Size of a Futures Contract

Futures Contracts vs. Forward Contracts

• They are traded on a central exchange.

• A clearinghouse.

– Credit risk is minimized.

• Futures contracts are standardized instruments.

• Gains and losses are marked to market daily.

– Adjusted at the end of each trading day based on the settlement price.

Size of a Futures Contract

• The amount of the underlying asset to be delivered under the contract.

– 5,000 bushels for the corn futures on the Chicago Board of Trade (CBOT).

– One million U.S. dollars for the Eurodollar futures on the Chicago Mercantile Exchange (CME).^a

• A position can be closed out (or offset) by entering into a reversing trade to the original one.

• Most futures contracts are closed out in this way rather than have the underlying asset delivered.

– Forward contracts are meant for delivery.

aCME and CBOT merged on July 12, 2007.

Daily Settlements

• Price changes in the futures contract are settled daily.

• Hence the spot price rather than the initial futures price is paid on the delivery date.

• Marking to market nullifies any financial incentive for not making delivery.

– A farmer enters into a forward contract to sell 100,000 bushels of corn at $2.00 per bushel in November.

– Suppose the price of corn rises to $2.5 by November.

– The farmer has incentive to sell his harvest in the spot market for $2.5.

Daily Settlements (concluded)

• (continued)

– With marking to market, the farmer has transferred

$0.5 per bushel from his futures account to that of the clearinghouse by November.^a

– When the farmer makes delivery, he is paid the spot price, $2.5 per bushel.

– The farmer has little incentive to default.

– The net price remains $2.5 − 0.5 = 2 per bushel, the original delivery price.

aSee p. 505.

Daily Cash Flows

• Let F_i denote the futures price at the end of day i.

• The contract’s cash flow on day i is F_i − F_i−1.

• The net cash flow over the life of the contract is

(F₁ − F0) + (F₂ − F1) + · · · + (F_n − F_n−1)

= F_n − F0 = S_T − F0.

• A futures contract has the same accumulated payoff S_T − F0 as a forward contract.

• The actual payoff may vary because of the reinvestment of daily cash flows and how S_T − F0 is distributed.

Daily Cash Flows (concluded)

6 -

? ?

6

0 1 2 3 · · · n

F1 − F0 F2 − F1 F3 − F2 · · · Fn − Fn−1

Delivery and Hedging

• Futures price is the delivery price that makes the futures contract zero-valued.

• Delivery ties the futures price to the spot price.

• On the delivery date, the settlement price of the futures contract is determined by the spot price.

• Hence, when the delivery period is reached, the futures price should be very close to the spot price.^a

• Changes in futures prices usually track those in spot price, making hedging possible.

aBut since early 2006, futures for corn, wheat, and soybeans occasion- ally expired at a price much higher than that day’s spot price (Henriques, 2008).

Forward and Futures Prices

• Surprisingly, futures price equals forward price if interest rates are nonstochastic!^a

• This result “justifies” treating a futures contract as if it were a forward contract, ignoring its marking-to-market feature.

• The West Texas Intermediate (WTI) futures price was negative on April 20, 2020!^b

aCox, Ingersoll, & Ross (1981); see p. 164 of the textbook for proof.

bApril 21 was the last trading day for oil delivery in May to Cushing, Oklahoma.

Remarks

• When interest rates are stochastic, forward and futures prices are no longer theoretically identical.

– Suppose interest rates are uncertain and futures prices move in the same direction as interest rates.

– Then futures prices will exceed forward prices.

• For short-term contracts, the differences tend to be small.

• Unless stated otherwise, assume forward and futures prices are identical.

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 positions of the two parties are 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

dollars.

Forward Options

• What is delivered is now 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 as there is no marking to market.

Example

• Consider an American 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.

aRecall p. 482.

bThe counterparty will pay you $110 for the underlying asset.

Some Pricing Relations

• Let T be the delivery date, the current time be 0, and the option^a 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,^b

futures option = max(F_t − X, 0), (66) forward option = max(F_t − X, 0) e^{−r(T −t)}. (67)

aOn the futures or forward contract

bRecall p. 511.

Some Pricing Relations (concluded)

• A European futures option is worth the same as a European option on the underlying asset if T = t.

– Futures price equals spot price at maturity.

• This conclusion is model independent.

Put-Call Parity

The put-call parity is slightly different from the one in Eq. (31) on p. 228.

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.

aSee Theorem 12.4.4 of the textbook for proof.

Early Exercise

The early exercise feature is not valuable for forward options.^a

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.

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.

aSee Theorem 12.4.5 of the textbook for proof.

Black’s Model

• Formulas for European futures options:

C = F e^−rtN (x) − Xe^−rtN (x − σ√

t ), (68) P = Xe^−rtN (−x + σ√

t ) − F e^−rtN (−x), (69) where x =^{Δ ln(F/X)+(σ2/2) t}

σ√

t .

• The above formulas are related to those for options on a stock paying a continuous dividend yield.

• They are Eqs. (44) on p. 333 with q set to r and S replaced by F .

aBlack (1976). It is also called the Black-76 model.

Black’s Model (concluded)

• Volatility σ is that of the stock price.^a

• This observation incidentally proves Theorem 16 (p. 517).

• 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)}.^b

aContributed by Mr. Lu, Yu-Ming (R06723032, D08922008) on April 7, 2021.

bRecall Eqs. (66)–(67) on p. 514.

Binomial Model for Forward and Futures Options

• In a risk-neutral economy, futures price behaves like a stock paying a continuous dividend yield of r.

– The futures price at time 0 is (p. 490) F = Se^rT.

– From Lemma 10 (p. 301), 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 Se^rΔte^{r(T −Δt)} = Se^rT = F.

Binomial Model for Forward and Futures Options (continued)

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

– By Eq. (64) on p. 500, the futures price at time 0 is F = Se^{(r−q) T}.

– From Lemma 10 (p. 301), 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) Δt}e(r−q)(T −Δt) = Se^{(r−q) T} = F.

aContributed by Mr. Liu, Yi-Wei (R02723084) on April 16, 2014.

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) (70) by Eq. (45) on p. 335.

– The futures price moves from F to F u with

probability p_f and to F d with probability 1 − p_f. – The original u and d are used here.

• The binomial tree algorithm for forward options is identical except that Eq. (67) on p. 514 is the payoff.

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 pf per period.

aRecall Lemma 12 (p. 490).

Spot and Futures Prices under BOPM (concluded)

• Similarly, the stock price moves from S = F e^−rT to Sde^rΔt

with probability 1 − pf per period.

• Note that

S(ue^rΔt)(de^rΔt) = Se^2rΔt = S.

• This binomial model for S differs from the CRR tree.

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

Negative Probabilities Revisited

• As 0 < pf < 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. (64) on p. 500, recover 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 some 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

aThanks to a lively discussion on April 16, 2014.

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.

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 − DB < Y_A − DA or Y_B − YA < D_B − DA.

• Consider this alternative arrangement:

– A borrows dollars.

– B borrows yen.

– They enter into a currency swap with a bank (the swap dealer) as the intermediary.

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 interest rate is originally (YA + D_B)%.

• The new arrangement has a smaller total rate of (D_A + Y_B)%.

• So the total gain is ((DB − DA) − (YB − YA))%.

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

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. 533 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.

– PD 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. 533 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. (63) on p. 500, the forward contract maturing i years from now has a dollar value of

f_i = (SY^Δ _i) e^−qi − D_ie^−ri. (71) – 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. 536.

• 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.

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

• { 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. (72)

• In the discrete-time setting, a martingale means

E[ X_n+1 | X₁, X₂, . . . , X_n ] = X_n. (73)

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

• Identity (73) then says the expected fortune after the (n + 1)st gamble equals the fortune after the nth gamble regardless of what may have occurred before.

aThe origin of the name is somewhat obscure.

Martingales (concluded)

• A martingale is therefore a notion of fair games.

• Apply the law of iterated conditional expectations to both sides of Eq. (73) on p. 547 to yield

E[ X_n ] = E[ X₁ ] (74) for all n.

• Similarly,

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

Still a Martingale?

• Suppose we replace Eq. (73) on p. 547 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 (73) on p. 547?^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.

– It is not a martingale by the original definition.

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 | Z1, Z2, . . . , Z_n ]

= E[ Z_n + X_n+1 | Z1, Z₂, . . . , Z_n ]

= E[ Z_n | Z1, Z₂, . . . , Z_n ] + E[ X_n+1 | Z1, Z₂, . . . , Z_n ]

= Z_n + E[ X_n+1 ] = Z_n.

Probability Measure

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

• 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 (72)–(73) on p. 547, the

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

– But it need not be so.

aOnly certain sets such as the Borel sets receive probabilities (Feller, 1972).

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. 169.

– 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.

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

• Now,

E[ Z_n+1 − (n + 1) μ | X₁, X₂, . . . , X_n ]

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

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

= Z_n + μ − (n + 1) μ

= Z_n − nμ.

Example (concluded)

• Define

I_n =^Δ { X1, X₂, . . . , 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.

aRecall Eq. (37) on p. 269.

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

 S(i)

= E[ C(i + 1) | S(i) ] Rⁱ⁺¹

= pC_u + (1 − p) C_d Rⁱ⁺¹

= C(i) Rⁱ .

Martingale Pricing (continued)

• It is easy to show that E

 C(k) R^k

 S(i)

= C(i)

Rⁱ , i ≤ k.

• This simplified 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^b (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 May 3, 2006.

bRecall they are called simple claims.

Martingale Pricing (continued)

• In general, the discount process is a martingale in that^a E_i^π

 C(k) R^k

= C(i)

Rⁱ , i ≤ k. (75) – 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^b martingale (probability) measure.

aIn this general formulation, Asian options do qualify.

bTwo probability measures are said to be equivalent if they assign nonzero probabilities to the same set of states.

Martingale Pricing (continued)

• Equation (75) 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. (76)

– 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 (76) on p. 563 is the general formulation of risk-neutral valuation.

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

Martingale Pricing (concluded)

Theorem 17 A discrete-time model is arbitrage-free if and only if there exists an equivalent probability measure^a such that the discount process is a martingale.

aCalled the risk-neutral probability measure.

Futures Price under the BOPM

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

– The expected futures price in the next period is p_fF 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

aRecall Eq. (70) on p. 522.

bSee Exercise 13.2.11 of the textbook.

Futures Price under the BOPM (concluded)

• Futures prices do not form a martingale under the risk-neutral probability p = (R − d)/(u − d).^a

– The expected futures price in the next period equals F u R − d

u − d + F d u − R u − d

= F uR − ud

u − d + F ud − Rd u − d

= F R.

aRecall Eq. (34) on p. 255.

Martingale Pricing and Numeraire

• The martingale pricing formula (76) on p. 563 uses the money market account as numeraire.

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

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

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

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

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 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 (77)

to rule out arbitrage opportunities.

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.

Example (continued)

• By inequalities (77) on p. 570, 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.

Example (continued)

• It is easy to verify that C

P = p C₁

P1 + (1 − p) C₂ P2

with

p =^Δ (S/P ) − (S₂/P₂) (S₁/P₁) − (S2/P₂).

• By inequalities (77) on p. 570, 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 (39) on p. 273, S 

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

– Its probability measure pu/R differs from p.

• SN(x) for the call and SN(−x) for the put in the Black-Scholes formulas (p. 303) use stock as the numeraire as well.^a

aSee Exercise 13.2.12 of the textbook.