As mentioned earlier, we have to use an approximation of the caplet valua-tion formula in Thm. 7.2.1 to state a caplet price, because the formula is expressed as a power series. If we approximate the caplet price with the three first terms of the power series, we find that the caplet price is given by the time integral over the squared stochastic volatility. We are interested in the special case when λ(t) = a+be−Z(t), as defined in Eq. (5.1), for a nGOUS Z(t). Remember from Ch. 5 that we computed formulas for the characteristic function ofZ(t),ϕZt(θ), in Prop. 5.2.1, and for the joint characteristic function ofZ(t)at two different times,ΦZtZs(θ, ϑ), in Prop.
5.2.2. These formulas can be used to express the first and second moments ofσ2T. That is, by Eq. (5.7) and Prop. 5.2.3 in Ch. 5, it is easy to see that the first moment is given by
EQT+δ integral of the squared stochastic volatility is given by
EQT+δ
7.3. Caplet valuation formula for a geometric Brownian motion with an exponential nGOUS stochastic volatility where
EQT+δ
h
λ2(s)λ2(t)i
=a4+ 2a3b ϕZt(i) +ϕZs(i)
+a2b2 4ΦZtZs(i, i) +ϕZt(2i) +ϕZs(2i) + 2ab3 ΦZtZs(i,2i) + ΦZtZs(2i, i)
+b4ΦZtZs(2i,2i).
If we would want an explicit analytical formula for the approximated caplet price with more terms, e.g.mterms, we would have to compute the(m− 1)-th moment of 1)-the time integral of 1)-the squared stochastic volatility. As mentioned in the end of Ch. 5, that is possible to do, but requires a lot of time due to the exploding number of terms as the exponentmgrows. In addition to that the number of terms grows fast, each term is also time consuming to compute. That is, each term inC(X;l0)consists of several integrals which have to be solved, and asmgrows, the number of integrals in each term that has to be solved grows as well. By this we conclude that, in theory, it is possible to compute an explicit analytical approximation of C(X;l0)with an arbitrarily small error. However, the time consumption do-ing this would be so large that it is recommended to use numerical methods to computeC(X;l0)if the desired number of terms is bigger than three.
In the next chapter we will use another approach to derive a caplet valuation formula, which can be used to approximate the ATM caplet price for a geometric Brownian motion with exponential nGOUS stochastic volatility.
Chapter 8
CAPLET VALUATION WITH A BLACK-SCHOLES APPROACH
In the last chapter we found a caplet valuation formula for the LIBOR forward rate modeled by a general geometric Itô-Lévy process, by use of Fourier transformation. Then we applied that formula on the special case when the LIBOR forward rate is driven by a geometric Brownian motion with an exponential nGOUS stochastic volatility, as introduced in Ch. 5. We know that there is possible to derive caplet valuation formulas for geometric Brownian motions with deterministic volatility, by use of a Black-Scholes approach. Inspired by that we are going to derive a caplet valuation formula for a geometric Brownian motion with stochastic volatility. It turns out that it is not possible to state the expectations in this general formula explicitly, and thus we precede the calculations by rewriting functions as power series.
As a warning, the calculations turn out to be quite messy.
8.1 Caplet valuation formula by a Black-Scholes approach In this section we will use a Black-Scholes approach to derive a general caplet valuation formula for a geometric Brownian motion with stochastic volatility. In [Fil09] an equivalent formula is stated for volatilitiesλ(t, T) which are deterministic functions, without proof. Our version of this for-mula is stated in Prop. 8.1.1, with a proof which is inspired by the proof of the easier case in [Ben04], when the volatility is a constant.
So, we want to derive a caplet valuation formula from a LIBOR forward rate which is given by
L(T, T) =L(t, T) exp
λ◦WT+δ
, (8.1)
wheret≤T ≤ T,λ(t)∈ V([0,T]2)andWT+δ is aQT+δ-Brownian motion.
That is, a LIBOR forward rate prevailing at time T, applicable to the interval[T, T +δ]. The caplet valuation formula derived in this section is applicable to all stochastic volatilities as long as they are in the function spaceV([0,T]2)and satisfies Novikov’s conditon (Thm. 2.1.2).
Proposition 8.1.1 (Caplet valuation formula for a geometric Brownian mo-tion with stochastic volatility).Let L(T, T) be as given in Eq. (8.1) with stochastic volatilityλ(t, T) ∈ V([0,T]2). Then the general caplet price is
given by
(see Def. 5.3.2) and the natural filtration of theQT+δ-Brownian motion, and
d1,2=
Remark.Note thatΦ(·)is stochastic in Eq. (8.2), asλ(t, T)is a stochastic process.
Proof ([Ben04]). In Proof II of Prop. 3.3.1 we proved that the no-arbitrage price of a caplet at timetis given by
Cpl(t;T, T +δ, K, N) =
when the dynamics ofF(t;T, T +δ)follows a geometric Brownian motion with constant volatility. In the current case we have a stochastic volatility, and therefore extends the information flow to ensure adaptedness of the model. That is, define a filtration{Ft}t≤T such thatFtN ∪ Ftλ ⊆ Ftfor all t, where{FtN}t≤T is the filtration generated by the Brownian motion and {Ftλ}t≤T is the filtration generated by the stochastic volatility, both under QT+δ. The LIBOR is a simply compound forward rate according to Def.
3.3.1 and 3.2.1, and we can defineL(T, T) := F(T;T, T +δ). Combining these facts we have that
Cpl(t;T, T +δ, K, N) (8.3)
holds. Now we only have to focus on the expectation given above. First we see that
where we usedFt-independence in the first step and the tower rule of ex-pectations in the last step. Further we can rewrite the indicator function ac-cording to the model ofL(T, T)given in Eq. (8.1). Defineσ2T :=RT
t λ2(s, T)ds.
8.1. Caplet valuation formula by a Black-Scholes approach Then we have that
L(T, T)> K It is a known fact from stochastic analysis that Itô integrals are normally distributed, and we have that
Z T
whereN(µ, σ2)denotes the normal distribution with meanµand variance σ2. As normally distributed variables can be written asX=µ+σZ, where Z∼N(0,1), the inequality above becomes
Zσ >ln
where we definedd2for convenience. This gives us, from Eq. (8.4), that EQT+δ
For the second expectation above we then reach
−EQT+δ
where we in the last equality used the fact that the pdf of the standard normal distribution is symmetric about zero. By conventional notation we then have that−EQT+δ
expectation in Eq. (8.5), inserting the model ofL(T, T)in Eq. (8.1) and using the introduced notation, we get
EQT+δ
where we in the last step used the substitutionu=z−σ. Now we consider the lower limit in the last integral above.
−d2−σ=−
This leaves us with EQT+δ
Combining the results for the expectations in Eq. (8.5) with the caplet pricing formula in Eq. (8.3) gives us the desired result.
8.2 The Black-Scholes type caplet valuation formula as power