10.3 Dupire’s formula
10.3.2 Link between local and implied volatility
Dupire’s formula (10.11) can be rewritten in terms of market implied volatilities, observing that for every option,
C(T, K) = CBS(T, K, I(T, K)),
10.3. Dupire’s formula 153
Figure 10.3: Examples of local volatility surface. Left: artificial data; the im-plied volatility is of the form I(K) = 0.15 ×100K for all maturities (S0 = 100).
Right: local volatility computed from S&P 500 option prices with spline inter-polation.
where CBS(T, K, σ) denotes the Black-Scholes call price with volatility σ and I(T, K) is the implied volatility for strike K and maturity date T .
Substituting this expression into Dupire’s formula, we get
σ2(T, K) = 2
Suppose first that the implied volatility does not depend on the strike (no smile). In this case, the local volatility is also independent from the strike and equation (10.14) is reduced to
σ2(T ) = I2(T ) + 2I(T )T ∂I
The implied volatility is thus equal to the root of mean squared local volatility over the lifetime of the option.
To continue the study of equation (10.14), let us make a change of variable to switch from the strike K to the log-moneyness variable x = log(S/ ˜K), with I(T, K) = J (T, x). The equation (10.14) becomes
2J T∂J
∂T + J2− σ2
1 − x
J
∂J
∂x
2
− σ2J T∂2J
∂x2 +1
4σ2J2T2 ∂J
∂x
2
= 0.
Assuming that I and its derivatives remain bounded when T → 0, we obtain by sending T to 0:
J2(0, x) = σ2(0, x)
1 − x
J
∂J
∂x
2 . This differential equation can be solved explicitly:
J (0, x) =
Z 1 0
dy σ(0, xy)
−1
. (10.15)
We have thus shown that, in the limit of very short time to maturity, the implied volatility is equal to the harmonic mean of local volatilities. This result was established by Berestycki and Busca [7]. When the local volatility σ(0, x) is differentiable at x = 0, equation (10.15) allows to prove that (the details are left to the reader)
∂J (0, 0)
∂x =1 2
∂σ(0, 0)
∂x .
The slope of the local volatility at the money is equal, for short maturities, to twice the slope of the implied volatility.
This asymptotic makes it clear that the local volatility model, although it allows to calibrate the prices of all options on a given date, does not reproduce the dynamic behavior of these prices well enough. Indeed, the market implied volatility systematically flattens out for long maturities (see Figure 10.1), which results in the flattening of the local volatility surface computed from Dupire’s formula. Assuming that the model is correct and that the local volatility surface remains constant over time, we therefore find that the ATM slope of the implied volatility for very short maturities should systematically decrease with time, a property which is not observed in the data. This implies that the local volatility surface cannot remain constant but must evolve with time: σ(T, K) = σt(T, K), an observation which leads to local stochastic volatility models.
Chapter 11
Gaussian interest rates models
In this chapter, we provide an introduction to the modelling of the term struc-ture of interest rates. We will develop a pricing theory for securities that depend on default-free interest rates or bond prices. The general approach will exploit the fact that bonds of many different maturities are driven by a few common factors. Therefore, in contrast with the previous theory developed for a finite securities markets, we will be in the context where the number of traded assets is larger (in fact infinite) than the number of sources of randomness.
The first models introduced in the literature stipulate some given dynamics of the instantaneous interest rate process under the risk neutral measure Q, which is assumed to exist. The prices of bonds of all maturities are then deduced by computing the expected values of the corresponding discounted payoff under Q. We shall provide a detailed analysis of the most representative of this class, namely the Vasicek model. An important limitation of this class of models is that the yield curve predicted by the model does not match the observed yield curve, i.e. the calibration to the spot yield curve is not possible.
The Heath-Jarrow-Morton approach (1992) solves this calibration problem by taking the spot yield curve as the initial condition for the dynamics of the en-tire yield curve. The dynamics of the yield curve is driven by a finite-dimensional Broanian motion. In order to exclude all possible arbitrage opportunities, we will assume the existence of a risk neutral probability measure, for all bonds with all maturities. In the present context of a large financial market, This condition leads to the so-called Heath-Jarrow-Morton restriction which states that the dynamics of the yield curve is defined by the volatility process of zero-coupon bonds together with a risk premia process which is common to all bonds with all maturities.
Finally, a complete specification of an interest rates model requires the spec-ification of the volatility of bonds. As in the context of finite securities markets,
155
this is achieved by a calibration technique to the options markets. We therefore provide an introduction to the main tools for the analysis of fixed income deriva-tives. An important concept is the notion of forward neutral measure, which turns the forward price processes with pre-specified maturity into martingales.
In the simplest models defined by deterministic volatilities of zero-coupon bonds, this allows to express the prices of European options on zero-coupon bonds in closed form by means of a Black-Scholes type of formula. The structure of im-plied volatilities extracted from these prices provides a powerfull tool for the calibration of the yield curve to spot interest rates and options.