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# Binomial Model for Forward and Futures Options

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(1)

### Put-Call Parity

The put-call parity is slightly diﬀerent from the one in Eq. (22) on p. 204.

Theorem 14 (1) For European options on futures contracts,

C = P − (X − F ) e−rt.

(2) For European options on forward contracts, C = P − (X − F ) e−rT.

• See text for proof.

(2)

### Early Exercise

The early exercise feature is not valuable for and forward options.

Theorem 15 American forward options should not be

exercised before expiration as long as the probability of their ending up out of the money is positive.

• See text for proof.

Early exercise may be optimal for American futures options even if the underlying asset generates no payouts.

Theorem 16 American futures options may be exercised optimally before expiration.

(3)

### Black’s Model

a

• Formulas for European futures options:

C = F e−rtN (x) − Xe−rtN (x − σ√

t), (44) P = Xe−rtN (−x + σ√

t) − F e−rtN (−x), where x ≡ ln(F/X)+(σσt 2/2) t.

• Formulas (44) are related to those for options on a stock paying a continuous dividend yield.

• They are exactly Eqs. (30) on p. 307 with q set to r and S replaced by F .

a

(4)

### Black Model (concluded)

• This observation incidentally proves Theorem 16 (p. 451).

• For European forward options, just multiply the above formulas by e−r(T −t).

– Forward options diﬀer from futures options by a factor of e−r(T −t).a

aRecall Eqs. (42)–(43) on p. 448.

(5)

### Binomial Model for Forward and Futures Options

• Futures price behaves like a stock paying a continuous dividend yield of r.

– The futures price at time 0 is (p. 427) F = SerT.

– From Lemma 10 (p. 278), the expected value of S at time Δt in a risk-neutral economy is

SerΔt.

– So the expected futures price at time Δt is

(6)

### Binomial Model for Forward and Futures Options (continued)

• The above observation continues to hold if S pays a dividend yield!a

– By Eq. (40) on p. 435, the futures price at time 0 is F = Se(r−q) T.

– From Lemma 10 (p. 278), the expected value of S at time Δt in a risk-neutral economy is

Se(r−q) Δt.

– So the expected futures price at time Δt is Se(r−q) Δte(r−q)(T −Δt) = Se(r−q) T = F.

(7)

### Binomial Model for Forward and Futures Options (concluded)

• Now, under the BOPM, the risk-neutral probability for the futures price is

pf ≡ (1 − d)/(u − d) by Eq. (31) on p. 309.

– The futures price moves from F to F u with

probability pf and to F d with probability 1 − pf. – Note that the original u and d are used!

• The binomial tree algorithm for forward options is

(8)

### Spot and Futures Prices under BOPM

• The futures price is related to the spot price via F = SerT

if the underlying asset pays no dividends.

• Recall the futures price F moves to F u with probability pf per period.

• So the stock price moves from S = F e−rT to F ue−r(T −Δt) = SuerΔt

with probability pf per period.

(9)

### Spot and Futures Prices under BOPM (concluded)

• Similarly, the stock price moves from S = F e−rT to SderΔt

with probability 1 − pf per period.

• Note that

S(uerΔt)(derΔt) = Se2rΔt = S.

• So this binomial model is not the CRR tree.

• This model may not be suitable for pricing barrier options (why?).

(10)

### Negative Probabilities Revisited

• As 0 < pf < 1, we have 0 < 1 − pf < 1 as well.

• The problem of negative risk-neutral probabilities is solved:

– Build the tree for the futures price F of the futures contract expiring at the same time as the option.

– Let the stock pay a continuous dividend yield of q.

– By Eq. (40) on p. 435, calculate S from F at each node via

S = F e−(r−q)(T −t).

(11)

### Swaps

• Swaps are agreements between two counterparties to exchange cash ﬂows in the future according to a

predetermined formula.

• There are two basic types of swaps: interest rate and currency.

• An interest rate swap occurs when two parties exchange interest payments periodically.

• Currency swaps are agreements to deliver one currency against another (our focus here).

• There are theories about why swaps exist.a

(12)

### Currency Swaps

• A currency swap involves two parties to exchange cash ﬂows in diﬀerent currencies.

• Consider the following ﬁxed rates available to party A and party B in U.S. dollars and Japanese yen:

Dollars Yen A DA% YA% B DB% YB%

• Suppose A wants to take out a ﬁxed-rate loan in yen, and B wants to take out a ﬁxed-rate loan in dollars.

(13)

### Currency Swaps (continued)

• A straightforward scenario is for A to borrow yen at YA% and B to borrow dollars at DB%.

• But suppose A is relatively more competitive in the dollar market than the yen market, i.e.,

YB − YA < DB − DA.

• Consider this alternative arrangement:

– A borrows dollars.

– B borrows yen.

– They enter into a currency swap with a bank as the

(14)

### Currency Swaps (concluded)

• The counterparties exchange principal at the beginning and the end of the life of the swap.

• This act transforms A’s loan into a yen loan and B’s yen loan into a dollar loan.

• The total gain is ((DB − DA) − (YB − YA))%:

– The total interest rate is originally (YA + DB)%.

– The new arrangement has a smaller total rate of (DA + YB)%.

• Transactions will happen only if the gain is distributed so that the cost to each party is less than the original.

(15)

### Example

• A and B face the following borrowing rates:

Dollars Yen

A 9% 10%

B 12% 11%

• A wants to borrow yen, and B wants to borrow dollars.

• A can borrow yen directly at 10%.

• B can borrow dollars directly at 12%.

(16)

### Example (continued)

• The rate diﬀerential in dollars (3%) is diﬀerent from that in yen (1%).

• So a currency swap with a total saving of 3 − 1 = 2% is possible.

• A is relatively more competitive in the dollar market.

• B is relatively more competitive in the yen market.

(17)

### Example (concluded)

• Next page shows an arrangement which is beneﬁcial to all parties involved.

– A eﬀectively borrows yen at 9.5% (lower than 10%).

– B borrows dollars at 11.5% (lower than 12%).

– The gain is 0.5% for A, 0.5% for B, and, if we treat dollars and yen identically, 1% for the bank.

(18)

Party B Bank

Party A

Dollars 9% Yen 11%

Dollars 9%

Yen 11%

Yen 9.5%

Dollars 11.5%

(19)

### As a Package of Cash Market Instruments

• Assume no default risk.

• Take B on p. 467 as an example.

• The swap is equivalent to a long position in a yen bond paying 11% annual interest and a short position in a dollar bond paying 11.5% annual interest.

• The pricing formula is SPY − PD.

– PD is the dollar bond’s value in dollars.

– PY is the yen bond’s value in yen.

– S is the \$/yen spot exchange rate.

(20)

### As a Package of Cash Market Instruments (concluded)

• The value of a currency swap depends on:

– The term structures of interest rates in the currencies involved.

– The spot exchange rate.

• It has zero value when

SPY = PD.

(21)

### Example

• Take a 3-year swap on p. 467 with principal amounts of US\$1 million and 100 million yen.

• The payments are made once a year.

• The spot exchange rate is 90 yen/\$ and the term

structures are ﬂat in both nations—8% in the U.S. and 9% in Japan.

• For B, the value of the swap is (in millions of USD)

1

90 × 

11 × e−0.09 + 11× e−0.09×2 + 111× e−0.09×3



0.115 × e−0.08 + 0.115 × e−0.08×2 + 1.115 × e−0.08×3

= 0.074.

(22)

### As a Package of Forward Contracts

• From Eq. (39) on p. 435, the forward contract maturing i years from now has a dollar value of

fi ≡ (SYi) e−qi − Die−ri. (45) – Yi is the yen inﬂow at year i.

– S is the \$/yen spot exchange rate.

– q is the yen interest rate.

– Di is the dollar outﬂow at year i.

– r is the dollar interest rate.

(23)

### As a Package of Forward Contracts (concluded)

• For simplicity, ﬂat term structures were assumed.

• Generalization is straightforward.

(24)

### Example

• Take the swap in the example on p. 470.

• Every year, B receives 11 million yen and pays 0.115 million dollars.

• In addition, at the end of the third year, B receives 100 million yen and pays 1 million dollars.

• Each of these transactions represents a forward contract.

• Y1 = Y2 = 11, Y3 = 111, S = 1/90, D1 = D2 = 0.115, D3 = 1.115, q = 0.09, and r = 0.08.

• Plug in these numbers to get f1 + f2 + f3 = 0.074 million dollars as before.

(25)

### Stochastic Processes and Brownian Motion

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Of all the intellectual hurdles which the human mind has confronted and has overcome in the last ﬁfteen hundred years, the one which seems to me to have been the most amazing in character and the most stupendous in the scope of its consequences is the one relating to the problem of motion.

— Herbert Butterﬁeld (1900–1979)

(27)

### Stochastic Processes

• A stochastic process

X = { X(t) } is a time series of random variables.

• X(t) (or Xt) is a random variable for each time t and is usually called the state of the process at time t.

• A realization of X is called a sample path.

(28)

### Stochastic Processes (concluded)

• If the times t form a countable set, X is called a discrete-time stochastic process or a time series.

• In this case, subscripts rather than parentheses are usually employed, as in

X = { Xn }.

• If the times form a continuum, X is called a continuous-time stochastic process.

(29)

### Random Walks

• The binomial model is a random walk in disguise.

• Consider a particle on the integer line, 0, ±1, ±2, . . . .

• In each time step, it can make one move to the right with probability p or one move to the left with

probability 1 − p.

– This random walk is symmetric when p = 1/2.

• Connection with the BOPM: The particle’s position denotes the number of up moves minus that of down moves up to that time.

(30)

20 40 60 80 Time

-8 -6 -4 -2 2 4

Position

(31)

### Random Walk with Drift

Xn = μ + Xn−1 + ξn.

• ξn are independent and identically distributed with zero mean.

• Drift μ is the expected change per period.

• Note that this process is continuous in space.

(32)

### Martingales

a

• { X(t), t ≥ 0 } is a martingale if E[ | X(t) | ] < ∞ for t ≥ 0 and

E[ X(t) | X(u), 0 ≤ u ≤ s ] = X(s), s ≤ t. (46)

• In the discrete-time setting, a martingale means

E[ Xn+1 | X1, X2, . . . , Xn ] = Xn. (47)

• Xn can be interpreted as a gambler’s fortune after the nth gamble.

• Identity (47) then says the expected fortune after the (n + 1)th gamble equals the fortune after the nth gamble regardless of what may have occurred before.

aThe origin of the name is somewhat obscure.

(33)

### Martingales (concluded)

• A martingale is therefore a notion of fair games.

• Apply the law of iterated conditional expectations to both sides of Eq. (47) on p. 481 to yield

E[ Xn ] = E[ X1 ] (48) for all n.

• Similarly,

E[ X(t) ] = E[ X(0) ] in the continuous-time case.

(34)

### Still a Martingale?

• Suppose we replace Eq. (47) on p. 481 with E[ Xn+1 | Xn ] = Xn.

• It also says past history cannot aﬀect the future.

• But is it equivalent to the original deﬁnition (47) on p. 481?a

aContributed by Mr. Hsieh, Chicheng (M9007304) on April 13, 2005.

(35)

### Still a Martingale? (continued)

• Well, no.a

• Consider this random walk with drift:

Xi =

⎧⎨

Xi−1 + ξi, if i is even, Xi−2, otherwise.

• Above, ξn are random variables with zero mean.

aContributed by Mr. Zhang, Ann-Sheng (B89201033) on April 13, 2005.

(36)

### Still a Martingale? (concluded)

• It is not hard to see that

E[ Xi | Xi−1 ] =

⎧⎨

Xi−1, if i is even, Xi−1, otherwise.

– It is a martingale by the “new” deﬁnition.

• But

E[ Xi | . . . , Xi−2, Xi−1 ] =

⎧⎨

Xi−1, if i is even, Xi−2, otherwise.

– It is not a martingale by the original deﬁnition.

(37)

### Example

• Consider the stochastic process { Zn

n i=1

Xi, n ≥ 1 },

where Xi are independent random variables with zero mean.

• This process is a martingale because E[ Zn+1 | Z1, Z2, . . . , Zn ]

= E[ Zn + Xn+1 | Z1, Z2, . . . , Zn ]

= E[ Zn | Z1, Z2, . . . , Zn ] + E[ Xn+1 | Z1, Z2, . . . , Zn ]

(38)

### Probability Measure

• A probability measure assigns probabilities to states of the world.

• A martingale is deﬁned with respect to a probability measure, under which the expectation is taken.

• A martingale is also deﬁned with respect to an information set.

– In the characterizations (46)–(47) on p. 481, the

information set contains the current and past values of X by default.

– But it need not be so.

(39)

### Probability Measure (continued)

• A stochastic process { X(t), t ≥ 0 } is a martingale with respect to information sets { It } if, for all t ≥ 0,

E[ | X(t) | ] < ∞ and

E[ X(u) | It ] = X(t) for all u > t.

• The discrete-time version: For all n > 0, E[ Xn+1 | In ] = Xn, given the information sets { In }.

(40)

### Probability Measure (concluded)

• The above implies

E[ Xn+m | In ] = Xn for any m > 0 by Eq. (19) on p. 151.

– A typical In is the price information up to time n.

– Then the above identity says the FVs of X will not deviate systematically from today’s value given the price history.

(41)

### Example

• Consider the stochastic process { Zn − nμ, n ≥ 1 }.

– Zn n

i=1 Xi.

– X1, X2, . . . are independent random variables with mean μ.

• Now,

E[ Zn+1 − (n + 1) μ | X1, X2, . . . , Xn ]

= E[ Zn+1 | X1, X2, . . . , Xn ] − (n + 1) μ

= E[ Zn + Xn+1 | X1, X2, . . . , Xn ] − (n + 1) μ

= Zn + μ − (n + 1) μ

(42)

### Example (concluded)

• Deﬁne

In ≡ { X1, X2, . . . , Xn }.

• Then

{ Zn − nμ, n ≥ 1 } is a martingale with respect to { In }.

(43)

### Martingale Pricing

• The price of a European option is the expected discounted payoﬀ at expiration in a risk-neutral economy.a

• This principle can be generalized using the concept of martingale.

• Recall the recursive valuation of European option via C = [ pCu + (1 − p) Cd ]/R.

– p is the risk-neutral probability.

– \$1 grows to \$R in a period.

(44)

### Martingale Pricing (continued)

• Let C(i) denote the value of the option at time i.

• Consider the discount process

 C(i)

Ri , i = 0, 1, . . . , n

 .

• Then, E

C(i + 1) Ri+1

= pCu + (1 − p) Cd

Ri+1 = C(i) Ri .

(45)

### Martingale Pricing (continued)

• It is easy to show that E

C(k) Rk

= C

Ri, i ≤ k. (49)

• This formulation assumes:a

1. The model is Markovian: The distribution of the

future is determined by the present (time i ) and not the past.

2. The payoﬀ depends only on the terminal price of the underlying asset (Asian options do not qualify).

aContributed by Mr. Wang, Liang-Kai (Ph.D. student, ECE, Univer-

(46)

### Martingale Pricing (continued)

• In general, the discount process is a martingale in thata Eiπ

C(k) Rk

= C(i)

Ri , i ≤ k. (50) – Eiπ is taken under the risk-neutral probability

conditional on the price information up to time i.

• This risk-neutral probability is also called the EMM, or the equivalent martingale (probability) measure.

aIn this general formulation, Asian options do qualify.

(47)

### Martingale Pricing (continued)

• Equation (50) holds for all assets, not just options.

• When interest rates are stochastic, the equation becomes C(i)

M (i) = Eiπ

C(k) M (k)

, i ≤ k. (51)

– M(j) is the balance in the money market account at time j using the rollover strategy with an initial

investment of \$1.

– It is called the bank account process.

• It says the discount process is a martingale under π.

(48)

### Martingale Pricing (continued)

• If interest rates are stochastic, then M(j) is a random variable.

– M(0) = 1.

– M(j) is known at time j − 1.

• Identity (51) on p. 496 is the general formulation of risk-neutral valuation.

(49)

### Martingale Pricing (concluded)

Theorem 17 A discrete-time model is arbitrage-free if and only if there exists a probability measure such that the

discount process is a martingale.a

aThis probability measure is called the risk-neutral probability mea- sure.

(50)

### Futures Price under the BOPM

• Futures prices form a martingale under the risk-neutral probability.

– The expected futures price in the next period is pfF u + (1 − pf) F d = F

1 − d

u − d u + u − 1 u − d d

= F (p. 454).

• Can be generalized to

Fi = Eiπ[ Fk ], i ≤ k, where Fi is the futures price at time i.

• This equation holds under stochastic interest rates, too.

(51)

### Martingale Pricing and Numeraire

a

• The martingale pricing formula (51) on p. 496 uses the money market account as numeraire.b

– It expresses the price of any asset relative to the money market account.

• The money market account is not the only choice for numeraire.

• Suppose asset S’s value is positive at all times.

aJohn Law (1671–1729), “Money to be qualified for exchaning goods and for payments need not be certain in its value.”

bLeon Walras (1834–1910).

(52)

### Martingale Pricing and Numeraire (concluded)

• Choose S as numeraire.

• Martingale pricing says there exists a risk-neutral

probability π under which the relative price of any asset C is a martingale:

C(i)

S(i) = Eiπ

C(k) S(k)

, i ≤ k.

– S(j) denotes the price of S at time j.

• So the discount process remains a martingale.a

aThis result is related to Girsanov’s theorem.

(53)

### Example

• Take the binomial model with two assets.

• In a period, asset one’s price can go from S to S1 or S2.

• In a period, asset two’s price can go from P to P1 or P2.

• Both assets must move up or down at the same time.

• Assume

S1

P1 < S

P < S2

P2 (52)

(54)

### Example (continued)

• For any derivative security, let C1 be its price at time one if asset one’s price moves to S1.

• Let C2 be its price at time one if asset one’s price moves to S2.

• Replicate the derivative by solving

αS1 + βP1 = C1, αS2 + βP2 = C2,

using α units of asset one and β units of asset two.

(55)

### Example (continued)

• By Eqs. (52) on p. 502, α and β have unique solutions.

• In fact,

α = P2C1 − P1C2

P2S1 − P1S2 and β = S2C1 − S1C2 S2P1 − S1P2 .

• The derivative costs C = αS + βP

= P2S − P S2

P2S1 − P1S2 C1 + P S1 − P1S

P2S1 − P1S2 C2.

(56)

### Example (concluded)

• It is easy to verify that C

P = p C1

P1 + (1 − p) C2 P2 . – Above,

p ≡ (S/P ) − (S2/P2) (S1/P1) − (S2/P2). – By Eqs. (52) on p. 502, 0 < p < 1.

• C’s price using asset two as numeraire (i.e., C/P ) is a martingale under the risk-neutral probability p.

• The expected returns of the two assets are irrelevant.

(57)

### Brownian Motion

a

• Brownian motion is a stochastic process { X(t), t ≥ 0 } with the following properties.

1. X(0) = 0, unless stated otherwise.

2. for any 0 ≤ t0 < t1 < · · · < tn, the random variables X(tk) − X(tk−1)

for 1 ≤ k ≤ n are independent.b

3. for 0 ≤ s < t, X(t) − X(s) is normally distributed with mean μ(t − s) and variance σ2(t − s), where μ and σ = 0 are real numbers.

a

(58)

### Brownian Motion (concluded)

• The existence and uniqueness of such a process is guaranteed by Wiener’s theorem.a

• This process will be called a (μ, σ) Brownian motion with drift μ and variance σ2.

• Although Brownian motion is a continuous function of t with probability one, it is almost nowhere diﬀerentiable.

• The (0, 1) Brownian motion is called the Wiener process.

aNorbert Wiener (1894–1964).

(59)

### Example

• If { X(t), t ≥ 0 } is the Wiener process, then X(t) − X(s) ∼ N (0, t − s).

• A (μ, σ) Brownian motion Y = { Y (t), t ≥ 0 } can be expressed in terms of the Wiener process:

Y (t) = μt + σX(t). (53)

• Note that

Y (t + s) − Y (t) ∼ N (μs, σ2s).

(60)

### Brownian Motion as Limit of Random Walk

Claim 1 A (μ, σ) Brownian motion is the limiting case of random walk.

• A particle moves Δx to the left with probability 1 − p.

• It moves to the right with probability p after Δt time.

• Deﬁne

Xi

⎧⎨

+1 if the ith move is to the right,

−1 if the ith move is to the left.

– Xi are independent with

Prob[ Xi = 1 ] = p = 1 − Prob[ Xi = −1 ].

(61)

Brownian Motion as Limit of Random Walk (continued)

• Assume n ≡ t/Δt is an integer.

• Its position at time t is

Y (t) ≡ Δx (X1 + X2 + · · · + Xn) .

• Recall

E[ Xi ] = 2p − 1,

Var[ Xi ] = 1 − (2p − 1)2.

(62)

Brownian Motion as Limit of Random Walk (continued)

• Therefore,

E[ Y (t) ] = n(Δx)(2p − 1), Var[ Y (t) ] = n(Δx)2 

1 − (2p − 1)2  .

• With Δx ≡ σ√

Δt and p ≡ [ 1 + (μ/σ)√

Δt ]/2, E[ Y (t) ] = nσ√

Δt (μ/σ)√

Δt = μt, Var[ Y (t) ] = nσ2Δt 

1 − (μ/σ)2Δt

→ σ2t, as Δt → 0.

(63)

Brownian Motion as Limit of Random Walk (concluded)

• Thus, { Y (t), t ≥ 0 } converges to a (μ, σ) Brownian motion by the central limit theorem.

• Brownian motion with zero drift is the limiting case of symmetric random walk by choosing μ = 0.

• Note that

Var[ Y (t + Δt) − Y (t) ]

=Var[ Δx Xn+1 ] = (Δx)2 × Var[ Xn+1 ] → σ2Δt.

• Similarity to the the BOPM: The p is identical to the probability in Eq. (29) on p. 274 and Δx = ln u.

(64)

### Geometric Brownian Motion

• Let X ≡ { X(t), t ≥ 0 } be a Brownian motion process.

• The process

{ Y (t) ≡ eX(t), t ≥ 0 }, is called geometric Brownian motion.

• Suppose further that X is a (μ, σ) Brownian motion.

• X(t) ∼ N(μt, σ2t) with moment generating function E



esX(t)



= E [ Y (t)s ] = eμts+(σ2ts2/2) from Eq. (20) on p 153.

(65)

### Geometric Brownian Motion (concluded)

• In particular,

E[ Y (t) ] = eμt+(σ2t/2), Var[ Y (t) ] = E 

Y (t)2 

− E[ Y (t) ]2

= e2μt+σ2t



eσ2t − 1 .

(66)

0.2 0.4 0.6 0.8 1 Time (t) -1

1 2 3 4 5 6 Y(t)

(67)

### A Case for Long-Term Investment

a

• Suppose the stock follows the geometric Brownian motion

S(t) = S(0) eN (μt,σ2t) = S(0) etN (μ,σ2/t ), t ≥ 0, where μ > 0.

• The annual rate of return has a normal distribution:

N

μ, σ2 t

.

• The larger the t, the likelier the return is positive.

• The smaller the t, the likelier the return is negative.

a

(68)

### Continuous-Time Financial Mathematics

(69)

A proof is that which convinces a reasonable man;

a rigorous proof is that which convinces an unreasonable man.

— Mark Kac (1914–1984) The pursuit of mathematics is a divine madness of the human spirit.

— Alfred North Whitehead (1861–1947), Science and the Modern World

(70)

### Stochastic Integrals

• Use W ≡ { W (t), t ≥ 0 } to denote the Wiener process.

• The goal is to develop integrals of X from a class of stochastic processes,a

It(X) ≡

 t

0 X dW, t ≥ 0.

• It(X) is a random variable called the stochastic integral of X with respect to W .

• The stochastic process { It(X), t ≥ 0 } will be denoted by 

X dW .

aKiyoshi Ito (1915–2008).

(71)

### Stochastic Integrals (concluded)

• Typical requirements for X in ﬁnancial applications are:

– Prob[ t

0 X2(s) ds < ∞ ] = 1 for all t ≥ 0 or the stronger  t

0 E[ X2(s) ] ds < ∞.

– The information set at time t includes the history of X and W up to that point in time.

– But it contains nothing about the evolution of X or W after t (nonanticipating, so to speak).

– The future cannot inﬂuence the present.

(72)

### Ito Integral

• A theory of stochastic integration.

• As with calculus, it starts with step functions.

• A stochastic process { X(t) } is simple if there exist 0 = t0 < t1 < · · ·

such that

X(t) = X(tk−1) for t ∈ [ tk−1, tk), k = 1, 2, . . . for any realization (see ﬁgure on next page).

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J0 J1 J2 J3 J4 J5 : J

## = B

J

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### Ito Integral (continued)

• The Ito integral of a simple process is deﬁned as

It(X) ≡

n−1

k=0

X(tk)[ W (tk+1) − W (tk) ], (54) where tn = t.

– The integrand X is evaluated at tk, not tk+1.

• Deﬁne the Ito integral of more general processes as a limiting random variable of the Ito integral of simple stochastic processes.

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### Ito Integral (continued)

• Let X = { X(t), t ≥ 0 } be a general stochastic process.

• Then there exists a random variable It(X), unique almost certainly, such that It(Xn) converges in probability to It(X) for each sequence of simple

stochastic processes X1, X2, . . . such that Xn converges in probability to X.

• If X is continuous with probability one, then It(Xn) converges in probability to It(X) as

δn ≡ max

1≤k≤n(tk − tk−1)

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### Ito Integral (concluded)

• It is a fundamental fact that 

X dW is continuous almost surely.

• The following theorem says the Ito integral is a martingale.

– A corollary is the mean value formula E

  b

a

X dW



= 0.

Theorem 18 The Ito integral 

X dW is a martingale.

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### Discrete Approximation

• Recall Eq. (54) on p. 523.

• The following simple stochastic process { X(t) } can be used in place of X to approximate  t

0 X dW ,

X(s) ≡ X(t k−1) for s ∈ [ tk−1, tk), k = 1, 2, . . . , n.

• Note the nonanticipating feature of X.

– The information up to time s,

{ X(t), W (t), 0 ≤ t ≤ s },

cannot determine the future evolution of X or W .

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### Discrete Approximation (concluded)

• Suppose we deﬁned the stochastic integral as

n−1

k=0

X(tk+1)[ W (tk+1) − W (tk) ].

• Then we would be using the following diﬀerent simple stochastic process in the approximation,

Y (s) ≡ X(t k) for s ∈ [ tk−1, tk), k = 1, 2, . . . , n.

• This clearly anticipates the future evolution of X.a

aSee Exercise 14.1.2 of the textbook for an example where it matters.

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:

J

:

J

;

(a) (b)

:

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### Ito Process

• The stochastic process X = { Xt, t ≥ 0 } that solves Xt = X0 +

 t

0 a(Xs, s) ds +

 t

0 b(Xs, s) dWs, t ≥ 0 is called an Ito process.

– X0 is a scalar starting point.

– { a(Xt, t) : t ≥ 0 } and { b(Xt, t) : t ≥ 0 } are stochastic processes satisfying certain regularity conditions.

– a(Xt, t): the drift.

– b(Xt, t): the diﬀusion.

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### Ito Process (continued)

• A shorthanda is the following stochastic diﬀerential equation for the Ito diﬀerential dXt,

dXt = a(Xt, t) dt + b(Xt, t) dWt. (55) – Or simply

dXt = at dt + bt dWt.

– This is Brownian motion with an instantaneous drift at and an instantaneous variance b2t.

• X is a martingale if at = 0 (Theorem 18 on p. 525).

a

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### Ito Process (concluded)

• dW is normally distributed with mean zero and variance dt.

• An equivalent form of Eq. (55) is dXt = at dt + bt

dt ξ, (56)

where ξ ∼ N (0, 1).

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