Forward Price

Forward Price

• The payoff of a forward contract at maturity is S_T − X.

• Forward contracts do not involve any initial cash flow.

• The forward price is the delivery price which makes the forward contract zero valued.

– That is, f = 0 when F = X.

Forward Price (concluded)

• The delivery price cannot change because it is written in the contract.

• But the forward price may change after the contract comes into existence.

– The value of a forward contract, f , is 0 at the outset.

– It will fluctuate with the spot price thereafter.

– This value is enhanced when the spot price climbs and depressed when the spot price declines.

• The forward price also varies with the maturity of the contract.

Forward Price: Underlying Pays No Income

Lemma 9 For a forward contract on an underlying asset providing no income,

F = Se^rτ. (32)

• If F > Se^rτ:

– Borrow S dollars for τ years.

– Buy the underlying asset.

– Short the forward contract with delivery price F .

Proof (concluded)

• At maturity:

– Deliver the asset for F .

– Use Se^rτ to repay the loan, leaving an arbitrage profit of F − Se^rτ > 0.

• If F < Se^rτ, do the opposite.

Example

• r is the annualized 3-month riskless interest rate.

• S is the spot price of the 6-month zero-coupon bond.

• A new 3-month forward contract on a 6-month

zero-coupon bond should command a delivery price of Se^r/4.

• So if r = 6% and S = 970.87, then the delivery price is 970.87 × e^0.06/4 = 985.54.

Contract Value: The Underlying Pays No Income

The value of a forward contract is

f = S − Xe^−rτ.

• Consider a portfolio of one long forward contract, cash amount Xe^−rτ, and one short position in the underlying asset.

• The cash will grow to X at maturity, which can be used to take delivery of the forward contract.

• The delivered asset will then close out the short position.

• Since the value of the portfolio is zero at maturity, its

Forward Price: Underlying Pays Predictable Income

Lemma 10 For a forward contract on an underlying asset providing a predictable income with a PV of I,

F = (S − I) e^rτ. (33)

• If F > (S − I) e^rτ, borrow S dollars for τ years, buy the underlying asset, and short the forward contract with delivery price F .

• At maturity, the asset is delivered for F , and (S − I) e^rτ is used to repay the loan, leaving an arbitrage profit of F − (S − I) e^rτ > 0.

• If F < (S − I) e^rτ

Example

• Consider a 10-month forward contract on a $50 stock.

• The stock pays a dividend of $1 every 3 months.

• The forward price is (

50 − e^−r3/4 − e^−r6/2 − e^−3×r9/4)

e^r10×(10/12). – r_i is the annualized i-month interest rate.

Underlying Pays a Continuous Dividend Yield of q

The value of a forward contract at any time prior to T is f = Se^−qτ − Xe^−rτ. (34)

• Consider a portfolio of one long forward contract, cash amount Xe^−rτ, and a short position in e^−qτ units of the underlying asset.

• All dividends are paid for by shorting additional units of the underlying asset.

• The cash will grow to X at maturity.

• The short position will grow to exactly one unit of the

Underlying Pays a Continuous Dividend Yield (concluded)

• There is sufficient fund to take delivery of the forward contract.

• This offsets the short position.

• Since the value of the portfolio is zero at maturity, its PV must be zero.

• One consequence of Eq. (34) is that the forward price is

F = Se^{(r−q) τ}. (35)

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 CBT.

– One million U.S. dollars for the Eurodollar futures on the CME.

• 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.

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 a food processor 100,000 bushels of corn at $2.00 per bushel in November.

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

Daily Settlements (concluded)

• (continued)

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

– With marking to market, the farmer has transferred

$0.5 per bushel from his futures account to that of the food processor by November.

– 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 $2.00 per bushel, the original delivery price.

Delivery and Hedging

• 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 prices.

• This makes 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.

Daily Cash Flows

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

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

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

(F₁ − F⁰) + (F₂ − F¹) + · · · + (Fⁿ − Fⁿ−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 differ because of the reinvestment of daily cash flows and how S − F is distributed.

Forward and Futures Prices

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

– See text for proof.

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

aCox, Ingersoll, and Ross (1981).

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 position of the two parties will be at the prevailing futures price.

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.

– This futures contract has zero value.

• It works as if the put holder sold a futures contract for the strike price X minus the prevailing futures price 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.

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.

• If an offsetting position is then taken in the forward market, 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

futures option = max(F_t − X, 0), (37) forward option = max(F_t − X, 0) e^{−r(T −t)}. (38)

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 the options.

– Futures price equals spot price at maturity.

– This conclusion is independent of the model for the spot price.

Put-Call Parity

The put-call parity is slightly different from the one in Eq. (19) on p. 185.

Theorem 11 (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.

Early Exercise and Forward Options

The early exercise feature is not valuable.

Theorem 12 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 13 American futures options may be exercised optimally before expiration.

Black Model

• Formulas for European futures options:

C = F e^−rtN (x) − Xe^−rtN (x − σ√

t), (39) P = Xe^−rtN (−x + σ√

t) − F e^−rtN (−x), where x ≡ ln(F /X)+(σ²/2) t

σ√

t .

• Formulas (39) are related to those for options on a stock paying a continuous dividend yield.

• In fact, they are exactly Eqs. (25) on p. 272 with the dividend yield q set to the interest rate r and the stock price S replaced by the futures price F .

Black Model (concluded)

• This observation incidentally proves Theorem 13 (p. 407).

• 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)} based on Eqs. (37)–(38) on p. 404.

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. 384)

F = Se^rT.

– From Lemma 7 (p. 249), 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

Se^r∆te^r(T−∆t) = Se^rT = F.

Binomial Model for Forward and Futures Options (concluded)

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

p_f ≡ (1 − d)/(u − d) by Eq. (26) on p. 274.

– The futures price moves from F to F u with

probability p_f and to F d with probability 1 − p^f.

• The binomial tree algorithm for forward options is identical except that Eq. (38) on p. 404 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.

• The stock price moves from S = F e^−rT to F ue^{−r(T −∆t)} = Sue^r∆t with probability p_f per period.

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

with probability 1 − p^f per period.

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.

– Calculate S from F at each node via S = F e−(r−q)(T −t).

• Of course, this model may not be suitable for pricing

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).

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.

– That is, 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 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 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 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 the yen market.

Example (concluded)

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

– A effectively borrows yen at 9.5%.

– B borrows dollars at 11.5%.

– 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. 421 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 two-year swap on p. 421 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. (34) on p. 390, the forward contract maturing i years from now has a dollar value of

f_i ≡ (SYⁱ) e^−qi − Dⁱe^−ri. (40) – 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. 424.

• 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.

• A sample path defines an ordinary function of t.

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.

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 total 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.

• 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.

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. (41)

• In the discrete-time setting, a martingale means

E[ X_n+1 | X1, X₂, . . . , X_n ] = X_n. (42)

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

• Identity (42) 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. (42) on p. 435 to yield

E[ X_n ] = E[ X₁ ] (43) for all n.

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

Still a Martingale?

• Suppose we replace Eq. (42) on p. 435 with E[ X_n+1 | Xⁿ ] = X_n.

• It also says past history cannot affect the future.

• But is it equivalent to the original definition (42) on p. 435?^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, ξⁿ 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 | Xⁱ−1 ] =





X_i−1, if i is even, X_i−1, otherwise.

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

• But

E[ X_i | . . . , Xⁱ−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 { Zⁿ ≡

∑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 + E[ X ] = Z .

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 (41)–(42) on p. 435, the

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

– But it need not be so.

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 | Iⁿ ] = X_n, given the information sets { Iⁿ }.

Probability Measure (concluded)

• The above implies E[ X^n+m | Iⁿ ] = X_n for any m > 0 by Eq. (16) on p. 143.

– 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 { Zⁿ − nµ, n ≥ 1 }.

– Z_n ≡ ∑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 − nµ.

Example (concluded)

• Define

I_n ≡ { X¹, X₂, . . . , X_n }.

• Then

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

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 { C(i)

Rⁱ , i = 0, 1, . . . , n }

.

• Then, E

[ C(i + 1) Rⁱ⁺¹

 C(i) = C ]

= pC_u + (1 − p) C^d

Rⁱ⁺¹ = C

Rⁱ.

Martingale Pricing (continued)

• It is easy to show that E

[ C(k) R^k

 C(i) = C ]

= C

Rⁱ, i ≤ k. (44)

• 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 E_i^π

[ C(k) R^k

]

= C(i)

Rⁱ , i ≤ k. (45) – 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.

Martingale Pricing (continued)

• Equation (45) 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. (46)

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

investment of $1.

– So 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 (46) on p. 450 is the general formulation of risk-neutral valuation.

Martingale Pricing (concluded)

Theorem 14 A discrete-time model is arbitrage-free if and only if there exists a probability measure such that the

discount process is a martingale. This probability 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

p_fF u + (1 − p^f) F d = F

(1 − d

u − d u + u − 1 u − d d

)

= F (p. 410).

• 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

• The martingale pricing formula (46) on p. 450 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.

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.

• Assume

S₁

P₁ < S

P < S₂ P₂ 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)

• This yields

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 (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).

• The derivative’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.