Vanilla Models With Local Volatility

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

As we saw in post 6 of this series, European swaptions can be valued as European options on a forward swap rate. Consequently, a full term structure model specifying the dynamics of the yield curve as a whole is unnecessary for European swaption valuation. Instead, we simply need a terminal distribution for a single swap rate (or a model for said rate's evolution). Models of this nature are known as vanilla models, to distinguish them from full term structure models. If multiple rates are involved in the pricing of a European option, a vanilla model can be extended by copula methods to price such an option through the joint distribution of each rate involved.

In this post, we review one-factor diffusion models, where our ability to alter the terminal distribution of the underlying swap rate stems from a single place: a swap rate dependent diffusion term. Models of this nature are known as local volatility function (LVF) models. Particularly tractable examples of such models include: the constant elasticity of variance (CEV) model, the displaced diffusion model and quadratic models. At first, we will review these types of models, before reviewing efficient numerical methods for European options within the more general LVF model class.

The General Framework

Model dynamics

Denote the swap rate by $S(t)$ (note that the forward Libor rate $L(t, S, T)$ is simply a special case of a swap rate with underlying swap paying on just one date, namely the maturity date $T$). Let $W(t)$ be a one-dimensional Brownian motion under the measure $\mathbb{S}$ such that $S(\cdot)$ is a martingale. From now on, we will write $\mathbb{E}_t[\cdot] := \mathbb{E}^\mathbb{S}_t\left[ \cdot \right]$ to denote the expectation under the measure $\mathbb{S}$. We assume that $S(t)$ follows the one-dimensional SDE

where $\lambda > 0$ is constant and $\varphi(\cdot)$ is a real-valued function of the swap rate $S(t)$ satisfying some regularity conditions. In most cases, we would want $S(t) \ge 0$, which is easily seen to impose the condition

If this were not the case, there would always be a non-zero probability that $S(t)$ would go negative on the interval where $\varphi$ was non-zero. In some cases, however, we will knowingly violate condition (2) if it makes the model more tractable.

Volatility smile and implied density

The function $\varphi$ exists in order to match the distribution of $S(\cdot)$ to that observed through options observed in the market. Specifically, let $c \equiv c(t, S; T, K)$ represent the time-$t$ price of a $T$-maturity European call option (payer swaption) with strike $K$, where $S \equiv S(t)$. $c$ has time-$t$ price given by the expectation

The time-$t$ probability density function of $S(T)$ can be derived from the time-$t$ observed values of $c$ via

This result allows us to construct the time-$t$ marginal density of $S(T)$ from prices of $T$-maturity call options for a continuum of strikes $K$. In options markets, it is more common to express the strike dependency of option prices in terms of the implied volatility of the option. Specifically, for a given option price $c$ observed in the market, with strike $K$ and maturity $T$, we define the time-$t$ Black implied volatility function $\sigma_B(t,S;T,K)$ as the solution to

Notice that the above formula is the price of an option under the Black model, with constant volatility given by $\sigma_B(t, S; K, T)$.

At any given snapshot in time, both $t$ and $S(t)$ are fixed and we observe option prices on a strike-maturity grid. Hence, we consider $\sigma_B(t,S;K,T)$ a mapping from $K \mapsto \sigma_B(t, S; T, K)$ for fixed maturity $T$ and refer to this function as the $T$-maturity volatility smile. In most fixed income markets, this smile is a downwards sloping function of $K$, although it is not common for $\sigma_B$ to eventually increase for sufficiently large values of $K$.

Choosing $\varphi$

If we were to allow $\varphi$ to depend on time, a key result by Dupire shows that any arbitrage-free terminal distribution of $S(T)$ can be realised through a suitable choice of $\varphi \equiv \varphi(t, S)$, $t \in [0, T]$. Indeed, we do so by finding $\phi(S, K)$ through the observed volatility surface $\sigma_B(0, S(0); S, K)$, $(S, K) \in [0, T] \times [0, \infty)$.

Indeed, writing

we find that

However, unless the resulting $\varphi$ is monotonic in $S$, the resulting model will imply non-stationary volatility smile behaviour, contrary to empirical results. Consequently, non-monotonic specifications of $\varphi$ should be approached with caution.

Hence, the basic model (1) is best used in markets where the volatility smile is close to a montonic function of $K$. A classic choice for $\varphi$ that satisfies this condition is the constant elasticity of variance (CEV) specification

for some constant $p$.

The CEV model

The CEV model is given by the SDE

where $p > 0$ is a constant. There are a few key points regarding the CEV specification (3):

1. All solutions to (3) are non-explosive,
2. For $p \ge \frac{1}{2}$, (3) has a unique solution,
3. For $0 < p < 1$, $S = 0$ is an attainable boundary for (3), in the sense that $S(t)$ will reach this boundary with non-zero probability in finite time. For $p \ge 1$, $S=0$ is an unattainable boundary for (3),
4. For $0 < p \le 1$, $S(t)$ is a martingale. For $p > 1$, $S(t)$ is a supermartingale.

Through point 2, we know that solutions to (3) are unique for $p \ge \frac{1}{2}$. Hence, for $\frac{1}{2} \le p < 1$, if the solution ever reaches $S = 0$, then it will stay there, i.e. it will be absorbed. However, for $0 < p < \frac{1}{2}$, solutions are not unique, and there will be solutions that stay at 0 if they reach the origin, and others that jump out of it. Hence, it is important to supply a boundary condition at $S = 0$. In practice, we set $S = 0$ to be an absorbing barrier, in the sense that if $S(\tau) = 0$, then $S(u) = 0$ for all $u \ge \tau$, where $\tau$ is the first time at which $S(t) = 0$. This is the only arbitrage-free boundary condition.

Call option pricing

Pricing European call options under the CEV model requires evaluation of the expectation

for $S(\cdot)$ following (3). As a tedious exercise in integration, after applying a suitable transform to $S(\cdot)$ and inverting said transform, we find that, with $\mathcal{X}_\nu^2(\gamma)$ representing a non-central chi-squared distributed variable with $\nu$ degrees of freedom and non-centrality parameter $\gamma$, and defining the following variables:

for $0 < p < 1$, $K > 0$ and the absorbing boundary condition, we find that

To see this model in action, we set $S(0) = 6\%$ and $\lambda = 10\%$, fix $T = 10$ years and plot the Black implied volatilities implied volatilities generated by different values of $p$ as a function of moneyness $K/S(0)$.

Notice that, in the plot above, as $p$ approaches $1$, i.e. the CEV model approaches the Black model, the implied volatility approaches the flat implied volatility curve produced by the Black model with Black volatility equal to $\lambda$.