Monotonicity of Prices in Heston Model

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Monotonicity of Prices in Heston Model

Sidi Mohamed Ould Aly

To cite this version:

Sidi Mohamed Ould Aly. Monotonicity of Prices in Heston Model. International Journal of Theoretical and Applied Finance, World Scientific Publishing, 2013, 16 (3), pp.1350016.

�10.1142/S0219024913500167�. �hal-00678437v4�

c World Scientific Publishing Company

Monotonicity of Prices in Heston Model

S. M. OULD ALY

Universit Paris-Est, Laboratoire d’Analyse et de Mathmatiques Appliques 5, boulevard Descartes, 77454 Marne-la-Valle cedex 2, France

sidi-mohamed.ouldaly@univ-mlv.fr

Received (29 May 2012) Revised (04 October 2012)

In this article, we study the price monotonicity in the parameters of the Heston model for a contract with a convex pay-off function; in particular we consider European put options. We show that the price is increasing in the constant term in the drift of the variance process and decreasing in the coefficient of the linear term in the drift of variance process. We also show that the price is increasing in the correlation for small values of the stock and decreasing for the large values.

Keywords: Heston model; Monotonicity; Put options; Maximum principle; Correlation.

1. Introduction

The main attraction of the Black-Scholes model is the ability to express the price of European options in terms of a volatility parameter. Moreover, for convex pay- offs, these formulas are strictly increasing with respect to the volatility parameter, which can cover the risk associated with this parameter through the purchase or sale of options. However, following the rejection of deterministic volatility assumption by empirical studies, practitioners are increasingly convinced that the best way to model the dynamics of an underlying is to consider a model where the process of instantaneous variance is stochastic.

In a general stochastic volatility model, the variance process does not depend solely on its current value. For example, under the Heston model, the variance pro- cess is given as the unique solution of the following stochastic differential equation

dVt= (a−bVt)dt+σp

VtdWt, V0=v. (1.1) The options prices depend on the initial value of the variance process v and the parameters a, b, σ and ρ. These parameters are often calibrated to market price of derivatives, so they tend to change their values regularly. It is then important to know the impact they have on option prices.

The initial value of the variance process has a positive effect on prices of convex pay-off in a large class of stochastic volatility models. See for example, Bergman et al. (1996), Hobson (1998), Janson and Tysk (2002) and Kijima (2002). When

1

the volatility process is stochastic but bounded between two values mand M, El Karoui et al (1998) show that the price of an option is bounded between the Black- Scholes prices with volatilitiesmandM. In [17], Romano and Touzi show that the derivative of the value-function of an option with respect to the volatility under models such as Hull and White (1987) and Scott (1987) has a constant sign, and does not vanish before maturity. Henderson (2005) shows that convex option prices are decreasing in the market price of volatility risk. However, to our knowledge, the dependence of the European option price on the correlation parameter is not known in any stochastic volatility model.

In this article, we study the price monotonicity of European put options with respect to the parametersv,a,bandρ. We first show that the value function of put price is a classical solution of the Black-Scholes equation. Then using a Maximum principle we show that the price is increasing in the initial value of the variance process as well as in the constant term in the drift of the variance process and decreasing in the coefficient of the linear term in the drift of variance process. We also show that the price is increasing in the correlation for small values of the stock and decreasing for the large values.

This paper is organized as follows: In section 2 we recall some properties of the put price in the Heston model. In section 3 we study the monotonicity of the price with respect to the parameters of the drift of the variance process. The section 4 is devoted to the study of the monotonicity with respect to the correlation.

2. Preliminaries

Under a complete filtered probability space (Ω,F,(Ft)t≥0,P) satisfying the usual conditions, we consider the Heston stochastic volatility model for a price process S_t, defined by the following stochastic differential equations

dSt

S_t =√ VtdW_t¹,

dVt= (a−bVt)dt+σ√

VtdW², dhW¹, W²it=ρdt, (2.1) where a, b, σ > 0 and |ρ| < 1. Let (S, V) be the solution of this equation with initial value (S₀=s, V₀=v). One can writeS^s as

St=s exp Z t

0

pVsdW_s¹−1 2

Z t 0

Vsds

(2.2) The process (St)_t≥0is a local martingale; it is even a true martingale, by Mijatovi´c and Urusov [14]. Thereby, using the Call-Put parity, all the results of this paper hold for Call options.

We consider an European put option on S with strike K and maturity t. Its current price is given, for (s, v)∈R^∗+×R+, by

P(t, s, v) =E[(K−St)+|S0=s;V0=v]. (2.3)

If we replace the put pay-off by a functiong∈ C²(R) such thatxg⁰andx²g⁰⁰are bounded, then Ekstr¨om, Tysk 2010 (cf. [4] Theorem.2.3) show that the function

u(t, s, v) :=E[g(St)|S0=s, V0=v] (2.4) is a classical solution of the pricing equation. In particular, it satisfiesu∈ C R+3

∩ C^1,2,2 R^∗+

3

∩ C^1,0,1 R^∗+×R^∗+×R+

. In addition, a probabilistic representation of the derivative ofuwith respect tov is given as

∂u

∂v(t, s, v) =E Z t

0

e^−bτ( ˆS_τ^s)²∂²u

∂s²(t−τ,Sˆ_τ^s,Vˆ_τ^v)dτ

, (2.5)

where ( ˆS^s,Vˆ^v) is the unique solution starting from (s, v) to the stochastic differen- tial equation

dSˆ^s_t

Sˆ_t^s =ρσdt+

qVˆ_t^vdW_t¹, dVˆ_t^v=

a+^σ₂² −bVˆ_t^v dt+σ

qVˆ_t^vdW_t².

(2.6) Obviously, the European put pay-off does not satisfy the assumptions of this theorem. Nevertheless, Propositions 3.1 and 3.2 of [4] (which require only g to be continuous and bounded) ensure that P ∈ C(R³+) ∩ C^1,2,2 R^∗+

3

so that LP(t, s, v) = 0, ∀(t, s, v)∈R^∗+×R^∗+×R^∗+,

P(0, s, v) = (K−s)₊,∀(s, v)∈R^∗+×R⁺, (2.7) where

Lϕ=−∂ϕ

∂t +

(a−bv) ∂

∂v+1 2s²v ∂²

∂s²+1 2σ²v ∂²

∂v² +ρσsv ∂²

∂v∂s

ϕ. (2.8)

3. Monotonicity with respect to the parametersv, a and b

In this section we study the monotonicity properties of the put price with respect to the parameters v,a andb. We first give an extension of the result of [4] to the European put pay-off.

Theorem 3.1. In addition to (2.7), we haveP ∈ C^1,0,1(R+×R^∗+×R^∗+). Further- more, the derivative ofP with respect tov is given by

∂P

∂v(t, s, v) =E Z t

0

e^−bτh(t−τ,Sˆ_τ^s,Vˆ_τ^v)dτ

, (3.1)

where( ˆS^s,Vˆ^v)is the solution starting from (s, v)to (2.6) andhis defined onR^∗+ 3

by

h(τ, x, y) =Ey





K q

(1−ρ²)Rτ 0 V_udu

N⁰





−log(x/K)−ρRτ 0

√V_udW_u²+¹₂Rτ 0 V_udu q

(1−ρ²)Rτ 0 V_udu







, (3.2) whereN is the cumulative distribution function of the standard normal law.

Remark 3.1. Note that the functionh(t, s, v) is simplys²∂ssP(t, s, v). As a direct consequence of this theorem, we have for anyt, s >0, the functionv7−→P(t, s, v) is increasing.

Proof. Writing S_t=sexp

ρ Z t

0

pV_sdW_s²+p 1−ρ²

Z t 0

pV_sdWˆ_s²−1 2

Z t 0

V_sds

, (3.3) where the Brownian motion ˆW² in independent fromW², we have

P(t, s, v) =KE[N(d1)]−sE

e^ρR0t

√VudW_u²−^ρ₂²Rt

0VuduN(d2)

, (3.4) where

d1= −log(_K^s)−ρRt 0

√V_udW_u²+¹₂Rt 0V_udu q

(1−ρ²)Rt 0V_udu

(3.5)

and

d₂=d₁− s

(1−ρ²) Z t

0

V_udu. (3.6)

We can write∂ss2P(t, s, v), using this stochastic representation ofP, as

∂²P

∂s² =E_v





K/s² q

(1−ρ²)Rt 0V_udu

N⁰





−log(s/K)−ρRt 0

√V_udW_u²+¹₂Rt 0V_udu q

(1−ρ²)Rt 0V_udu







. (3.7) Set

h(t, s, v) =s²∂²P

∂s²(t, s, v). (3.8)

The main purpose of the assumption (xg⁰ and x2g⁰⁰ are bounded) is to give a stochastic representation of the second derivative of P with respect to s and to ensure that it is continuous and bounded. Here we see that we have a stochastic representation of∂ssP given by (3.7). Following the procedure of [4] (cf Proposition 4.1, 4.2), we only need to show that the function

(t, s, v)7−→H(t, s, v) :=E Z t

0

e^−bτh(t−τ,Sˆ_τ^s,Vˆ_τ^v)dτ

(3.9) is continuous on R^∗+×R+×R^∗+ and bounded by an integrable random variable.

For this, we consider a sequence (t_n, s_n, v_n)−→(t, s, v) and show thatH(t_n, s_n, v_n) converges toH(t, s, v). As ( ˆS_τ^sn,Vˆ_τ^vn) converges to ( ˆS^s_τ,Vˆ_τ^v) in probability, we only need to find an upper bound of H(t, s, v) by an integrable random variable and conclude by applying the dominated convergence theorem.

To obtain the desired upper bound, we first note that for anyx, y ∈ Rand 0≤τ ≤twe have

h(x, y, t−τ)≤Ey





K

√ 2π

q

(1−ρ²)Rt−τ 0 Vudu



=:M(t−τ, y). (3.10) We can easily see that for any 0≤y₁≤y₂, we have

M(t−τ, y1)≥M(t−τ, y2). (3.11) On the other hand, by the comparison theorem, we have

Vˆ_τ^v ≥ V_τ^v, a.s. (3.12)

It follows that

M(t−τ,Vˆ_τ^v) ≤ M(t−τ, V_τ^v), a.s. (3.13) Then,

E h

h( ˆS_τ^s,Vˆ_τ^v, t−τ)i

≤E[M(t−τ, V_τ^v)] =E



EV_τ^v





K

√2π q

(1−ρ²)Rt−τ 0 Vudu









=Ev





K

√2π q

(1−ρ²)Rt τV_udu



. (3.14)

The last line follows from the Markov property of the processV. It follows that E

Z t 0

e^−bτhv( ˆS_τ^s,Vˆ_τ^v, t−τ)dτ

≤ Z t

0

Ev





K

√2π q

(1−ρ²)Rτ 0 Vudu



dτ . (3.15) We have, by Dufresne [3],

E_v



 1 qRτ

0 Vudu



<+∞, ∀τ >0. (3.16) Moreover, for anyv≥0, we have

τ→0lim τ²³Ev



 1 qRτ

0 V_udu



= 0. (3.17)

It follows that for anyv≥0, Z t 0

E



 1 qRτ

0 V_u^vdu



dτ <+∞. (3.18)

The rest of the proof of the Theorem is identical to Proposition 3.1 of [4] by using this upper bound. Thus, the functionH is continuous onR+×R^∗+×R^∗+.

Monotonicity with respect to a and b

We now study the monotonicity properties of the put price with respect to the parameters a and b. Note that the paths of the variance process are increasing with respect to a and decreasing with respect tob. This means that increasinga generates higher volatility which will increase the Put price. To verify this claim, we will let the put price vary in terms ofaandb: We write

P^a,b(t, s, v) =E h

(K−S_t^a,b)+

S₀^a,b=s;V₀^a,b=vi

, (3.19)

where (S^a,b, V^a,b) is the unique solution starting with (s, v) of the stochastic differ- ential equations

dS_t^a,b S^a,b_t =

q

V_t^a,bdW_t¹, dV_t^a,b = (a−bV_t^a,b)dt+σ

q

V_t^a,bdW², dhW¹, W²i_t=ρdt.

(3.20) The following maximum principle will be crucial for the proof of the main result of this section. The proof of this theorem can be found in the appendix.

Theorem 3.2 (Maximum Principle). Fort >0, let

µ^∗_t = sup{µ >0 : ES_t^µ <∞}. (3.21) Let Lbe the operator defined by (2.8) and ϕ∈ C^1,2,2(R^∗+

3)∩ C(R³+)so that

∀t, M >0, ∃λ < µ^∗_t : sup

τ≤t, s≤M, v∈R

|ϕ(t, s, v)| ≤M^λ. (3.22) Supposeϕsatisfies

Lϕ(t, s, v) ≤ 0 (resp <0),∀(t, s, v)∈]0,+∞[×R^∗+×R^∗+,

ϕ(0, s, v) ≥ 0, ∀(s, v)∈R^∗+×R^∗+. (3.23) Thenϕ ≥ 0 (respϕ > 0) on R^∗+

3.

We establish the monotonicity of P with respect to a and b in the following result

Proposition 3.1. Let a2> a1 andb1< b2. We have

P^a1,b(t, s, v)< P^a2,b(t, s, v), ∀b≥0, ∀(t, s, v)∈R^∗+

3 (3.24)

and

P^a,b1(t, s, v)> P^a,b2(t, s, v), ∀a≥ σ²

2 , ∀(t, s, v)∈R^∗+

3. (3.25) Proof. For anya, b≥0, let

L^a,bϕ=−rϕ−∂ϕ

∂t +

rs∂.

∂s+ (a−bv)∂.

∂v +1 2s²v∂².

∂s² +1 2σ²v∂².

∂v² +ρσsv ∂².

∂v∂s

ϕ (3.26)

We can easily check that

L^a2,b(P^a2,b−P^a1,b)(t, s, v) =−(a2−a₁)^∂P_∂v^a1,b,∀(t, s, v)∈R^∗+ 3, (P^a2,b−P^a1,b)(0, s, v) = 0, ∀(s, v)∈R^∗+

2. (3.27) and

L^a,b2(P^a,b1−P^a,b2)(t, s, v) =−(b2−b1)v ^∂P_∂v^a,b1,∀(t, s, v)∈R^∗+ 3, (P^a,b1−P^a,b2)(0, s, v) = 0, ∀(s, v)∈R^∗+

2. (3.28) We have, by Theorem 3.1, the function∂_vP^a1,band∂_vP^a,b1 are positive. Then, by Theorem 3.2, that (P^a2,b−P^a1,b)>0 and (P^a,b1−P^a,b2)>0.

4. Monotonicity with respect to the correlation

This section focuses on the monotonicity properties of the price of the European put with respect to the correlation. Note that the method we used in the previous section to establish the monotonicity with respect tov,aandbcan not be applied here. Indeed, the idea of this method was to differentiate (2.7) with respect to the parameter considered and obtain a differential system as (Lu <0 on Cand u≥0 on∂C), which gives the sign ofuby applying the maximum principle; while if we differentiate (2.7) with respect toρ, we obtain the system

L˜^∂P_∂ρ(t, s, v) =−σsv_∂s∂v^∂P (t, s, v),∀(t, s, v)∈]0, T]×R^∗+×R^∗+,

∂P

∂ρ(0, s, v) = 0, ∀(s, v)∈R^∗+×R^∗+. (4.1) As the sign of ∂svP is not necessarily constant, this does not allow us to deduce the sign of the derivative of P with respect toρusing the maximum principle. To analyze the impact ofρ in the priceP, we will study the sign of the derivative of P with respect to ρ. This derivative can be obtained by differentiating (3.4) with respect to ρ:

∂P

∂ρ =E





Z log(^K_s)

−∞

e^xN⁰





x−ρRt 0

√VudW_u²+¹₂It

q

(1−ρ²)Rt 0Iudu





ρx−Rt 0

√VudW_u²+^ρ₂It

p(1−ρ²)³It

! dx



, (4.2) whereIt:=Rt

0Vudu. The sign ^∂P_∂ρ is not obvious, however the following figure shows that there is a change of monotonicity depending on the value of the strike price.

We see that ^∂P_∂ρ is positive fors < K= 1 and negative for fors >1.

In order to determine if this change of monotonicity is unique, we will study in details the sign of the derivative of P with respect to ρ fors very large and very small. For this we define the quantities

s^ρ₀(t, v) = inf

s >0 : ∂P

∂ρ(t, s, v)≤0

(4.3) and

s^ρ_∞(t, v) = supn

s >0 : P˙ρ(t, s, v)≥0o

. (4.4)

Fig. 1. ˙Pρfors∈[0.4,2.5] (K= 1,v0= 0.1,b= 3,σ= 0.2 andt= 0.5).

Having s^ρ₀ > 0 (resp s^ρ_∞ < +∞) means that ^∂P_∂ρ is positive (resp negative) for s small (respslarge). We next present the main result of this section.

Theorem 4.1. For anyt, v >0 andρ∈]−1,1[, we have

0< s^ρ₀(t, v)≤s^ρ_∞(t, v)<+∞ (4.5) Proof. We use the results obtained in [16], where it is shown that forRsufficiently large, we have

lnP

−1 2It+

Z t 0

pVudW_u¹> R

∼ −µ⁺R (4.6)

and

lnP

−1 2It+

Z t 0

pVudW_u¹<−R

∼ −µ⁻R, (4.7)

withµ⁺= inf{p >0, T^∗(p) =t}(>1),µ⁻= inf{p >0, T^∗(−p) =t}and T^∗(p) = sup

t >0, E^Q exp

p²−p 2

Z t 0

Vudu

<+∞

, (4.8)

where underQthe processV satisfies the stochastic differential equation dVt= (a−(b−ρσp)Vt)dt+σp

VtdW_t^Q. (4.9)

We can easily see that, forksufficiently large, we have lnP(t, e^k, v)∼ −µ⁻k, ln P(t, e^−k, v)−1−e^−k

∼ −µ⁺k (4.10) and

k→+∞lim

∂_ρP(t, e^k, v)

kP(t, e^k, v) =−∂µ⁻

∂ρ , lim

k→+∞

∂_ρP(t, e^−k, v)

k(P(t, e^−k, v)−1−e^−k) =−∂µ⁺

∂ρ . (4.11)

By the comparison theorem, the processV is increasing with respect toρunderQ (see also [15]) forp >0 and decreasing forp <0. This means that forp >1,T^∗(p) (as a function ofρ) is decreasing and for anyp⁰>0,ρ7−→T^∗(−p⁰) is increasing. On the other hand,p7−→T^∗(p) is decreasing nearµ⁺ andp7−→T^∗(−p) is increasing nearµ⁻. It follows thatµ⁺is decreasing with respect toρandµ⁻is increasing with respect to ρ. This means that, for ksufficiently large, we have

∂_ρP(t, e^−k, v)>0 (4.12) and

∂_ρP(t, e^k, v)<0. (4.13) Thus (4.5).

So far we confirmed that 0< s^ρ₀ ≤s^ρ_∞<+∞, which means that the Put price is increasing in the correlation for small values of the stock price and decreasing for large values. The question is whethers^ρ₀=s^ρ_∞, which means that there is only one points^ρ(t, v) so that the derivative ofP with respect toρis positive fors≤s^ρand negative for s > s^ρ. All numerical experiments seem to confirm this intuition. In the next sections, we will show thats^ρ₀=s^ρ_∞ for short and long maturities.

4.1. Small-Time Asymptotic Behavior

We study here the monotonicity with respect to the correlation for short maturities.

The main result of this section is the following Proposition Proposition 4.1. For any ρ∈]−1,1[and any v∈R⁺∗, we have

t→0lim sign ∂P

∂ρ(t, e^−x, v) =sign(x). (4.14) Consequently,

t→0lims^ρ₀(t, v) = lim

t→0s^ρ_∞(t, v) = 1. (4.15) Proof. Let (S, V) be the unique solution of (2.6) starting with (s, v). By Forde and Jacquier (cf [6]), we have

t→∞lim tlogE(K−St)+=−Λ^∗(log(K

s)), fors > K (4.16) and

t→∞lim tlogE(St−K)+=−Λ^∗(log(K

s)), fors < K, (4.17) where Λ^∗ is the Fenchel-Legendre transform of the function Λ defined by

Λ(p) = ^vp

σ(√

1−ρ²cot(¹₂σp√

1−ρ²)−ρ), for p∈]p−, p₊[,

Λ(p) =∞, for p∈R\]p₋, p+[,

(4.18)

withp₋ and p₊ are given by

p−= arctan

√

1−ρ² ρ

1 2σp

1−ρ² 1ρ<0−π σ1ρ=0+

−π+ arctan √

1−ρ² ρ

1 2σp

1−ρ² 1ρ>0, (4.19)

p₊=

π+ arctan

√1−ρ² ρ

1 2σp

1−ρ² 1_ρ<0+π σ1_ρ=0+

arctan √

1−ρ² ρ

1 2σp

1−ρ² 1_ρ>0. (4.20) The function Λ^∗ is given by

Λ^∗(x) =xp^∗(x)−Λ(p^∗(x)), (4.21) wherep^∗(x) is the unique solution of

x= Λ⁰(p^∗(x)) (4.22)

and Λ⁰ is given by

Λ⁰(p) = v

σ(p

1−ρ²cot(¹₂σpp

1−ρ²)−ρ)+ σvp(1−ρ²) csc²(¹₂σpp 1−ρ²) 2σ(p

1−ρ²cot(¹₂σpp

1−ρ²)−ρ)². (4.23) Let Σ_t(x) be the Black-Scholes implied volatility, defined as the unique solution of

P(t, K e^−x, v) =PBS(t, K e^−x, K; Σt(x)), (4.24) where

PBS(t, s, k,Σ) =KN(−log(s/k) +tΣ/2

√tΣ )−sN(−log(s/k)−tΣ/2

√tΣ ). (4.25)

By Theorem 2.4 of [6], we have

limt→0Σt(x) = |x|

p2Λ^∗(x). (4.26)

WritingP(t, s, v) in terms of the Black-Schole implied volatility as in (4.24) and noting that the dependence of the right term of (4.24) with respect to ρ is only through Σ and using the fact that the Black-Scholes put price is is increasing with respect to the implied volatility, we see thatp(t, s, v) and Σ_t(log(K/s))) have the same monotonicity with respect to the correlation. Therefore

sign ˙Pρ(t, s, v) = sign ∂Σt(log(^K_s)))

∂ρ . (4.27)

The implied volatility is differentiable with respect to the correlation. Moreover, using Lemma 4.1 below, we have

t→0lim

∂Σt(x)

∂ρ = −|x|

2Λ^∗(x)p 2Λ^∗(x)

∂Λ^∗(x)

∂ρ . (4.28)

Let’s consider the derivative of Λ^∗(x) with respect toρ, forx∈R. This derivative is given by

∂Λ^∗(x)

∂ρ = ∂p^∗(x)

∂ρ (x−Λ⁰(p^∗(x)))−∂Λ

∂ρ(p^∗(x)) =−∂Λ

∂ρ(p^∗(x)) (as Λ⁰(p^∗(x)) =x)

=

−vp^∗(x)

√2ρ 1−ρ²

cot(θ^∗(x))−(1−ρ²)θ^∗(x) csc²(θ^∗(x)) + 1

2σ(p

1−ρ²cot(θ^∗(x))−ρ)² , (4.29) where

θ^∗(x) := 1

2σp^∗(x)p

1−ρ². (4.30)

Using Lemma 4.2 below, which ensures that, for anyx∈R, 2ρ

p1−ρ² cot(θ^∗(x))−(1−ρ²)θ^∗(x) csc²(θ^∗(x))

+ 1>0, (4.31) we have

sign ∂Λ^∗(x)

∂ρ = sign (−vp^∗(x)). (4.32)

On the other hand, asp^∗(x) has the same sign asx, we deduce that fortsufficiently small, we have

sign ˙P_ρ= sign ∂Σ_t(x))

∂ρ = sign ∂Λ^∗(x)

∂ρ = sign (log(K s)).

Lemma 4.1. For any x6= 0, we have

t→0lim

∂Σt(x)

∂ρ = −|x|

2Λ^∗(x)p 2Λ^∗(x)

∂Λ^∗(x)

∂ρ . (4.33)

Lemma 4.2. For any ρ∈[−1,1]andx∈R, we have 2ρ

p1−ρ² cot(θ^∗(x))−(1−ρ²)θ^∗(x) csc²(θ^∗(x))

+ 1>0. (4.34)

4.2. Large-Time Asymptotic Behavior

It is known that for long maturities the implied volatility curve in a stochastic volatility model flattens, so it does not depend on the strike. Under Heston model, Forde et al [7] showed that (under the assumptionb−ρσ >0) the implied volatility can be written as

Σ²_t(x) = 8V^∗(0) +a1(x)/t+o(t), (4.35) where V^∗ and a₁ are given below. The main result of this section is the following result

Proposition 4.2. For anyρ∈]−1,1[such thatb−ρσ >0 and for anyv >0, we have

t→+∞lim s^ρ₀(t, v) = lim

t→+∞s^ρ_∞(t, v) = +∞. (4.36) Proof. We will use the notations of [7]. Under the assumptionb−ρσ >0, we have, for anyp∈]p−, p+[,

V(p) = lim

t→∞t⁻¹logE[exp (p(X_t−x₀))] = a

σ²(b−ρσp−d(−ip)), (4.37) where

d(−ip) =p

(b−ρσp)²+σ²(p−p²) (4.38) and

p_± :=

−2bρ+σ±p

σ²+ 4b²−4bσρ

. (4.39)

Let’s consider the functionp^∗:R−→]p₋, p+[ defined by

p^∗(x) :=

σ−2bρ+ (aρ+xσ)

σ²+4b²−4bρσ x²σ²+2xaρσ+a²

1/2

2σ(1−ρ²) , for x∈R. (4.40)

Fortsufficiently large andx∈R, we have (cf. [7]) 1

S₀E(St−S0e^−x)+= 1 + A(0)

√2πtexp (−(1−p^∗(0))x−V^∗(0)t) (1 +O(1/t)), (4.41) whereV^∗ is the Fenchel-Legendre transform ofV defined by

V^∗(x) := sup{px−V(p), p∈]p₋, p+[} (4.42) andAis the function defined in a neighborhood of 0 by

A(x) = −1

pV⁰⁰(p^∗(x))

U(p^∗(x))

p^∗(x)(1−p^∗(x)), (4.43) where

U(p) :=

2d(−ip)

b−ρσp+d(−ip) _σ^2a₂

expv aV(p)

. (4.44)

Similarly, the Black-Scholes implied volatility can be written as (cf. [7], Theorem 3.2)

Σ²_t(x) = 8V^∗(0) +a₁(x)/t+o(t), (4.45) where

a1(x) =−8 log

−A(0)p

2V^∗(0)

+ 4(2p^∗(0)−1)x. (4.46) In particular, for anyx∈R, we have

t→+∞lim Σ²_t(x) = 8V^∗(0). (4.47)

Now using (4.45) we show, in a similar way as Lemma 4.1, that

t→+∞lim

∂Σt(x)

∂ρ = 8∂V^∗(0)

∂ρ . (4.48)

We have

∂V^∗(0)

∂ρ =−∂V

∂ρ(p^∗(0)) + (x−V⁰(p^∗(0)))∂p^∗(0)

∂ρ =−∂V

∂ρ(p^∗(0))

=−a

σ² −σp^∗(0) + σp^∗(0)(b−ρσp^∗(0))

p(b−ρσp^∗(0))²+σ²(p^∗(0)−p^∗(0)²)

!

= −_2ρ^ap^∗(0)(1−2p^∗(0))

p(b−ρσp^∗(0))²+σ²(p^∗(0)−p^∗(0)²)1ρ6=0. (4.49) The first two lines follow from the fact thatV⁰(p^∗(0)) = 0. Forρ= 0, we have

∂V^∗(0)

∂ρ

_ρ=0=−a

2σ −1 + b

pb²+σ²/4

!

( > 0). (4.50) Lemma 4.3 below ensures that the functionϕ defined in (4.55) is increasing.

Note that, for anyρ∈[−1,1], we have

ϕ(ρ) =p^∗(0). (4.51)

Asϕ(0) = ¹₂, we deduce that for any ρ6= 0, (ϕ(ρ)−1/2) has the same sign as ρ. This means that

ϕ(ρ)−1/2

ρ >0. (4.52)

Therefore, we have

∂V^∗(0)

∂ρ > 0, ∀ρ∈[−1,1]. (4.53)

It follows that for anyx∈R,

t→+∞lim

∂Σt(x)

∂ρ >0. (4.54)

It follows that fortsufficiently large, the put price is increasing with respect to the correlation. Thus (4.36).

Lemma 4.3. The functionϕ defined by

ρ∈]−1,1[7−→ ϕ(ρ) := σ−2bρ+ρp

σ²+ 4b²−4bρσ

2σ(1−ρ²) (4.55)

is increasing.

5. Appendix

Appendix A. Proof of Theorem 3.2 Suppose∃(t, s, v)∈R^∗+

3so thatϕ(t, s, v)<0. Consider (S^s, V^v) the unique solution of the stochastic differential equations

dS_t^s S_t^s =p

V_t^vdW_t¹,

dV_t^v= (a−bV_t^v)dt+σp

V_t^vdW², dhW¹, W²it=ρdt, S₀^s=s, V₀^v=v.

(A.1) Let’s define theF-stopping times

τ= inf

u∈[0, t] :ϕ(t−u, S_u^s, V_u^v)≥ ϕ(t, s, v) 2

(A.2) and

¯ τ_n= inf

u∈[0, t] :S_u^s∧V_u^v∈ 1

n, n c

∧t. (A.3)

We haveP(τ < t) = 1. Applying the Itˆo formula to the process (ϕ(t−u, S_u^s, V_u^v))_u≤t between 0 andτ∧τ¯_n, we have

ϕ(t−τ∧τ¯n, S^s_τ∧¯τn, V_τ∧¯^v_τn) =ϕ(t, s, v) + Z τ∧¯τn

0

S_u^sp

V_u^v∂sϕ(t−u, S_u^s, V_u^v)dW_u¹+ σ

Z τ∧¯τn

0

pV_u^v∂_vϕ(t−u, S_u^s, V_u^v)dW_τ² +

Z τ∧¯τ_n 0

Lϕ(t−u, S_u^s, V_u^v)du. (A.4) AsS andV are in ]0, n], we have

ϕ(t, s, v) =−E Z τ∧¯τn

0

Lϕ(t−u, S_u^s, V_u^v)du+E

ϕ(t−τ∧τ¯n, S_τ∧¯^s _τn, V_τ∧¯^v _τn)

≥E

ϕ(t−τ∧¯τ_n, S_τ∧¯^s _τ

n, V_τ∧¯^v_τ

n)

≥ ϕ(t, s, v)

2 P(τ≤τ¯n) +E

ϕ(t−τ¯n, S_τ^s_¯n, V_τ¯^v_n)1τ >¯τ_n

. (A.5)

Writing

{τ >τ¯n}={sup

u≤τ

Vu≥n} ∪ {sup

u≤τ

Su≥n}, (A.6)

we have E

ϕ(t−τ¯n, S_τ^s_¯n, V_τ¯^v_n)1τ >¯τ_n

≤n^λ

P

sup

u≤t

Vu≥n

+P

sup

u≤t

Su≥n

. (A.7) Now using Doob’s martingale inequality, we have

P

sup

u≤t

Su≥n

≤ ES

µ∗ t+λ

2

t

n^µ

∗t+λ 2

=⇒n^λP

sup

u≤t

Su≥n

≤ ES

µ∗ t+λ

2

t

n^µ

∗t−λ 2

−→_n→∞0. (A.8)

Similarly, applying Doob’s martingale inequality to the martingalee^bt(Vt−^a_b) and taking into account the fact that

EV^p <∞, ∀p >0, (A.9) we obtain

n→∞lim n^λP

sup

u≤t

Vu≥n

= 0. (A.10)

Therefore

n→∞lim E

ϕ(t−τ¯_n, S_τ^s_¯

n, V_τ¯^v

n)1_{τ >¯τn}

= 0. (A.11)

This means that

ϕ(t, s, v)≥ ϕ(t, s, v)

2 . (A.12)

Hence the contradiction (ϕ(t, s, v) is supposed to be negative). Thusϕ≥0.

Now assumeLϕ <0. Let’s take (t, s, v) witht >0. Applying the Itˆo formula to the process (ϕ(t−u, S_u^s, V_u^v))_u≤tbetween 0 andt∧τ¯n, we have

ϕ(t− t

2∧τ¯n, S^st 2∧¯τn, Vt^v

2∧¯τn) =ϕ(t, s, v) + Z ₂^t∧¯τ_n

0

S_u^sp

V_u^v∂sϕ(t−u, S_u^s, V_u^v)dW_u¹+ σ

Z ^t₂∧¯τ_n 0

pV_u^v∂vϕ(t−u, S_u^s, V_u^v)dW_τ² +

Z ₂^t∧¯τ_n 0

Lϕ(t−u, S_u^s, V_u^v)du. (A.13) We get, by the same way as before,

ϕ(t, s, v)≥ −E Z ^t₂

0

Lϕ(t−u, S_u^s, V_u^v)du >0. (A.14)

Thusϕ(t, s, v)>0.

Appendix B. Proof of Lemma 4.1:

The put priceP is given, in terms of the Black-Scholes implied volatility, by P(t, s, v) =KN

−log(s/K) +₂^tΣ²_t

√tΣt

−sN

−log(s/K)−^t₂Σ²_t

√tΣt

. (B.1) Differentiating this expression on both sides with respect toρ, we can write ˙P_ρ as

P˙_ρ(t, s, v) =KN⁰

−log(s/K) +₂^tΣ²_t

√tΣt

√ t∂Σ_t

∂ρ (−log(s/K)). (B.2) On the other hand, by (4.26), we know that

limt→0Σt(x) = |x|

p2Λ^∗(x). (B.3)

Moreover the function ρ 7−→ |x|/p

2Λ^∗(x)(ρ) is C¹ on ]−1,1[. We claim that t7−→ ^∂Σ_∂ρ^t is bounded near 0. This is equivalent to say that

P˙ρ(t, s, v) K√

t N⁰ (−log(s/K) +₂^tΣ²_t)/(√

tΣt) is bounded. (B.4) Writing

P =E

K−s exp

ρ Z t

0

pV_udW_u¹+p 1−ρ²

Z t 0

pV_udW_u²−I_t/2

+

, (B.5) we can write ˙Pρ as

P˙_ρ(t, s, v) =E

"

− Z t

0

pV_udW_u¹+ ρ p1−ρ²

Z t 0

pV_udW_u²

!

S_t1_K≥St

#

. (B.6) Applying the H¨older inequality, withp >1, we have

P˙_ρ(t, s, v)≤

"

E

−S_t Z t

0

pV_udW_u¹+ Stρ p1−ρ²

Z t 0

pV_udW_u²

p#^1/p

×[P(K≥S_t)]^p−1p , (B.7) whereP(K≥St) can be written as

P(K≥St) = ∂P

∂K =N

−log(s/K) +₂^tΣ²_t

√tΣt

+KN⁰

−log(s/K) +₂^tΣ²_t

√tΣt

√ t∂Σt

∂K. (B.8) On the other hand, for anyy >0, we have

N(−y)≤ 1 y

exp −y²/2

√2π . (B.9)

It follows that for anys > Kandt sufficiently small, we have N

−log(s/K) +^t₂Σ²_t

√tΣt

≤

√tΣ_t

log(s/K)−₂^tΣ²_tN⁰

−log(s/K) +₂^tΣ²_t

√tΣt

. (B.10) Then, for s > K, there exists a constantM >0 such that, fort sufficiently small, we have

P(K≥St)≤M √ t N⁰

−log(s/K) +₂^tΣ²_t

√tΣt

. (B.11)

It follows that P˙ρ(t, s, v)

√tN⁰_−log(s/K)+t 2Σ²_t

√tΣ_t

≤M2 [E|Yt|^p]^1/p× √

tN⁰

−log(s/K) +₂^tΣ²_t

√tΣ_t

−1 p

, (B.12)

where

Yt= − Z t

0

pVudW_u¹+ ρ p1−ρ²

Z t 0

pVudW_u²

!

× exp

ρ Z t

0

pV_udW_u¹+p 1−ρ²

Z t 0

pV_udW_u²−I_t/2

. (B.13) Setx= log(s/K). Fort small,

−x+₂^tΣt

√tΣt

∼ −x

√tΣt

. (B.14)

We choosepso that

p=p(t) =c

t. (B.15)

For this particularp, we have √

tN⁰

−log(s/K) +^t₂Σ²_t

√tΣt

−1 p

∼M3 exp

−t c

log√

t− x² 2tΣ_t

≤M4. (B.16) We next show that [E|Yt|^p(t)]^1/p(t) is bounded fort close to 0. For this, we use the usual inequality

|y| ≤e^y+e^−y, ∀y∈R. (B.17) We get

|Y| ≤ Y1(t) +Y2(t), (B.18)

where

Y₁(t) = exp (ρ−1) Z t

0

pV_udW_u¹+ρ+ 1−ρ² p1−ρ²

Z t 0

pV_udW_u²−I_t/2

!

(B.19) and

Y2(t) = exp (1 +ρ) Z t

0

pVudW_u¹+1−ρ−ρ² p1−ρ²

Z t 0

pVudW_u²−It/2

!

. (B.20) BothY1 andY2 can be written as

Y_i(t) = exp

α_i Z t

0

pV_udW_u¹+β_i Z t

0

pV_udW_u²−1 2I_t

, i= 1,2. (B.21) In particular, we have

EY_i^p =E exp

αip Z t

0

pVudW_u¹+βip Z t

0

pVudW_u²−p 2It

=E exp

α_ip Z t

0

pV_udW_u¹+β_i²p²−p 2 I_t

. (B.22)

By [16] and the fact that Z t

0

pV_udW_u¹= (V_t−v−at+bI_t)/σ, (B.23) we have, forpsufficiently large,

EY_i^p= exp (−αip(v+at)/σ+aϕ(t) +vψ(t)), (B.24) where

ψ(t) = b σ² +

p2λⁱ₂(p)σ²−b²

σ² tan (g(t, p)), ϕ(t) = b

σ²t+ 2

σ²(log cosg(0, p)−log cosg(t, p)) (B.25) and

g(t, p) =

p2λⁱ₂(p)σ²−b²

2 t+ arctan( λⁱ₁(p)σ²−b

p2λⁱ₂(p)σ²−b²), (B.26) with

λⁱ₁(p) =α_ip/σ and λⁱ₂(p) =β²_ip²−p

2 +α_ibp/σ. (B.27)

It follows that

[EY_i^p]^1/p= exp

−αi(v+at)/σ+aϕ(t)

p +vψ(t) p

. (B.28)

In particular, forp=p(t) =c/t, we have, fortsufficiently small, g(t, p(t)) ∼ cβiσ

2 + arctan(αi

β_i). (B.29)

Similarly, we have aϕ(t)

p(t) +vψ(t)

p(t) ∼ vβ_ic σ tan

cβ_iσ

2 + arctan(α_i βi

)

. (B.30)

Note that the coefficientc in (B.15) was chosen so that

−π/2<cβiσ

2 + arctan(αi

β_i)< π/2, for i= 1,2. (B.31) We finally have, for i= 1,2,

t→0lim[EY_i^p]^1/p= exp vβic

σ tan cβiσ

2 + arctan(αi

β_i)

<+∞. (B.32) It follows that, using (B.12) and (B.16), the claim (B.4) is verified. We proceed similarly fors < K, by using the call price instead of the put price.

Appendix C. Proof of Lemma 4.2:

Let’s set

η(x) = 2ρ

p1−ρ² cot(θ^∗(x))−(1−ρ²)θ^∗(x) csc²(θ^∗(x))

+ 1 (C.1)

andη(x) =ϕ(θ^∗(x)), whereϕis defined by ϕ(θ) = 2ρ

p1−ρ²

cos(θ)

sin(θ) −(1−ρ²) θ sin²(θ)

+ 1. (C.2)

For anyx∈R,p^∗(x)∈[p₋, p+], we have θ(ρ) :=1

2σp

1−ρ² p₋ ≤θ^∗(x)≤ 1 2σp

1−ρ² p₊ =: ¯θ(ρ), ∀x∈R. (C.3) So we only need to show thatϕis positive on [θ,θ].¯

We can easily see thatϕisC¹ on θ,θ¯

\{0}, its derivative is given by ϕ⁰(θ) = 2ρ

p1−ρ²

(ρ²−2) sin(θ) + 2(1−ρ²)θcos(θ)

sin³(θ) . (C.4)

A simple study of the sign of the function

θ7−→(ρ²−2) + 2(1−ρ²)cos(θ)

sin(θ), (C.5)

shows that it reaches its maximum on θ,θ¯

at 0 and this maximum is equal to (−ρ²)<0. We deduce that

ρϕ⁰(θ)≤0, ∀θ∈ θ,θ¯

. (C.6)

We only have two possible situations:

Case ρ >0 : In this case, we have θ=−π+ arctan

p1−ρ² ρ

!

et ¯θ= arctan

p1−ρ² ρ

!

. (C.7) On the other hand, the functionϕis decreasing on

θ,θ¯

. In particular, we have, for anyθ∈

θ,θ¯ , ϕ(θ)≥ϕ(¯θ) = 2ρ

p1−ρ²

ρ

p1−ρ² −(1−ρ²) arctan(

p1−ρ²

ρ )(−1− ρ² 1−ρ²)

! + 1

= 1 +ρ²

1−ρ² − 2ρ

p1−ρ²arctan

p1−ρ² ρ

! . We do the following change of variables

0≤y=

p1−ρ²

ρ ⇐⇒ ρ= 1

p1 +y². (C.8)

We obtain

ϕ(θ)≥ϕ(¯θ) =1 y

2 +y²

y −2 arctany

>0, (C.9) as the minimum of the function (y7−→ ^2+y_y² −2 arctany) is reached at the pointy₀=

q

3+√ 17

2 and is≈0.78.

Case ρ <0 : In this case, we have θ= arctan

p1−ρ² ρ

!

et ¯θ=π+ arctan

p1−ρ² ρ

!

. (C.10) The functionϕis increasing on

θ,θ¯

. In particular, we have, for any θ∈ θ,θ¯

,

ϕ(θ)≥ϕ(θ) = 2ρ p1−ρ²

ρ

p1−ρ² −(1−ρ²) arctan(

p1−ρ²

ρ )(−1− ρ² 1−ρ²)

! + 1

= 1 +ρ²

1−ρ² − 2ρ

p1−ρ²arctan

p1−ρ² ρ

!

. (C.11)

We do the following change of variables 0≤z=−

p1−ρ²

ρ ⇐⇒ ρ= −1

√1 +z². (C.12) Thusϕ(θ)≥ϕ(θ) = ¹_z

2+z²

z −2 arctanz

>0.

Appendix D. Proof of Lemma 4.3 The functionϕis defined on [−1,1], by

ϕ(ρ) =σ−2bρ+ρp

σ²+ 4b²−4bρσ

2σ(1−ρ²) , (D.1)

forρ∈]−1,1[, and

ϕ(−1) = σ+ 4b

4(σ+ 2b), ϕ(1) = 4b−σ

4(2b−σ). (D.2)

The functionϕisC¹on [−1,1], and its derivative is given, by ϕ⁰(ρ) = −2b+p

σ²+ 4b²−4bρσ

2σ(1−ρ²) − 2bρσ

2σ(1−ρ²)p

σ²+ 4b²−4bρσ+ 2ρ

1−ρ²p^∗(0), (D.3) forρ∈]−1,1[, and

ϕ⁰(−1) = 2b²σ²

(σ+ 2b)³ + σ²

2(σ+ 2b), ϕ⁰(1) =

2b²σ²

(2b−σ)³ + σ² 2(2b−σ)

. (D.4)