### Futures Options

*• The underlying of a futures option is a futures contract.*

*• Upon exercise, the option holder takes a position in the*
futures contract with a futures price equal to the

option’s strike price.

**– A call holder acquires a long futures position.**

**– A put holder acquires a short futures position.**

*• The futures contract is then marked to market.*

*• And the futures position of the two parties will be at the*
prevailing futures price (thus zero-valued).

### Futures Options (concluded)

*• It works as if the call holder received a futures contract*
*plus cash equivalent to the prevailing futures price F*_{t}*minus the strike price X:*

*F*_{t}*− X.*

**– This futures contract has zero value.**

*• It works as if the put holder sold a futures contract for*
*X − F*_{t}

### Forward Options

*• Similar to futures options except that what is delivered*
is a forward contract with a delivery price equal to the
option’s strike price.

* – Exercising a call forward option results in a long*
position in a forward contract.

* – Exercising a put forward option results in a short*
position in a forward contract.

*• Exercising a forward option incurs no immediate cash*
ﬂows: There is no marking to market.

### Example

*• Consider a call with strike $100 and an expiration date*
in September.

*• The underlying asset is a forward contract with a*
delivery date in December.

*• Suppose the forward price in July is $110.*

*• Upon exercise, the call holder receives a forward*
contract with a delivery price of $100.^{a}

*• If an oﬀsetting position is then taken in the forward*
market,^{b} *a $10 proﬁt in December will be assured.*

*• A call on the futures would realize the $10 proﬁt in July.*

### Some Pricing Relations

*• Let delivery take place at time T , the current time be 0,*
and the option on the futures or forward contract have
*expiration date t (t ≤ T ).*

*• Assume a constant, positive interest rate.*

*• Although forward price equals futures price, a forward*
*option does not have the same value as a futures option.*

*• The payoﬀs of calls at time t are, respectively,*^{a}

*futures option = max(F*_{t}*− X, 0),* (62)
*forward option = max(F*_{t}*− X, 0) e*^{−r(T −t)}*. (63)*

### Some Pricing Relations (concluded)

*• A European futures option is worth the same as the*
corresponding European option on the underlying asset
if the futures contract has the same maturity as both
options.

**– Futures price equals spot price at maturity.**

*• This conclusion is independent of the model for the spot*
price.

### Put-Call Parity

^{a}

The put-call parity is slightly diﬀerent from the one in Eq. (30) on p. 221.

**Theorem 13 (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}*.*

aSee Theorem 12.4.4 of the textbook for proof.

### Early Exercise

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

**Theorem 14 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 Theorem 12.4.5 of the textbook for proof.*

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

**Theorem 15 American futures options may be exercised**

### Black’s Model

^{a}

*• Formulas for European futures options:*

*C* = *F e*^{−rt}*N (x) − Xe*^{−rt}*N (x − σ√*

*t),* (64)
*P* = *Xe*^{−rt}*N (−x + σ√*

*t) − F e*^{−rt}*N (−x),*
*where x* =^{Δ} ^{ln(F/X)+(σ}^{2}^{/2) t}

*σ**√*

*t* .

*• Formulas (64) are related to those for options on a stock*
paying a continuous dividend yield.

*• They are exactly Eqs. (42) on p. 322 with q set to r*
*and S replaced by F .*

a

### Black’s Model (concluded)

*• This observation incidentally proves Theorem 15*
(p. 501).

*• 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. (62)–(63) on p. 498.

### 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. 474)**
*F = Se*^{rT}*.*

**– From Lemma 9 (p. 290), the expected value of S at***time Δt in a risk-neutral economy is*

*Se*^{rΔt}*.*

**– So the expected futures price at time Δt is**

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

*• The above observation continues to hold even if S pays a*
dividend yield!^{a}

**– By Eq. (60) on p. 484, the futures price at time 0 is**
*F = Se*^{(r−q) T}*.*

**– From Lemma 9 (p. 290), 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**

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

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

*p*_{f} = (1^{Δ} *− d)/(u − d)*
by Eq. (43) on p. 324.

**– The futures price moves from F to F u with**

*probability p*_{f} *and to F d with probability 1 − p*_{f}.
**– Note that the original u and d are used!**

*• The binomial tree algorithm for forward options is*

### Spot and Futures Prices under BOPM

*• The futures price is related to the spot price via*
*F = Se*^{rT}

if the underlying asset pays no dividends.^{a}

*• Recall the futures price F moves to F u with probability*
*p*_{f} per period.

*• So the stock price moves from S = F e** ^{−rT}* to

*F ue*

^{−r(T −Δt)}*= Sue*

^{rΔt}*with probability p*f per period.

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

*• Similarly, the stock price moves from S = F e** ^{−rT}* to

*Sde*

^{rΔt}with probability 1 *− p*f per period.

*• Note that*

*S(ue*^{rΔt}*)(de*^{rΔt}*) = Se*^{2rΔt}*= S.*

*• So this binomial model for S is not the CRR tree.*

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

### Negative Probabilities Revisited

*• As 0 < p*f *< 1, we have 0 < 1 − p*_{f} *< 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. (60) on p. 484, calculate S from F at each*
node via

*S = F e**−(r−q)(T −t)**.*

### 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}

### 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 *D*A% *Y*A%
B *D*B% *Y*B%

*• Suppose A wants to take out a ﬁxed-rate loan in yen,*

### Currency Swaps (continued)

*• A straightforward scenario is for A to borrow yen at*
*Y*_{A}*% and B to borrow dollars at D*_{B}%.

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

*Y*_{B} *− D*B *< Y*_{A} *− D*A or *Y*_{B} *− Y*A *< D*_{B} *− D*A*.*

*• Consider this alternative arrangement:*

**– A borrows dollars.**

**– B borrows yen.**

**– They enter into a currency swap with a bank (the**

### 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 ((D*_{B} *− D*_{A}) *− (Y*_{B} *− Y*_{A}))%:

* – The total interest rate is originally (Y*A

*+ D*B)%.

**– The new arrangement has a smaller total rate of**
*(D*_{A} *+ Y*_{B})%.

*• Transactions will happen only if the gain is distributed*

### 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%.*

### 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.*

### 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.

Party B Bank

Party A

Dollars 9% Yen 11%

Dollars 9%

Yen 11%

Yen 9.5%

Dollars 11.5%

### As a Package of Cash Market Instruments

*• Assume no default risk.*

*• Take B on p. 517 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 SP*_{Y} *− P*_{D}.

* – P*D is the dollar bond’s value in dollars.

**– P**_{Y} is the yen bond’s value in yen.

**– S is the $/yen spot exchange rate.**

### 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*

*SP*_{Y} *= P*_{D}*.*

### Example

*• Take a 3-year swap on p. 517 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*

*+ 111*

^{−0.09×2}*× e*

^{−0.09×3}*−*

0*.115 × e** ^{−0.08}* + 0

*.115 × e*

*+ 1*

^{−0.08×2}*.115 × e*

^{−0.08×3}= 0*.074.*

### As a Package of Forward Contracts

*• From Eq. (59) on p. 484, the forward contract maturing*
*i years from now has a dollar value of*

*f*_{i}*= (SY*^{Δ} _{i}*) e*^{−qi}*− D*_{i}*e*^{−ri}*.* (65)
**– Y**_{i}*is the yen inﬂow at year i.*

**– S is the $/yen spot exchange rate.**

**– q is the yen interest rate.**

**– D**_{i}*is the dollar outﬂow at year i.*

**– r is the dollar interest rate.**

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

*• For simplicity, ﬂat term structures were assumed.*

*• Generalization is straightforward.*

### Example

*• Take the swap in the example on p. 520.*

*• 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.*

*• Y*1 *= Y*2 *= 11, Y*3 *= 111, S = 1/90, D*1 *= D*2 *= 0.115,*
*D*3 *= 1.115, q = 0.09, and r = 0.08.*

*• Plug in these numbers to get f* *+ f* *+ f* *= 0.074*

*Stochastic Processes and Brownian Motion*

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)

### Stochastic Processes

*• A stochastic process*

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

*• X(t) (or X*_{t}*) 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.*

### 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 = { X*_{n}*}.*

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

### 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.

20 40 60 80 Time

-8 -6 -4 -2 2 4

Position

### Random Walk with Drift

*X*_{n}*= μ + X*_{n−1}*+ ξ*_{n}*.*

*• ξ** _{n}* are independent and identically distributed with zero
mean.

*• Drift μ is the expected change per period.*

*• Note that this process is continuous in space.*

### 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.* (66)

*• In the discrete-time setting, a martingale means*

*E[ X*_{n+1}*| X*_{1}*, X*_{2}*, . . . , X*_{n}*] = X*_{n}*.* (67)

*• X** _{n}* can be interpreted as a gambler’s fortune after the

*nth gamble.*

*• Identity (67) then says the expected fortune after the*
*(n + 1)th gamble equals the fortune after the nth*

### Martingales (concluded)

*• A martingale is therefore a notion of fair games.*

*• Apply the law of iterated conditional expectations to*
both sides of Eq. (67) on p. 531 to yield

*E[ X*_{n}*] = E[ X*_{1} ] (68)
*for all n.*

*• Similarly,*

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

### Still a Martingale?

*• Suppose we replace Eq. (67) on p. 531 with*
*E[ X*_{n+1}*| X*_{n}*] = X*_{n}*.*

*• It also says past history cannot aﬀect the future.*

*• But is it equivalent to the original deﬁnition (67) on*
p. 531?^{a}

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

### Still a Martingale? (continued)

*• Well, no.*^{a}

*• Consider this random walk with drift:*

*X** _{i}* =

⎧⎨

⎩

*X*_{i−1}*+ ξ*_{i}*, if i is even,*
*X*_{i−2}*,* *otherwise.*

*• Above, ξ** _{n}* are random variables with zero mean.

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

### Still a Martingale? (concluded)

*• It is not hard to see that*

*E[ X*_{i}*| X** _{i−1}* ] =

⎧⎨

⎩

*X*_{i−1}*, if i is even,*
*X*_{i−1}*, otherwise.*

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

*• But*

*E[ X*_{i}*| . . . , X*_{i−2}*, X** _{i−1}* ] =

⎧⎨

⎩

*X*_{i−1}*, if i is even,*
*X*_{i−2}*, otherwise.*

### Example

*• Consider the stochastic process*

*Z** _{n}* =

^{Δ}

*n*
*i=1*

*X*_{i}*, n ≥ 1*

*,*

*where X** _{i}* are independent random variables with zero
mean.

*• This process is a martingale because*
*E[ Z*_{n+1}*| Z*1*, Z*2*, . . . , Z** _{n}* ]

= *E[ Z*_{n}*+ X*_{n+1}*| Z*1*, Z*_{2}*, . . . , Z** _{n}* ]

= *E[ Z*_{n}*| Z*1*, Z*_{2}*, . . . , Z*_{n}*] + E[ X*_{n+1}*| Z*1*, Z*_{2}*, . . . , Z** _{n}* ]

### 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.

*• Second, a martingale is deﬁned with respect to an*
information set.

**– In the characterizations (66)–(67) on p. 531, the**

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

### Probability Measure (continued)

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

*E[ | X(t) | ] < ∞ and*

*E[ X(u) | I*_{t}*] = X(t)*
*for all u > t.*

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

### Probability Measure (concluded)

*• The above implies*

*E[ X*_{n+m}*| I*_{n}*] = X*_{n}*for any m > 0 by Eq. (26) on p. 166.*

**– A typical I**_{n}*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.

### Example

*• Consider the stochastic process { Z*_{n}*− nμ, n ≥ 1 }.*

**– Z*** _{n}* =

^{Δ}

_{n}*i=1* *X** _{i}*.

* – X*1

*, X*2

*, . . . are independent random variables with*

*mean μ.*

*• Now,*

*E[ Z*_{n+1}*− (n + 1) μ | X*_{1}*, X*_{2}*, . . . , X** _{n}* ]

= *E[ Z*_{n+1}*| X*_{1}*, X*_{2}*, . . . , X** _{n}* ]

*− (n + 1) μ*

= *E[ Z*_{n}*+ X*_{n+1}*| X*1*, X*2*, . . . , X** _{n}* ]

*− (n + 1) μ*

= *Z*_{n}*+ μ − (n + 1) μ*

### Example (concluded)

*• Deﬁne*

*I** _{n}* =

^{Δ}

*{ X*1

*, X*

_{2}

*, . . . , X*

_{n}*}.*

*• Then*

*{ Z*_{n}*− nμ, n ≥ 1 }*
is a martingale with respect to *{ I*_{n}*}.*

### Martingale Pricing

*• Stock prices and zero-coupon bond prices are expected*
to rise, while option prices are expected to fall.

*• They are not martingales.*

*• Why is then martingale useful?*

*• Recall a martingale is deﬁned with respect to some*
*information set and some probability measure.*

*• By modifying the probability measure, we can convert a*
price process into a martingale.

### Martingale Pricing (continued)

*• The price of a European option is the expected*
discounted payoﬀ 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 = [ pC** _{u}* + (1

*− p) C*

_{d}*]/R.*

**– p is the risk-neutral probability.**

**– $1 grows to $R in a period.**

### Martingale Pricing (continued)

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

*• Consider the discount process*
*C(i)*

*R*^{i}*, i = 0, 1, . . . , n*

*.*

*• Then,*
*E*

*C(i + 1)*
*R*^{i+1}

* C(i)*

= *pC** _{u}* + (1

*− p) C*

_{d}*R** ^{i+1}* =

*C(i)*

*R*

^{i}*.*

### Martingale Pricing (continued)

*• It is easy to show that*
*E*

*C(k)*
*R*^{k}

* C(i)*

= *C(i)*

*R*^{i}*, i ≤ k.* (69)

*• 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-

### Martingale Pricing (continued)

*• In general, the discount process is a martingale in that*^{a}
*E*_{i}^{π}

*C(k)*
*R*^{k}

= *C(i)*

*R*^{i}*, i ≤ k.* (70)
**– E**_{i}* ^{π}* 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.

### Martingale Pricing (continued)

*• Equation (70) holds for all assets, not just options.*

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

*M (i)* *= E*_{i}^{π}

*C(k)*
*M (k)*

*, i ≤ k.* (71)

**– 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.*

### 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.**^{a}

*• Identity (71) on p. 547 is the general formulation of*
risk-neutral valuation.

aBecause the interest rate for the next period has been revealed then.

### Martingale Pricing (concluded)

**Theorem 16 A discrete-time model is arbitrage-free if and***only if there exists a probability measure*^{a} *such that the*

*discount process is a martingale.*

aThis measure is called the risk-neutral probability measure.

### Futures Price under the BOPM

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

**– The expected futures price in the next period is**^{a}
*p*f*F u + (1 − p*f*) F d = F*

1 *− d*

*u − d* *u +* *u − 1*
*u − d* *d*

*= F.*

*• Can be generalized to*

*F*_{i}*= E*_{i}^{π}*[ F*_{k}*], i ≤ k,*
*where F*_{i}*is the futures price at time i.*

*• This equation holds under stochastic interest rates, too.*^{b}

a

### Martingale Pricing and Numeraire

^{a}

*• The martingale pricing formula (71) on p. 547 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.”

### 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)* *= E*_{i}^{π}

*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 (1960).

### Example

*• Take the binomial model with two assets.*

*• In a period, asset one’s price can go from S to S*1 or
*S*_{2}.

*• In a period, asset two’s price can go from P to P*1 or
*P*2.

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

*• Assume*

*S*1

*P*1 *<* *S*

*P* *<* *S*2

*P*2 (72)

### Example (continued)

*• For any derivative security, let C*_{1} be its price at time
*one if asset one’s price moves to S*_{1}.

*• Let C*_{2} be its price at time one if asset one’s price
*moves to S*_{2}.

*• Replicate the derivative by solving*

*αS*_{1} *+ βP*_{1} = *C*_{1}*,*
*αS*2 *+ βP*2 = *C*2*,*

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

### Example (continued)

*• By Eqs. (72) on p. 553, α and β have unique solutions.*

*• In fact,*

*α =* *P*_{2}*C*_{1} *− P*1*C*_{2}

*P*2*S*1 *− P*1*S*2 *and β =* *S*_{2}*C*_{1} *− S*1*C*_{2}
*S*2*P*1 *− S*1*P*2 *.*

*• The derivative costs*
*C* = *αS + βP*

= *P*2*S − P S*2

*P*2*S*1 *− P*1*S*2 *C*1 + *P S*1 *− P*1*S*

*P*2*S*1 *− P*1*S*2 *C*2*.*

### Example (continued)

*• It is easy to verify that*
*C*

*P* *= p* *C*_{1}

*P*_{1} + (1 *− p)* *C*_{2}
*P*_{2} *.*
**– Above,**

*p* =^{Δ} *(S/P ) − (S*_{2}*/P*_{2})
*(S*_{1}*/P*_{1}) *− (S*_{2}*/P*_{2})*.*
**– By Eqs. (72) on p. 553, 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.*

### Example (concluded)

*• In the BOPM, S is the stock and P is the bond.*

*• Furthermore, p assumes the bond is the numeraire.*

*• In the binomial option pricing formula (38) on p. 262,*
*the S*

*b(j; n, pu/R) term uses the stock as the*
numeraire.

**– It results in a diﬀerent probability measure pu/R.**

*• In the limit, SN(x) for the call and SN(−x) for the put*
in the Black-Scholes formula (p. 292) use the stock as
the numeraire.^{a}

### 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 ≤ t**_{0} *< t*_{1} *< · · · < t** _{n}*, the random variables

*X(t*

*)*

_{k}*− X(t*

*)*

_{k−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 σ*^{2}*(t − s), where μ*
*and σ = 0 are real numbers.*

a

### 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 σ*^{2}.

*• 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.*

*• If condition 3 is replaced by “ X(t) − X(s) depends only*
*on t − s,” we have the more general Levy process.*^{b}

aNorbert Wiener (1894–1964). He received his Ph.D. from Harvard

### 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).* (73)

*• Note that*

*Y (t + s) − Y (t) ∼ N (μs, σ*^{2}*s).*

### 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 right with probability p*
*after Δt time.*

*• It moves Δx to the left with probability 1 − p.*

*• Deﬁne*

*X** _{i}* =

^{Δ}

⎧⎨

⎩

+1 *if the ith move is to the right,*

*−1 if the ith move is to the left.*

**– X*** _{i}* are independent with

Brownian Motion as Limit of Random Walk (continued)

*• Assume n* *= t/Δt is an integer.*^{Δ}

*• Its position at time t is*

*Y (t)* *= Δx (X*^{Δ} _{1} *+ X*_{2} + *· · · + X*_{n}*) .*

*• Recall*

*E[ X** _{i}* ] =

*2p − 1,*

*Var[ X** _{i}* ] = 1

*− (2p − 1)*

^{2}

*.*

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*^{Δ} *√*

*Δt and p* *= [ 1 + (μ/σ)*^{Δ} *√*

*Δt ]/2,*^{a}
*E[ Y (t) ] = nσ√*

*Δt (μ/σ)√*

*Δt = μt,*
*Var[ Y (t) ] = nσ*^{2}*Δt*

1 *− (μ/σ)*^{2}*Δt*

*→ σ*^{2}*t,*
*as Δt → 0.*

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.*

*• Similarity to the the BOPM: The p is identical to the*
*probability in Eq. (41) on p. 285 and Δx = ln u.*

*• Note that*

*Var[ Y (t + Δt) − Y (t) ]*

*=Var[ Δx X*_{n+1}*] = (Δx)*^{2} *× Var[ X** _{n+1}* ]

*→ σ*

^{2}

*Δt.*

### Geometric Brownian Motion

*• Let X* =^{Δ} *{ X(t), t ≥ 0 } be a Brownian motion process.*

*• The process*

*{ Y (t)* *= e*^{Δ} ^{X(t)}*, t ≥ 0 },*
is called geometric Brownian motion.

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

*• X(t) ∼ N(μt, σ*^{2}*t) with moment generating function*
*E*

*e*^{sX(t)}

*= E [ Y (t)*^{s}*] = e*^{μts+(σ}^{2}^{ts}^{2}^{/2)}

### Geometric Brownian Motion (concluded)

*• In particular,*^{a}

*E[ Y (t) ] = e*^{μt+(σ}^{2}^{t/2)}*,*
*Var[ Y (t) ] = E*

*Y (t)*^{2}

*− E[ Y (t) ]*^{2}

*= e*^{2μt+σ}^{2}^{t}

*e*^{σ}^{2}^{t}*− 1*
*.*

aRecall Eqs. (29) on p. 176.

0.2 0.4 0.6 0.8 1 Time (t) -1

1 2 3 4 5 6 Y(t)

### A Case for Long-Term Investment

^{a}

*• Suppose the stock follows the geometric Brownian*
motion

*S(t) = S(0) e*^{N (μt,σ}^{2}^{t)}*= S(0) e*^{tN (μ,σ}^{2}^{/t )}*,* *t ≥ 0,*
*where μ > 0.*

*• The annual rate of return has a normal distribution:*

*N*

*μ,* *σ*^{2}
*t*

*.*

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

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

a

*Continuous-Time Financial Mathematics*

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*

### Stochastic Integrals

*• Use W* =^{Δ} *{ W (t), t ≥ 0 } to denote the Wiener process.*

*• The goal is to develop integrals of X from a class of*
stochastic processes,^{a}

I_{t}*(X)* =^{Δ}

_{t}

0 *X dW, t ≥ 0.*

*• I*_{t}*(X) is a random variable called the stochastic integral*
*of X with respect to W .*

*• The stochastic process { I*_{t}*(X), t ≥ 0 } will be denoted*
by

*X dW .*

### Stochastic Integrals (concluded)

*• Typical requirements for X in ﬁnancial applications are:*

**– Prob[** _{t}

0 *X*^{2}*(s) ds < ∞ ] = 1 for all t ≥ 0 or the*
stronger _{t}

0 *E[ X*^{2}*(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.**

### 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 = t*_{0} *< t*_{1} *< · · ·*

such that

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

J_{0} J_{1} J_{2} J_{3} J_{4} J_{5}
: J

## = B

J

### Ito Integral (continued)

*• The Ito integral of a simple process is deﬁned as*

I_{t}*(X)* =^{Δ}

*n−1*

*k=0*

*X(t*_{k}*)[ W (t** _{k+1}*)

*− W (t*

_{k}*) ],*(74)

*where t*

_{n}*= t.*

**– The integrand X is evaluated at t**_{k}*, not t** _{k+1}*.

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

### Ito Integral (continued)

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

*• Then there exists a random variable I*_{t}*(X), unique*
almost certainly, such that I_{t}*(X** _{n}*) converges in
probability to I

_{t}*(X) for each sequence of simple*

*stochastic processes X*1*, X*2*, . . . such that X** _{n}* converges

*in probability to X.*

*• If X is continuous with probability one, then I*_{t}*(X** _{n}*)
converges in probability to I

_{t}*(X) as*

*δ** _{n}* = max

^{Δ}

*1≤k≤n**(t*_{k}*− t**k−1*)

### 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}

**Theorem 17 The Ito integral**

*X dW is a martingale.*

*• A corollary is the mean value formula*

*E*

_{b}

*a*

*X dW*

*= 0.*

### Discrete Approximation

*• Recall Eq. (74) on p. 575.*

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

0 *X dW ,*

*X(s)* *= X(t*^{Δ} _{k−1}*) for s ∈ [ t*_{k−1}*, t*_{k}*), 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 .*

### Discrete Approximation (concluded)

*• Suppose, unlike Eq. (74) on p. 575, we deﬁned the*
stochastic integral from

*n−1*

*k=0*

*X(t*_{k+1}*)[ W (t** _{k+1}*)

*− W (t*

_{k}*) ].*

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

*Y (s)* *= X(t*^{Δ} _{k}*) for s ∈ [ t*_{k−1}*, t*_{k}*), k = 1, 2, . . . , n.*

*• This clearly anticipates the future evolution of X.*^{a}

:

J

:

J

;

(a) (b)

:

### Ito Process

*• The stochastic process X = { X*_{t}*, t ≥ 0 } that solves*
*X*_{t}*= X*_{0} +

_{t}

0 *a(X*_{s}*, s) ds +*

_{t}

0 *b(X*_{s}*, s) dW*_{s}*, t ≥ 0*
is called an Ito process.

* – X*0 is a scalar starting point.

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

**– a(X**_{t}*, t): the drift.*

### Ito Process (continued)

*• A shorthand*^{a} is the following stochastic diﬀerential
equation^{b} *(SDE) for the Ito diﬀerential dX** _{t}*,

*dX*_{t}*= a(X*_{t}*, t) dt + b(X*_{t}*, t) dW*_{t}*.* (75)
**– Or simply**

*dX*_{t}*= a*_{t}*dt + b*_{t}*dW*_{t}*.*

**– This is Brownian motion with an instantaneous drift***a*_{t}*and an instantaneous variance b*^{2}* _{t}*.

*• X is a martingale if a** _{t}* = 0.

^{c}

aPaul Langevin (1872–1946) in 1904.

b *X*

### Ito Process (concluded)

*• From calculus, we would expect* _{t}

0 *W dW = W (t)*^{2}*/2.*

*• But W (t)*^{2}*/2 is not a martingale, hence wrong!*

*• The correct answer is [ W (t)*^{2} *− t ]/2.*

*• An equivalent form of Eq. (75) is*
*dX*_{t}*= a*_{t}*dt + b*_{t}*√*

*dt ξ,* (76)

*where ξ ∼ N (0, 1).*

### Euler Approximation

*• Deﬁne t*_{n}*= nΔt.*^{Δ}

*• The following approximation follows from Eq. (76),*

*X(t* *n+1*)

= *X(t**n*) +*a( **X(t**n*)*, t**n*) Δ*t + b( **X(t**n*)*, t**n*) Δ*W (t**n*)*.* (77)

*• It is called the Euler or Euler-Maruyama method.*

*• Recall that ΔW (t** _{n}*) should be interpreted as

*W (t*

*)*

_{n+1}*− W (t*

*n*

*),*

*not W (t* ) *− W (t* )!

### Euler Approximation (concluded)

*• With the Euler method, one can obtain a sample path*
*X(t* 1*), X(t*2*), X(t*3*), . . .*

from a sample path

*W (t*_{0}*), W (t*_{1}*), W (t*_{2}*), . . . .*

*• Under mild conditions, X(t*_{n}*) converges to X(t** _{n}*).

### More Discrete Approximations

*• Under fairly loose regularity conditions, Eq. (77) on*
p. 584 can be replaced by

*X(t* * _{n+1}*)

= *X(t*_{n}*) + a( X(t*_{n}*), t*_{n}*) Δt + b( X(t*_{n}*), t** _{n}*)

*√*

*Δt Y (t*_{n}*).*

* – Y (t*0

*), Y (t*

_{1}

*), . . . are independent and identically*distributed with zero mean and unit variance.

### More Discrete Approximations (concluded)

*• An even simpler discrete approximation scheme:*

*X(t* * _{n+1}*)

= *X(t*_{n}*) + a( X(t*_{n}*), t*_{n}*) Δt + b( X(t*_{n}*), t** _{n}*)

*√*

*Δt ξ.*

**– Prob[ ξ = 1 ] = Prob[ ξ = −1 ] = 1/2.**

**– Note that E[ ξ ] = 0 and Var[ ξ ] = 1.**

*• This is a binomial model.*

*• As Δt goes to zero, X converges to X.*^{a}