Exercises 6.1 Mixing solution and Hull-White decomposition For general SVMs, there is no closed-form formula for European call options.

To circumvent this diculty, we can use asymptotic methods as described in this chapter. However, for long maturity date, such methods are no longer applicable and one needs to rely on Monte-Carlo (MC) simulation. When the forward conditional to the instantaneous volatility is a log-normal It^o process, the MC simulation can be considerably simplied: This is called the Mixing solution [29].

Let us consider the following SVM dened by dft = atft



dZt +

p 1 ???? 2dWt



da = b(at)dt + (at)dZt with Wt and Zt two independent Brownian motions and the initial conditions ft=0 = f0 and a0 = .

1. Using It^o's formula, prove that d ln ft = ????

1 2a2t dt + at



dZt +

p 1 ???? 2dWt



2. Conditional to the path of the second Brownian Zt (with ltration FZ), ft is a log-normal process with mean mt and variance Vt. Prove that

~ f0  E[ftjFZ] = f0e????1 2 2 R t 0 a2 sds+

R t 0 asdZs Vt = (1 ???? 2)

Z t 0

a2s ds 3. Deduce that the fair value C at time t of a European call option with strike K and maturity date T  t is C = EP[CBS(K; T;
s (1 ???? 2)
1 T ???? t Z T t a2s dsj ~ f0; t)jFt]
(6.67)
with CBS the Black-Scholes formula as given by (3.1). This formula is called the mixing solution. For  = 0, we obtain the Hull-White decomposition [109]:
C = EP[CBS(K; T;
s 1 T ???? t Z T t a2s dsjf0; t)jFt] (6.68)
From (6.67) and (6.68), we see that the MC pricing of a call option only requires the simulation of the volatility at. The forward ft has been integrated out.