Approximate hedging for nonlinear transaction costs on the volume of traded assets

Abstract

This paper is dedicated to the replication of a convex contingent claim h(S ₁) in a financial market with frictions, due to deterministic order books or regulatory constraints. The corresponding transaction costs can be rewritten as a nonlinear function G of the volume of traded assets, with G′(0)>0. For a stock with Black–Scholes midprice dynamics, we exhibit an asymptotically convergent replicating portfolio, defined on a regular time grid with n trading dates. Up to a well-chosen regularization h ⁿ of the payoff function, we first introduce the frictionless replicating portfolio of $h^{n}(S^{n}_{1})$, where S ⁿ is a fictitious stock with enlarged local volatility dynamics. In the market with frictions, a suitable modification of this portfolio strategy provides a terminal wealth that converges in $\mathbb{L}^{2}$ to the claim h(S ₁) as n goes to infinity. In terms of order book shapes, the exhibited replicating strategy only depends on the size 2G′(0) of the bid–ask spread. The main innovation of the paper is the introduction of a “Leland-type” strategy for nonvanishing (nonlinear) transaction costs on the volume of traded shares, instead of the commonly considered traded amount of money. This induces lots of technicalities, which we overcome by using an innovative approach based on the Malliavin calculus representation of the Greeks.

Appendix

1.1 A.1 Convergence to the buy-and-hold price

Suppose that h(x)=(x−K)⁺ is the convex payoff function associated to the call option. We want to prove that the prices $(\widehat {C}^{n}(0,S_{0}))$ converge to S ₀ as n goes to ∞. To this end, observe that $\widehat{S}^{n,(0,x)}$, x=S ₀, admits the Doléans-Dade form $\widehat{S}^{n,(0,x)}_{t}:=\exp (Z^{n}_{t}-(1/2)\langle Z^{n}\rangle_{t})$. We introduce for any M>0 the stopping time τ ^n,M:=inf{t: 〈Z ⁿ〉 _t ≥M}. We easily show that $\mathbb{Q}[\langle Z^{n}\rangle_{T}< M]\to0$ as n→∞, so that $\langle Z^{n}\rangle_{T\wedge\tau^{n,M}}\to M$ as n→∞. Indeed, we may assume without loss of generality that Z ⁿ is bounded.

Observe that h may be uniformly approximated by a sequence of smooth convex functions g ^m with bounded first derivatives. Therefore, we may assume without loss of generality that h is smooth. When the payoff function is h(x)=x, we easily get that $x=\widehat{C}^{n}(t,x)=\mathbb{E}\widehat{S}^{n,(t,x)}_{T}$, so that the supermartingale $\widehat{S}^{n,(t,x)}$ is a martingale. Using the Itô decomposition, we deduce from the convexity of g ^m that for all stopping times τ, $\mathbb{E}g^{m}(\widehat {S}^{n,(0,x)}_{T})\ge\mathbb{E}g^{m}(\widehat{S}^{n,(0,x)}_{T\wedge \tau ^{n,M}})$ and $\mathbb{E}h(\widehat{S}^{n,(0,x)}_{T})\ge\mathbb {E}h(\widehat {S}^{n,(0,x)}_{T\wedge\tau^{n,M}})$ as m goes to ∞. Therefore, by Fatou’s lemma,

$$\liminf_n \mathbb{E}h(\widehat{S}^{n,(0,x)}_{T})\ge\mathbb {E}\liminf_n h\big(\widehat {S}^{n,(0,x)}_{T\wedge\tau^{n,M}}\big). $$

Using the time change $Z^{n}_{t}=B_{\langle Z^{n} \rangle_{t}}$, where B is a Brownian motion in a new filtration (see Theorem II.42 in [19]), we deduce that

$$\mathbb{E}\liminf_n h\big(\widehat{S}^{n,(0,x)}_{T\wedge\tau ^{n,M}}\big)\ge \int_\mathbb{R}h\big(xe^{\sqrt{M}z-\frac{1}{2}M}\big)\varphi(z)\,dz, $$

where φ is the density of the standard Gaussian distribution. As M goes to ∞, using the explicit expression on the right-hand side in this inequality as the Black–Scholes price of the call of maturity 1 and strike $\sqrt{M}$, we get $\mathbb {E}\liminf_{n} h(\widehat{S}^{n,(0,x)}_{T\wedge\tau^{n,M}})\ge x$. On the other hand, $\mathbb{E}h(\widehat{S}^{n,(0,x)}_{T\wedge \tau ^{n,M}})\le\mathbb{E}(\widehat{S}^{n,(0,x)}_{T\wedge\tau^{n,M}})=x$ since h(x)≤x. Therefore, we obtain $\mathbb{E}h(\widehat {S}^{n,(0,x)}_{T})\to x$ as n→∞.  □

1.2 A.2 Proof of Proposition 3.6

Note that we cannot immediately conclude about the existence of a solution of (3.7) because the operator is not uniformly parabolic on (0,∞)×(0,1). This is why we bring the problem back to another one whose domain satisfies the required uniform parabolicity.

Fix $n\in\mathbb{N}$. Recall from Lemma 3.2 that $\widehat {S}^{n}$ is the unique solution of the stochastic equation

$$\begin{aligned} \widehat{S}^{n,(t,x)}_s =& x + \int_t^s \hat{\gamma }_n(\widehat {S}^{n,(t,x)}_u) \,dW_u , \quad t\le s \le1,\ (t,x)\in[0,1]\times (0,\infty), \end{aligned}$$

where we use the superscript (t,x) in order to emphasize the initial condition. Introducing $\hat{\gamma}_{n}^{m}:x\mapsto\sqrt{\sigma^{2}x^{2}+\sigma \gamma_{n} |x|+m^{-1}}$, we denote by $\widehat{S}^{n,m}$ the solution of

$$\begin{aligned} \widehat{S}^{n,m,(t,x)}_s =& x + \int_t^s \hat{\gamma }_n^m(\widehat{S}^{n,m,(t,x)}_u) \,dW_u , \quad t\le s \le1,\ (t,x)\in [0,1]\times(0,\infty), \end{aligned}$$

for any m>0. Since $\| \hat{\gamma}_{n}^{m}-\hat{\gamma}_{n} \|_{\infty}\le m^{-1/2}$ for m>0, we obtain $\widehat{S}^{n,m,(t,x)}_{1} \to \widehat{S}^{n,(t,x)}_{1}$ in $\mathbb{L}^{2}$ as m goes to ∞, uniformly in (t,x)∈[0,1]×(0,∞). We deduce that the functions $\widehat{C}^{n,m}(t,x) := \mathbb{E}[ h^{n}(\widehat {S}^{n,m,(t,x)}_{1}) \mid \mathcal{F}_{t}]$ converge uniformly, as m→∞, to the function $\widehat{C}^{n}(t,x) := \mathbb{E}[ h^{n}(\widehat{S}^{n,(t,x)}_{1}) \mid \mathcal{F}_{t}]$.

Applying Lemma 5.3.3 from [9] with Condition (A′) together with the estimate |∇h ⁿ|≤L implies that

$$\begin{aligned} \big|\widehat{C}^{n,m}(t,x) - \widehat{C}^{n,m}(u,y)\big|\le L \sqrt {\mathbb{E}\big| \widehat{S}^{n,(x,t)}_1 - \widehat {S}^{n,(u,y)}_1\big|^2} \le K \sqrt{(x-y)^2+|t-u|} \end{aligned}$$

for m>0, 0≤t,u≤1, and x,y≤|R| for a given R∈(0,∞), where the constant K depends on n, m, and R. We deduce that $\widehat{C}^{n,m}$ is continuous for any m>0, and hence so is $\widehat{C}^{n}$.

Fix m>0. We use arguments of Sect. 6.3 in [9] and try to follow their notation. Let us consider the sets

$$\begin{array}{rl@{\qquad}rl} Q_m :=& (0,1)\times\bigg(\dfrac{1}{m},m\bigg), & B_m &:= \{ 1\} \times\bigg(\dfrac{1}{m},m\bigg) , \\ T_m :=& \{0\} \times\bigg(\dfrac{1}{m},m\bigg) , & S_m &:= [0,1) \times\bigg\{ \dfrac{1}{m},m\bigg\} . \end{array} $$

For each y∈S _m , it is easy to observe that there exists a closed ball $K_{y}^{m}$ such that $K_{y}^{m}\cap Q_{m}=\emptyset$ and $K_{y}^{m}\cap \overline{Q_{m}}=\{y\}$. It follows that the function W _y proposed in [9, Eq. (2.4) in Chap. 6] defines a barrier for each y∈S _m . Besides, $\widehat{C}^{n}$ and h ⁿ are continuous, and $\hat{\sigma}_{n}$ is Lipschitz on $\overline{Q_{m}}$. By [9, Theorem 6.3.6] we deduce that the Dirichlet problem

$$\begin{array}{rl@{\quad}l} \displaystyle u_t(t,x)+\frac{1}{2}\hat{\sigma }_n^2(x)x^2u_{xx}(t,x)&=0, &(t,x)\in Q_m \cup T_m,\\ \displaystyle u(T,x) &= h^n(x),& x\in B_m,\\ \displaystyle u(t,x) &= \widehat{C}^n(t,x),&(t,x)\in S_m, \end{array} $$

admits a unique solution u ^n,m, which is continuous on $\overline {Q_{m}}$ with continuous derivatives $u^{n,m}_{t}$, $u^{n,m}_{xx}$ on Q _m ∪T _m . Moreover, [9, Theorem 6.5.2] implies that u ^n,m has the stochastic representation

$$\begin{aligned} u^{n,m}(t,x) =& \mathbb{E}\Big( \widehat{C}^n \big(\tau^m,\widehat {S}^{n,(t,x)}_{\tau^m}\big){\mathbf{1}_{\{\tau^m< 1\}}} + h^n\big(\widehat {S}^{n(t,x)}_1\big){\mathbf{1}_{\{\tau^m=1\}}} \Big) , \quad(t,x)\in Q_m, \end{aligned}$$

where τ ^m is the first time when $\widehat{S}^{n,(t,x)}$ exits Q _m . The definition of $\widehat{C}^{n}$ implies

$$\begin{aligned} u^{n,m}(t,x) =& \mathbb{E}\widehat{C}^n \big(\tau^m,\widehat{S}^{n,(t,x)}_{\tau ^m}\big) = \mathbb{E}h^n\big(\widehat{S}^{n(t,x)}_1\big) = \widehat{C}^{n}(t,x) , \quad(t,x)\in Q_m. \end{aligned}$$

As m→∞, we deduce that $\widehat{C}^{n}$ solves the PDE (3.7). Moreover, the function $\bar{C}^{n}: (t,y) \mapsto\widehat{C}^{n}(t,e^{y})$ solves the uniformly parabolic PDE

$$\begin{array}{rl@{\quad}l} \displaystyle v_t(t,y)+\frac{1}{2}\hat{\sigma }^2_n(e^y)v_{yy}(t,y)-\frac{1}{2}\hat{\sigma }_n^2(e^y)v_{y}(t,y)&=0,& (t,y)\in[0,1)\times\mathbb{R},\\ \displaystyle v(1,y)&=h(e^y),& y\in\mathbb{R}. \end{array} $$

By [9, Theorem 6.3.6], $\bar{C}^{n}$ is also the unique solution of the same PDE restricted to an arbitrary smooth bounded domain. Moreover, [9, Theorem 6.5.2] implies that $\bar{C}^{n}$ has a unique probabilistic representation. We deduce that $\widehat{C}^{n}$ is the unique solution of (3.7).  □

1.3 A.3 Classical properties of the Malliavin derivative

In this section, we simply recall classical properties of Malliavin calculus, which are widely used in the derivation of sensitivity estimates in this paper. For a more detailed presentation of the Malliavin calculus and its application in finance, see [10].

We first detail its relation with the first variation process. Let X be a one-dimensional Itô process with dynamics

$$\begin{aligned} d X_t =& b(X_t) \,dt+\sigma(X_t) \,dW_t, \end{aligned}$$

with differentiable drift and diffusion coefficients. Its first variation process ∇X solves the stochastic differential equation

$$\begin{aligned} d \nabla X_t =& b'(X_t)\,dt + \sigma'(X_t) \,dW_t, \quad\nabla X_0 = 1. \end{aligned}$$

Then the Malliavin derivative of X can be computed via the relation

$$\begin{aligned} D_s X_t =& \nabla X_t (\nabla X_s)^{-1}\sigma(X_s), \quad0\le s \le t \le T. \end{aligned}$$

Besides, if g is a $C^{1}_{b}$ function, we have

$$\begin{aligned} D_s g(X_t) =& g'(X_t)\nabla X_t (\nabla X_s)^{-1}\sigma(X_s), \quad0\le s \le t \le T. \end{aligned}$$

We finally recall the integration-by-parts formula. For a given Malliavin-differentiable random variable ϕ and a stochastic process u, we have

$$\begin{aligned} \mathbb{E}\int_0^\infty(D_t \phi) u_t \,dt =& \mathbb{E}\phi\int_0^\infty u_s \,dW_s, \end{aligned}$$

where the last stochastic integral is of Skorokhod type and coincides with the classical Itô integral whenever u is $\mathbb{F}$-adapted.