### 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. 437)
*F = Se*^{rT}*.*

*– From Lemma 10 (p. 275), the expected value of S at*
*time ∆t in a risk-neutral economy is*

*Se*^{r∆t}*.*

*– So the expected futures price at time ∆t is*

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

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

– By Eq. (39) on p. 445, the futures price at time 0 is
*F = Se*^{(r−q) T}*.*

*– From Lemma 10 (p. 275), the expected value of S at*
*time ∆t in a risk-neutral economy is*

*Se*^{(r−q) ∆t}*.*

*– So the expected futures price at time ∆t 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 − d)/(u − d)*
by Eq. (30) on p. 302.

*– 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.

*• 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 −∆t)}*= Sue*

^{r∆t}### Spot and Futures Prices under BOPM (concluded)

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

*Sde*

^{r∆t}*with probability 1 − p*_{f} per period.

*• Note that*

*S(ue*^{r∆t}*)(de*^{r∆t}*) = Se*^{2r∆t}*6= S.*

*• So the binomial model 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 now*
solved:

– Suppose the stock pays a continuous dividend yield
*of q.*

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

*– By Eq. (39) on p. 445, calculate S from F at each*
node via

### 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,*
and B wants to take out a fixed-rate loan in dollars.

### 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} *− 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 as 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. 477 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. 477 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.*

### As a Package of Forward Contracts

*• From Eq. (38) on p. 445, the forward contract maturing*
*i years from now has a dollar value of*

*f*_{i}*≡ (SY*_{i}*) e*^{−qi}*− D*_{i}*e*^{−ri}*.* (43)
*– 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. 480.*

*• 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*_{1} *+ f*_{2} *+ f*_{3} *= 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.* (44)

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

*E[ X*_{n+1}*| X*_{1}*, X*_{2}*, . . . , X*_{n}*] = X*_{n}*.* (45)

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

*nth gamble.*

*• Identity (45) 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.

### Martingales (concluded)

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

*• Apply the law of iterated conditional expectations to*
both sides of Eq. (45) on p. 491 to yield

*E[ X*_{n}*] = E[ X*_{1} ] (46)
*for all n.*

*• Similarly,*

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

### Still a Martingale?

*• Suppose we replace Eq. (45) on p. 491 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 (45) on*
p. 491?^{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}*≡*

X*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.

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

– In the characterizations (44)–(45) on p. 491, 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. (19) on p. 152.*

*– 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}*≡* P_{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}*≡ { X*_{1}*, X*_{2}*, . . . , X*_{n}*}.*

*• Then*

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

### Martingale Pricing

*• Recall that the price of a European option is the*
expected discounted future payoff at expiration in a
risk-neutral economy.

*• 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*

½ *C(i)*

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

¾
*.*

*• Then,*
*E*

· *C(i + 1)*
*R*^{i+1}

¯¯

¯*¯ C(i) = C*

¸

= *pC*_{u}*+ (1 − p) C*_{d}

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

*C*

*R*^{i}*.*

### Martingale Pricing (continued)

*• It is easy to show that*
*E*

· *C(k)*
*R*^{k}

¯¯

¯*¯ C(i) = C*

¸

= *C*

*R*^{i}*, i ≤ k.* (47)

*• 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- sity of Wisconsin-Madison) and Mr. Hsiao, Huan-Wen (B90902081) on

### 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.* (48)
*– 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 (48) 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.* (49)

*– 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.*

*• Identity (49) on p. 506 is the general formulation of*
risk-neutral valuation.

### 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.

### Futures Price under the BOPM

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

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

µ*1 − d*

*u − d* *u +* *u − 1*
*u − d* *d*

¶

*= F*
(p. 464).

*• Can be generalized to*

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

### Martingale Pricing and Numeraire

^{a}

*• The martingale pricing formula (49) on p. 506 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).

### 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}

a

### 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}
to rule out arbitrage opportunities.

### 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)

*• This yields*

*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 (concluded)

*• It is easy to verify that*
*C*

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

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

*p ≡* *(S/P ) − (S*_{2}*/P*_{2})
*(S*_{1}*/P*_{1}*) − (S*_{2}*/P*_{2})*.*

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

### 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 6= 0 are real numbers.*

aRobert Brown (1773–1858).

### 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.*

aNorbert Wiener (1894–1964).

### 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).* (50)

*• Note that*

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