The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 626 Wednesday, February 4, 2004 1:33 PM

626 The Mathematics of Financial Modeling and Investment Management

• uis

Ru^11  Q ∫t()sd 

t =– ----------------log (Λ

u)= – ----------------log E

t t e 

(ut– ) (ut– )  

As noted above, this formula does not imply

u

u

• Q

(ut)Rt = Et ∫is()sd

t

Relationships for Bond and Option Valuation

We have established the formula

u –u

• (ut–)Rt = EQ ∫ ()sd
  t e
  e tis

in a rather intuitive way as the expectation under risk-neutral probabil-

ity of discounted final bond values. However, this formula can be

derived formally as a particular case of the general expression for the

price of a security that we determined in Chapter 15 on arbitrage pric-

ing in continuous time:

T T T

• r ud –ru ud
  St = Et e

Q ∫t u

ST + ∫e ∫t dDs

t

considering that, for zero-coupon bonds, the payoff rate is zero and that

we assume ST = 1.

We used risk-neutral probabilities for the following reason. The factor

uis()sd

e ∫t

represents capital appreciation pathwise. However, the formula

u

u –∫t is()sd

Λt = e

which gives the price at time t of a bond of face value 1 maturing at u in

a deterministic environment, does not hold pathwise in a stochastic