Taking expectation and simplifying equation (5) :
-Dτ + Ds Çr — δ —— ) + Dυ (a — βvt) +
+Dss — + Dvv72 — + DsvP7vt - Dr
(6)
Provided that the assumption of continuous differentiability holds and the initial (terminal) conditions
are well specified, the value function of a derivative asset can be determined by solving p.d.e. (6). Epps
(2004b) shows how to obtain the solution for a special case of a Europeamstyle D : D (st,vτ, 0) = eβsτ. No
closed-form solution is available for an Americamstyle derivative security.
4. Joint Density of (sτ, υτ) under P
4.1. Joint Ch.F.
Consider time и ∈ (t,T]. Let the conditional joint ch.f. of (st, vτ) be:
Φ(u) ≡ Ê [el^sτ+^ vτ ) ∣Λt] = Φ(ζ 1,ζ 2; su, vu ,T — и).
Above, conditioning on just the three state variables su,vu,T — и vs. the whole information set Eu is
motivated by the Markovian property of Brownian motions.
By the tower property:
Φ(t) = Ê ∣ei½sτ+ζzvτ)∣Et] = Ê [Ê p‰sτ+ζzvτ)∣Eu] ∣Et] = E[Φ(u) ∖Et].
Therefore, taking и arbitrarily close to t, it follows that Ê [dΦ (t) ∣Et] = 0.
By Ito’s lemma:
dΦ (ζ 1,ζ 2; St,Vt,T ) = —Φτ dt + Φs⅛St + Φv dvt + ∣Φss^ <s)t + ∣Φvv d {v)t + Φsv d (s,v')t.
Using equations (3) and (4), taking expectation, factoring out dt and simplifying, the conditional joint
ch.f. Φ (ζι,ζ2; St,vt,τ) solves p.d.e.:
0 = —φt + φs r^ — δ--— ) + φv (a — βvt) + φss -— + φvv 72 ■— + φsv P7vt.
(7)
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