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Lecture Notes
Financial Mathematics 1
Gerald Trutnau
Department of Mathematical Sciences
Seoul National University
Version: June 3, 2015
Non-Corrected version
This text is a summary of the lecture
Financial Mathematics 1 held atSeoul National University, Spring Term 2015
Please email all misprints and mistakes to
1
The main source for this lecture is
- Lamberton, Damien; Lapeyre, Bernard: Introduction to stochastic calculus applied tofinance. Second edition. Chapman & Hall/CRC Financial Mathematics Series, Boca Ra-ton, FL, 2008.
Here are some additional references:
1. Baxter, Martin, Andrew Rennie: Financial Calculus: An Introduction to DerivativePricing, Cambridge University Press, 1996.
2. Bjork, Tamas: Arbitrage Theory in Continuous Time, Oxford University Press.
3. Elliott, Robert J.; Kopp, P. Ekkehard: Mathematics of financial markets. Secondedition. Springer Finance. Springer-Verlag, New York, 2005.
4. Hull, John: Options, Futures, and Other Derivatives, 6th ed., Prentice Hall, 2006.
5. Karatzas, Ioannis and Shreve, Steven: Methods of mathematical finance - Springer,1998.
6. Shreve, Steven E.: Stochastic Calculus for finance I, II, Springer, 2004.
7. Wilmott, Paul; Dewynne, Jeff; Howison, Sam: Option Pricing: Mathematical ModelsAnd Computation, Oxford Financial Press; 1994.
2
1 Introduction to options
An option gives its holder the right (but not the obligation) to buy or sell a certainamount of financial asset, by a certain date for a certain strike price.
The writer (=seller) of the option has to specify:
(i) type of option
call : option to buy
put : option to sell
(ii) underlying asset (stock, bond, currency, etc. ...)
(iii) the amount of underlying asset
(iv) the expiration date (=maturity)American option: can be exercised at any time before maturity
European option: can be exercised only at maturity
(v) the exercise price (=strike price = price at which the transaction is done, if theoption is exercised)
The price of an option is also called premium.
In an organized market, the premium is quoted by the market. Otherwise, the problem isto price the option, but even if the option is traded on a market, it can be interesting todetect possible abnormalities in the market.
Example 1.1 European call option
Ct = its price at time t < TT = expiration dateK = exercise priceSt = stock price at time t
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At time T , the buyer of the European call option has the right to buy the stock for K.Thus the value of the call at maturity is
(ST −K)+ = max(0, ST −K) =
0, if ST ≤ K
(”holder does not exercise the option”)
ST −K, if ST > K
(”holder buys the stock for K and sells it back on
the market for ST”)
Two questions arise:
(i) How much should the buyer pay for the option (at time 0) ?
”C0 = premium =?”
(”Problem of pricing the option”)
(ii) The writer of the call, who earns the premium initially, must be able to deliver astock at price K at time T . How should the writer generate an amount of (ST −K)+
at time T ?
(”Problem of hedging the option”)
We can only answer under additional assumptions:
basic assumption: absence of arbitrage opportunity in the market, i.e.
”there is no riskless profit available in the market”
A rigorous mathematical description will be given in chapter 2. Here as an example con-sider:
Arbitrage and put/call parity:
Let Ct (resp. Pt) be the value at time t of an European call (resp. put) on the samestock with value St at time t ∈ [0, T ], and
T = maturity of both, put and call
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K = exercise price of both, put and call
At time T
CT = (ST −K)+
PT = (K − ST )+
ThusCT − PT = (ST −K)+ − (K − ST )+ = ST −K (1)
Assume additionally: it is possible to borrow or to invest money throughout [0, T ] atconstant rate r.The last is called continuous compounding. Thus a capital X deposited in the bank (ormoney market account) at time t accumulates to
X · limn→∞
(1 +
r
n
)n(T−t)
︸ ︷︷ ︸”periodic compounding withn periods during time T−t”
= X · er(T−t)
a cash sum K at time T can be generated by depositing Ke−r(T−t) at time t.
Claim: under absence of arbitrage opportunity, the call/put prices on the stockS must satisfy (1) with appropriate discounting of K for all t < T , i.e.
Ct − Pt = St −Ke−r(T−t), ∀t < T.
(”put/call parity”)Assume, for instance
Ct − Pt > St −Ke−r(T−t) for some t < T. (2)
Then, a writer could do the following at time t:
He buys a share of stock and a put, and sells a call.
”net value of the operation” Ct − Pt − St =: Vt
If Vt ≥ 0: he invests Vt at rate r until T .
If Vt < 0: he borrows Vt at rate r until T .
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Two outcomes are possible:
1)ST > K︸ ︷︷ ︸
put notexercised !
: K − ST︸ ︷︷ ︸”he has to pay the
exercised call”
+ (Ct − Pt − St)er(T−t)︸ ︷︷ ︸”sum he gets/pays for
investing/borrowing at interest rate r”
+ ST︸︷︷︸”he sells his
share of stock”
>(2)
0
2)ST ≤ K︸ ︷︷ ︸
call notexercised !
: (K − ST )︸ ︷︷ ︸”he exercises
his put”
+ (Ct − Pt − St)er(T−t)︸ ︷︷ ︸”same, he clears his bank
account again”
+ ST︸︷︷︸”he sells his shareof stock again”
>(2)
0
In both cases he made a profit without using any initial capital.
(”arbitrage strategy”)
Black-Scholes model and its extensions:
- no arbitrage arguments lead to many interesting equations, but are not suffi-cient for deriving pricing formulas
- Black-Scholes (1973) were the first to suggest a model whereby one can derive anexplicit price for a European call that pays no dividend. According to their model,the writer of the option can hedge himself perfectly.
The Black-Scholes model and its extensions are based on stochastic calculus, and in par-ticular on Ito’s formula.
2 Discrete time models
In this chapter we present the main ideas of option theory in discrete-time models.
2.1 Discrete-time formalism
2.1.1 Assets
Let (Ω,F, P ) be a finite probability space, i.e. Ω = ω1, . . . , ωN0, N0 ∈ N, P(Ω) the powerset of Ω, i.e. the set of all subsets of Ω. Consider an increasing sequence of σ-algebras inΩ:
∅,Ω = F0 ⊂ F1 ⊂ · · · ⊂ FN = P(Ω) = F
N = ”time horizon” (often the maturity of the options)
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Fn = ”information available at time n”
We assume P (ω) > 0 for all ω ∈ Ω. For any time n = 0, . . . , N , we consider (d + 1)positive Fn-measurable random variables S0
n, S1n, . . . , S
dn (in this chapter all r.v.’s are
real-valued), which are financial assets
Sn := ( S0n︸︷︷︸
riskless
asset
, S1n, . . . , S
dn︸ ︷︷ ︸
risky assets
) ”vector of stock prices at time n”
We set S00 = 1 and
βn :=1
S0n
, n = 0, . . . , N ”discount factor from period n to 0”
Example 2.1 If the return of the riskless asset (e.g. bond) is constant over one periodand equal to r, then
S0n = S0
n−1 + S0n−1 · r = S0
n−1(1 + r) = · · · = S00︸︷︷︸
=1
(1 + r)n = (1 + r)n.
Thus βn = 1(1+r)n
. So, if the amount of βn is invested in the riskless asset S00 at time 0,
then the amount of 1 (e.g. one dollar) will be available at time n.
2.1.2 Strategies
A trading strategy is a (discrete) stochastic process
φ = (φn)n=0,...,N = ((φ0n, φ
1n, . . . , φ
dn))n=0,...,N
in Rd+1, where for each i = 0, ..., d and n = 0, . . . , N
φin = number of shares of asset i held in the portfolio at time n.
φ is assumed to be predictable, i.e.
∀i = 0, . . . , d
φi0 ∈ F0
φin ∈ Fn−1, if n ≥ 1
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Illustration :
at time 0: S0 = (1, S10 , ..., S
d0) and the positions φ0
0, φ10, ..., φ
d0 ∈ F0 are given
at time n− 1, n ≥ 1: select the positions φ0n, φ
1n, ..., φ
dn ∈ Fn−1
(Sn−1 is known) (based on information at time n− 1)
”the positions of the portfolio at time n are decided with respect to the informationavailable at time n− 1”
The value of the portfolio at time n is the scalar product
Vn(φ) = φn · Sn =d∑i=0
φinSin n = 0, . . . , N.
The discounted value is
Vn(φ) = βn(φn · Sn) = φn · Sn,
with βn =1
S0n
and Sn︸︷︷︸”vector of
discounted prices”
= (1, βnS1n, . . . , βnS
dn), n = 0, . . . , N.
By considering discounted prices, the price of the non-risky asset becomes the monetaryunit (a so called numeraire). A strategy is called self financing, if
φn+1 · Sn = φn · Sn ∀n = 0, . . . , N − 1 (3)
This means that the positions in the portfolio are changed without changing the totalvalue of the portfolio (”portfolio is only redistributed”).
Remark 2.2 (3) is equivalent to
Vn+1(φ)− Vn(φ)︸ ︷︷ ︸ = φn+1 · Sn+1 − φn · Sn = φn+1 (Sn+1 − Sn)︸ ︷︷ ︸ n = 0, . . . , N − 1. (4)
”net gain from period n to n+ 1” only due to ”price moves”
Proposition 2.3 Equivalent are
(i) The strategy φ is self-financing.
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(ii) For any n = 1, . . . N
Vn(φ) = V0(φ) +n∑j=1
φj ·∆Sj (∆Sj := Sj − Sj−1)
(iii) For any n = 1, . . . , N
Vn(φ) = V0(φ) +n∑j=1
φj ·∆Sj (∆Sj := Sj − Sj−1)
Proof (i) ⇐⇒ (ii) is (4) summed up. (i) ⇐⇒ (iii) holds because:
(3) ⇐⇒ φn · Sn = φn+1 · Sn ⇐⇒ Vn+1(φ)− Vn(φ) = φn+1(Sn+1− Sn) n = 0, . . . N − 1.
The proposition shows that if one follows a self-financing strategy, then the discountedvalue of the portfolio (hence its value) is completely defined by the initial wealth and thestrategy ((φ1
n, . . . , φdn))n=0,...,N (this is only true because ∆S0
j = 0). More precisely, thefollowing holds.
Proposition 2.4 For any predictable process ((φ1n, . . . , φ
dn))n=0,...,N and any F0-measurable
random variable V0, there exists a unique predictable process (φ0n)n=0,...,N such that the
strategy φ = (φ0, φ1, . . . , φd) is self-financing and such that V0(φ) = V0.
Proof Suppose φ0 = (φ0n)n=0,...,N such as in the statement of the proposition exists. Then
V0(φ) = φ00 +
d∑i=1
φi0 Si0 = V0, hence
φ00 = V0 −
d∑i=1
φi0 Si0, (5)
and for n ≥ 1
(∗)
Vn(φ) = φ0
n +∑d
i=1 φin S
in (by definition)
Vn(φ) = V0(φ) +∑n
j=1
∑di=1 φ
ij (Sij − Sij−1) (by Proposition 2.3 (iii))
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By (∗)
φ0n = V0(φ)︸ ︷︷ ︸
=V0
+n−1∑j=1
d∑i=1
φij (Sij − Sij−1) +d∑i=1
φin (−Sin−1) (6)
Therefore, if we define φ0 through (5), and (6), then φ0 satisfies all the desired properties.
2.1.3 Admissible strategies and arbitrage
The quantities φin were not supposed to be positive, in fact, they were allowed to haveany value.
φ0n < 0 means that we have borrowed the amount |φ0
n| in the riskless asset.
For i ≥ 1 : φin < 0 means that we are ”short” a number φin of asset i.
In this model borrowing and short-selling are allowed, but we shall restrict the class ofstrategies by the following definition.
Definition 2.5 A strategy φ is admissible if it is self-financing and if Vn(φ)(ω) ≥ 0 forall ω ∈ Ω, n = 0, . . . , N (Thus the investor must be able to pay back his debts at anytime).
Definition 2.6 An arbitrage strategy is an admissible strategy φ with zero initial valueand non-zero final value, i.e. V0(φ)(ω) = 0 for all ω ∈ Ω, but P (VN(φ) > 0) > 0.
2.2 Martingales and arbitrage opportunities
2.2.1 Martingales and martingale transforms
As before let (Ω,F, P ) be finite probability space with F = P(Ω), and P (ω) > 0 ∀ω ∈ Ω,(Fn)n=0,...N be a filtration on F, i.e. Fi ⊂ F are sub-σ-algebras and
F0 ⊂ F1 ⊂ · · · ⊂ FN ⊂ F.
In this section 2.2.1 we do not assume F0 = ∅,Ω,FN = F.We say that (Xn)n=0,...,N is adapted (to (Fn)n=0,...N) if Xn ∈ Fn ∀n = 0, . . . , N .
Definition 2.7 An adapted sequence (Mn)n=0,...,N (automatically integrable since |Ω| <∞ and real-valued !) of real-valued r.v.’s is
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- a martingale, if E[Mn+1 | Fn] = Mn n = 0, . . . , N − 1
- a supermartingale, if E[Mn+1 | Fn] ≤Mn n = 0, . . . , N − 1
- a submartingale, if E[Mn+1 | Fn] ≥Mn n = 0, . . . , N − 1
Remark 2.8 (i) Definition 2.7 can be generalized to multidimensions : e.g. a sequenceof Rd-valued r.v.’s (Mn)n=0,...,N is a martingale, if each component is a real-valuedmartingale.
(ii) Let i = 1, . . . , d. If the price of asset i at time n (Sin)n=0,...,N is a martingale, then
E[(Sin+1 − Sin)2] ≤ minS∈Fn
E[(Sin+1 − S)2] n = 0, . . . , N − 1
”using information Fn, Sin is the best guess for Sin+1 in the mean square error”
(All Fn-measurable r.v.’s are in L2(Ω,Fn, P ), since |Ω| <∞ and since they are supposedto be real-valued).
Definition 2.9 A sequence (Hn)n=0,...N of r.v.’s is called predictable, if H0 ∈ F0, andHn ∈ Fn−1 for all n ≥ 1.
Proposition 2.10 Let (Mn)n=0,...,N be a martingale, (Hn)n=0,...,N be predictable, bothw.r.t. the filtration (Fn)n=0,...,N . Then
Xn := H0M0 +n∑k=1
Hk (Mk −Mk−1)︸ ︷︷ ︸=:∆Mk
n = 0, . . . , N
is a martingale w.r.t. (Fn).
Remark: Xn in 2.10 is a discrete version of the stochastic integral∫ t
0
Hs dMs
that we will consider later.
Proof (of 2.10) (Xn) is adapted. For n ≥ 0
E[Xn+1 −Xn | Fn] = E[Hn+1(Mn+1 −Mn) | Fn]
= Hn+1 E[Mn+1 −Mn | Fn]︸ ︷︷ ︸=0 (Mn Martingale)
(since Hn+1 ∈ Fn)
= 0
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thus (Xn) is a martingale.
Proposition 2.11 Let (Mn)n=0,...,N be an adapted sequence of r.v.’s. Equivalent are :
(i) (Mn) is a martingale.
(ii) For any predictable sequence (Hn), we have
E[N∑k=1
Hk (Mk −Mk−1)︸ ︷︷ ︸=∆Mk
] = 0 (7)
Proof Let (Mn) be a martingale, and (Hn) any predictable process. Then by 2.10 (sincea martingale has constant expectation)
E[H0M0 +N∑k=1
Hk(Mk −Mk−1)] = E[H0M0]
thus (7) holds. Conversely, suppose that (Mn) is adapted, and that (ii) holds. Let A ∈ Fjand j ∈ 0, . . . , N − 1 be fixed but arbitrary. Then (ii) holds in particular for
Hn = Hj,An :=
0 if n ∈ 0, . . . , N − 1 − j + 11A if n = j + 1
and so (7) impliesE[Mj+1 1A] = E[Mj 1A].
Since A ∈ Fj, and j ∈ 0, . . . , N − 1 were arbitrary, it follows that
E[Mj+1 | Fj] = Mj j = 0, . . . , N − 1
and the result follows.
2.2.2 Viable financial markets
We suppose again the assumptions of 2.1.1 to hold. A probability measure P ∗ on (Ω,F)is called equivalent to P , if P ∗(ω) > 0 for all ω ∈ Ω.Notation: P ∗ ≈ P .
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Definition 2.12 A market is viable, if there is no arbitrage opportunity.
Theorem 2.13 (Fundamental Theorem of Asset Pricing) The market is viable, ifand only if there exists a probability measure P ∗ equivalent to P such that the discountedprices of assets (Sin)n=0,...,N are P ∗-martingales for i = 1, . . . , d.
Proof Assume: ∃P ∗ ≈ P , such that (Sin)n=0,...,N are P ∗-martingales. For any self-financingstrategy φ = (φn)
Vn(φ) =2.3
V0(φ) +n∑j=1
φj ·∆Sj
=d∑i=0
[φi0 S
i0 +
n∑j=1
φij ·∆Sij
].︸ ︷︷ ︸
P ∗−martingales ∀i by assumption and 2.10
Therefore (Vn(φ))n=0,...,N is a P ∗-martingale and so
E∗[Vn(φ)] = E∗[V0(φ)︸ ︷︷ ︸=V0(φ)
], n = 0, . . . , N . (E∗ = ”expectation w.r.t. P ∗”)
If φ is admissible and V0(φ) = 0, it follows:
• VN(φ)(ω) ≥ 0 for all ω (by definition, since φ is admissible)
• E∗[VN(φ)] = 0 (since V0(φ) = 0)
But0 = E∗[VN(φ)] =
∑ω∈Ω
VN(φ)(ω)P ∗(ω)
implies VN(φ)(ω) = 0 for all ω since P ∗(ω) > 0 for all ω ∈ Ω. Therefore VN(φ)(ω) = 0 forall ω and so an arbitrage strategy cannot exist.
The converse is a little tricky. Thus assume the market is viable. Want to show : ∃P ∗ ≈ Psuch that (Sin)n is a P ∗-martingale ∀i = 1, . . . , d. Define
Γ := X r.v. | X ≥ 0, and P (X > 0) > 0.
Note: Γ is a convex set, i.e. t ∈ [0, 1], X, Y ∈ Γ ⇒ X + t(Y −X) ∈ Γ ∀t ∈ [0, 1].Now:
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the market is viable ⇐⇒ ∀φ admissible with V0(φ) = 0, we have VN(φ) /∈ Γ.
For any predictable process((φ1
n, . . . , φdn))n=0,...,N
define
Gn(φ) : =n∑j=1
(φ1j ∆S1
j + · · ·+ φdj ∆Sdj ).
Note that
Gn(φ) =2.3(iii)
2.4
Vn(φ0, φ1, . . . , φd) (∗)
for uniquely determined predictable φ0, such that (φ0, φ1, . . . , φd) is self-financing andV0(φ0, . . . , φd) = 0.
Lemma 2.14 Suppose the market is viable. Then for any predictable process (φ1, . . . , φd)we have GN(φ) /∈ Γ.
Proof (of Lemma 2.14) Assume to the contrary GN(φ) ∈ Γ.
(a) If Gn(φ) ≥ 0, for all n = 1, ..., N , then add φ0 to (φ1, . . . , φd) as in (∗). It follows, that(φ0, φ1 . . . , φd) is admissible, V0(φ0, φ1 . . . , φd) = G0(φ) = 0, but VN(φ0, φ1 . . . , φd) =GN(φ) ∈ Γ. Thus (φ0, φ1 . . . , φd) is an arbitrage strategy and the market is notviable. This is a contradiction.
(b) Suppose not all Gn(φ) ≥ 0. Then
n := supk ≥ 1 | P (Gk(φ) < 0) > 0 ≤ N−1 (since GN(φ) ∈ Γ, hence GN(φ) ≥ 0).
From the definition of n, we get
P (Gn(φ) < 0) > 0 and ∀j ∈ n+ 1, ..., N we have Gj(φ) ≥ 0.
Define
ψj :=
0, if j ∈ 1, ..., n1Gn(φ)<0 φj, if j ∈ n+ 1, ..., N.
Then ψ is predictable, because Gn(φ) < 0 ∈ Fn ⊂ Fj−1 if j ∈ n + 1, ..., N andφj ∈ Fj−1. Moreover
Gj(ψ) =
0 if j ∈ 1, ..., n1Gn(φ)<0( Gj(φ)︸ ︷︷ ︸
≥0, since j≥n+1
− Gn(φ)), if j ∈ n+ 1, ..., N.
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Thus Gj(ψ) ≥ 0 for any j = 1, . . . , N and GN(ψ) > 0 on the set Gn(φ) < 0, whichhas strictly positive probability. It follows GN(ψ) ∈ Γ. Therefore by (a) applied toψ instead of φ we get again the contradiction that the market is not viable.
We have seen that the assumption GN(φ) ∈ Γ leads to a contradiction in all cases. It cantherefore not hold and so GN(φ) /∈ Γ.
To continue the proof of Theorem 2.13, we let
V := GN(φ) | φ predictable process in Rd ⊂ RΩ = RN0 ,
”subspace of RN0”.
Since the market is viable by assumption, Lemma 2.14 implies V ∩ Γ = ∅. Let
K := X ∈ Γ |∑ω∈Ω
X(ω) = 1
Clearly K is convex, compact, 0 /∈ K, and V ∩K = ∅. Now, we need another lemma.
Lemma 2.15 Let K ⊂ RN0 be a compact and convex set, 0 /∈ K, and let V be a subspaceof RN0. If V ∩K = ∅, then there exists a linear map ξ : RN0 −→ R, with
(1) ξ(x) > 0 ∀x ∈ K
(2) ξ(x) = 0 ∀x ∈ V
(Note that (1) and (2) imply V ⊂ ξ−1(0) (hyperplane) and ξ−1(0) ∩K = ∅).
We will prove Lemma 2.15 later below. By Lemma 2.15 there exists (λ(ω))ω∈Ω, such that
(i)∑
ω∈Ω λ(ω)X(ω) > 0 for all X ∈ K (because any linear map ξ : RN0 −→ R has the
form ξ(z1, . . . , zN0) =∑N0
i=1 λizi for some (λ1, ..., λN0) ∈ RN0)
(ii)∑
ω∈Ω λ(ω) GN(φ)(ω) = 0 for all φ = (φ1, . . . , φd) predictable.
Choosing X(ω) = 0 for all but one ω in (i), implies that λ(ω) > 0. Thus λ(ω) > 0 for allω ∈ Ω. Therefore
P ∗(ω) :=λ(ω)∑ω∈Ω λ(ω)
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defines a probability measure on Ω with P ∗ ≈ P . We have
E∗[N∑j=1
φj ∆Sj] =∑ω∈Ω
GN(φ)(ω)P ∗(ω) =1∑
ω∈Ω λ(ω)
∑ω∈Ω
λ(ω)GN(φ)(ω) =(ii)
0
for any predictable process (φ1, . . . , φd). Therefore, choosing all but one coordinate φi of(φ1, . . . , φd) to be 0, we obtain for any predictable process (φin)n=0,...,N in R and for anyi = 1, . . . d, that
E∗[N∑j=1
φij ∆Sij] = 0.
Now Proposition 2.11 implies that ((Sin)n=0,...,N , P∗) is a martingale for any i.
It remains to show Lemma 2.15. For the proof of Lemma 2.15, we need:
Lemma 2.16 Let C ⊂ RN0 be closed, convex, and 0 /∈ C. Then there is α > 0 and alinear map ξ : RN0 −→ R with
ξ(x) ≥ α ∀x ∈ C
Proof (of Lemma 2.16) Let Bλ(0) := x ∈ RN0 | ‖x‖ ≤ λ, ‖x‖ = (x, x)1/2 = Euclideannorm. Then we can find some λ > 0 with Bλ(0) ∩ C 6= ∅. Since Bλ(0) ∩ C is compact,and x 7→ ‖x‖ is continuous, there exists a unique x0 ∈ Bλ(0) ∩ C, with
infx∈Bλ(0)∩C
‖x‖ = ‖x0‖ = dist(0, Bλ(0) ∩ C) = ”orthogonal projection of 0 on C”.
If x ∈ Bλ(0)c ∩ C, then ‖x‖ > λ ≥ ‖x0‖, thus
‖x‖ ≥ ‖x0‖ ∀x ∈ C.
In particular, since C is convex, for any x ∈ C
‖x0 + t(x− x0)‖2 ≥ ‖x0‖2, ∀t ∈ [0, 1].
Thus
(x0, x− x0) ≥ − t2‖x− x0‖2 0 (as t 0),
and soξ(x) := (x0, x) ≥ ‖x0‖2︸ ︷︷ ︸
6=0since 0/∈C
:= α > 0, ∀x ∈ C.
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Proof (of Lemma 2.15) Set
C = K − V := x ∈ RN0 | x = y − z; y ∈ K, z ∈ V .
Then C is convex, closed, and 0 /∈ C. It follows hence from Lemma 2.16 that there existsξ : RN0 −→ R linear and α > 0 such that ξ(x) ≥ α for all x ∈ C. This implies
ξ(x) = ξ(y)− ξ(z) ≥ α ∀y ∈ K ∀z ∈ V.
If we choose z = 0, we obtain ξ(y) ≥ α > 0 for all y ∈ K and replacing z by λz; λ ∈ R,we obtain ξ(y)− λξ(z) ≥ α for all λ ∈ R, hence ξ(z) = 0 for all z ∈ V .
2.3 Complete markets and option pricing
2.3.1 Complete markets
A European option can be characterized by its payoff h at maturity, which is a non-negative FN -mb r.v. For instance, for the asset S1, h = (S1
N −K)+ in case of a call withstrike price K, and h = (K − S1
N)+ in case of a put with strike price K. More generally,we define:
Definition 2.17 (i) A contingent claim (with non-negative payoff) is a non-negativereal-valued FN -measurable random variable h.
(ii) The contingent claim defined by h is called attainable, if there exists an admissiblestrategy φ worth h at time N , i.e. VN(φ) = h.
Remark 2.18 For contingent claims in a viable market we have:
∃ self-financing φ with VN(φ) = h =⇒ h is attainable.
Proof Since the market is viable by 2.13 ∃P ∗ ≈ P such that (Sin)n=0,...,N , i = 1, . . . , d areP ∗-martingales. Thus the martingale transform (Vn(φ))n=0,...,N is also a P ∗-martingale by2.10. It follows
Vn(φ) = E∗[ VN(φ)︸ ︷︷ ︸= 1
S0N
h≥0
| Fn] ≥ 0 ∀n ≤ N.
Thus Vn(φ) = S0nVn(φ) ≥ 0 for any n ≤ N , and so φ is admissible.
17
Definition 2.19 The market is complete, if every contingent claim is attainable.
Remark 2.20 The assumption of completeness of a market is restrictive and not as usefulas the no-arbitrage assumption. It is, however, useful for pricing and hedging contingentclaims (see also model of Cox-Ross-Rubinstein below).
Theorem 2.21 Let the market be viable. Then it is complete, if and only if there existsa unique measure P ∗ ≈ P under which the discounted prices (Sin)n=0,...,N , 1 ≤ i ≤ d, areP ∗-martingales. (Here the probability P ∗ will become the computing tool whereby one canderive pricing formulas and hedging strategies).
Proof ”⇒ ” Assume the market is viable and complete: Then
∀h ≥ 0 FN -mb ∃ admissible strategy φ with VN(φ) = h,
and furtherh
S0N
=︸︷︷︸completeness
VN(φ) =︸︷︷︸self-financing
V0(φ) +N∑j=1
φj ·∆Sj. (8)
Since the market is viable: ∃P 1 ≈ P under which the (Sin)n=0,...,N are martingales. Suppose∃P 2 with the same properties. Then by (8) and 2.10
Ei[h
S0N
]= Ei[VN(φ)] = Ei[V0(φ)] = V0(φ), i = 1, 2.
The last equality holds since V0(φ) ∈ F0 = ∅,Ω. Thus
E1
[h
S0N
]= E2
[h
S0N
].
Choose h = 1A S0N , A ∈ FN = F ⇒ P 1 = P 2, and uniqueness is proved.
”⇐ ” Let the market be viable but incomplete: Then
∃h ≥ 0 FN -mb r.v., that is not attainable.
Note: h 6= 0 otherwise h would be attainable. Define
V := Y r.v. | Y = u0 +N∑n=1
φn ·∆Sn, u0 ∈ F0, (φn)n=1,...,N predictable n = 1, . . . , N.
18
We have h /∈ V , because otherwise by Proposition 2.4 and Remark 2.18, h would beattainable. Thus V ⊂⊂ X | X r.v. ∼= RN0 is a true linear subspace. Let P ∗ ≈ P be anequivalent martingale measure. Then, there exists a r.v. X 6= 0 with E∗[XY ] = 0 for allY ∈ V (exercise). Define
P ∗∗(ω) :=
(1 +
X(ω)
2 supω∈Ω |X(ω)|
)P ∗(ω).
Since E∗[X] = E∗[X · 1Ω︸︷︷︸∈V
] = 0, we get P ∗∗(Ω) = 1. Clearly, P ≈ P ∗∗ and P ∗∗ 6= P ∗.
Moreover for any predictable process (φn)n=1,...,N = ((φ1n, . . . , φ
dn))n=1,...,N we get
E∗∗[
N∑n=1
φn ·∆Sn
]= 0. (exercise)
Thus by 2.11 (Sin)n=0,...,N , i = 1, . . . , d is a P ∗∗-martingale and so there is more than oneequivalent martingale measure.
2.3.2 Pricing and hedging contingent claims in complete markets
Assume the market is viable and complete and let P ∗ be unique equivalent martingalemeasure (also called risk-neutral measure). Let h be a contingent claim and φ an
admissible strategy with VN(φ) = h. We know from section 2.3.1, that(Vn(φ)
)n=0,...,N
is
a P ∗-martingale, and that
V0(φ) = E∗[VN(φ)] = E∗[h
S0N
],
as well as
Vn(φ) = S0nVn(φ) = S0
n E∗[h
S0N
∣∣∣ Fn] n = 0, . . . , N. (9)
Thus, at any time n = 0, . . . , N , the value of the admissible strategy that generates h is
given by S0n E∗[hS0N
∣∣∣ Fn] (”the value of the option at time n”). Therefore, if the investor
sells the option for the price of
E∗[h
S0N
]at time n = 0, and then follows the admissible strategy φ he can exactly generate thenecessary payoff h at time n = N (”the investor is perfectly hedged”). In particular,
E∗[hS0N
]is the fair price of the option at time n = 0.
19
Remark 2.22 In a complete market the calculaton of the fair option price only requiresthe knowledge of the risk-neutral measure P ∗ (not P ). We could have started with only
(Ω,F) and (Fn)n=0,...,N .
If these are specified, we do not need to find the true probability P (e.g. by empiricalobservations, statistics) in order to price the option!
2.3.3 Introduction to American options
Assume the market is viable and complete, and let P ∗ be the unique equivalent martingalemeasure (risk-neutral measure). An American option can be exercised at any time n =0, . . . , N . Thus it should be defined by an adapted sequence of non-negative r.v.’s
Zn ≥ 0, Zn ∈ Fn, n = 0, . . . , N.
Example 2.23
American call on S1 with strike price K payoff at time n is Zn = (S1n −K)+
American put on S1 with strike price K payoff at time n is Zn = (K − S1n)+.
In order to determine fair price for the option with possible payoff Zn proceed backwards
price at time N : UN = ZN
price at time N − 1 (if the option is not directly exercised (at time N − 1), thewriter must deliver ZN at time N , so (at time N − 1) he needs the amount that isnecessary to deliver ZN at time N):
UN−1 = max
ZN−1︸ ︷︷ ︸”option directly
exercised”
, S0N−1 E
∗[ZNS0N
∣∣∣ FN−1
]︸ ︷︷ ︸
value of the option at time N − 1 thatgenerates ZN at time N” see (9)
By induction for n = 1, . . . , N :
Un−1 = max
(Zn−1, S
0n−1 E
∗[UnS0n
∣∣∣ Fn−1
]).
20
Assume a constant return at rate r for riskless asset, i.e.
S0n = (1 + r)n.
Then
Un−1 = max
(Zn−1,
1
1 + rE∗[Un
∣∣∣ Fn−1
]),
and so for the discounted price of the American option Un := UnS0n
, it holds
Un−1 = max( Zn−1︸︷︷︸:=
Zn−1
S0n−1
,E∗[Un | Fn−1]).
Proposition 2.24 (Un)n=0,...,N is a P ∗-supermartingale. It is the smallest P ∗-supermartingalethat dominates (Zn)n=0,...,N .
Proof As max of adapted r.v.’s Un is adapted, and
Un−1 = E∗[Un−1 | Fn−1] = E∗[max(Zn−1,E∗[Un | Fn−1]) | Fn−1]
≥ E∗[E∗[Un | Fn−1] | Fn−1] = E∗[Un | Fn−1],
thus the first claim holds. Let (Tn)n=0,...,N be another P ∗-supermartingale that dominates(Zn)n=0,...,N . By backward induction :
TN ≥ ZN = UN
Assume Tn ≥ Un n ≤ N , then
Tn−1 ≥ E∗[Tn | Fn−1] ≥ E∗[Un | Fn−1],
henceTn−1 ≥ max(Zn−1,E∗[Un | Fn−1]) = Un−1.
2.4 Cox-Ross-Rubinstein (CRR) model
The CRR model is the discrete version of the Black-Scholes model. In this model Sn =(S0
n, S1n), n = 0, . . . , N , with riskless asset
S0n = (1 + r)n, for some r > −1,
21
and the risky asset is supposed to satisfy
S1n+1 =
S1n(1 + a) or
S1n(1 + b) where − 1 < a < b,
for n = 0, ..., N − 1, with S10 = c > 0 for a given constant c. The possible states of
Tn :=S1n
S1n−1
∈ 1 + a, 1 + b, n = 1, . . . , N,
are then given by
Ω = ω = (x1, . . . , xN) | xi ∈ 1 + a, 1 + b, i = 1, . . . , N.
Each N -tuple in ω = (x1, . . . , xN) ∈ Ω represents the successive values of the ratio Tn(ω),n = 1, . . . , N , on Ω, i.e. Tn(ω) = xn for any n.
Definition: Let n ∈ N and Xi : Ω→ R, i = 1, ..., n, be maps. Then
σ(X1, ..., Xn)
is defined to be the smallest σ-algebra A in Ω for which the maps Xi are A/B(R)-mb,i = 1, ..., n. σ(X1, ..., Xn) is called the σ-algebra generated by (the maps) X1, ..., Xn.
Consider the filtration
F0 := ∅,Ω, Fn := σ(T1, . . . , Tn), n = 1, . . . , N,
on F := P(Ω). We have (exercise):
Fn = σ(S10 , . . . , S
1n), S1
n = S10
n∏k=1
Tk, n = 0, . . . , N.
It holdsP ((x1, . . . , xN)) = P (T1 = x1, . . . , Tn = xn).
(In particular, knowing P is equivalent to knowing the joint distribution of T1, . . . , Tn, seelater). We suppose that P (ω) > 0 for all ω ∈ Ω.
22
1. The discounted price (Sn) is a martingale under a probability measure P on (Ω,F), ifand only if E[Tn+1 | Fn] = 1+r, n = 0, . . . , N−1, where E denote the expectation w.r.t. P
Proof Clearly (Sn) is adapted. Thus
(Sn)P -martingale ⇐⇒ (S1n)P -martingale
⇐⇒ E[S1n+1 | Fn] = S1
n
⇐⇒ E[ S1
n+1
S1n
| Fn]
= 1 (1
S1n
∈ Fn and S1n > 0)
⇐⇒ E[Tn+1 | Fn] = 1 + r (S1n+1
S1n
=S1n+1
S1n
· 1
1 + r= Tn+1 ·
1
1 + r)
2. If the market is arbitrage-free, then r ∈ (a, b).
Proof If there is no arbitrage opportunity, then ∃P ∗ ≈ P , and (Sn) is a P ∗-martingale.By 1. applied with P = P ∗
E∗[Tn+1 | Fn] = 1 + r,
hence
E∗[Tn+1 | Fn] = 1 + r = (1 + a)P ∗(Tn+1 = 1 + a) + (1 + b)P ∗(Tn+1 = 1 + b)
= 1 + aP ∗(Tn+1 = 1 + a) + b P ∗(Tn+1 = 1 + b) ∈ (1 + a, 1 + b).
3. By 2., if r /∈ (a, b), then there are arbitrage strategies.
E.g. if r ≤ a, then at time n = 0: borrow the amount of S10 in the riskless asset to
finance a share of stock S10 , i.e.
φ0 = (−S10 , 1), and keep the strategy up to time N , i.e.
φn = (−S10 , 1), n = 0, . . . , N , (i.e. at time N : pay back the loan and sell risky asset).
Then φ = (φn)n is predictable, self-financing, and for all n = 0, . . . , N
Vn(φ) = φn · Sn = S1n − S1
0(1 + r)n ≥︸︷︷︸(r≤a)
S10
(n∏k=1
Tk − (1 + a)n
)︸ ︷︷ ︸
≥0
≥ 0.
23
Thus φ is admissible, V0(φ) = 0, and
P (VN(φ) > 0) ≥ P (N∏k=1
Tk − (1 + a)N > 0)
= 1− P ((1 + a, . . . , 1 + a))︸ ︷︷ ︸∈(0,1)
> 0.
So φ is an arbitrage strategy.If r ≥ b, we can make a riskless profit by short-selling the risky asset, e.g.
φn = (S10 ,−1), n = 0, . . . , N is an arbitrage strategy (exercise).
4. From now on suppose: r ∈ (a, b). Set p = b−rb−a ∈ (0, 1). Then
(S1n) is a P -martingale ⇐⇒
(Tn)n=1,...,N is independent and
identically distributed (i.i.d) with
P (T1 = 1 + a) = p = 1− P (T1 = 1 + b)
(10)
Proof ”⇐ ”:
E[Tn+1 | Fn] =︸︷︷︸Tn+1 independent
of σ(T1,...,Tn)=Fn
E[Tn+1] =︸︷︷︸id
n=1,...,N−1
(1 + a)p+ (1 + b)(1− p)
= 1 + ap+ b(1− p) = 1 + (a− b)p+ b
= 1 + r − b+ b = 1 + r.
Thus (S1n) is a P -martingale by 1.
”⇒ ”: If E[Tn+1 | Fn] = 1 + r n = 0, . . . , N − 1, then
1 + r = (1 + a) E[1Tn+1=1+a | Fn] + (1 + b) E[1Tn+1=1+b | Fn]︸ ︷︷ ︸=1−E[1Tn+1=1+a | Fn]
.
Thus, we obtain E[1Tn+1=1+a | Fn] = p = 1−E[1Tn+1=1+b | Fn], n = 0, . . . , N − 1. Byinduction we get (exercise)
P (T1 = x1, . . . , Tn = xn) =n∏i=1
pi,
24
where pi = p, if xi = 1 + a and pi = 1 − p, if xi = 1 + b. Thus T1, . . . , TN is i.i.d, andP (T1 = 1 + a) = p as desired.
If the l.h.s. or the r.h.s. of (10) holds, then (S1
n) is a P -martingale, and P is uniquelydetermined. Thus the market is viable (free of arbitrage) and complete, and P = P ∗. Wewill assume this from now on.
5. Let Cn (resp. Pn) the value at time n = 0, ..., N , of a European call (resp. put) on ashare of the stock S1 with strike price K and maturity N .
(a) The put/call parity equation holds
Cn − Pn = S1n −K(1 + r)n−N , n = 0, ..., N.
Proof By (9) we know that the value (fair price) of these options at any time n isgiven by
Cn = S0n E∗
[(S1
N −K)+
S0N
∣∣∣ Fn] , (resp. Pn = S0n E∗
[(K − S1
N)+
S0N
∣∣∣ Fn])Then with S0
n = (1 + r)n we obtain
Cn − Pn = (1 + r)n−N E∗[(S1N −K)+ − (K − S1
N)+ | Fn]
= (1 + r)n−N E∗[S1N −K | Fn]
= (1 + r)n E∗[S1N | Fn]−K(1 + r)n−N
= S1n −K(1 + r)n−N (since (S1
n)n=0,...,N is a martingale)
as desired.
(b) Derive an explicit formula for the price of a call. Show: we can write Cn = c(n, S1n),
where c is a function that depends on K, a, b, r, and N .
Proof For n = 0, ..., N , write
S1N = S1
n
N∏i=n+1
Ti.
25
Then S1n is Fn-mb and
∏Ni=n+1 Ti is independent of Fn = σ(T1, . . . , Tn). Hence, using
Proposition 4.12 from the probabilistic part of the lecture with Φ(x, y) = (xy−K)+,we obtain
Cn = (1 + r)n−N E∗[(S1n
N∏i=n+1
Ti −K)+
︸ ︷︷ ︸=Φ(S1
n,∏Ni=n+1 Ti)
| Fn] = (1 + r)n−Nϕn(S1n),
where
ϕn(x) = E∗[Φ(x,N∏
i=n+1
Ti)] = E∗[(x ·N∏
i=n+1
Ti −K)+]
=N−n∑j=0
(N − nj
)pj(1− p)N−n−j
(x · (1 + a)j(1 + b)N−n−j −K
).
Thus c(n, x) = (1 + r)n−Nϕn(x).
(c) c(n, x) satisfies the recursive equations
c(n, x) =p c(n+ 1, x(1 + a)) + (1− p)c(n+ 1, x(1 + b))
1 + r, n = 0, . . . , N − 1.
Proof We have
ϕn(x) = E∗[(x ·N∏
i=n+1
Ti −K)+]
= E∗[
(x ·N∏
i=n+1
Ti −K)+(1Tn+1=1+a + 1Tn+1=1+b)
]
= E∗(x(1 + a)
N∏i=n+2
Ti −K
)+
1Tn+1=1+a
+E∗
(x(1 + b)N∏
i=n+2
Ti −K
)+
1Tn+1=1+b
=︸︷︷︸
Tn+1 indep.
of Tn+2,...,TN
pϕn+1(x(1 + a)) + (1− p)ϕn+1(x(1 + b)).
26
Here the last equality follows as in the proof of Propodition 4.6. of the Probabilisticbackground.
6. If (Hn) is the replicating (=hedging) strategy of a call, then H1n = ∆(n, S1
n−1) at timen = 1, ..., N , where ∆ is expressed in terms of c.
Proof We have
Vn(H) = Hn · Sn = H0n(1 + r)n +H1
nS1n =
(9)Cn =
5.c(n, S1
n).
Since S1n = S1
n−1(1 + a) or S1n = S1
n−1(1 + b), we obtain
H0n(1 + r)n +H1
nS1n−1(1 + a) = c(n, S1
n−1(1 + a)) (∗)H0n(1 + r)n +H1
nS1n−1(1 + b) = c(n, S1
n−1(1 + b)) (∗∗)
(∗∗)− (∗) H1n =
c(n, S1n−1(1 + b))− c(n, S1
n−1(1 + a))
S1n−1(b− a)
,
thus
∆(n, x) =c(n, x(1 + b))− c(n, x(1 + a))
x(b− a).
7. Study the asymptotics in the N -period model as N → ∞. In particular derive theasymptotics of put and call prices.
For this, let T > 0. Divide the (continuous time) interval [0, T ] in N subintervals oflength T
Nand let R > 0 be the instantaneous interest rate (continuous compounding), i.e.
eRT = limN→∞
1 +RT
N︸︷︷︸=:rN
N
.
Let σ > 0 be given (we will see later: σ “=” limN→∞ var(
log(S1N))).
Let −1 < aN < bN , and aN , bN be given as unique solutions to
log
(1 + aN1 + rN
)= −σ
√T
N, log
(1 + bN1 + rN
)= σ
√T
N.
27
(a) Suppose YN = XN1 + · · ·+XN
N with
XNi , 1 ≤ i ≤ N i.i.d., XN
i ∈
−σ√T
N, σ
√T
N
, E[XN
1 ] = µN , and NµN →N→∞
µ ∈ R.
For the characteristic function P YN of P YN we have for any ξ ∈ R:
P YN (ξ) = E[exp(iξYN)] =indep.
N∏j=1
E[exp(iξXNj )] =
i.d.E[exp(iξXN
1 )]N
= E[1 + iξXN1 −
ξ2
2(XN
1 )2 + rest︸︷︷︸=o( 1
N )
]N
=
(1 +
iξNµN − ξ2
2σ2T +N · o
(1N
)N
)N
−→N→∞
exp(iξµ− ξ2
2σ2T ) = E[exp(iξY )], Y ∼ N(µ, σ2T ).
Thus the characteristic functions P YN converge pointwise to the characteristic functionof N(µ, σ2T ) which is obviously continuous in ξ = 0. It follows from Levy’s continuitytheorem that P YN −→
N→∞N(µ, σ2T ) weakly.
(b) Calculate the asymptotics of the fair put/call prices.
For given N ≥ 1 the (fair) put price at time n = 0 is:
P(N)0 = (1 + rN)−N E∗[(K − S1
0
N∏j=1
Tj)+]
= E∗[(K(1 + rN)−N − S1
0
N∏j=1
(Tj
1 + rN
))+],
= E∗[(K(1 + rN)−N − S1
0eYN)+]
with
YN :=N∑j=1
log( Tj
1 + rN
).
28
Assuming Tj ∈ 1 + aN , 1 + bN, we have that
XNj := log
(Tj
1 + rN
)∈
−σ√T
N, σ
√T
N
, 1 ≤ j ≤ N.
Furthermore (XNj )1≤j≤N is i.i.d. under P ∗ (cf. 4.) and
E∗[XNj ] =
(−σ√T
N
)p+
(σ
√T
N
)(1− p) = (1− 2p)σ
√T
N,
where p = bN−rNbN−aN
. It follows
(1− 2p) =2− eσ
√TN − e−σ
√TN
eσ√
TN − e−σ
√TN
,
and
N · E∗[XNj ] −→
N→∞µ = −σ
2T
2.
Thus (YN)N≥1, (XNj )1≤j≤N , satisfy the conditions of 7.(a). Define
ψ(y) := (K e−RT − S10 e
y)+ ∈ Cb(R) (for call not bounded!)
Using that x 7→ (x− const)+ is Lipschitz continuous with Lipschitz constant one, we get∣∣∣P (N)0 − E∗[ψ(YN)]
∣∣∣ =∣∣∣E∗[(K(1 + rN)−N − S1
0 exp(YN))+ −
(K e−RT − S1
0 exp(YN))+
]∣∣∣
≤ K∣∣∣(1 + rN)−N − e−RT
∣∣∣ −→N→∞
0,
and since P ∗YN −→ N(µ, σ2T ) weakly and ψ ∈ Cb(R), we obtain
limN→∞
P(N)0 = lim
N→∞E∗[ψ(YN)] =
∫Rψ(y)N(µ, σ2T )(dy) =
1√2πσ2T
∫ ∞−∞
ψ(y) e−(y−µ)2
2σ2T dy.
The first zero of the decreasing function ψ is given by
Z0 = log
(Ke−RT
S10
)= log
(K
S10
)−RT.
29
Thus
limN→∞
P(N)0 =
1√2πσ2T
∫ Z0
−∞ψ(y) e−
(y−µ)2
2σ2T dy
=1√
2πσ2T
∫ Z0
−∞e−
(y−µ)2
2σ2T dy ·Ke−RT − 1√2πσ2T
∫ Z0
−∞ey−
(y−µ)2
2σ2T dy · S10 .
In the first integral of the last expression we make the substitution z = y−µσ√T
. Noting that
y − (y − µ)2
2σ2T=−4µy − (y − µ)2
2σ2T=
(y + µ)2
2σ2T,
we make the substitution z = y+µ
σ√T
in the second integral. By this, we obtain
limN→∞
P(N)0 =
1√2π
∫ Z0−µσ√T
−∞e−
z2
2 dz ·Ke−RT − 1√2π
∫ Z0+µ
σ√T
−∞e−
z2
2 dz · S10
= Ke−RT Φ
log(KS10
)−RT + σ2T
2
σ√T
− S10 Φ
log(KS10
)−RT − σ2T
2
σ√T
where Φ(x) = 1√
2π
∫ x−∞ e
− z2
2 dz. Therefore the asymptotic put price is
limN→∞
P(N)0 = E∗[ψ(Y )] = Ke−RT Φ(−d2)− S1
0 Φ(−d1)
with
d1 =− log
(KS10
)+RT + σ2T
2
σ√T
, d2 = d1 − σ√T .
The asymptotic call price now follows from the put/call parity:
limN→∞
C(N)0 = lim
N→∞
(P
(N)0 + S
(N)0︸︷︷︸
≡S10=c
−K(1 + rN)−N︸ ︷︷ ︸→Ke−RT
)= Ke−RT
(Φ(−d2)− 1︸ ︷︷ ︸∫−d2−∞ −
∫∞−∞=−
∫∞−d2
=−∫ d2−∞ by symmetry
)− S1
0
(Φ(−d1)− 1︸ ︷︷ ︸
=−∫ d1−∞
)
= S10Φ(d1)−Ke−RTΦ(d2).
Remark 2.25 The only non-observable parameter is σ! Its interpretation as variancesuggests it should be estimated by statistical methods.
30
3 Brownian motion and Stochastic Differential Equa-
tions (SDEs)
3.1 General notions
Definition 3.1 Let (Ω,A, P ) be a probability space, and (E,E) a measurable space. Astochastic process is a family of r.v.s (Xt)t≥0 from (Ω,A, P ) with values in (E,E) (E iscalled state space). Thus
∀t ≥ 0 Xt : (Ω,A) −→ (E,E) measurable.
∀ω ∈ Ω the map X.(ω) : [0,∞) −→ E
t 7→ Xt(ω),
is called a sample path (or a trajectory).
Remark 3.2 - t stands for time
- a stochastic process can be considered as map
X := X.(·) : R+ × Ω −→ E.
We always assume that X is B(R+)⊗A︸ ︷︷ ︸product σ−algebra
\E measurable.
- we will mainly work with stochastic processes on a finite time interval [0, T ].
Definition 3.3 Let (Ω,A, P ) be a probability space. A filtration on A is an increasingfamily (Ft)t≥0 of sub-σ-algebras of A, i.e.
Fs ⊂ Ft ⊂ A ∀s ≤ t.
A process (Xt)t≥0 is adapted to (Ft)t≥0, if Xt is Ft-mb ∀t ≥ 0, i.e. Xt ∈ A := ω ∈Ω | Xt(ω) ∈ A ∈ Ft ∀A ∈ E, t ≥ 0.
For any t ≥ 0, Ft is interpreted as information (knowledge) at time t. To express that(Ft)t>0 is a filtration on (Ω,A, P ) we shortly write (Ω,A, (Ft)t>0, P ).
Remark 3.4 We will only work with filtrations (Ft)t≥0 such that F0 contains all P -zerosets of A, i.e.
A′ ⊂ A ∈ A and P (A) = 0⇒ A′ ∈ F0. (11)
31
A filtration as in Remark 3.4 can always be constructed from an ordinary one. Indeed, let
NP := A′ ⊂ Ω | A′ ⊂ A ∈ A, P (A) = 0,
andAP := A ∪ A′ | A ∈ A, A′ ∈ NP,
then one can show that AP is a σ-algebra on Ω, and that P extends to AP by the formulaP (A∪A′) = P (A), if A ∈ A, A′ ∈ NP (exercise). (Ω,AP , P ) is called the completion of Aw.r.t. P . Then let
Ft := σ(Ft ∪NP ) t ≥ 0.
Then (Ft)t≥0 is a filtration on AP that satisfies (11) with A replaced by AP .A useful consequence of (11) is the following:
X = Y P -a.s. and Y Ft-mb⇒ X Ft-mb (exercise)
Gt := σ(Xs; s ≤ t) is called the filtration generated by (Xt)t≥0. If (Xt)t≥0 is adapted tosome filtration (Ft)t≥0, then Gt ⊂ Ft,∀t ≥ 0 (since Xs ∈ Ft for any s ≤ t and (Gt)t≥0 isthe smallest filtration for which this is the case). In general (σ(Xs; s ≤ t))t≥0 does notsatisfy (11), but by the above consideration the completed filtration Gt := σ(σ(Xs; s ≤t)∪NP ), t ≥ 0, satisfies (11). This (Gt)t≥0 is called the natural filtration of (Xt). Obviously(Xt)t≥0 is also adapted to (Gt)t≥0.
Definition 3.5 A stopping time w.r.t. the filtration (Ft)t≥0 is a map τ : Ω→ [0,∞] suchthat
τ ≤ t ∈ Ft, ∀t ≥ 0.
In particular τ is A-measurable. The σ-algebra associated with τ is
Fτ := A ∈ A | A ∩ τ ≤ t ∈ Ft ∀t ≥ 0
It is called the σ-algebra of the ”τ -past” or of the ”events prior to τ”. Fτ is interpreted asinformation available before τ .
For technical reasons it is also convenient to consider right-continuous filtrations (Ft)t≥0,i.e. filtrations (Ft)t≥0 that satisfy Ft = Ft+ := ∩ε>0Ft+ε for any t ≥ 0. By considering theright-continuous filtration (Ft+)t≥0 instead of (Ft)t≥0, we may often assume that (Ft)t≥0
is right-continuous.(Ft)t>0 is said to satisfy the “usual conditions” if it is right-continuous, (Ω,A, P ) acomplete probability space, and N ∈ A such that P (N) = 0 implies N ∈ F0. From nowon we assume whenever necessary the usual conditions.
32
Proposition 3.6 (Let (Ft)t≥0 satisfy the usual conditions)
(1) If S is a stopping time, S is FS-mb.
(2) Let S be a stopping time, P (S < ∞) = 1, (Xt)t≥0 be continuous (i.e. P (t 7→Xt is continuous) = 1), and adapted to (Ft)t≥0. Then XS is FS-mb, where
(XS)(ω) :=
XS(ω)(ω), if S(ω) <∞,0 otherwise.
(3) If S, T are stopping times with S ≤ T P -a.s, then FS ⊂ FT .
(4) If S, T are stopping times, then S ∧ T := inf(S, T ) is a stopping time. In particular∀t ≥ 0, t fixed S ∧ t is a stopping time.
(5) τ : Ω→ [0,∞] is a stopping time, if and only if it is a weak stopping time, i.e.
τ < t ∈ Ft, ∀t ≥ 0.
Proof Exercise, cf. e.g. lecture SDEs.
3.2 Brownian motion
Definition 3.7 A real-valued stochastic process (Bt)t≥0 is called a (standard) Brownianmotion (BM for short, or BM starting in zero), if :
(1) P (ω ∈ Ω | B0(ω) = 0) = 1 (”start in zero”, ”standard”).
(2) ∀0 ≤ s < t, the r.v. Bt−Bs is normally distributed with mean 0 and variance t− s,i.e. for any b ∈ R
P (Bt −Bs ≤ b) =1√
2π(t− s)
∫ b
−∞e−
x2
2(t−s) dx.
(3) (Bt)t≥0 has independent increments, i.e. for any 0 ≤ t1 < · · · < tn, the r.v.’s
Bt1 , Bt2 −Bt1 , . . . , Btn −Btn−1
are independent.
33
(4) Almost all sample paths of (Bt)t≥0 are continuous, i.e.
P (ω | t 7→ Bt(ω) is continuous) = 1.
Remark 3.8 The construction, i.e. existence of a BM is not trivial to show (see forinstance lecture SDEs). Here, we accept the existence of a BM as in 3.7.
Definition 3.9 Let (Gt)t≥0 be a filtration on (Ω,A, P ). An (Gt)-adapted process (Bt)t≥0
is called a (Gt)t≥0-BM (or BM w.r.t. (Gt)t≥0), if
(1) (Bt)t≥0 is a BM.
(2) ∀s ≤ t: Bt −Bs is independent of Gs.
Remark 3.10 Any BM (Bt)t≥0 is an σ(Bs; s ≤ t)-BM. One can also check that any BM(Bt)t≥0 is a BM w.r.t. its natural filtration.
3.3 Continuous-time martingales
Definition 3.11 Let (Ω, (Ft)t≥0,A, P ) be a filtered probability space. An (Ft)-adaptedfamily (Mt)t≥0 of integrable r.v.’s (i.e. E[|Mt|] <∞ for any t ≥ 0) is a
- a martingale, if Ms = E[Mt | Fs] for all s ≤ t.
- a supermartingale, if Ms ≥ E[Mt | Fs] for all s ≤ t.
- a submartingale, if Ms ≤ E[Mt | Fs] for all s ≤ t.
Proposition 3.12 If (Xt)t≥0 is a standard (Ft)-BM, then:
(1) (Xt)t≥0 is an (Ft)-martingale.
(2) (X2t − t)t≥0 is an (Ft)-martingale.
(3) (eσXt−σ2
2t)t≥0 is an (Ft)-martingale for any σ ∈ R.
Proof
(1) One easily checks E[|Xt|] <∞ for any t ≥ 0 and ∀s ≤ t
E[Xt | Fs]−Xs =Xs∈Fs
E[Xt −Xs | Fs] =3.11(2)
E[Xt −Xs] =3.7(2)
0.
34
(2) We have E[X2t ] <∞ for any t ≥ 0 and for any s ≤ t
E[X2t −X2
s | Fs] = E[(Xt −Xs)2 + 2Xs(Xt −Xs) | Fs]
= E[(Xt −Xs)2 | Fs] + 2Xs E[(Xt −Xs) | Fs]︸ ︷︷ ︸
=0 by (1)
=3.11(2)
E[(Xt −Xs)2] =
3.7(2)t− s,
thus using X2s ∈ Fs, we obtain E[X2
s | Fs]− s = E[X2t − t | Fs] for any s ≤ t.
(3) is left as an exercise.
The (sub)martingale property remains true for bounded (!) stopping times.
Theorem 3.13 (optional sampling theorem) Let (Mt)t≥0 be a continuous martingalew.r.t. (Ft)t≥0. If τ1, τ2 are two bounded stopping times with τ1 ≤ τ2, then Mτ2 is integrableand
E[Mτ2 | Fτ1 ] = Mτ1 P -a.s.
(The above equation holds with ” ≥ ”, if (Mt)t≥0 is a submartingale).
Remark 3.14 Theorem 3.13 implies:
∀ bounded stopping times τ, Mτ is integrable, and E[Mτ ] = E[M0].
(In fact: If we preassume continuity and adaptedness of (Mt)t≥0 as in 3.13, this is alsoequivalent to the statement in 3.13).
Proposition 3.15 Let (Xt)t≥0 be an (Ft)-BM. Define
Ta := infs ≥ 0 | Xs = a, a ∈ R (inf ∅ := +∞).
Then Ta is a stopping time, finite P -a.s. and its distribution is characterized by its Laplacetransform
E[e−λTa ] = e−√
2λ|a|, λ ≥ 0.
Proof By 3.4 we may assume that t 7→ Xt(ω) is continuous ∀ω ∈ Ω. Let first a > 0.Then
Ta ≤ t = sups≤t
Xs ≥ a =⋂
ε>0,ε∈Q
sups≤t
Xs > a− ε =⋂
ε>0,ε∈Q
⋃s∈Qs≤t
Xs > a− ε ∈ Ft.
35
Thus Ta is a stopping time. Hence Ta ∧ n is a bounded stopping time ∀n ≥ 1, and so
E[eσXTa∧n−σ2
2Ta∧n] =
3.12(3)+3.141 ∀n ≥ 0, σ ∈ R. (12)
From now on, we let σ > 0. Then, we have 0 ≤ eσXTa∧n−σ2
2Ta∧n ≤ eσa P -a.s. ∀n ≥
1 (integrable majorant) and
limn→∞
1Ta<∞eσXTa∧n−
σ2
2Ta∧n = 1Ta<∞e
σXTa−σ2
2Ta =
XTa=a1Ta<∞e
σa−σ2
2Ta
and since Xt ≤ a on Ta =∞, for σ > 0
limn→∞
1Ta=∞eσXTa∧n−
σ2
2Ta∧n = lim
n→∞1Ta=∞e
σXn−σ2
2n = 0.
Thus for (any) σ > 0
1 =(12)
limn→∞
E[eσXTa∧n−σ2
2Ta∧n] =
LebesgueE[ lim
n→∞eσXTa∧n−
σ2
2Ta∧n] = E[1Ta<∞e
σa−σ2
2Ta ]
This implies
E[1Ta<∞e−σ
2
2Ta ] = e−σa (13)
Letting σ 0, we obtain again using Lebesgue E[1Ta<∞] = 1, hence
P (Ta <∞) = 1,
and so, by (13)
E[e−σ2
2Ta ] = e−σa.
If a < 0, then since Wt := −Xt is also a (standard) (Ft)-BM, and
Ta = infs ≥ 0 | Xs = a = infs ≥ 0 | Ws = −a︸︷︷︸>0
,
we conclude as before E[e−σ2
2Ta ] = e−σ(−a). Now, we let λ := σ2
2, then the assertion follows
for any λ > 0. The assertion for λ = 0 is trivial.
Theorem 3.16 (Doob’s inequality) Let (Mt)t∈[0,T ] be a continuous martingale. Then
E[ supt∈[0.T ]
|Mt|2] ≤ 4E[M2T ]
Proof See textbooks on probability theory or martingale theory.
36
3.4 Stochastic integral and Ito claculus
Let (Wt)t≥0 be a standard (Ft)-BM on a filtered probability space (Ω,A, (Ft)t≥0, P ). Fora certain class (that we will precise later) of adapted process (Ht)t≥0 we want to definethe stochastic integral ∫ t
0
Hs dWs, t ≥ 0 (14)
From the next theorem and the following remark, we can see that it is not possible todefine (14) in the Lebesgue-Stieltjes sense.
Theorem 3.17 Let τn = 0 = tn0 < tn1 < · · · < tnN(τn) <∞, tni ∈ R, n ≥ 1, be a sequence
of partitions of [0,∞) with
|τn|︸︷︷︸”mesh of τn”
:= sup1≤i≤N(τn)
|tni − tni−1| −→n→∞
0, tnN(τn) −→n→∞
∞.
Define
Snt :=∑tni∈τn
tni+1≤t
(Wti+1−Wti)
2.
Then Snt −→n→∞
t in L2(Ω,A, P ). If additionally τn ⊂ τn+1, then
P (Snt −→n→∞
t ∀t ≥ 0) = 1.
Proof L2-convergence: We have
E[Snt ] =∑tni∈τn
tni+1≤t
E[(Wti+1−Wti)
2] =∑tni∈τn
tni+1≤t
(tni+1 − tni ) −→n→∞
t,
and
var(Snt ) =∑tni∈τn
tni+1≤t
var((Wtni+1
−Wtni)2)
=∑tni∈τn
tni+1≤t
(E[(Wtni+1
−Wtni)4]︸ ︷︷ ︸
=3(tni+1−tni )2 (exercise)
−(tni+1 − tni )2)
= 2∑tni∈τn
tni+1≤t
(tni+1 − tni )2 −→n→∞
0.
Thus Snt − E[Snt ] −→n→∞
0 in L2, hence also Snt − t −→n→∞
in L2.
P -a.s. convergence: Proof only for τn = dyadic partition, i.e.
tni =i
2n, i = 0, . . . , n · 2n = N(τn), n ≥ 1.
37
Then
var(Snt ) = 2
i+12n≤t∑
i=0
(1
2n
)2
≤ 2k·2n−1∑i=0
(1
2n
)2
=2k
2n(for t ≤ k).
Thus using Chebyshev’s inequality, i.e.
P (|Snt − E[Snt ]| ≥ ε) ≤ 1
ε2var(Snt ) ∀ε > 0,
we get∞∑n=1
P (|Snt − E[Snt ]| ≥ ε) ≤ 2k
ε2<∞ ∀ε > 0
Thus (Snt − E[Snt ])n≥1 converges fast in probability to zero. Consequently (cf. lectureTIM1, Lemma 7.7), (Snt − E[Snt ])n≥1 converges P -a.s. to zero. Since limn→∞ E[Snt ] = t bythe above, we have hence shown that
P ( limn→∞
Snt = t) = 1 for any t ≥ 0.
This means: ∀t ≥ 0 ∃Ωt with P (Ωt) = 1 and
limn→∞
Snt (ω) = t ∀ω ∈ Ωt (dependence on t !).
Let Q+2 := m · 2−n ; m,n ∈ N ∪ 0 be the positive dyadic rationals and
Ω :=⋂t∈Q+
2
Ωt.
Then P (Ω) = 1, since P ((⋂t∈Q2
Ωt)c) = P (
⋃t∈Q2
Ωct) ≤
∑t∈Q2
P (Ωct) = 0 and
Snt (ω)n→∞−→ t ω ∈ Ω and t ∈ Q+
2 .
Let t ∈ [0,∞) \Q+2 . Then ∃(t+k )k≥1, (t
−k )k≥1 ⊂ Q+
2 , with t+k t, t−k t as k ∞ and forall ω ∈ Ω, n, k ≥ 1, we get
Snt−k
(ω) ≤ Snt (ω) ≤ Snt+k
(ω).
Thust−k = lim
n→∞Snt−k
(ω) ≤ lim infn→∞
Snt (ω) ≤ lim supn→∞
Snt (ω) ≤ limn→∞
Snt+k
(ω) = t+k
for any k ≥ 1. Letting k →∞ we get: limn→∞ Snt (ω) = t for all ω ∈ Ω, t ≥ 0 (”Sandwich
argument”).
38
Remark 3.18 Theorem 3.17 shows that P -a.e. path of BM is of infinite variation. Indeed,suppose (Wt(ω))t≥0 is of finite variation, then∑
tni∈τn
tni+1≤t
(Wtni+1(ω)−Wtni
(ω))2 ≤ maxtni∈τn
tni+1≤t
|Wtni+1(ω)−Wtni
(ω)|
︸ ︷︷ ︸<ε, for |τn|<δ
by uniform continuity on [0,t]
∑tni∈τn
tni+1≤t
|Wtni+1(ω)−Wtni
(ω)|
︸ ︷︷ ︸<∞ by assumption
⇒ 〈W 〉t(ω) := limn→∞
Snt (ω) = 0, ∀t ≥ 0.
But by Theorem 3.17 this only possible for ω in some P -zero set, i.e. ω /∈ Ω. Therefore,as we already said, we can not define (14) pathwise as an Lebesgue-Stieltjes integral.However, the quadratic variation of (Wt)t≥0 along (τn) is a well behaved function (positive,continuous and increasing). This fact can be exploited in order to develop a new approachfor the definition of (14).
3.4.1 Construction of the stochastic integral
Let (Wt)t≥0 be a standard (Ft)-BM on
(Ω,A, (Ft)t≥0, P ).
Fix T > 0.
Definition 3.19 (Ht)t∈[0,T ] is called a simple process, if
Ht(ω) =
p∑i=1
φi(ω)1(ti−1,ti](t),
where 0 = t0 < t1 < · · · < tp = T , and φi, i = 1, ..., p, is Fti−1-measurable and bounded.
The stochastic integral of H is defined for all t ∈ [0, T ] by
I(H)t :=
p∑i=1
φi(Wti∧t −Wti−1∧t)
⇒ (I(H)t)t∈[0,T ] is continuous, (Ft)-adapted and (square) integrable.
We have
I(H)t =k∑i=1
φi(Wti −Wti−1) + φk+1(Wt −Wtk), if t ∈ (tk, tk+1]. (15)
39
One writes ∫ t
0
Hs dWs := I(H)t, t ∈ [0, T ].
Proposition 3.20 If (Ht)t∈[0,T ] is a simple process, then
(1)(∫ t
0
Hs dWs
)t∈[0,T ]
is a continuous (Ft)-martingale.
(2) E[( ∫ t
0
Hs dWs
)2]
= E[ ∫ t
0
H2s ds
], t ∈ [0, T ].
(3) E[
supt≤T
∣∣ ∫ t
0
Hs dWs
∣∣2] ≤ 4E[ ∫ T
0
H2s ds
].
Proof
(1) Let s < t. Include s, t in 0 = t0 < · · · < tp = T .
0 = t0 < · · · < tp = T, p ∈ p, p+ 1, p+ 2.
For n ≤ p we can use (15), and write
Mn :=
∫ tn
0
Hu dWu =n∑i=1
φi(Wti −Wti−1)
with φi ∈ Gi−1 := Fti−1, i = 1, ..., p.
Further Xj := Wtj , j = 0, ..., p is a (Gj)-martingale since (Wt)t≥0 is an (Ft)-martingale. Thus (Mn) is a martingale transform (see Proposition 2.10). Then∃j < l ≤ p with s = tj, t = tl, and so
E[ ∫ t
0
Hu dWu | Fs]
= E[Ml | Gj] = Mj =
∫ t
0
Hu dWu.
We already know that( ∫ t
0Hu dWu
)t≥0
is continuous adapted and integrable, thus
(1) follows.
(2) Using the notations of (1), there exists n ≤ p, with t = tn. Then
E[M2n] =
n∑i,j=1
E[φi φj(Xi −Xi−1)(Xj −Xj−1)]. (16)
40
If i < j, then
E[φi φj(Xi −Xi−1)(Xj −Xj−1)]
= E[E[φi φj(Xi −Xi−1)(Xj −Xj−1) | Gj−1]
]= E
[φi φj(Xi −Xi−1)E[(Xj −Xj−1) | Gj−1]︸ ︷︷ ︸
=0
]= 0, (17)
and for j > i we get the same. For i = j, we have by independence of the Brownianincrements
E[(φi)2(Xi −Xi−1)2] = E
[E[(φi)
2(Xi −Xi−1)2 | Gi−1]]
= E[(φi)
2 E[(Xi −Xi−1)2 | Gi−1]︸ ︷︷ ︸=E[(Wti
−Wti−1)2]=ti−ti−1
]. (18)
Since t = tn for some n ∈ 0, . . . , p, we obtain
E[( ∫ t
0
Hs dWs
)2]
= E[M2n] =
(16),(17),(18)
E[n∑i=1
(φi)2(ti − ti−1)] = E
[ ∫ t
0
H2s ds
].
(3) Since( ∫ t
0Hs dWs
)t∈[0,T ]
is a continuous martingale, (3) is just Doob’s inequality
together with (2).
Remark 3.21 Define∫ T
t
Hs dWs :=
∫ T
0
Hs dWs −∫ t
0
Hs dWs, t ≤ T.
If A ∈ Ft, then s 7→ 1A1s>tHs is still simple process, since
∀s ≤ T 1A 1s>t︸ ︷︷ ︸1(t,T ](s)
Hs =
p∑i=1
φi1A1(ti−1,ti]∩(t,T ](s)
Here
φi1A ∈
Fti−1
if t ≤ ti−1
Ft if t ∈ (ti−1, ti)
Ft if t ≥ ti
and (ti−1, ti] ∩ (t, T ] =
(ti−1, ti] if t ≤ ti−1
(t, ti] if t ∈ (ti−1, ti)
∅ if t ≥ ti.
41
Further, one can easily see from the definition of stochastic integral that∫ T
0
1A1s>tHs dWs = 1A
∫ T
t
Hs dWs. (19)
Define
H :=
(Ht)t∈[0,T ] | H is (Ft)-adapted and E[ ∫ T
0
H2s ds
]<∞
.
Obviously H contains all simple processes.
Proposition 3.22 Let (Wt)t≥0 be an (Ft)-BM. There exists a unique linear map
J : H −→Mc = continuous (Ft)-martingales on [0, T ],
H 7→ (J(H)t)t∈[0,T ]
such that:
(1) If (Ht)t∈[0,T ] is a simple process, then
P (I(H)t = J(H)t ∀t ∈ [0, T ]) = 1.
(2) If t ∈ [0, T ], then
E[J(H)2t ] = E
[ ∫ t
0
H2s ds
]for any H ∈ H.
The uniqueness of J is in the following sense: Let J ′ be another map with the sameproperties as J , then
P (J(H)t = J ′(H)t ∀t ∈ [0, T ]) = 1 for any H ∈ H.
Define: ∫ t
0
Hs dWs := J(H)t for t ≥ 0, H ∈ H. (”stochastic integral”)
Proposition 3.23 Let H ∈ H. Then:
(1)
E[
supt∈[0,T ]
∣∣ ∫ t
0
Hs dWs
∣∣2] ≤ 4E[ ∫ T
0
H2s ds
](20)
42
(2) If τ ≤ T is an (Ft)-stopping time, then
P(∫ τ
0
Hs dWs =
∫ T
0
1s≤τHs dWs
)= 1. (21)
Proof of 3.22 (and 3.23). We shall accept: H ∈ H⇒ ∃(Hn)n∈N simple processes with
E[ ∫ T
0
(Hs −Hns )2 ds
]−→n→∞
0.
For a sketch of proof of this fact see e.g. [Karatzas/Shreve, Brownian motion and stochasticcalculus (2nd edition 1991), 2.5 Problem, p.135]. For H ∈ H and (Hn)n∈N like above, and∀p ≥ 1
E[
supt∈[0,T ]
∣∣∣I(Hn+p)t − I(Hn)t
∣∣∣2] ≤ 4E[ ∫ T
0
(Hn+ps −Hn
s )2 ds]
(22)
⇒ ∃ subsequence (nk) with
E[
supt∈[0,T ]
∣∣∣I(Hnk+1)t − I(Hnk)t
∣∣∣2] 12 ≤ 1
2k
⇒ E[ ∞∑k=1
supt∈[0,T ]
∣∣∣I(Hnk+1)t − I(Hnk)t
∣∣∣]≤
∞∑k=1
E[
supt∈[0,T ]
∣∣∣I(Hnk+1)t − I(Hnk)t
∣∣∣2] 12 ≤ 1 <∞
⇒ P( ∞∑k=1
supt∈[0,T ]
∣∣∣I(Hnk+1)t − I(Hnk)t
∣∣∣ <∞) = 1
⇒ ∃J(H)t := limk→∞
I(Hnk)t uniformly on [0, T ] P -a.s.
In particular: (J(H)t)t∈[0,T ] is continuous and adapted. Taking the limit in (22) along thesubsequence (n+ p nk), we obtain with Fatou’s lemma
E[
supt∈[0,T ]
∣∣∣J(H)t − I(Hn)t
∣∣∣2] ≤ 4E[ ∫ T
0
(Hs −Hns )2 ds
]. (23)
(23) implies: J(H)t does not depend on the choice of approximating sequence (Hn)n∈Nand 3.22 (1) holds. Moreover: (23) ⇒ I(Hn)t → J(H)t in L2 for any t ∈ [0, T ]. ThenJ(H) is a martingale, since for any s ≤ t P -a.s.
E[J(H)t | Fs] = E[ limk→∞
I(Hnk)t | Fs] = limk→∞
E[I(Hnk)t | Fs] = limk→∞
I(Hnk)s = J(H)s.
43
3.22 (2) holds, because
E[J(H)2t ] =
(23)limn→∞
E[I(Hn)2t ] =
3.20limn→∞
E[ ∫ t
0
(Hns )2 ds
]= E
[ ∫ t
0
H2s ds
].
(20) holds, because
E[
supt∈[0,T ]
|J(H)t|2]
= limk
E[
supt∈[0,T ]
∣∣∣ I(Hnk)t
∣∣∣2] ≤ limk
4E[ ∫ T
0
(Hnks )2 ds
]= 4E
[ ∫ T
0
(Hs)2 ds]
and the uniqueness follows, since for any n ≥ 1
E[ supt∈[0,T ]
|J(H)t − J ′(H)t|2]
≤ 2(E[ sup
t∈[0,T ]
∣∣∣J(H)t − I(Hn)t
∣∣∣2] + E[ supt∈[0,T ]
∣∣∣I(Hn)t − J ′(H)t
∣∣∣2])
≤ 16E[ ∫ T
0
(Hs −Hns )2 ds
]−→n→∞
0.
Finally, we want to show (21): Start with stopping times of the form
τ =n∑i=1
ti1Ai , 0 < t1 < · · · < tn = T, Ai ∈ Fti , pairwise disjoint , 1 ≤ i ≤ n, P (∪ni=1Ai) = 1.
τ is indeed a stopping time by Proposition 3.6(5), since
1s>τ = 1s>∑ni=1 ti1Ai =
n∑i=1
1s>ti1Ai︸ ︷︷ ︸=
0 if s ≤ ti
1Ai ∈ Fti ⊂ Fsif s > ti.
44
Hence 1s>τHs ∈ H and also 1s>ti1AiHs ∈ H if H ∈ H, i = 1, ..., n. Then∫ T
0
1s>τHs dWs =n∑i=1
∫ T
0
1s>ti1AiHs dWs
= limk→∞
n∑i=1
∫ T
0
1s>ti1AiHnks dWs
=(19)
limk→∞
n∑i=1
1Ai
∫ T
ti
Hnks dWs
=n∑i=1
1Ai
∫ T
ti
Hs dWs =
∫ T
τ
Hs dWs,
and so ∫ T
0
1s≤τHs dWs =
∫ τ
0
Hs dWs.
For general stopping time τ ≤ T we have that
τn := T1τ=T +2n−1∑k=0
(k + 1)T
2n1τ∈[ kT
2n,(k+1)T
2n) ( τ as n→∞)
is of the previous form and by continuity∫ τn
0
Hs dWs −→n→∞
∫ τ
0
Hs dWs P -a.s.
Further
E[∣∣∣ ∫ T
0
1s≤τnHs dWs −∫ T
0
1s≤τHs dWs
∣∣∣2] =3.22(2)
E[ ∫ T
0
1τ<s≤τnH2s ds
]−→n→∞
0
by Lebesgue. Thus ∫ T
0
1s≤τnHs dWs −→∫ T
0
1s≤τHs dWs
in L2(Ω,A, P ), hence P -a.s. along a subsequence, say (τnk). Consequently∫ τ
0
Hs dWs = limk→∞
∫ τnk
0
Hs dWs = limk→∞
∫ T
0
1s≤τnkHs dWs =
∫ T
0
1s≤τHs dWs
P -a.s. and (21) is shown.
45
Define
H := (Hs)s∈[0,T ] | H is (Ft)-adapted and
∫ T
0
H2s ds <∞ P -a.s..
In the following proposition we extend the stochastic integral to integrand processes inH. In this case the stochastic integral is not necessarily a martingale.
Proposition 3.24 There exists a unique linear map
J : H −→ C = continuous processes on [0, T ]
such that:
(1) Extension property: If (Ht)t∈[0,T ] is a simple process, then
P(J(H)t = I(H)t ∀t ∈ [0, T ]
)= 1.
(2) Continuity property: If (Hn)n∈N ⊂ H, then
P( ∫ T
0
(Hns )2 ds ≥ ε
)−→n→∞
0 ∀ε > 0⇒ P(
supt∈[0,T ]
|J(Hn)t| ≥ ε)−→n→∞
0 ∀ε > 0.
For H ∈ H, we set as before∫ t
0
Hs dWs := J(H)t, t ∈ [0, T ].
Proof Existence: Let H ∈ H and
Tn := infs ∈ [0, T ] |
∫ s
0
H2u du ≥ n
, inf ∅ :=∞.
Then Tn ∞ P -a.s. as n∞. Since Tn ≤ t⇔∫ t
0H2u du ≥ n, we get
Tn ≤ t =∫ t
0
H2u du ≥ n
.
Thus it is enough to show that( ∫ t
0H2u du
)t∈[0,T ]
is adapted in order to show that Tn is a
stopping time. If H ∈ H this is true, since ∃Hn simple with∫ t
0
(Hnu )2 du︸ ︷︷ ︸
adapted
−→n→∞
∫ t
0
H2u du P -a.s. for t = T hence ∀t ∈ [0, T ].
46
If H ∈ H then this is also true since∫ t
0
(|Hu| ∧ k)2 du︸ ︷︷ ︸adapted, since (|Hu|∧k)u∈[0,T ]
is in H
−→k→∞
∫ t
0
H2u du P -a.s. ∀t ∈ [0, T ].
Furthermore
E[ ∫ T
0
H2s1s≤Tn ds
]= E
[ ∫ T∧Tn
0
H2s ds
]≤ n,
thusHns := Hs1s≤Tn is in H ∀n ≥ 1.
Note that 1s≤Tn is Fs-mb since Tn is a stopping time. Furthermore for any n, k ≥ 1
∫ t
0
Hns dWs =
(21)
∫ T
0
1s≤tHns dWs =
∫ T
0
1s≤t 1s≤Tn
=Hn+ks︷ ︸︸ ︷
1s≤Tn+kHs︸ ︷︷ ︸=Hn
s since Tn≤Tn+k
dWs
=
∫ T
0
1s≤t∧TnHn+ks dWs =
(21)
∫ t∧Tn
0
Hn+ks dWs.
Thus on Tn > T we have P -a.s.
J(Hn)t = J(Hn+k)t ∀t ≤ T. (24)
Since Tn > T =∫ T
0H2u du < n
Ω (up to some P -zero set) we can define almost
surely a process J(H) by localization, i.e. for n ∈ N:
∀t ≤ T J(H)t := J(Hn)t on∫ T
0
H2u du < n
.
Then,
- J(H)t is (almost surely) continuous since J(Hn)t is continuous for any n, and J(H)is well defined by (24)
- the extension property holds by construction, since if H is a simple process(bounded !), then P -a.s. H = Hn = Hn0 for ∀n ≥ n0 for some n0 (since Tn0 = ∞for some n0 ⇒ Tn =∞ ∀n ≥ n0). Thus
J(H)t = J(Hn0)t = I(H)t ∀t ≤ T P -a.s.
47
Next we show the continuity property. For any N ∈ N
P(
supt∈[0,T ]
|J(H)t| ≥ ε)≤ P
(∫ T
0
H2s ds ≥
1
N
)+ P
(1
∫ T0 H2
s ds<1N supt∈[0,T ]
|J(H)t| ≥ ε)
(25)
and
T 1N
= infs ∈ [0, T ] |
∫ s
0
H2u du ≥
1
N
≤ T1 and
∫ T
0
H2u du <
1
N
= T 1
N> T.
Hence ∀t ≤ T we have P -a.s.
1T 1N>TJ(H)t = 1T 1
N>T1T1>TJ(H)t = 1T 1
N>T1T1>TJ(H1)t
= 1T 1N>T
∫ t
0
1s≤T1Hs dWs = 1T 1N>T
∫ t∧T 1N
0
1s≤T1Hs dWs
=(21)
1T 1N>T
∫ T
0
1s≤T11s≤t∧T 1NHs dWs
=(21)
1T 1N>T
∫ t
0
1s≤T 1NHs dWs.
Therefore
P(1T 1
N>T sup
t∈[0,T ]
|J(H)t| ≥ ε)
= P(
1T 1N>T sup
t∈[0,T ]
∣∣ ∫ t
0
1s≤T 1NHs dWs
∣∣ ≥ ε)
≤Chebychev′sineq. and (20)
4
ε2E[ ∫ T
0
H2s1s≤T 1
N ds]≤ 4
ε2N.
Thus by (25)
P(
supt∈[0,T ]
|J(H)t| ≥ ε)≤ P
(∫ T
0
H2s ds ≥
1
N
)+
4
ε2Nfor any N ∈ N.
As a result, the continuity property follows. Next we show linearity. First note that theextension property and the continuity property imply (exercise)
J(H) = J(H) ∀H ∈ H.
Thus linearity holds for J on H. Let H,K ∈ H, then with Hns = 1s≤TnHs and Kn
s =
1s≤TnKs, Tn := infs ∈ [0, T ] |
∫ s0K2u du ≥ n
, we have∫ T
0
(Hns −Hs)
2 ds,
∫ T
0
(Kns −Ks)
2 ds −→n→∞
0 P -a.s.
48
and the convergence is also in probability. Then for α, β ∈ R, by the continuity propertyJ(αHn+βKn)→ J(αH+βK), J(Hn)→ J(H) and J(Kn)→ J(K) in probability, henceP -a.s. along a common subsequence, say (nl). It follows P -a.s.
J(αH + βK)t = liml→∞
J(αHnl + βKnl)t = liml→∞
(αJ(Hnl)t + βJ(Knl)t)
= liml→∞
(αJ(Hnl)t + βJ(Knl)t) = αJ(H)t + βJ(K)t.
The uniqueness of J follows similarly (exercise).
Summary: For
(Wt)t≥0 (Ft)-BM,
(Ht)t≥0 (Ft)-adapted process with P(∫ T
0
H2s ds <∞
)= 1,
we can define the stochastic integral(∫ T
0
Hs ds <∞)t∈[0,T ]
,
and we know that the stochastic integral is a martingale, if
E[ ∫ T
0
H2s ds
]<∞.
Moreover, it holds (using adequate stopping)
E[ ∫ T
0
H2s ds
]<∞ ⇔ E
[supt∈[0,T ]
∣∣∣ ∫ t
0
Hs dWs
∣∣∣2] <∞,and in either of these cases one has
E[( ∫ T
0
Hs dWs
)2]= E
[ ∫ T
0
H2s ds
].
3.4.2 Ito calculus
Definition 3.25 Let (Wt)t≥0 be an (Ft)-BM on (Ω,F, P ). An R-valued process (Xt)t∈[0,T ]
is called an Ito process, if
P(Xt = X0 +
∫ t
0
Hs dWs +
∫ t
0
Ks ds ∀t ∈ [0, T ])
= 1,
where
49
- X0 is F0-measurable,
- (Ht)t∈[0,T ], and (Kt)t∈[0,T ] are (Ft)-adapted processes,
-∫ T
0|Ks| ds <∞ P -a.s.
-∫ T
0H2s ds <∞ P -a.s.
The following proposition implies that the decomposition of an Ito process is unique.
Proposition 3.26 If (M)t∈[0,T ] is a continuous martingale, such that
Mt =
∫ t
0
Ks ds, t ∈ [0, T ], and P(∫ T
0
|Ks| ds <∞)
= 1,
thenP (Mt = 0 ∀t ∈ [0, T ]) = 1.
(”A continuous martingale with finite variation is P -a.s. constant”.)
Proof Exercise.
Proposition 3.26 implies with adequate stopping:
- an Ito process decomposition is unique, i.e. if P -a.s for all t ≤ T
Xt = X0 +
∫ t
0
Hs dWs +
∫ t
0
Ks ds = Xt = X0 +
∫ t
0
Hs dWs +
∫ t
0
Ks ds,
thenP (X0 = X0) = 1, H = H and K = K ds⊗ dP -a.e.
- if (Xt)t∈[0,T ] is a martingale, i.e. X0 +∫ t
0Hs dWs +
∫ t0Ks ds is a martingale, then
K = 0 ds⊗ dP -a.e.
Theorem 3.27 (1-dimensional Ito formula for Ito processes) Let (Xt)t∈[0,T ] be anIto process of the form
Xt = X0 +
∫ t
0
Hs dWs +
∫ t
0
Ks ds,
50
and f ∈ C2(R) (f is twice continuously differentiable function on R). Then P -a.s.
f(Xt) = f(X0) +
∫ t
0
f ′(Xs) dXs +1
2
∫ t
0
f ′′(Xs) d〈X,X〉s
where
〈X,X〉t := 〈X〉t :=
∫ t
0
H2s ds, t ∈ [0, T ],
and ∫ t
0
f ′(Xs) dXs :=
∫ t
0
f ′(Xs)Hs dWs +
∫ t
0
f ′(Xs)Ks ds, t ∈ [0, T ].
Likewise, if f ∈ C1,2(R+ × R) (twice continuously differentiable w.r.t. x ∈ R and oncecontinuously differentiable w.r.t. t ∈ R+ = [0,∞)), then
f(t,Xt) = f(0, X0) +
∫ t
0
f ′s(s,Xs) ds+
∫ t
0
f ′x(s,Xs) dXs +1
2
∫ t
0
f ′′x (s,Xs) d〈X,X〉s.
Proof See e.g. Karatzas/Shreve.
3.4.3 Examples: Ito formula in practice
For f(x) = x2, Xt = Wt, Ks ≡ 0, Hs ≡ 1:
W 2t = W 2
0︸︷︷︸=0
+
∫ t
0
2Ws dWs +1
2
∫ t
0
2 d 〈W,W 〉s︸ ︷︷ ︸=∫ s0 1 du=s
,= 2
∫ t
0
Ws dWs + t
and so
W 2t − t = 2
∫ t
0
Ws dWs.
Since E[ ∫ T
0W 2s ds
]=︸︷︷︸
Fubini
∫ T0E[W 2
s ] ds =∫ T
0s ds = T 2
2< ∞, it follows from Proposition
3.22 that( ∫ t
0Ws dWs
)t≥0
is a martingale. Thus we get a representation of the martingale
(W 2t − t)t≥0 from Proposition 3.12(2) as a stochastic integral.
Now, we want to find the solutions (if any) (St)t≥0 of
St = x0 +
∫ t
0
Ss(µ ds+ σ dWs), µ, σ, x0 ∈ R fixed.
51
In symbolic form this is written asdSt = St(µ dt+ σ dWt),
S0 = x0.(26)
Precisely, we are looking for an adapted and continuous process (St)t≥0 such that for P -a.e
ω ∈ Ω, (∫ t
0Ss(ω) ds) exists in the Lebesgue-Stieltjes sense, (
∫ t0Ss dWs) exists as stochastic
integral, and
St = x0 +
∫ t
0
σSs dWs +
∫ t
0
µSs ds P -a.s. ∀t ≥ 0.
In order to obtain a hint on how a solution might look like, let us suppose that wecan apply Ito’s formula with f(x) = log(x) /∈ C2(R) and that log(x0) is defined. Since〈S〉s =
∫ s0σ2S2
u du, this gives
log(St) = log(x0) +
∫ t
0
1
SsdSs −
1
2
∫ t
0
1
S2s
d〈S〉s
= log(x0) +
∫ t
0
σ dWs +
∫ t
0
µ ds− 1
2
∫ t
0
σ2 ds
= log(x0) + σWt +(µ− σ2
2
)t,
thus
St = x0 exp(σWt +
(µ− σ2
2
)t).
Since our procedure was not justified, we proceed backwards and show that this (St) reallyprovides a solution for any x0 ∈ R. We have
St = f(t,Wt) with f(t, x) = x0 exp(σx+
(µ− σ2
2
)t), t ≥ 0,
and since f ∈ C1,2(R+ × R) the time dependent Ito formula applies:
St = f(t,Wt)
= f(0,W0) +
∫ t
0
∂tf(s,Ws)︸ ︷︷ ︸=(µ−σ2
2
)Ss
ds+
∫ t
0
∂xf(s,Ws)︸ ︷︷ ︸=σSs
dWs +1
2
∫ t
0
∂xxf(s,Ws)︸ ︷︷ ︸=σ2Ss
d 〈W 〉s︸ ︷︷ ︸=s
= x0 +
∫ t
0
σSs dWs +
∫ t
0
µSs ds. (27)
52
Moreover, (St) is adapted and continuous. One can also check that (∫ t
0µSs ds) exists path-
wise P -a.s. since (St) is continuous, and that (∫ t
0σSs dWs) exists. Therefore the stochastic
differential equation (26) admits at least one solution.
Remark 3.28 One could also have checked that (St) is a solution by applying the Itoformula to the Ito process Zt = σWt +
(µ− σ2
2
)t with the C2-function f(x) = x0 exp(x).
We will establish uniqueness of a solution to (26). First we need an integration by partsformula.
Proposition 3.29 Let (Xt) and (Yt) be two Ito processes,
Xt = X0 +
∫ t
0
Hs dWs +
∫ t
0
Ks ds
and
Yt = Y0 +
∫ t
0
H ′s dWs +
∫ t
0
K ′s ds.
Then
XtYt = X0Y0 +
∫ t
0
Xs dYs +
∫ t
0
YsdXs + 〈X, Y 〉t,
with the convention
〈X, Y 〉t =
∫ t
0
HsH′s ds, t ≥ 0.
Proof By Ito’s formula
(1):
(Xt + Yt)2 = (X0 + Y0)2 +
∫ t
0
2(Xs + Ys) d(Xs + Ys) +
∫ t
0
(Hs +H ′s)2 ds,
(2):
X2t = X2
0 +
∫ t
0
2Xs dXs +
∫ t
0
H2s ds
(3):
Y 2t = Y 2
0 +
∫ t
0
2Ys dYs +
∫ t
0
(H ′s)2 ds
53
(1)− ((2) + (3)) : 2XtYt = 2X0Y0 + 2
∫ t
0
XsdYs + 2
∫ t
0
Ys dXs + 2
∫ t
0
HsH′s ds
Now, we return to the uniqueness question. Let (Xt) be another solution to (26) withS0 = X0 = x0 6= 0. Define
Zt :=x0
St= exp
(− σWt +
(− µ+
σ2
2
)t)
= exp(σ′Wt +
(µ′ − (σ′)2
2
)t),
where σ′ = −σ and µ′ = −µ+ σ2. We know from (27) applied to (Zt) that
Zt = 1 +
∫ t
0
Zs(σ′ dWs + µ′ ds) = 1 +
∫ t
0
Zs(−σ dWs + (−µ+ σ2)ds).
From 3.29 we getd(XtZt) = Xt dZt + Zt dXt + d 〈X,Z〉t,
where
〈X,Z〉t =
∫ t
0
(σXs)(−σZs) ds = −∫ t
0
σ2XsZs ds.
Thus
d(XtZt) = XtZt(−σdWt + (−µ+ σ2)dt) + ZtXt(σdWt + µdt)− σ2XtZtdt = 0,
andXtZt = X0Z0 = x0 · 1 = x0 ∀t ≥ 0.
Therefore
Xt = x0Z−1t = x0
Stx0
= St P -a.s. ∀t ≥ 0.
⇒ P (Xt = St ∀t ≥ 0) = 1, i.e. X and S are indistinguishable.
We obtain:
Theorem 3.30 Let σ, µ, x0 ∈ R, T > 0, and (Wt)t≥0 be a BM. Then there exists a uniqueIto process (St)t∈[0,T ] that satisfies
St = x0 +
∫ t
0
Ss(σdWs + µds) P -a.s. ∀t ∈ [0, T ].
This process is given by
St = x0 exp(σWt +
(µ− σ2
2
)t), t ∈ [0, T ]
54
Proof We have proven the theorem for x0 6= 0. For x0 = 0 uniqueness can also be shown.
Remark 3.31 (i) The process (St) of 3.30 will model the evolution of the stock pricein the Black-Scholes model.
(ii) If µ = 0 then (St) is martingale (see 3.12).
Remark 3.32 Let U ⊂ R be an open set and (Xt)t∈[0,T ] and Ito process with P (Xt ∈U ∀t ∈ [0, T ]) = 1. If f ∈ C2(U), then the Ito formula holds:
f(Xt) = f(X0) +
∫ t
0
f ′(Xs) dXs +1
2
∫ t
0
f ′′(Xs) d〈X,X〉s P -a.s. ∀t ≥ 0.
This result allows us to apply e.g. Ito’s formula to log(Xt), if (Xt)t∈[0,T ] is a strictly positiveprocess.
55
4 The Black-Scholes model
4.1 Description of the model
4.1.1 The behavior of prices
Consider continuous-time model with:
- one riskless asset: (S0t ) (e.g a bond)
- one risky asset: (St) (e.g. a stock price)
We assume that (S0t ) solves the ODE
dS0t = rS0
t dt
S00 = 1,
(28)
where r > 0 is the (instantaneous) interest rate. Then
S0t = S0
0ert = ert, t ≥ 0.
We further suppose, that the behavior of the stock price is determined by the SDE
dSt = σS0 dBt + µSt dt, S0 > 0. (29)
and (Bt)t≥0 is a standard BM, σ and µ are constants. σ is the so-called ”volatility of(St)”. The model will be considered for t ∈ [0, T ], T is the maturity of an option on (St).We know that (29) has the (pathwise) unique solution
St = S0 exp(σBt +
(µ− σ2
2
)t), t ≥ 0.
Because of the explicit representation of (St), such a solution is called a closed-formsolution.
Remarks on St:
(0) (St) is called geometric (or also exponential) Brownian motion.
(1) log(StS0
)is normally distributed for any t ≥ 0, i.e. St
S0is lognormal.
(2) (St)t≥0 is continuous.
56
(3) S0 is the stock price at time 0, so usually considered as strictly positive constant.
(4) the relative increments are independent, i.e. if s ≤ t, then
StSs
= exp(σ(Bt −Bs) +
(µ− σ2
2
)(t− s)
),
or equivalently the relative increment St−SsSs
is independent of Fs, where (Ft)t≥0 isthe natural filtration of (Bt)t≥0 (see Definition 3.11+ Remark 3.10)
(5) the relative increments are stationary, i.e. if s ≤ t, then
St − SsSs
andSt−s − S0
S0
have same law.
Hypotheses (2), (4) and (5) are the hypotheses of Black and Scholes on the behavior ofthe stock price.
4.1.2 Self-financing strategies
Let (Ft)t≥0 be the natural filtration of (Bt)t≥0. A strategy is an R2-valued (Ft)-adaptedprocess
φ = (φt)t∈[0,T ] = (H0t , Ht)t∈[0,T ],
where
H0t = number of units (quantities) of riskless asset held at time t,
Ht = number of units (quantities) of risky asset held at time t.
The value of the portfolio at time t is
Vt(φ) = H0t S
0t +HtSt.
In discrete time (see Remark 2.2) we characterized self-financing strategies through theequality
Vn+1(φ)− Vn(φ) = φn+1(Sn+1 − Sn).
In continuous time the last is expressed through the SDE
dVt(φ) = H0t dS
0t +Ht dSt.
To give a meaning to this SDE, we need of course conditions on the strategy φ = (H0, H).These are stated in the following definition.
57
Definition 4.1 A self-financing strategy is a pair of (Ft)-adapted processes φ =((H0
t )t∈[0,T ], (Ht)t∈[0,T ]
)satisfying:
(i) ∫ T
0
(|H0s |+H2
s ) ds <∞ P -a.s.
(ii) ∀t ∈ [0, T ], we have
H0t S
0t +HtSt︸ ︷︷ ︸
=Vt(φ)
= H00S
00 +H0S0︸ ︷︷ ︸=V0(φ)
+
∫ t
0
H0s dS
0s +
∫ t
0
Hs dSs P -a.s.
By (i) both integrals in (ii) are well defined, since:∫ T
0
H0s dS
0s =
∫ T
0
H0s re
rs ds and
∫ T
0
|H0s re
rs| ds ≤ rerT∫ T
0
|H0s | ds︸ ︷︷ ︸
<∞
and ∫ T
0
Hs dSs = σ
∫ T
0
HsSs dWs + µ
∫ T
0
HsSs ds,
where the first integral exists since∫ T
0
H2s (ω)S2
s (ω)︸ ︷︷ ︸pathwise boundedsince continuous
ds ≤ C(ω)︸ ︷︷ ︸<∞
∫ T
0
H2s (ω) ds <∞
for P -a.e. ω ∈ Ω. Similarly, applying the Cauchy-Schwarz inequality the secondintegral also exists P -a.s.
DefineSt := e−rtSt, t ≥ 0 ”discounted price of risky asset”.
Proposition 4.2 (cf. Proposition 2.3) Let φ = (H0t , Ht)t∈[0,T ] with values in R2 be a
pair of (Ft)-adapted processes. Suppose∫ T
0(|H0
s | + |Hs|2) ds < ∞ P -a.s., and let Vt(φ) =
H0t S
0t +HtSt, and Vt(φ) := e−rtVt(φ), t ∈ [0, T ]. Then φ is a self-financing strategy, if and
only if
Vt(φ) = V0(φ) +
∫ t
0
Hs dSs P -a.s. ∀t ∈ [0, T ]. (30)
58
Proof We have to show that under the given assumptions, we have: (30) ⇔ 4.1(ii).Suppose φ is self-financing, i.e 4.1(ii) holds. Using integration by parts
d Vt(φ)︸ ︷︷ ︸=e−rtVt(φ)
= −re−rtVt(φ) dt+ e−rt dVt(φ) + d〈e−r(·), V·(φ)〉t︸ ︷︷ ︸=0
Thus
dVt(φ) =︸︷︷︸4.1(ii)
−re−rt(H0t e
rt +HtSt) dt+ e−rt(H0t d(ert) +Ht dSt)
= Ht(−re−rtSt dt+ e−rt dSt)
= Ht(St d(e−rt) + e−rt dSt) = Ht dSt
and (30) holds. The converse holds similarly, applying the integration by parts formula toVt(φ) = ertVt(φ).
Remark 4.3 In discrete time we imposed predictability on the strategies. In continuoustime the predictability condition is included in some sense in the definition of stochasticintegral. In the study of complete discrete markets we considered equivalent martingalemeasures. By this, we were able to find replicating strategies. The main tools for the samein continuous time are the Girsanov transformation and Ito’s representation theorem thatwe introduce in the next section.
4.2 Change of probability. Representation of martingales
4.2.1 Equivalent probabilities
Let (Ω,A, P ) be a probability space, and Q be another probability measure on (Ω,A).Then define
QA P (in words: Q is absolutely continuous w.r.t P (on A))
:⇔ A ∈ A and P (A) = 0 ⇒ Q(A) = 0
and
Q ≈A P (in words: Q and P are equivalent (on A))
:⇔ QA P and P A Q.
59
Theorem 4.4 (Radon-Nikodym special case) Let P,Q be probability measures on(Ω,A). Then
Q P ⇔ ∃ r.v. Z : Ω→ R+ with Q(A) =
∫A
Z dP ∀A ∈ A.
Z is called the (Radon-Nikodym) density of Q w.r.t. P and one writes Z = dQdP
. In partic-ular, if Q P , then: Q ≈ P ⇔ P (Z > 0) = 1.
Proof ”⇐” and last statement obvious. ”⇒” see Radon-Nikodym theorem (probabilisticpart of lecture).
4.2.2 The Girsanov theorem
Let (Ω,F, (Ft), P ) be a filtered probability space and (Bt)t∈[0,T ] be a standard (Ft)-BM.Then we have:
Theorem 4.5 (Girsanov transformation) Let (θt)t∈[0,T ] be (Ft)-adapted with∫ T
0θ2s ds <
∞ P -a.s. Suppose
Lt := exp(−∫ t
0
θs dBs −1
2
∫ t
0
θ2s ds), t ∈ [0, T ],
is an (Ft)-martingale w.r.t. P . Let dP (LT ) := LT dP . Then P (LT ) is a probability measurethat is equivalent to P and
Wt := Bt +
∫ t
0
θs ds, t ∈ [0, T ],
is a standard (Ft)-BM under P (LT ).
Remark 4.6 The most well-known sufficient condition on (Lt)t∈[0,T ] to be a martin-gale is Novikov’s condition, i.e.
E[
exp(1
2
∫ T
0
θ2 dt)]
<∞.
60
4.2.3 Representation of Brownian martingales
Let (Bt)t∈[0,T ] be a standard BM on (Ω,F, P ) with natural filtration (Ft)t∈[0,T ]. The fol-lowing is known as Ito’s representation theorem:
Theorem 4.7 Let (Mt)t∈[0,T ] be a square-integrable martingale w.r.t. (Ft) (= natural fil-
tration of (Bt)). Then there exists an (Ft)-adapted process (Ht)t∈[0,T ] with E[ ∫ T
0H2s ds
]<
∞ and
Mt = M0 +
∫ t
0
Hs dBs P -a.s. ∀t ∈ [0, T ]. (31)
In particular: (Mt) is continuous.
Proof See for instance Karatzas/Shreve.
If U is an FT -measurable r.v. and E[U2] <∞, i.e. U is square-integrable, then
U = E[U ] +
∫ T
0
Hs dBs, P -a.s.
for some (Ft)-adapted process with E[ ∫ T
0H2s ds
]<∞. For the proof it suffices to consider
the square-integrable ”Brownian martingale”
Mt := E[U |Ft], t ∈ [0, T ],
and to apply Theorem 4.7.
4.3 Pricing hedging options in the Black-Scholes model
4.3.1 A probability measure under which (St) is a martingale
Consider the model introduced in 4.1. We have
dSt = d(e−rtSt) = e−rt dSt − re−rtSt dt= e−rt(σSt dBt + µSt dt)− re−rtSt dt= St(σ dBt + (µ− r) dt)
= σSt(dBt +µ− rσ
dt)
If we set Wt = Bt + µ−rσt, then
dSt = σSt dWt (32)
61
and from 4.5 applied with θt ≡ µ−rσ
((θt)t∈[0,T ] satisfies Novikov’s condition 4.6) we knowthat
Wt = Bt +
∫ t
0
µ− rσ
dt, t ∈ [0, T ]
is a BM under
dP (LT ) = exp(− µ− r
σBT −
(µ− r)2
2σ2T)dP.
By Theorem 3.30, the unique solution to (32) is given
St = S0 exp(σWt −
σ2
2t).
Since (Wt) is a Brownian motion under P ∗ := P (LT ), we know that (St) is a (squareintegrable) P ∗-martingale.
4.3.2 Pricing
A contingent claim (or option) is defined to be a non-negative FT -measurable r.v. h,e.g.
h = f(ST ) with f(x) =
(x−K)+ ”European call”,
(K − x)+ ”European put”.
Definition 4.8 A strategy φ = (H0t , Ht)t∈[0,T ] is called admissible, if it is self-financing,
and if the discounted value Vt(φ) = H0t + HtSt of the corresponding portfolio is non-
negative for all t, and such that
supt∈[0,T ]
Vt(φ) ∈ L2(Ω,FT ,P∗).
An option (or contingent claim) h is said to be replicable, if there exists an admissiblestrategy φ with
VT (φ) = h. (”φ replicating strategy”)
In particular, if h is replicable then h ∈ L2(P ∗). For a European option (put or call), wealways have h ∈ L2(P ∗), since
((ST −K)+)2 ≤ S2T ∈ L1(P ∗), and ((K − ST )+)2 ≤ K2 ∈ L1(P ∗).
Theorem 4.9 In the Black-Scholes model any contingent claim h, with h ∈ L2(Ω,FT , P∗)
is replicable and the value at time t of any replicating portfolio is
Vt(φ) = Vt = E∗[e−r(T−t)h | Ft]. (33)
62
Remark: In this case one also says that E∗[e−r(T−t)h | Ft] is the value of the option h attime t ∈ [0, T ]. Note that it is independent of φ.Proof Assume first ∃ admissible strategy φ = (H0, H) that replicates the option h, i.e.
VT (φ) = h.
We haveVt := Vt(φ) = H0
t S0t +HtSt,
and for the discounted value Vt = e−rtVt we have
Vt = H0t +HtSt.
Since φ is in particular self-financing, we get
Vt =4.2V0 +
∫ t
0
Hu dSu =(32)
V0 +
∫ t
0
HuσSu dWu.
Since φ is admissible, we have supt≤T Vt ∈ L2(P ∗), hence supt≤T
∣∣∣ ∫ t0 HuσSu dWu
∣∣∣ ∈L2(P ∗). Therefore, (
∫ t0HuσSu dWu)t∈[0,T ], and hence (Vt)t∈[0,T ] is a square-integrable mar-
tingale w.r.t. P ∗ (cf. Assignments). It follows
Vt = E∗[VT | Ft],
thusVt = E∗[e−r(T−t)h | Ft].
We have hence proved: If φ is a replicating strategy, then (33) holds.To complete the proof, we have to show that any option h ∈ L2(Ω,FT , P
∗) is replicable,i.e. ∃ admissible strategy φ = (H0, H) such that VT (φ) = h.Since h ∈ L2(Ω,FT , P
∗) it follows that
Mt := E∗[e−rTh | Ft], t ∈ [0, T ],
is a square integrable (P ∗, (Ft))-martingale. We have
(Ft) natural filtration of (Bt) w.r.t. P =⇒P ∗≈P
(Ft) natural filtration of (Wt) w.r.t. P ∗.
Thus by Ito’s representation theorem 4.7 ∃(Kt)t∈[0,T ] (Ft)-adapted, with E∗[ ∫ T
0K2s ds
]<
∞ and
Mt = M0 +
∫ t
0
Ks dWs P ∗-a.s t ∈ [0, T ].
63
Define
φ = (H0t , Ht) :=
(Mt −
Kt
σ,Kt
σSt
).
Then clearly φ is (Ft)-adapted,∫ T
0(|H0
s |+ |Hs|2) ds <∞ P -a.s. and
Vt(φ) = H0t +HtSt = Mt = M0 +
∫ t
0
Ks dWs. (?)
Since V0(φ) = M0 and ∫ t
0
Hs dSs =(32)
∫ t
0
HsσSs dWs =
∫ t
0
Ks dWs,
we get
Vt(φ) = V0(φ) +
∫ t
0
Hs dSs =⇒4.2
φ is self-financing.
FurthermoreVt = Vt(φ) =
(?)ertMt = E∗[e−r(T−t)h | Ft].
This implies Vt ≥ 0, supt≤T Vt(φ) ∈ L2(P ∗) (Doob’s inequality applied to (Vt(φ))t∈[0,T ]),and VT = h, hence φ is admissible and replicating.
Remark 4.10 Suppose h in Theorem 4.9 can be written as h = f(ST ). Then
ST = S0 exp(σBT +
(µ− σ2
2
)T)
= S0 exp(σBt +
(µ− σ2
2
)t)
︸ ︷︷ ︸=St
exp(σ(BT −Bt︸ ︷︷ ︸
=WT−Wt−µ−rσ (T−t)
) +(µ− σ2
2
)(T − t)
)
= St exp(σ(WT −Wt) +
(r − σ2
2
)(T − t)
)Now
Vt =4.9
E∗[e−r(T−t)f(ST ) | Ft]
= E∗[e−r(T−t)f
(St︸︷︷︸
Ft-mb
exp(σ(WT −Wt︸ ︷︷ ︸
indep. of Ft
) +(r − σ2
2
)(T − t)
))︸ ︷︷ ︸
=φt(St,WT−Wt)
| Ft]
= F (t, St),
64
by Proposition 7.6 of the Probabilistic Background, where
F (t, x) = E∗[e−r(T−t)f
(x exp
(σ(WT −Wt) +
(r − σ2
2
)(T − t)
))]. (34)
Under P ∗ we have WT −Wt ∼ N(0, T − t). Thus for t < T
F (t, x) = e−r(T−t)1√
2π(T − t)
∫Rf(xe(r−σ
2
2)(T−t)eσy)e−
y2
2(T−t) dy
=z= y√
T−t
e−r(T−t)1√2π
∫Rf(xe(r−σ
2
2)(T−t)eσz
√T−t)e−
z2
2 dz. (∗)
Explicit calulcation of F in case of calls and puts: In case of a European call, we havef(x) = (x−K)+, thus
F (t, x) =(∗)
1√2π
∫ ∞−∞
(xeσz
√T−t−σ
2
2)(T−t) −Ke−r(T−t)
)+e−
z2
2 dz
= E∗[(xeσ
√θZ−σ
2θ2 −Ke−rθ
)+],
where Z ∼ N(0, 1) and θ := T − t. Set
d1 :=log(xK
)+(r + σ2
2
)θ
σ√θ
, d2 := d1 − σ√θ.
Then
F (t, x) = E∗[(xeσ
√θZ−σ
2θ2 −Ke−rθ
)1Z≥−d2
]=
∫ ∞−d2
(xeσ
√θz−σ
2θ2 −Ke−rθ
)e− z22√2π
dz
=
∫ d2
−∞
(xe−σ
√θz−σ
2θ2 −Ke−rθ
)e− z22√2π
dz
=
∫ d2
−∞xe−
(z−σ√θ)2
2dz√2π−Ke−rθN(d2),
with
N(d) :=1√2π
∫ d
∞e−
x2
2 dx.
Finally, with the change of the variable z = y + σ√θ, one gets
F (t, x) = xN(d1)−Ke−rθN(d2) (35)
and similarl for a European put, i.e. f(x) = (K − x)+, we obtain
F (t, x) = Ke−rθN(−d2)− xN(−d1).
65
4.3.3 Hedging calls and puts
In the proof of Theorem 4.9, we used Ito’s representation theorem to show the existenceof a replicating strategy and in Remark 4.10, we derived a pricing formula for Europeanoptions, or more generally for options of the form h = f(ST ). After having sold the optionfor the price F (0, S0), the writer of the option needs an explicit replicating strategy,so that he can generate explicitly the replicating portfolio. We will construct an explicitreplicating strategy in case h is given by h = f(ST ): At any time t a replicating portfoliomust satisfy
Vt = e−rtF (t, St), (where F is as in (34)).
One can show (not elementary): f ∈ Cb(R) =⇒ F ∈ C∞([0, T ) × R). This also holds forf(x) = (x − K)+ and f(x) = (K − x)+. We hence fix f such that F ∈ C∞([0, T ) × R)and define
F (t, x) := e−rtF (t, xert).
Then F (t, St) = e−rtF (t, St) = Vt, and for t < T by Ito’s formula
F (t, St) = F (0, S0) +
∫ t
0
∂xF (u, Su) dSu +1
2
∫ t
0
∂xxF (u, Su) d〈S〉u +
∫ t
0
∂tF (u, Su) du,
where d〈S〉u := d〈S, S〉u =(32)
σ2S2u du. Therefore
Vt = F (t, St) = F (0, S0) +
∫ t
0
σ∂xF (u, Su)Su dWu +
∫ t
0
Ku du
with∫ T
0|Ku| du < ∞ P ∗-a.s. Since Vt is a martingale under P ∗, we have
∫ t0Ku du = 0
P ∗-a.s. by Assignment no. 9, Exercise 2. It follows
Vt = F (t, St) = F (0, S0) +
∫ t
0
∂xF (u, Su) dSu = V0 +
∫ t
0
∂xF (u, Su) dSu.
Thus by (30), the natural candidate for H is
Ht = ∂xF (t, St) = ∂xF (t, St) (continuous in t).
Since F (t, St) = Vt = H0t +HtSt we then have to put
H0t = F (t, St)− ∂xF (t, St)St (continuous in t).
Then φ = (H0, H) is self-financing and Vt(φ) = Vt = F (t, St) as desired.
66
Remark 4.11 We have just shown (without Ito’s representation theorem), that in caseh = f(ST ), we can find an explicit replicating strategy.
Remark 4.12 We have, with the notations of Remark 4.10
∂xF (t, x) = N(d1) for a European call
and∂xF (t, x) = −N(−d1) for a European put.
∂xF (t, x) is called ”delta of the option”. More generally, if Vt = Ψ(t, St), then we havethe following terminologies :
∂xΨ(t, St) is called ”delta of the portfolio” (measures the sensitivity of the portfolioVt w.r.t. stock price fluctuations)
∂xxΨ(t, St) is called ”gamma of the portfolio”,
∂tΨ(t, St) is called ”theta of the portfolio”,
∂σΨ(t, St) is called ”vega of the portfolio”.
4.4 Implied volatility and volatility models
One of the main features of the Black-Scholes model :
”hedging and pricing formulas only depend on one parameter σ.”
The drift µ disappears by Girsanov, and r,K, S0 are given. The volatility σ is hence alsosaid to be non-observable. In practice, two methods are used to evaluate σ :
1. The historical method : In the present model σ2T is the variance of log(ST/S0
)The r.v.’s
log(ST/S0
), log
(S2T/ST
), . . . , log
(SNT/S(N−1)T
)are independent and identically distributed. Therefore σ can be estimated by statisticalmethods using stock prices of the past (for example by calculating empirical variances(?)).
2. The implied volatility method : Some options are quoted on organized markets.The price of options (calls and puts) is an increasing function of σ, so we can invert the
67
Black-Scholes pricing formula and associate an ”implied volatililty” to each quoted option.More precisely, let Cobs(S0, K, T ) be the observed option price of a benchmark (reference)option on the market. Then find σ that solves
Cobs(S0, K, T ) = F (0, S0) = Fσ(0, S0, K, T ).
Imperfection of the Black-Scholes model : Important differences between historicaland implied volatility are observed.
The implied volatility appears to depend on the strike price and time to maturity. In fact,if one plots a graph of implied volatility in dependence of strike prices one sees a ”volatilitysmile” (graph is convex). The graphs becomes ”more and more” convex the shorter thetime to maturity is. This contradicts the Black-Scholes model or in other terms : TheBlack-Scholes model does not describe the market correctly !? Nonetheless, the model isregarded as a standard reference. One possibility to construct a more consistent model isto consider a time dependent volatility (σt)t∈[0,T ].
(?) Empirical variances : given stock prices S0, ST , S2T , . . . , SNT , let
δi := log(S(i+1)T
SiT
), i = 0, . . . , N − 1
Empirical mean : δ :=1
N
N−1∑i=0
δi
Empirical variance : σhist :=1√T
( 1
N − 1
N−1∑i=0
(δi − δ)2) 1
2(”unbiased estimator”)
4.5 American options
4.5.1 Pricing American options
Definition 4.13 A trading strategy with consumption is defined as an adapted R2-valued process φ = (H0
t , Ht)t∈[0,T ] with :
1.∫ T
0(|H0
t |+H2t )dt <∞ a.s.
2. It holds ∀t ≤ T :
H0t S
0t +HtSt = H0
0S00 +H0S0 +
∫ t
0
H0u dS
0u +
∫ t
0
Hu dSu − Ct,
68
where (Ct)t∈[0,T ] is adapted, continuous and non-decreasing with C0 = 0 (Ct =”cumulative consumption up to time t”).
An American option is defined as an adapted non-negative process (ht)t∈[0,T ]. For sim-plicity
ht = ψ(St), ψ : R+ −→ R+ continuous. ψ(x) ≤ Ax+B, A,B ≥ 0 constants.
A trading strategy with consumption φ = (H0t , Ht) is said to hedge the American
option Ψ(St), if
Vt(φ) = H0t S
0t +HtSt ≥ ψ(St) a.s. ∀t ∈ [0, T ] (?)
Φψ := φ trading strategy with consumption, and (?) holds
Theorem 4.14 Define u : [0, T ]× R+ −→ R by
u(t, x) := supτ∈Jt,T
E∗[e−r(τ−t) ψ(x eσ(Wτ−Wt)+(r−σ
2
2)(τ−t))
]where Jt,T := τ stopping time | t ≤ τ ≤ T. Then there exists a strategy φ ∈ Φψ with
Vt(φ) = u(t, St), ∀t ∈ [0, T ].
Moreover, it holds :Vt(φ) ≥ u(t, St), ∀t ∈ [0, T ], φ ∈ Φψ.
Thus u(t, St) is the minimal portfolio value of a hedging strategy with consumption for anAmerican option.
Proof (only brief sketch)
1. One shows: e−rtu(t, St) = Snell envelope of e−rtψ(St) (i.e. the smallest right-continuous P ∗-supermartingale that dominates e−rtψ(St))
2. One shows: Vt(φ) is a P ∗-supermartingale ∀φ ∈ Φψ (because of consumption, i.e.−Ct !) ⇒ Vt(φ) ≥ u(t, St), ∀φ ∈ Φψ.
3. In order to show ∃φ ∈ Φψ with Vt(φ) = u(t, St) use the the Doob-Meyer decompo-sition and Ito’s representation theorem.
In view of Theorem 4.14 we call
u(t, St) = ”price of the American option at time t ∈ [0, T ]”.
69
Remark 4.15 Let τ be a stopping time τ ≤ T . Let φ be an admissible strategy in thesense of Definition 4.8. Suppose Vτ (φ) = ψ(Sτ ). Then V0(φ) = E∗[e−rτψ(Sτ )]
Proof Using the optional sampling theorem for bounded stopping times, we have
V0(φ) = E∗[e−rτVτ (φ) | F0] =no add. inf.
E∗[e−rτVτ (φ)] =assumption
E∗[e−rτψ(Sτ )].
Thus the quantity
u(0, S0) = supτ∈J0,T
E∗[e−rτψ(S0eσWτ+(r−σ
2
2)τ︸ ︷︷ ︸
=Sτ
)]
of Theorem 4.14 is the minimal initial capital that hedges all possible exercises. Thefollowing proposition shows that the American call price is equal to the European callprice.
Proposition 4.16 Let ψ in Theorem 4.14 be given by ψ(x) = (x−K)+, x ∈ R. Then
u(t, x) = F (t, x) (F as in (34) f = ψ)
Proof We only show the statement for t = 0 (the proof for t > 0 is the same). Letting S0 =x it is enough to show that E∗[e−rτ (Sτ −K)+] ≤ E∗[e−rT (ST −K)+] = E∗[(ST −e−rTK)+]for any stopping time τ ∈ [0, T ]. We have
E∗[(ST − e−rTK)+ | Fτ ] ≥ E∗[ST − e−rTK | Fτ ] = Sτ − e−rTK
since (St) is an ((Ft), P∗)-martingale. We get since τ ≤ T , and r ≥ 0,
E∗[(ST − e−rTK)+ | Fτ ]︸ ︷︷ ︸≥0
≥ Sτ − e−rτK
⇒ E∗[(ST − e−rTK)+ | Fτ ] ≥ (Sτ − e−rτK)+
Taking expectation, the result follows.
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4.5.2 Perpetual puts, critical price
The price of an American option is not equal to the one of an European put option, andthere is no closed form solution for
u(t, x) = supτ∈Jt,T
E∗[(Ke−r(τ−t) − x exp σ(Wτ −Wt)−
σ2
2(τ − t))+
](36)
Therefore one has to use numerical methods (see later) Here we only derive some quali-tative properties of u. For this, it is enough to consider u for t = 0, i.e.
u(0, x) = supτ∈J0,T
E∗[(Ke−rτ − x exp σWτ −
σ2
2τ)+
](37)
(For general t we then replace T by T − t and shift τ ∈ J0,T−t τ ∈ Jt,T , and τ τ − tin the E∗-expectation. Since Wτ−t ∼ Wτ −Wt we obtain indeed u(t, x)). On a probabilityspace (Ω,F, P ) consider a standard BM (Bt)t≥0 (with infinite time horizon). Let
J0,∞ : = τ | τ stopping time w.r.t. the natural filtration of (Bt)J0,T : = τ | τ ∈ J0,∞, τ ≤ T.
Then
u(0, x) = supτ∈J0,T
E∗[(Ke−rτ − x exp σWτ −
σ2
2τ)+
]≤ sup
τ∈J0,∞E∗[(Ke−rτ − x exp σWτ −
σ2
2τ)+1τ<∞
]︸ ︷︷ ︸
=:u∞(x)=value of ”perpetual put (can be exercised at any finite time)”
(38)
Remark 4.17 Perpetual puts are (of course) not traded in any real market. We nowderive an explicit expression for u∞, hence for an upper for u(0, x).
Proposition 4.18 We have
u∞(x) =
K − x if x ≤ x∗ := Kγ
1+γ, γ := 2r
σ2
(K − x∗)(xx∗
)−γif x > x∗.
Proof Since z 7→ (a − z)+ is convex, by (38) we see that u∞ is convex, decreasing on[0,∞], and
u∞(x) ≥ (K − x)+, ∀x,
u∞(x) ≥ E[(Ke−rT − x expσBT −
σ2
2T)+
], ∀T > 0, ∀x ≥ 0,
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which implies u∞(x) > 0,∀x ≥ 0. Then, since u∞(0) = K,
x∗ := supx ≥ 0 | u∞(x) = K − x < K,
is well defined, and we have
u∞(x) = K − x, ∀x ≤ x∗, u∞(x) > (K − x)+, ∀x > x∗. (39)
Let
X∗t := x exp(σBt −
σ2
2t), x ∈ [0,∞)
By the properties of the Snell envelope, we have
u∞(x) = E[(Ke−rτx − x expσBτx −
σ2
2τx)+1τx<∞
],
whereτx := inft ≥ 0 | e−rtu∞(Xx
t ) = e−rt(K −Xxt )+,
and inf ∅ :=∞. τx is hence an optimal stopping time. By (39)
τx = inft ≥ 0 | Xxt ≤ x∗ = inf
t ≥ 0 | σBt +
(r − σ2
2
)t ≤ log
(x∗x
)(40)
For any z ≥ 0, letτx,z := inft ≥ 0 | Xx
t ≤ z.
Then τx = τx,x∗ . Define
φ(z) := E[e−rτx,z(K −Xxτx,z)
+1τx,z<∞].
τx,x∗ optimal ⇒ maxz≥0 φ(z) = φ(x∗) = u∞(x).
Calculation of φ(x∗) : (First calculate φ explicitly; then maximize φ)
If z ≥ x, then τx,z = 0, hence φ(z) = (K − x)+. (41)
If z < x, then τx,z = inft ≥ 0 | Xxt = z,
hence
φ(z) = (K − z)+ E[e−rτx,z1τx,z<∞]
= (K − z)+ E[e−rτx,z ], (with e−r·∞ := 0) (42)
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So if 0 = z < x, then τx,z =∞, hence φ(z) = 0. As in (40) for 0 < z < x,
τx,z = inft ≥ 0 | σBt +
(r − σ2
2
)t ≤ log
(zx
)τx,z = inf
t ≥ 0 | Bt + µt ≤ 1
σlog(zx
)with µ = r
σ− σ
2. For any b ∈ R define
Tb := inft ≥ 0 | Bt + µt = b.
Then from (41) and (42)(K − x)+, if z ≥ x
(K − z)E[exp−rT 1σ
log( zx
)], if z ∈ (0, x) ∩ [0, K) (also ok for z = x)
0, if z ∈ (0, x) ∩ [K,∞) or 0 = z < x
The maximum of φ is K, if x = 0 (⇒ u∞(0) = K), and for x > 0 it is attained for z ≤ x.Since by definition x∗ < K, the maximum of φ for x > 0 is attained in [0, x]∩ [0, K). Nowwe use the formula
E[e−αTb ] = expµb− |b|√µ2 + 2α
We obtain for z ∈ (0, x] ∩ [0, K)
φ(z) = (K − z) expµσ
log(zx
)−∣∣∣ 1σ
log(zx
)︸ ︷︷ ︸≤0 since z≤x
∣∣∣√µ2 + 2r
= (K − z) exp 1
σlog(zx
)−( rσ− σ
2+
√( rσ− σ
2
)2
+ 2r︸ ︷︷ ︸=( r
σ+σ
2)2
)
= (K − z) exp
log(zx
)· 2r
σ2
= (K − z)
(zx
) 2rσ2
= (K − z)(zx
)γ.← formula also correct for 0 = z < x!
Thus for x > 0, u∞ is obtaied by maximizing
φ(z) = (K − z)(zx
)γ, z ∈ [0, x] ∩ [0, K).
Now
φ′(z) = −zγ
xγ+ (K − z) γ
zγ−1
xγ=zγ−1
xγ(Kγ − (1 + γ)z).
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Thus for 0 < z = Kγ1+γ
< x we have
u∞(x) = φ( Kγ
1 + γ
)= (K − x∗)
(x∗x
)γ,
and for x ≤ Kγ1+γ
= z we have
u∞(x) = φ(x) = K − x
Remark 4.19 Consider again the value of an American put with maturity T at time Twhere S0 = x, i.e.
u(t, x) = supτ∈Jt,T
E∗[(Ke−r(τ−t) − x exp σ(Wτ −Wt)−
σ2
2(τ − t))+
]Exactly as at the beginning of the proof of 4.18 one can show : ∀t ∈ [0, T ) ∃S(t) ∈ [0, K)with
u(t, x) = K − x, ∀x ≤ S(t),
u(t, x) > (K − x)+, ∀x > S(t). (43)
By the discussion after (37) it follows from (38) that
u(t, x) ≤ u∞(x) t ∈ [0, T ),
thus S(t) ∈ [x∗, K) ∀t ∈ [0, T ).
S(t) = ”critical price”
Interpretation :
if St ≤ S(t) ”exercise immediately the option”
if St > S(t) ”keep the option”
References
[1] Karatzas, I.; Shreve, S. E. Brownian motion and stochastic calculus. Graduate Textsin Mathematics, 113. Springer-Verlag, New York, (1991).
[2] Lamberton, D.; Lapeyre, B.: Introduction to stochastic calculus applied to finance.Second edition. Chapman & Hall/CRC Financial Mathematics Series, Boca Raton,FL, 2008.
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