtfs. are obtained from (76). Let x = 132. In the case of theCall, eq. (88) gives

θ1t = N(d1) ≈ 0.670272 stocks

andθ0t S0

t = C (t, x)− N(d1)x ≈ −69.6184 e.

Example 4.6

Greeks for a EU Put





4.3 Generalized Black-Scholes model

• More realistic model, than standard B-S, requires arbitrary number of assetsand random-sources and more general drift and volatility processes.• Volatility smiles can be explained by:

I Stochastic volatility models (Ex. Heston)

I Local volatility models

• Trading dates: T = [0,T ]• Random sources: A m dimensional Brownian motion W = (W 1, . . . ,W m),(W i and W j indep. for i 6= j) is defined on a complete filtered probabilityspace (Ω,P,F , Ftt∈T), where P is the a priori probability measure and thefiltration is generated by W and (and the null-sets). F = FT





• 1 + N assets:A Risk-free bank account, with price process S0 satisfying S0

0 > 0 and

dS0t = rtS

0t dt, (94)

where rt is the spot interest rate at time t. r is adapted.N risky assets with price processes S1, . . . ,SN , where

dS it = S i

tµitdt + S i

t

m∑

j=1

σijt dW jt . (95)

Here µ is the drift process and σ the volatility process, which are supposed tobe adapted. We will suppose that they are sufficiently integrable.





• Recall: The cross variations are given by

< S0 , S0 >= 0, < S0 , S i >= 0, (96)

< S i , S j >t=

∫ t

0S isS j

s

m∑

k=1

σiks σjks ds, (97)

where 1 ≤ i ≤ N and 1 ≤ j ≤ N.• Recall also Ito’s Lemma: Let Xt = (X 1

t , . . . ,Xnt ) where the X i are Ito

processes and let (t, x) 7→ F (t, x) be a real C 1,2 function on [0,T ]× Rn.Then

dF (t,Xt) =∂F

∂t(t,Xt) dt +

n∑

i=1

∂F

∂xi(t,Xt) dX i

t +1

2

n∑

i,j=1

∂2F

∂xi∂xi(t,Xt)d < X i , X j >t .

(98)


• The discounted prices S = S/S0 satisfy

S0t = 1 (99)

and for 1 ≤ i ≤ N

dS it = S i

t(µit − rt)dt + S it

m∑

j=1

σijt dW jt . (100)

• Interpretation of r , µ, σ: With the conditional variance defined byvar [X |Ft ] = E

[(X − E [X |Ft ])

2|Ft

]we have:

E[S0τ − S0

t |Ft

]

τ − t→ rtS

0t ,

var [S0τ − S0

t |Ft ]

τ − t→ 0 (101)

and

E[S iτ − S i

t |Ft

]

τ − t→ µitS

it ,

var [S iτ − S i

t |Ft ]

τ − t→ (S i

t)2m∑

j=1

(σijt )2, 1 ≤ i ≤ N,

(102)when τ ↓ t. So, rt and µit are the (instantaneous) conditional expected ratesof return at date t per time unit of the risk-free asset and the risky asset nr. irespectively.

• Excess rate of return above risk-free is µit − rt cf. (102)• Portfolio θ: θit is the number of units held in the portfolio of asset i at timet. θt = (θ0

t , θ1t , . . . , θ

Nt ) is Ft-measurable.

• V(θ) is the price process and V(θ) the discounted price process of a prtf. θ,

Vt(θ) = θt · St and Vt(θ) = θt · St , t ∈ T. (103)

• Gains and discounted gains

Gt(θ) =

∫ t

0θs · dSs and Gt(θ) =

∫ t

0θs · dSs , t ∈ T. (104)

• The prtf. θ is said to be self-financed if one of the two following(equivalent) conditions is satisfied: ∀t ∈ T,

Vt(θ) = V0(θ) + Gt(θ) or Vt(θ) = V0(θ) + Gt(θ). (105)

We will suppose portfolios to be sufficiently integrable. These are theadmissible portfolios. M is the market defined by the admissible portfoliosand the price process S .




4.4 Arbitrage and e.m.m.

• An Arbitrage Portfolio is a self-financed admissible prt. θ s.t.i) V0(θ) = 0ii) VT (θ) ≥ 0iii) EP [VT (θ)] > 0.

Theorem 4.7The market M is Arbitrage Free iff for all admissible (not necessarily self-fin.)portfolios θ the following holds true:

GT (θ) ≥ 0 ⇒ GT (θ) = 0. (106)

Proof:


• A Market Price of Risk is an adapted Rm-valued process γ s.t.

µt − rt = σtγt , ∀t ∈ T. (107)

(Does not always ∃, integrability conditions).• Arbitrage Free Market M ⇒ ∃ Market Price of Risk γ . We shall establishthis fact heuristically. For an admissible prtf. θ, formula (104) gives

Gt(θ) =

∫ t

0as · ((µs − rs)ds + σsdWs) , t ∈ T, (108)

where the N-dim. r.v. at is given by ait = θit Sit . Let σ

′t be the transpose of σt

and suppose that σ′tat = 0 (or equivalently at is orthogonal to the range of

σt). Then the integral w.r.t. W vanish, so

Gt(θ) =

∫ t

0as · (µs − rs)ds, t ∈ T. (109)

Suppose that the integrand 6= 0 on a set with strictly positive dt dP measure.Then we can chose θ s.t. the integrand is strictly positive on this set (left asexercise). This shows (see Theorem 4.7) that ∃ Arbitrage Opportunity.Therefore, in an Arbitrage Free Market we must have that, a.e. (t, ω) and∀x ∈ RN

(∀j ∈ 1, . . . ,NN∑

i=1

x itσ

ijt = 0) ⇒ x · (µt − rt) = 0. (110)

So, in an Arbitrage Free Market, µt − rt is in the range of σt . So (107) has asolution γt . This solution can be choosen (Exercise !) s.t. γt is Ft-meas.Consequently, ∃ a market price of risk process γ solution of (107).• An Equivalent Martingale Measure is a probability measure Q such thatQ ∼ P and S is a Q-martingale, i.e.

EQ

[|S i

t |]<∞, ∀t ∈ T and 0 ≤ i ≤ N (111)

andSt = EQ

[ST | Ft

], ∀t ∈ T. (112)

• Recall Girsanov’s

Theorem 4.8Let γ be a Rm-valued Ftt∈T-adapted process s.t. ξ is a P-martingale, where

ξt = exp

(∫ t

0(−γs · dWs −

1

2‖γs‖2

Rmds)

), t ∈ T. (113)

Then W 1, . . . , W m are independent Brownian motions w.r.t. Q, whereEQ [X ] = EP [ξTX ] and

Wt = Wt +

∫ t

0γsds. (114)

• Novikov’s criterion is a sufficient (but not necessary) condition guaranteeingthat ξ is a martingale:If L is a continuous local P-martingale such that

EP

[exp(

1

2< L , L >T )

]<∞,

then exp(L− < L , L >) is a P-martingale.

• Let M be Arbitrage Free. Then ∃ a market price of risk process γ (solutionof (107)). Suppose that γ can be chosen sufficiently integrable, so that ξdefined by (113) is a P-martingale (cf. Novikov’s criterion). Formulas (100),(107) and (114) then give

dS it = S i

t(µit−rt)dt+S it

m∑

j=1

σijt dW jt = S i

t

m∑

j=1

σijt (dW jt +γjt dt) = S i

t

m∑

j=1

σijt dW jt

S is then a Q-mart., so Q is an e.m.m.• On the other hand if ∃ γ then M is Arbitrage Free. More precisely

Theorem 4.9Suppose that there is a market price of risk process γ (solution of (107)) andlet γ and σ be sufficiently integrable. Then dQ = ξT dP is an e.m.m., themarket M is Arbitrage Free and

dS it = S i

t

m∑

j=1

σijt dW jt , 1 ≤ i ≤ N. (115)

• G(θ) is a Q-mart. for an admissible prtf. θ (in the situation of Th. 4.9). Infact

dGt(θ) =N∑

i=1

θit dS it =

N∑

i=1

S it

m∑

j=1

θit σijt dW j

t =m∑

j=1

(N∑

i=1

θit S it σ

ijt ) dW j

t .

If θ is admissible and self-fin. then V(θ) is a Q-mart.




4.5 Pricing derivatives

In this subsection, let the hypotheses of Theorem 4.9 be satisfied.• A EU derivative X is hedgeable if there exists an admissible self-fin. prtf. θs.t. VT (θ) = X . Such prtf. is called a hedging prtf. of X• Exercise Show that Vt(θ) = Vt(η) for all t ∈ T, when θ and η are twohedging prtf. of X• Arbitrage price Πt(X ) at t of a hedgeable EU derivative X is defined byΠt(X ) = Vt(θ), where θ is a hedging prtf. Πt(X ) is the discounted price.

Theorem 4.10Let X be a hedgeable EU derivative. Then Π(X ) is a Q-mart., so

Πt(X )/S0t = EQ

[X/S0

T | Ft

], ∀t ∈ T. (116)

Proof: QED





• The market M is complete when all sufficiently integrable derivatives arehedgeable.

Theorem 4.11Under the hypotheses of Theorem 4.9 the following statements are equivalent:i) M is completeii) σ

′t(ω)σt(ω) is invertible a.e. (t, ω) (and integrability conditions)

iii) The e.m.m Q is uniqueiv) The market price of risk process γ is unique.


Example: Deterministic S0 and σ : Price of a Call• Pay-off X of a EU Call on stock nr 1 with maturity T and strike K > 0 :

X = (S1T − K )+. (117)

• Let the bank-account S0 and the volatility matrix σ be deterministic Th.4.10 gives:

Πt(X ) = S0t /S0

TEQ

[(S1

T − K )+ | Ft

], ∀t ∈ T. (118)

By integration of Eq. (115)

S1T =

S0T

S0t

S1t exp

∫ T

t

m∑

j=1

σ1js dW j

s −1

2

∫ T

t

m∑

j=1

(σ1js )2 ds

. (119)

• Introduce Integrated variance for stock nr. 1 Υ1t,t′ =

∫ t′

t

∑mj=1(σ1j

s )2 ds andthe r.v.

Yt,T = − 1√Υ1

t,T

∫ T

t

m∑

j=1

σ1js dW j

s .

• Yt,T is N (0, 1) under Q and by (119)

S1T =

S0T

S0t

S1t exp

(−√

Υ1t,T Yt,T −

1

2Υ1

t,T

). (120)

The independence of S1t and Yt,T , and eq. (118) give

Πt(X ) = S0t /S0

T

∫ ∞

−∞

(S0T

S0t

S1t exp

(−√

Υ1t,T y − 1

2Υ1

t,T

)− K

)

+

exp(−y 2

2)

dy√2π.

(121)Let

d+(a, b) =1√b

(ln a +1

2b) d−(a, b) =

1√b

(ln a− 1

2b) (122)

Formulas (120) and (122) give

Πt(X ) = S0t /S

0T

∫ d−(S0T S1

tS0t K

,Υ1t,T )

−∞

(S0T

S0t

S1t exp

(−√

Υ1t,T y − 1

2Υ1

t,T

)− K

)exp(−y 2

2)dy√2π.

(123)

Proceeding now as in the case of (80) we obtain

Πt((S1T − K )+) = S1

t N

(d+(

S0TS1

t

S0t K

,Υ1t,T )

)− K

S0t

S0T

N

(d−(

S0TS1

t

S0t K

,Υ1t,T )

). (124)




4.6 Examples of Generalized Black-Scholes model

We here introduce basic examples of Generalized B-S models, with stochasticinterest rates or with stochastic volatility.

Non-deterministic Interest Rate• We here only mention the one factor model of Ho and Lee; (General studyin the Fixed Income course):

drt = at dt + btdWt , (125)

where at and bt are deterministic. If the volatility process σ in (95) isdeterministic, then one can establish an explicit pricing formula for a EU Callon a stock, analogous to the B-S formula (83) (see Fixed Income course).


1-dim. local volatility models; Dupire’s formula• Bank Account price S0, Stock price S1 and 1-dim. P-B.m. W

dS0t = rS0

t dt and dS1t = S1

t (r dt + σ(t,S1t )dWt 0 ≤ t ≤ T ,

where (t, x) 7→ σ(t, x) is a continuous function, called local volatility.• C (T ,K ) = e−rTE [(S1

T − K )+] is the price at t = 0 of a Call of maturity Tand strike K .• Given observations of prices C (T ,K ), Dupire’s formula gives the localvolatility by

1

2K 2σ(T ,K )2 ∂2

∂K 2C (T ,K ) =

∂

∂TC (T ,K ) + rK

∂

∂KC (T ,K ).

Proof: One can use local times and Tanaka’s formula.

Simple example of stochastic volatility and volatility smile• Bank Account price S0, Stock prices S1, S2 and 2-dim. B.m. W

dS0t = 0, dS1

t = S1t atdW 1

t and dS2t = S2

t dW 2t , 0 ≤ t ≤ T = 1,

where S10 = S2

0 = 1 and a is a stoch. proc. defined by

∀t ∈ [0,T/2] : at = 1 and ∀t ∈]T/2,T ] :

at = 1/2 if S2

T/2 ≤ e−1/4

at = 2 if S2T/2 > e−1/4

• Price p(K ) at t = 0 of a EU Call on S1 as function of K ; (Let cBS(K ,Υ) bethe BS-price in the case of deterministic volatility, see (124) with r = 0,t = 0, τ = T − t = 1 and integrated variance Υ, )

p(K ) =1

2(cBS(K , 5/8) + cBS(K , 5/2))

• Volatility smile for the Call: K 7→ σimpl(K ) where

p(K ) = cBS(K , (σimpl(K ))2)

2 4 6 8

1.25

1.3

1.35

1.4

1.45

Figure: Volatility smile : K 7→ σimpl(K )




The Heston model This a two factor model (m = 2), with constant interestrate r , one stock (N = 1) and with stochastic volatility σ. The model isspecified by real numbers a > 0, k > 0, δ > 0 and ρ ∈ [−1, 1] and by areal-valued process v called the variance process:

dvt = k(a− vt) dt + δ√

vt d(ρW 1t +

√1− ρ2 W 2

t ) (126)

Note that zt = ρW 1t +

√1− ρ2 W 2

t defines a Brownian motion. It hascorrelation ρ with W 1.• vt ≥ 0 a.s. and vt > 0 a.s. if v0 > 0 and δ2 ≤ 2ak. Square-root process (seeExercises).• Let σ11

t =√

vt , σ12 = 0. Then (95) gives

dS1t = S1

t µ1t dt + S1

t

√vtdW 1

t . (127)

• Market price of risk γ = (γ1, γ2); Eq. (107) gives: γ1t = (µ1

t − rt)/√

vt andγ2t arbitrary. So, according to Theorem 4.11, the market M is not complete.





• For convenience introduce λt = δ√

vt(ργ1t +

√1− ρ2γ2

t ). Girsanov’stransformation and eqs. (127) and (126) give

dS1t = S1

t

√vtdW 1

t , where dWt = γt dt + dWt , (128)

and

dvt = (k(a− vt)− λt) dt + δ√

vt dzt , where zt = ρW 1t +

√1− ρ2 W 2

t .(129)

Note that z is a Q-Bm. and that

S1t = S1

0 exp(

∫ t

0

√vsdW 1

s −1

2

∫ t

0vs ds). (130)

• λ is an arbitrary process (real-valued, adapted, . . . ) if |ρ| 6= 1. It can inprinciple be specified by studying the price of any volatility dependentderivative, completing the market. For a real world calibration one obtainsλ 6= 0. The price of a derivative is then obtained by formula (116).


• In order to give examples, as simple as possible, we chose λ = 0 in the sequel, soaccording to (129)

dvt = k(a− vt) dt + δ√

vs dzt . (131)

This equation is equivalent to

vt − a = (v0 − a)e−kt + δ

∫ t

0

e−k(t−s)√vs dzs . (132)

• EQ [vt ] and covQ [vt1 , vt2 ]; With tmin = min(t1, t2) :

EQ [vt ] = a + (v0 − a)e−kt , covQ [vt1 , vt2 ] = δ2e−k(t1+t2)(ae2ktmin − 1

2k+ (v0 − a)

ektmin − 1

k).

(133)

• Integrated variance Υt =∫ t

0vs ds; Formula (133) gives:

EQ [Υt ] = at + (v0 − a)1− e−kt

k(134)

and

varQ [Υt ] =δ2

2k3

(2akt + 2v0 − 5a + 4k(a− v0)te−kt + 4ae−kt + (a− 2v0)e−2kt

).

(135)

Example 4.12 (A Toy Derivative and volatility smile)We consider the EU derivative with maturity T , where 0 ≤ T ≤ T , and pay-offXT = − ln(S1

T/S10 ). According to (130)

XT = −∫ T

0

√vsdW

1s +

1

2ΥT .

The price P(0,T ) at t = 0 is according to (133)

P(0,T ) = EQ

[e−rTXT

]= e−rTEQ [ΥT/2] =

e−rT

2(aT + (v0 − a)

1− e−kT

k). (136)

The corresponding B-S price PBS(0,T , σ) of XT , i.e. the price of XT if the stock price isgiven by (57) (left as exercise):

PBS(0,T , σ) =e−rT

2σ2T . (137)

Implicit volatility for XT , σimp(T ); By definition solution of

PBS(0,T , σimp(T )) = P(0,T ). (138)

One has

(σimp(T ))2 = a + (v0 − a)1− e−kT

kT. (139)

The curve T 7→ σimp(T ) is the “volatility smile” for the family of derivatives XT .

2 4 6 8 10

0.145

0.155

0.16

0.165

0.17

Figure: Volatility smile : T 7→ σimpl(T ), a = 1/50, v0 = 3/100 and k = 1

Example 4.13 (Binary Option)

We consider an EU binary option with maturity T and strike K , where0 ≤ T ≤ T . The pay-off is then H(S1

T − K ) where H(y) = 0 for y ≤ 0 andH(y) = 1 for y > 0. The Black-Scholes price at t = 0 is (see Exercise)

pBS(T ,K , σ2T ) = e−rTN(d−(S0TS1

0

K,ΥT )).

The price in the Heston model p(T ,K ) :

p(T ,K ) = EQ

[e−rTH(S1

T − K )]. (140)

Using the integrated variance as time one proves, using the Dambis,Dubins-Schwarz theorem, that:

p(T ,K ) = EQ

[EQ

[e−rTH(S1

T − K ) | ΥT

]]= EQ [pBS(T ,K ,ΥT )] . (141)

An approximation up to second degree in powers of ΥT − EQ [ΥT ] is obtainedfrom:

pBS(T ,K ,ΥT ) = pBS(T ,K ,EQ [ΥT ]) +∂pBS(T ,K ,EQ [ΥT ])

∂ΥT(ΥT − EQ [ΥT ])

+1

2

∂2pBS(T ,K ,EQ [ΥT ])

∂Υ2T

(ΥT − EQ [ΥT ])2 + higher degree.

(142)

Thus

p(T ,K ) = pBS(T ,K ,EQ [ΥT ])+1

2

∂2pBS(T ,K ,EQ [ΥT ])

∂Υ2T

varQ [ΥT ]+higher degree.

(143)The second degree term is obtained from

∂2

∂Υ2N(d−) = N ′(d−)(d ′′2 − d−(d ′−)2),

where d ′− and d ′′− are the first and second derivatives w.r.t. Υ.

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