Fixed Income Probability Measures

Tutorial Contents

1. Fixed Income Notation
2. Fixed Income Probability Measures
3. Multi-Currency Markets
4. The HJM Framework
5. Basic Fixed Income Instruments
6. Caps, Floors and Swaptions
7. CMS Swaps, Caps and Floors
8. Bermudan Swaptions
9. Structured Notes and Exotic Swaps
10. Callable Libor Exotics and TARNs
11. Volatility Derivatives
12. Yield Curves and Curve Construction
13. Managing Yield Curve Risk
14. Vanilla Models With Local Volatility
15. Vanilla Models With Stochastic Volatility
16. Introduction to Term Structure Models
17. One-Factor Short Rate Models I
18. One-Factor Short Rate Models II
19. Multi-Factor Short Rate Models
20. The Libor Market Model I
21. The Libor Market Model II
22. Risk Management and Sensitivity Computations
23. P&L Attribution
24. Payoff Smoothing
25. Vegas in the Libor Market Model

Crucial to the modeling of interest rate derivatives is the risk-neutral pricing framework. This framework is expressed in the language of stochastic processes and two particularly important concepts are that of an equivalent martingale measure and a numeraire. In this post, we will introduce the theory behind these concepts and then give some common and useful measures.

Risk-Neutral Pricing

In this section, we review the key results from stochastic calculus and modern asset pricing theory. The results we uncover will give us a framework for pricing assets in a way that is consistent with the idea of no-arbitrage, a key assumption to mathematical finance. The following is technical and it is not crucial that you understand the theory inside out. The only crucial result is that of the equivalent martingale measure and corresponding numeraire. Keep that in mind if you find this section too advanced.

Setup

Consider an economy with continuous and frictionless trading, taking place on a finite time horizon $[0,T]$. We assume the existence of $p$ traded, dividend-free assets, whose prices are characterized by the vector-valued stochastic process $X(t) = (X_1(t),\ldots, X_p(t))^\intercal$. Here, a stochastic process is simply a continuous set of vector-valued random variables, indexed by the time $t$.

To model uncertainty and the flow of information, we introduce a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Here, $\Omega$ is a sample space with elements $\omega$ representing states of the world (or outcomes) that may materialise. $\mathcal{F}$ represents a specific $\sigma$-algebra on $\Omega$ (we will later see which one), effectively a family of subsets of outcomes from $\Omega$ that satisfies certain conditions. $\mathbb{P}$ represents a probability measure on the measure space $(\Omega, \mathcal{F})$. The details of the above setup require an understanding of the field of measure theory. Again, it is not important you understand this thoroughly. Just know that for each $A \in \mathcal{F}$ we find that $A \subset \Omega$ and that $\mathbb{P} : A \in \mathcal{F} \rightarrow [0, 1]$, i.e. $\mathbb{P}$ is a function that takes sets of outcomes and maps them to a number between 0 and 1 in such a way that each of these mappings can be considered a probability and $\mathbb{P}$ can be considered a probability measure.

We model the flow of information through time via a filtration $\mathbb{F} = \{\mathcal{F}(t) : t \in [0,T]\}$, a family of sub-$\sigma$-algebras of $\mathcal{F}$ satisfying $\mathcal{F}(s) \subset \mathcal{F}(t) \subset \mathcal{F}$ whenever $s \le t$. The best way to think about $\mathcal{F}(t)$ is as the information available at time $t$. As time passes, more and more information is revealed, so that $\mathcal{F}(t)$ incorporates all of the previous information up to time $s \le t$, plus any additional information gained from $s$ to $t$.

We will assume that the process $X(t)$ is adapted to $\mathcal{F}(t)$ for all $t$. Practically speaking, this just means that $X(t)$ is fully observable at time $t$. Finally, we require that the filtration $\mathbb{F}$ satisifes the usual conditions, which I will leave to the interested reader to explore further.

Let $\mathbb{E}^\mathbb{P}[\cdot]$ represent the expectation operator for the measure $\mathbb{P}$. When conditioning on information up to time $t$, we will use the notation $\mathbb{E}_t^\mathbb{P}[\cdot] := \mathbb{E}^\mathbb{P}[\cdot \vert \mathcal{F}(t)]$.

The above setup is fairly flexible. For our purposes, we focus on the situation where information is completely governed by a $d$-dimensional vector-valued Brownian Motion $W(t) = (W_1(t),\ldots,W_d(t))^\intercal$, where $W_i$ is independent of $W_j$ for all $i \neq j$. Again, it is not important to understand the technical details of Brownian motion on any particularly deep level. For now it suffices to say: $W(0) = 0$, $W_i(t) - W_i(s) \sim \mathcal{N}(0, t-s)$ for $t \ge s$ and that each of these increments are independent of one another for non-overlapping times.

Above, we mentioned that $\mathcal{F}$ represents a specific $\sigma$-algebra on $\Omega$. In fact, it is the one generated by $W$, i.e., for all $t$ we have $\mathcal{F}(t) = \sigma \left(\{W(u), u \in [0,t] \} \right)$ and $\mathcal{F} = \mathcal{F}(T)$.

We will assume that $X(t)$ is described by the vector-valued Ito process

where $\mu(s,\omega) \in \mathbb{R}^p$ and $\sigma(t,\omega)\in \mathbb{R}^{p\times d}$ are both themselves $\mathbb{F}$ adapted processes.

An incredibly powerful and useful concept that we require for our framework is that of a martingale. Before giving the precise definition, I will note that the concept of a martingale is really a simple one. A process is a martingale if the best guess for where the process will be in the future is where it is today. In this way, it represents a fair game, in the sense that if the martingale's current value represented the cost of entering into a bet, and it's future value your payoff from the bet, then you would on average expect to come out flat from playing the game.

Definition (Martingale)

Let $Y(t)$ be an adapted vector-valued process with $\mathbb{E}^P\left[\left\vert Y(t) \right\vert\right] < \infty$ for all $t \in [0,T]$. We say that $Y(t)$ is a martingale under the measure $\mathbb{P}$ if, for all $0 \le s \le t \le T$,

The second important concept that we require is known as Ito's Lemma. Ito's lemma is essentially the stochastic version of the differential of a function. If we consider a simple scalar-valued function $f(x,y)$ then the standard differential of $f$ is clearly just given by

This follows as the first order limit to the Taylor expansion

where H.O.T. represents the higher-order terms in the expansion, $\Delta f = :f(x + \Delta x, y + \Delta y) - f(x,y)$ and $\Delta x, \Delta y > 0$.

This is all well and good while $x$ and $y$ are deterministic. If we were to assume that $x = t$ was our deterministic time variable, but $y = X(t)$ was some stochastic variable of the form (2.1), then we have a different story. The notion of a differential breaks, since our higher order terms are no longer well-behaved. In particular, we have $(\text{d}X(t))^2 = \sigma(t,\omega)^2 (\text{d}W(t))^2 + \text{H.O.T.}$, where it can be shown that $(\text{d}W(t))^2 = \text{d}t$ in quadratic mean. Thus, we see that in the first order limit, the stochastic case leads to differential

Ito's Lemma extends this result to scalar-valued functions of multiple variables.

Theorem (Ito's Lemma)

Let $f(t,x)$, $x = (x_1,\ldots,x_p)^\intercal$, denote a continuous function, let $f(t,x) \in \mathbb{R}$ for all $t \in [0,T]$, $x \in \mathbb{R}^p$, and let $f \in C^{1,2}([0,T]\times \mathbb{R}^p)$. Let $X(t)$ be given by (2. 1) and define a scalar process $Y(t) := f(t,X(t))$. Let $\nabla_x f$ represent the $p$-dimensional vector of partial derivatives of $f$ with respect to the non-time variables. Likewise, let $H_x f$ represent the Hessian of $f$ with respect to the same variables. Then $Y(t)$ is an Ito process with stochastic differential given by

Trading Gains and Arbitrage

Consider an investor engaging in a trading strategy involving the $p$ assets $X_1,\ldots,X_p$. Let the trading strategy be characterized by a predictable adapted process $\phi(t,\omega) = (\phi_1(t,\omega), \ldots, \phi_p(t,\omega))^\intercal$, with $\phi_i(t,\omega)$ denoting the holding at time $t$ in the $i$th asset $X_i$. The value $\pi(t)$ of the trading strategy at time $t$ is thus given by

Clearly, over a small time interval $[t,t+\delta]$, the gain from trading is given by $\phi(t) \cdot (X(t+\delta) - X(t))$, suggesting that

is an appropriate model for trading gains in an infinitesimal interval.

A trading strategy $\pi$ is said to be self-financing if, for any $t \in [0,T]$,

This relationship expresses the concept that changes in portfolio value are solely caused by trading gains or losses, with no funds being added or withdrawn.

Definition (Arbitrage)

An arbitrage opportunity is a self-financing strategy $\phi$ for which $\pi(0) = 0$ and, for some $t \in [0,T]$,

In words, an arbitrage opportunity is a strategy that creates something from nothing, a free lunch. In a well-functioning market, arbitrage opportunities cannot exist. If they did, they would surely be exploited until the market returned to equilibrium at a level that eliminated the opportunity. No arbitrage will form a key requirement for all of the markets that we will consider.

Equivalent Martingale Measures, Complete Markets and Arbitrage

Currently, the idea of an arbitrage opportunity is an abstract one that bears no relation to the rest of our setup. However, one of the key insights of modern asset pricing is that under certain conditions, we can encapsulate no-arbitrage via the concept of an equivalent martingale measures.

First, note that two probability measures $\mathbb{P}$ and $\hat{\mathbb{P}}$ on the same measure space $(\Omega, \mathcal{F})$ are equivalent if $\mathbb{P}(A) = 0 \iff \hat{\mathbb{P}}(A) = 0$ for all $A \in \mathcal{F}$. Shown via the Radon-Nikodymn Theorem, two equivalent measures are connected through a process $R$ with $\mathbb{E}^\mathbb{P}[R] = 1$ such that $\hat{\mathbb{P}}(A) = \mathbb{E}^\mathbb{P}[R 1_A]$ for all $A \in \mathcal{F}$, known as the Radon-Nikodym derivative. $R$ is often denoted by $\tfrac{\text{d}\hat{\mathbb{P}}}{\text{d}\mathbb{P}}$. Now, an equivalent martingale measure for the process $Z(t)$ is simply an equivalent measure $\hat{\mathbb{P}}$ such that under $\hat{\mathbb{P}}$, $Z(t)$ is a martingale.

With these concept in place, we now introduce the important concept of a numeraire.

Definition (Numeraire)

A numeraire is a traded asset (a linear combination of our $p$ assets) with strictly positive price that pays no dividends, used to normalise the asset process. Denoting the numeraire value at time $t$ by $N(t)$, the normalized asset process is given by $X^N(t) := \tfrac{X(t)}{N(t)}$.

We say that a measure $\mathbb{Q}^N$ is an equivalent martingale measure induced by $N$ if $X^N(t)$ is a martingale under $\mathbb{Q}^N$. We call a trading strategy $\phi$ permissible if

is a martingale under $\mathbb{Q}^N$.

The last concept that we introduce is that of a complete market. Roughly, a complete market is one where the assets $X(t)$ are the only ones that matter, in the sense that all other assets can be generated as a linear combination of the $X_i$s.

Definition (Complete Market)

A complete market is one in which all finite-variance $\mathcal{F}(T)$-measurable random variables $V(T)$ can be replicated, in the sense that there exists a permisible trading strategy $\phi$ such that $V(T) = \phi(T) \cdot X(T) = \pi(T)$.

Via the fundamental theorems of asset pricing, we are now in a position to link all of these concepts, in a way that gives us a practical way of pricing derivatives.

Theorem (The Fundamental Theorems of Asset Pricing)

Restricting attention to permissible trading strategies, the following two equivalencies hold:

1. Our economy is arbitrage-free if and only if for each numeraire $N$, there exists at least one equivalent martingale measure $\mathbb{Q}^N$.
2. An arbitrage-free market is complete if and only if for each numeraire $N$, the risk neutral measure $\mathbb{Q}^N$ is unique.

Using the above, if we assume that our market is both arbitrage-free and complete then for each numeraire $N$, there is a corresponding equivalent martingale measure $\mathbb{Q}^N$. Hence, we can unambigously talk about the numeraire, measure pair $(N,\mathbb{Q}^N)$. Under this pair, any $T$-maturity derivative security paying out the random payoff $V(T)$ at time $T$ can be expressed via

or

where $\mathbb{E}_t^N[\cdot] := \mathbb{E}^{\mathbb{Q}^N}[\cdot \vert \mathcal{F}(t)]$.

Clearly, since $N$ is arbitrary in the above, if have another numeraire pair $(M, \mathbb{Q}^M)$ then

If we consider the time $T$ payoff $V(T) = Y(T)M(T)$ then this shows that

By comparison, this shows that the Radon-Nikodym derivative for the measure change between measures $\mathbb{Q}^N$ and $\mathbb{Q}^M$ is characterised by the process

Girsanov's Theorem

We have established that there is a close link between the concept of arbitrage and the existence and uniqueness of an equivalent martingale measure. We saw that there is even an easy way to characterize the Radon-Nikodym derivative linking two equivalent martingale measures via their corresponding numeraires. From a practical perspective, this does not tell us a huge amount about how we convert our asset price dynamics when acting under different numeraires. However, in the setup that we have introduced above, there is a practical way of switching between measures.

Consider two measures, $\mathbb{P}$ and $\mathbb{P}(\theta)$ related by the density $\eta^\theta(t) = \mathbb{E}_t^{\mathbb{P}}\left[\tfrac{\text{d}\mathbb{P}(\theta)}{\text{d}\mathbb{P}} \right]$, where $\eta^\theta(t)$ is an exponential martingale given by the Ito process

The $d$-dimensional process $\theta$ is known as the market price of risk. Assuming that $\eta^\theta$ defines a martingale, we have the following important result:

Theorem (Girsanov's Theorem)

Suppose that $\eta^\theta(t)$ defined above is a martingale, then for all $t \in [0, T]$

is a Brownian motion under the measure $\mathbb{P}(\theta)$.

Fixed Income Probability Measures

As we have seen, selecting an equivalent martingale measure is largely a matter of choosing a numeraire. Here, we go through a number of important numeraires used in fixed income pricing. Throughout, we assume that our market precludes arbitrage and is complete, and we use $V(t)$ to denote the time $t$ price of a derivative security making an $\mathcal{F}(T)$-measurable payment of $V(T)$.

The Risk Neutral Measure

The most fundamental numeraire in the fixed-income world (and even more generally in derivatives pricing) is the continuously compounded money market account $\beta(t)$, satisfying the locally deterministic SDE

where $r(t)$ is the same short rate that we saw in Section 1. Solving this SDE shows that

We denote the equivalent martingale measure associated with $\beta$ by $\mathbb{Q}$. Under this measure, we find that

Utilising the explicit formula (2.3), we find that the above is equivalent to

As we saw in the Section 1, a ZCB pays out 1 unit at its maturity $T$. Hence, if we set $V(T) = 1$ then $V(t) = P(t,T)$ for $t \le T$ and we arrive at the bond pricing formula

This result is important because it shows that specification of the dynamics of the short-rate $r(t)$ under $\mathbb{Q}$ is enough to determine ZCB prices at all times and maturities. Models based on such a direct specification are known as short rate models. Note the similarity between (2.5) and (1.1). Clearly, if $r(t)$ is deterministic then the two expressions will agree and $r(u) = f(t,u)$for all $u \ge t$. However, if $r(t)$ is random, these two results diverge, and we find that $f(t,u) \neq \mathbb{E}_t^\mathbb{Q}[r(u)]$. However, if we were to replace $f(t,u)$ by the equivalent futures rate $q(t,u)$ then the above does hold. The exact result goes as follows.

Lemma

Assuming that mark-to-market takes place continuously, under regularity conditions on $r$, futures rate $F(\cdot, T, T+\tau)$ is a $\mathbb{Q}$-martingale, and

Proving this result is relatively simple. We exploit the fact that the mark-to-market process leads to daily cash proceeds proportional to $\text{d}F(t,T,T+\tau) = F(t+\text{d}t, T, T+\tau) - F(t,T,T+\tau)$. Deflating each of these back to the current time $t$ and noting that the futures value is always zero shows that holding the futures contract until an arbitrary time $s \in [t,T]$ leads to the expression $V(t) = 0 = \mathbb{E}_t^\mathbb{Q}\left[\int_t^s \beta(s)^{-1} \text{d}F(s,T,T+\tau) \right]$. Since this holds for arbitrary time horizons, we deduce that

which implies that $F$ is a $\mathbb{Q}$-martingale. The result follows immediately.

Taking the limit as $\tau \searrow 0$ on both sides of the above then gives our result for the instantaneous futures rate, i.e.

The T-Forward Measure

The $T$-forward measure $\mathbb{Q}^T$ uses the $T$-bond as numeraire. We will denote the conditional expectation operator under the $T$-forward measure by $\mathbb{E}_t^T[\cdot]$, so that

By comparing (2.4) and (2.6), it should be clear that switching from the risk-neutral to the $T$-forward measure decouples the expectation of the discount factor $D(t,T) := \tfrac{\beta(t)}{\beta(T)}$ from the expectation of the payoff $V(T)$. This is often convenient when pricing IR derivatives. From equation (2.2), we see that the change of measure process from the risk-neutral to the $T$-forward measure is given by density

Since the $(T+\tau)$-bond is another traded asset and has price $P(t,T+\tau)$, we find that by definition, the time $t$ forward bond price $P(t,T, T+\tau) = \tfrac{P(t,T+\tau)}{P(t,T)}$ is a martingale under the $T$-forward measure. This is, in fact, where the name $T$-forward measure comes from.

While the forward bond price $P(t, T, T+\tau)$ is a martingale under the $T$ forward measure, we find that the forward Libor rate $L(t,T,T+\tau)$ is a martingale under the $(T+\tau)$-forward measure $\mathbb{Q}^{T+\tau}$. This follows immediately from the relation

Using this fact, we find that

a result analogous to that of the futures rate result under the risk-neutral measure.

The Spot Measure

When working on a full tenor structure $0 = T_0 < \cdots < T_N$ of forward rates, it can be inconvenient to work with a particular forward measure $T_n$. Evolving through time, once $t > T_n$, our numeraire is invalidated and we must switch measure to one that is still live. Another way to avoid this issue is by introducing a numeraire that extends to arbitrary horizons by construction. While the money market account $\beta$ accomplishes this, it can be awkward to work with an inherently continuous construct when dealing with a discrete tenor structure. An alternative is the discrete-time equivalent to $\beta$.

At time $0$, we invest £$1$ in $\tfrac{1}{P(0,T_1)}$ $T_1$-bonds. Clearly, at $T_1$, this investment returns

Reinvesting (rolling) this amount at $T_1$ into $T_2$-bonds then returns

at $T_2$. While $t \in (T_1, T_2]$, this investment clearly has value $\frac{1}{P(0,T_1)}\cdot\frac{P(t,T_2)}{P(T_1,T_2)}$. By repeating this strategy at each date in the tenor structure, we arrive at an investment with time $t$ value $B(t)$ given by

where we have again used the shorthand $L_n(t) = L(t,T_n,T_{n+1})$.

Since we constructed $B(t)$ through a permissible trading strategy and $B(t) > 0$ for all $t$, $B(t)$ is a valid numeraire. The measure generated by $B(t)$ is called the spot (Libor) measure, denoted by $\mathbb{Q}^B$ with associated conditional expectation $\mathbb{E}_t^B[\cdot]$. Then

where

It is clear that the spot measure resembles the risk neutral measure, since the risk-neutral measure is in some sense its continuous counterpart. In fact, if the futures rate is marked-to-market discretely, on dates $T_0,\ldots,T_N$, it turns out to be a martingale under the spot measure.

Terminal and Hybrid Measures

The big advantage of the spot measure that we discussed was that it remains alive throughout the span of the tenor structure. This property is key for valuing securities which may mature at any date in the tenor structure (e.g. barriers or range accruals).

Clearly, picking the $T$-forward measure corresponding to the final maturity in the tenor structure, i.e. $T = T_N$ also satisfies this neat property. The measure induced by $P(t,T_N)$, $\mathbb{Q}^N$ is called the terminal measure. As usual,

One interpretation of the term $\tfrac{V(T)}{P(T,T_N)}$ is as the time $T_N$ proceeds of rolling the payoff $V(T)$ into the $T_N$-bond. Alternatively, we could roll $V(T)$ into the spot numeraire asset $B(T)$, leading to $T_N$ payoff $V(T) \cdot \tfrac{B(T_N)}{B(T)}$. This gives rise to the equivalent formula

Clearly, this idea generalises. For instance, we can define a numeraire such as

This asset corresponds to the strategy of initially investing into the $T$-bond and then rolling our proceeds into the spot measure at $T$. Letting $\hat{\mathbb{Q}}^T$ denote the corresponding measure with conditional expectation operator $\hat{\mathbb{E}}_t^T$, we can write

The associated measure to such numeraire is known as a hybrid measure. Clearly, many such hybrid measures exist and can be useful for different situations.

Swap Measures

Since the annuity factor $A_{k,m}(t)$ of Section 1 is just a linear combination of ZCBs, it qualifies as a numeraire asset. The associated measure $\mathbb{Q}^{k,m}$ is known as a swap measure or annuity measure. We have

where $\mathbb{E}_t^{k,m}[\cdot]$ denotes the expectation operator under $\mathbb{Q}^{k,m}$.

In the absence of arbitrage, the forward swap rate $S_{k,m}(t)$, which we have seen is defined by

is clearly a martingale under $\mathbb{Q}^{k,m}$, giving the measure its name.