CONCLUSIONES PARCIALES

The purpose here is to determine the order of the error arising due to approximating the solution up to the first two leading terms in the asymptotic expansion, that is, P0+

√

εP1.

4.9.1 Regularization of the Payoff function

The payoff h(XT)defined as

h(XT):=max{XT−K, 0},

is a piecewise continuous function. If X :=log S, such that h(ST) =max

n

eST _{−K, 0}o_,

then the payoff function has a discontinuous derivative with respect to stock price S at matu- rity when S≈log K (that is X≈K– at-the-money). It requires a smooth and bounded payoff function to analyse the error.

Consider a small time to maturity of order of a very small parameter, ζ, and denote the regularized price from equation (4.14), by Pε,ζ _{and its regularized first-order correction from}

equation (4.57), by Pζ_{. Then, the regularized problem takes the form}

Lε_Pε,ζ _{=0, (4.106)}

where the operatorLε_{is defined as}

Lε _:= 1 εL0+ 1 √ ε L₁+ L₂, with the regularized payoff, hζ_(X

T)as the terminal condition. In particular, hζ(XT)is taken

to be the Black-Scholes price at time T−ζ

hζ_(X

T) =CBS(T−ζ, x, K, T; ¯σ). (4.107)

Thus, unlike h(XT), hζ(XT)is a smooth function ofC∞-class, for 0 < ζ  1. Consequently,

the regularized first-order correction to problem (4.106) is given as,

Pζ _{= P}ζ

0 +p

ζ

Chapter 4. 65

where P₀ζ and pζ₁are defined as

P₀ζ =CBS(t−ζ, x, K, T; ¯σ) and, (4.109) pζ 1 = −[T−t]  V2x2 ∂2 ∂x2 +V3x 3 ∂3 ∂x3  Pζ 0. (4.110)

4.9.2 Accuracy of the Approximation

To determine the magnitude of the error, |Pε_{−P|, it requires to compute first, |P}ε_−Pε,ζ_|,

|Pε,ζ_−Pζ_|and|Pζ_{−P|such that one can easily obtain|P}ε_{−P|from the inequality}

|Pε_{−P| ≤ |P}ε_−Pε,ζ_{| + |P}ε,ζ_−Pζ_{| + |P}ζ_{−P|, (4.111)}

with a requirement that Pε _≈Pε,ζ_{, P}ε,ζ _≈Pζ _{and P}ζ _≈P.

Lemma 4.9.1. Suppose a fixed point(t, x, y)with t ≤ T, then there exist small parameters ¯ζ1 > 0,

¯ε1>0 and a constant c₁∗>0 which might depend on t, T, x and y such that,

|Pε_{(t, x, y) −P}ε,ζ_{(t, x, y)| ≤c}∗

1ζ, (4.112)

for all 0<ζ < ζ¯1and 0<ε< ¯ε1.

The proof of equation (4.112) can easily be developed from the concept of risk neutral valu- ation, see AppendixC, SectionC.2.

Lemma 4.9.2. Suppose a fixed point(t, x, y)with t ≤ T, then there exist small parameters ¯ζ2 > 0,

¯ε2>0 and a constant c2∗>0 which might depend on t, T, x and y such that,

|P(t, x) −Pζ_{(t, x)| ≤c}∗

2ζ. (4.113)

The proof is given in AppendixC, SectionC.3

Lemma 4.9.3. Suppose a fixed point(t, x, y)with t ≤ T, then there exist small parameters ¯ζ3 > 0,

¯ε3>0 and a constant c₃∗>0 which might depend on t, T, x and y such that,

|Pε,ζ_{(t, x) −P}ζ_{(t, x)| ≤c}∗ 3  ε|log ζ| +ε r ε ζ +ε  .

defined15as ζ =min_¯

ζ1, ¯ζ2, ¯ζ3 and ε=min{¯ε1, ¯ε2, ¯ε3}, then from equation (4.111),

|Pε_{−P| ≤ |P}ε_−Pε,ζ_{| + |P}ε,ζ_−Pζ_{| + |P}ζ _−P|. ≤ [c∗₁+c₂∗]ζ+c₃∗  ε|log ζ| +ε r ε ζ +ε  . ≤2 max{c∗₁, c∗₂}ζ+c∗₃  ε|log ζ| +ε r ε ζ +ε  .

Now, if ζ ≈ε, it can be deduced that

|Pε_{−P| ≤c}∗

4[ε+ε|log ε|],

for some constant c∗₄ > 0. Hence, at a fixed point t < T and x, y ∈ _{R, the accuracy of the}

approximation of call prices is given by

lim

ε↓0

|Pε_{(t, x, y) −P(t, x)|}

ε|log ε|1+l

=0, (4.114)

for any l >0. Therefore, the error generated due to first-order approximation is of order ε. The above methods are efficient for only fast-mean reverting volatility models and prove not to work for the slow case. Using the method of Monte Carlo simulations with antithetic variates, [117] analysed results by [41] and [55] for fast-mean and non-fast mean reverting volatilities respectively, together with the classical Black-Scholes price. By comparing dif- ference rates16for different times-to-maturity of both, at-the-money (ATM) and out-of-the-money (OTM) options, through numerical experiments, they showed that:

• difference rates for non-fast mean reverting volatility are much higher than those of the fast mean-reverting volatility.

• first-order approximation is not reliable for non-fast mean reverting volatility but the converse is true.

• the difference rates for the first-order approximation prices increase with depth of OTM for a given time to maturity.

• first-order price approximation accuracy decreases with time to maturity.

For some particular maturities and OTM options, the first-order approximation reflected relatively large errors. However, [117] improved the accuracy of the prices by approximating the price up to the εP2(t, x, y)-term in the asymptotic expansion. An explicit form of P2 is

15_{Refer to the proofs in AppendixCfor the significance of(ζ}¯_{1, ¯ζ2, ¯ζ3})and(¯ε1, ¯ε2, ¯ε3). 16_{The ratio of the difference between analytic value and Monte Carlo value to Monte Carlo value.}

Chapter 4. 67

derived in AppendixB.1. In the fast mean-reverting setting, the difference between first and second order approximations for near ATM options and options with longer maturities is almost negligible. The second order approximation improves the accuracy for long-maturity options or ATM options in the case of non-fast mean-reverting volatility.