Definition 4.1. Viscosity sub/super/solution for the non-local eikonal equation)
1. American Call basket options with dividends
The American Call option on a basket of assets is the right to sell a certain basket of assets, i.e., a fixed weighted sum of underlyings, at strike priceK during the time of the validity of the contract. Strike price, underlyings, and maturity time are written in the contract. The term “American” refers to the early exercise right of the option holder. This early exercise right leads to the feature of free boundaries in the typical continuous diffusion models of financial markets. Let us consider such a (for simplicity complete) market withnrisky assetsS= (S1, . . . , Sn) which satisfy
dSi= (r(S)−δi(S))Sidt+σi(S)SidWi (1.1) in the risk-neutral measure, and whereS→r(S),S →σi(S), and S→δi(S) are bounded Lipschitz-continuous functions which model interest rates, volatilities, and dividends, respectively. Dividends are always nonnegative. LetT > 0 be the maturity time. We consider (1.1) on the time interval [0, T] and assume thatW is
This work was completed with the support of SFB 359 (DFG) and BMBF.
ann-dimensional Brownian motion which satisfies for all 0≤t≤T ρij(t) =
t 0
[Wi, Wj](s)ds, (1.2)
with constantρij modeling the correlations of the returns of the assets (we shall consider the extension to variable correlations below). In order to introduce the basket volatility we make the following observation on stochastic sums. IfF1(t) = S1(t) +ã ã ã+Sn(t) and assuming that [F1](t)>0, then
Z(t) = t
0
1
[F1](u)dF1(u) (1.3)
is a Brownian motion by Levy’s theorem. We have
dF1=σB(S)dZ(t), (1.4)
where we call
σB(S) =.
ij
ρijσiσjSiSj (1.5) the basket volatility. We want to show that the value function of an American basket Call with dividends increases as the basket volatility increases. The value function is defined as the solution (δt, S)→VC(δt, S) of the obstacle equation
max ∂u
∂δt ưLSu, fưu
= 0, (1.6)
in (0, T]×Rn+×Rm+, and satisfies the initial condition (K is the strike price) VC(0, S) =fC(S) =
i
Si−K +
. (1.7)
Here,δtdenotes the time to maturityT−t(hence the minus sign in the diffusion equation in (1.6)), and
LSu= 1
2σσT :DS2u+Sδ,rã ∇Su−ru, (1.8) whereSδ,r := ((r−δ1)S1, . . . ,(r−δn)Sn) andDS2u=
SiSj ∂2u
∂Si∂Sj
. The volatil- ity matrixσσT, the volatilitiesσi, and covariances (ρij) are related by
σσT =σiρijσj. (1.9)
Indicating the dependence of the basket volatilityσB(S) =3
ijρijσiσjSiSj on R= (ρij) by writingσBRwe observe that in our model of constant correlations we have
σRB≤σRB iffR≤R. (1.10) As usual we say thatR≤R iff R−R is positive. Next recall that the exercise region is the contact setE={(t, S)|VC(t, S=fC(S)}. Our main Theorem then is the following.
American Call Options 263 Theorem 1.1. The American Call option basket value function VC is monotone with respect to the basket volatility, i.e.,
σB↑ ⇒ VCσB ↑ (1.11)
where
σB(S) =.
ij
ρijσiσjSiSj.
This means that the exercise region shrinks with increasing basket volatility.
Remark 1.2. If dividends δi = 0, then the American basket Call option value functionVC equals the European basket Call option value function.
The proof of Theorem 1 is easily reduced to the case of European basket Call option. The reason is that the value function of American Call options is the limit of the value functions of certain Bermudean Call options (as is well known). If 0< T1<ã ã ã< Tn=T is the tenor structure of a Bermudean option and
Dk={T1, . . . , Tn}, (1.12) then the value of a Bermudean Call option at timet= 0 (we consider the value at timet= 0 w.l.o.g.) is
VBC(0, x) = sup
τ∈Dk
Ex(e−rτfC(S(τ))), (1.13) where the expectation is taken w.r.t. the risk neutral measure.
If ∆ := maxk∈{T1,...,Tn}Tk+1−Tk, then ∆→0 impliesVBC↑VC. Hence, the proof of Theorem 3 reduces to main stochastic comparison theorems of stochastic sums with convex nondecreasing data. If data are convex but not nondecreasing, then mean stochastic comparison results hold only with additional restrictions.
One possible restriction is no drift at all, but this means that constant drift terms are also allowed (proof by coordinate transformation). Hence, in context of an American Put option, we can allow for a flat yield curve.
Theorem 1.3. Let rbe a constant function. The American basket Put option value function VP is monotone with respect to the basket volatility, i.e.,
σB ↑ ⇒ VPσB ↑. (1.14)
The elegance of stating theorems in terms of the basket volatility has to be paid off by some restriction of the model w.r.t. correlations. However, our results can be extended to financial market models where the assets S = (S1, . . . , Sn) satisfy
dSi
Si =ài(S)dt+σ(S)dW, (1.15)
and whereσis ann×nmatrix-valued bounded continuous function. Note that in the latter model correlations between the returns of the assets may depend onS.
We state the following corollaries.
Corollary 1.4. In the framework of the more general model class (1.15), the Amer- ican Call option basket value functionVC is monotone with respect to σσT, i.e.,
σσT ↑ ⇒ VCσB ↑. (1.16)
This means that the exercise region shrinks with increasing volatility matrixσσT. Similarly,
Corollary 1.5. Let the interest rates r be constant. In the framework of the more general model class (1.15) the American Put basket option value function VC is monotone with respect toσσT, i.e.,
σσT ↑ ⇒ VPσB ↑. (1.17)
This means that the exercise region shrinks with increasing basket volatility ma- trixσσT.
This paper is organized as follows. In the next section 2 we state two mean stochastic comparison results. In Section 3 we recall some results on the WKB- expansion of the fundamental solution of parabolic equations. Finally, in Section 4 we provide a simplified proof of Hajek’s result in the univariate case. In Section 5 we provide the main ideas of our extensions stated in Section 2.