Pricing and hedging for a sticky diffusion

TL;DR

The paper analyzes pricing and hedging in a sticky diffusion extension of Black–Scholes, where the risky asset price $S$ spends positive time at a threshold $\zeta$ via local time and a stickiness parameter $\rho$. It establishes that No Arbitrage and No Free Lunch with Vanishing Risk (NFLVR) hold if and only if the interest rate $r=0$, and under this condition derives an arbitrage-free pricing equation with a replication representation obtained through martingale methods, while noting non-uniqueness of the replication strategy in the sticky setting. The authors show that every locally bounded payoff replicable in the standard Black–Scholes model is also replicable in the sticky model, and they prove price monotonicity: convex payoffs have prices that decrease with increasing stickiness; the replicable payoff class expands with $\rho$. Numerical experiments assess discrete-time hedging and model-mismatch effects, revealing that hedging errors decay at a rate similar to the classic BS model as hedging becomes finer, but misrepresenting stickiness (or using smooth SDEs) can cause irreducible hedging errors, highlighting practical importance of accurately modeling stickiness.

Abstract

We introduce a financial market model featuring a risky asset whose price follows a sticky geometric Brownian motion and a riskless asset that grows with a constant interest rate $r\in \mathbb R $. We prove that this model satisfies No Arbitrage (NA) and No Free Lunch with Vanishing Risk (NFLVR) only when $r=0 $. Under this condition, we derive the corresponding arbitrage-free pricing equation, assess replicability and representation of the replication strategy. We then show that all locally bounded replicable payoffs for the standard Black--Scholes model are also replicable for the sticky model. Last, we evaluate via numerical experiments the impact of hedging in discrete time and of misrepresenting price stickiness.

Figures (3)

• Figure 1: Simulated sample paths of the sticky (dark blue) geometric Brownian motion of parameters $(\mu=0,\sigma=0.25,\rho=1,\zeta=10)$ and the non-sticky (light blue) geometric Brownian motion of parameters $(\mu=0,\sigma=0.25)$. The approximation method used for the simulation is from AnaLejVil.
• Figure 2: Price and delta of the Black-Scholes and sticky Black-Scholes. Finite-elements approximations of the price (left) and delta (right) curves of a Call option of strike $K=11$ and $T=10$ in the $(r,\sigma)=(0,0.25)$ standard Black-Scholes model (blue), the $(r,\sigma,\rho,\zeta)=(0,0.25,1.0,10.0)$ sticky Black-Scholes model (green) and the $(r,\sigma,\rho,\zeta)=(0,0.25,2.0,10.0)$ sticky Black-Scholes model (orange).
• Figure 3: (top): Parts of risky asset in the payoff replication strategy for one sample path of $S$ with $\rho=0$, for (dark blue) $N=100$ and (light blue) $N=2000$. (bottom): Same, but with $\rho=1$.

Theorems & Definitions (38)

• Example 2.1: standard Black--Scholes model
• Remark 2.2
• Definition 3.2: (No) Arbitrage
• Definition 3.3: (No) Free Lunch with Vanishing Risk
• Definition 3.4: (No) Unbounded Profit with Bounded Risk
• Theorem 3.5: Theorem 12.1.2 of Delbaen2006
• Theorem 3.6: one-dimensional version of Theorem 12.3.5 of Delbaen2006
• Proposition 3.7
• proof
• Theorem 3.8