For No Arbitrage Opportunities to Exist, Investment Strategy Must be a Martingale. (Didactic 2)

We postulate the existence of a stochastic discount factor (SDF) that prices all assets in this economy and denote it by M. This process is also termed the pricing kernel, the pricing operator, or the state price density. It is well known that assuming that no dynamic arbitrage trading strategies can be implemented by trading in the financial securities issued in the economy is roughly equivalent to the existence of a strictly positive SDF. For no arbitrage opportunities to exist, the product of M with the value process of any investment strategy must be a martingale. Technically, recall that a martingale is a family of random variables ξ(t) such that the mathematical expectation of the increment ξ(t₂)−ξ(t₁) (for arbitrary t₁ < t₂), conditioned on the past values ξ(s) (s ≤ t₁), is zero. The drift of a martingale is thus zero. This is different from the Markov process, which is better known in the physical community, defined by the independence of the next increment on past values.

Under an adequate definition of the space of admissible trading strategies, the product MV is a martingale, where V is the value process of any admissible self-financing trading strategy implemented by trading on financial securities. Then,

V(t) = E_tV(s) M(s)/M(t) —– (1)

where s is a future date and E_t[x] denotes the mathematical expectation of x taken at time t. In particular, we require that a bank account and zero-coupon discount bonds of all maturities satisfy this condition.

A security is referred to as a (floating-rate) bank account, if it is “locally riskless”. Thus, the value at time t, of an initial investment of B(0) units in the bank account that is continuously reinvested, is given by the following process

B(t) = B(0) exp{∫₀^t r(s)ds} —– (2)

where r(t) is the instantaneous nominal interest rate.

We further assume that, at any time t, riskless discount bonds of all maturity dates trade in this economy and let P (t,s) denote the time t price of the s maturity bond. We require that P (s,s) = 1, that P (t,s) > 0 and that ∂P(t,s)/∂s exists.

Instantaneous forward rates at time t for all times-to-maturity x > 0, f(t,x), are defined by

f(t,x)=−∂logP(t,t+x)/∂x —– (3)

which is the rate that can be contracted at time t for instantaneously riskless borrowing or lending at time t + x. We require that the initial forward curve f (0,x), for all x, be continuous.

Equivalently, from the knowledge of the instantaneous forward rates for all times- to-maturity between 0 and time s − t, the price at time t of a bond with maturity s

can be obtained by

P(t,s) = exp {−∫₀^s−t f(t,x)dx} —– (4)

Forward rates thus fully represent the information in the prices of all zero-coupon bonds. The spot interest rate at time t, r(t), is the instantaneous forward rate at time t with time-to-maturity 0,

r(t) = f(t,0) —– (5)

For convenience, we model the dynamics of forward rates. Clearly, we could as well model the dynamics of bond prices directly, or even the dynamics of the yields to maturity of the zero-coupon bonds. We use forward rates with fixed time-to-maturity rather than fixed maturity date. The model of HJM starts from processes for forward rates with a fixed maturity date. This is different from what we do. If we use a “hat” to denote the forward rates modeled by HJM,

f^ˆ(t,s) = f (t,s−t) —– (6)

or, equivalently,

f^ˆ(t,x) = f(t,t+x) —– (7)

for fixed s. Brace and Musiela define forward rates in the same fashion. Miltersen, Sandmann and Sondermann, and Brace, Gatarek and Musiela use definitions of forward rates similarly, albeit for non-instantaneous forward rates. Modelling forward rates with fixed time-to-maturity is more natural for thinking of the dynamics of the entire forward curve as the shape of a string evolving in time. In contrast, in HJM, forward rate processes disappear as time reaches their maturities. Note, however, that we still impose the martingale condition on bonds with fixed maturity date, since these are the financial instruments that are actually traded.