Futures Options

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

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

• Formulas for European futures options:

C = F e^−rtN (x) − Xe^−rtN (x − σ√

t), (64) P = Xe^−rtN (−x + σ√

t) − F e^−rtN (−x), where x =^{Δ ln(F/X)+(σ2/2) 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 pf 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 − pf 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 < 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. (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 DA% YA% B DB% YB%

• 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 − 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

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 (YA + DB)%.

– 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.

– 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. 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^−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. (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_ie^−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.

• 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 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

• { 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 | X₁, X₂, . . . , 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[ X₁ ] (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 | 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 ]

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.

– 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

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.

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)

= 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

 C(i)

= C(i)

Rⁱ , 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)

Rⁱ , 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 pfF u + (1 − pf) 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

• 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 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 (72)

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 Eqs. (72) on p. 553, 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 . – Above,

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

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

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

2. for any 0 ≤ t₀ < t₁ < · · · < 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 σ²(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 σ².

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

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

• 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, σ²s).

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 + X₂ + · · · + 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) ] = nσ²Δt 

1 − (μ/σ)²Δt

→ σ²t, 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 ] → σ²Δt.

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, σ²t) with moment generating function E



e^sX(t)



= E [ Y (t)^s ] = e^{μts+(σ2ts2/2)}

Geometric Brownian Motion (concluded)

• In particular,^a

E[ Y (t) ] = e^μt+(σ2t/2), Var[ Y (t) ] = E 

Y (t)² 

− E[ Y (t) ]²

= e^2μt+σ2t



e^σ2t − 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

• Suppose the stock follows the geometric Brownian motion

S(t) = S(0) e^{N (μt,σ2t)} = S(0) e^{tN (μ,σ2/t )}, t ≥ 0, where μ > 0.

• The annual rate of return has a normal distribution:

N



μ, σ² t

 .

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

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

a

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

0 E[ X²(s) ] ds < ∞.

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

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

– The future cannot influence the present.

Ito Integral

• A theory of stochastic integration.

• As with calculus, it starts with step functions.

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

such that

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

J₀ J₁ J₂ J₃ J₄ J₅ : 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 X1, X2, . . . 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 − tk−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 = X₀ +

 _t

0 a(X_s, s) ds +

 _t

0 b(X_s, s) dW_s, t ≥ 0 is called an Ito process.

– X0 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 b²_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(tn) +a( X(tn), tn) Δt + b( X(tn), tn) ΔW (tn). (77)

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

• Recall that ΔW (t_n) should be interpreted as W (t_n+1) − W (tn),

not W (t ) − W (t )!

Euler Approximation (concluded)

• With the Euler method, one can obtain a sample path X(t 1), X(t2), X(t3), . . .

from a sample path

W (t₀), W (t₁), W (t₂), . . . .

• 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 (t0), Y (t₁), . . . 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