5.4 Numerical scrutinies for modelling volatility
5.4.2 Consideration using implied normal-jump variation
This subsection shows a straightforward approach to model variation in the risk-free xed income market. As was noted in the introduction, the use of normal jump-diusion models is suitable in the market situation described in the previous subsection. Using these models, the combination of low rates and high volatility can be handled.
For the example of this section the data of March 31st, 2015 has been used. The swaption prices are quoted for ve dierent swap tenors (2, 5, 10,
20, and 30 years), for six dierent option expiries (3 months and 1, 5, 10, 20,and 30 years) and for eight degrees of moneyness dened as strike minus the forward swap rate (-100 bps, -50bps, -25bps, ATM, 25bps, 50bps, 100bps and 200bps). Black volatilities especially for short maturities and tenors, are high, as can be seen in Figure 5.3 in the appendix.
The parameters of the model in equation (5.3.6) are the same for all maturity and tenor combinations: λ = 0.075, a = −0.3 and b = 0.6. The values of implied volatilities using the pricing formula of Theorem II are shown in the Figure 5.4.
The values of Black volatility are shown in Figure 5 in the appendix.
Dierences between strikes as well as between consecutive expiries and tenors are high. Implied volatilities for rates near zero are highly sensitive. To apply these volatilities to interest risk analysis is dicult.
The implied volatilities of the normal-jump model are much less sensitive and dierences between consecutive values are modest. These more robust values are also more applicable for risk analysis, e.g. for calculation of risk measures. Figure 6 in the appendix shows the normal-jump implied volatility values divided by forward swap rates times 100, which are easier to compare to the Black model's volatilities.
Local volatility tting
As Figure 5.4 shows, the normal jump-diusion model cannot explain the skewness of volatility. The values of volatility are typically smallest near strikes. The volatility becomes higher as the strike becomes smaller or higher.
This is a common observation for implied volatility models. The case is sim-ilar when log-normal jump-diusion is used. The values of volatilities consti-tute smile or smirk depending on the choice of parameters (see Glasserman and Kou, 2003, p. 395).
This subsection gives one obvious next step for the analysis of this sec-tion and show a straightforward method to model the constant volatility. The method for calibrating volatility skewness has been considered, e.g. by
Cole-man et al. (1999), Andersen and Andreasen (2001), Benaim et al. (2008).
This example uses the pricing model of Theorem II and ts a model using an ordinary least square method (follows Coleman et al. (1999)):
mina,bΣi(σu(Ki)−σˆu(Ki))2, (5.4.1) where σu(Ki):s are volatilities for dierent strikes. σˆu(K) is dened by the functionaebK. The solutions have been found separately for dierent expiry and tenor. The function is extremely simple but it gives a reasonably good t. A more complicated function presented in Benaim (2008) gives a slightly better t. The values for parameters a and b are shown in Figure 7 in the appendix. It is possible to use the tted functions for extrapolation to complete the data.
5.5 Conclusion
The assumption of log-normality is typical for the HJM- and LIBOR- interest models. This assumption has three fundamental problems: it ignores the jumps of underlying time series, it assumes constant volatilities and it is unwieldy in the low (or negative) rate environment. This paper has presented an approach to solving all these problems by combining the jump-diusion model with normality assumption and local volatility tting. This approach is the rst presented in the literature to combine Bachelier's option pricing model and a jump process. Obviously, more complicated models using those elements could be developed, but the purpose of this paper has focused on nding as simple a method as possible for tackling those three problems.
.0.1 Appendix
Figure 5: Implied Black volatilities (source: Bloomberg)
Figure 6: Ratios of normal jump-diusion volatilities and forward swap rate as percents
Figure 7: Values for parameters a and b in section 4.2.1
.1 Symbol description
Et expectation given information available at time t Wt standard Brownian motion
Φ(y) the cumulative density function of standard normal distribution r(t) spot rate at time t
P(t, T) the price at time t of a zero-coupon bond which pays a principle of one at time T and no other payments
f(t, T) instantaneous forward rate
L(t, S, T)simple forward rate at time tfor period [S, T] δ xed accrual period
α(t, T)the drift of forward rate process σ(t, T) the volatility of forward rate process f instantaneous forward rate
Sm,n(T)forward swap rate at time T, with the payment datesTm, Tm+1, ..., Tn Am,n(T)annuity at time T, with the payment dates Tm, Tm+1, ..., Tn
CBSis swaption price under log-normal distributed underlying forward swap rate
CJ is swaption price when forward swap rate follows log-normal distribution with jumps
α0(t, T) the drift of swap rate process θ(t, T) the volatility of swap rate process Other necessary denitions are in the text.
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