No-arbitrage bounds for the forward smile given marginals

Abstract

We explore the robust replication of forward-start straddles given quoted (Call and Put options) market data. One approach to this problem classically follows semi-infinite linear programming arguments, and we propose a discretisation scheme to reduce its dimensionality and hence its complexity. Alternatively, one can consider the dual problem, consisting in finding optimal martingale measures under which the upper and the lower bounds are attained. Semi-analytical solutions to this dual problem were proposed by Hobson and Klimmek [Financ. Stochastics, 2015, 19, 189–214] and by Hobson and Neuberger [Math. Financ., 2012, 22, 31–56]. We recast this dual approach as a finite-dimensional linear program, and reconcile numerically, in the Black–Scholes and in the Heston model, the two approaches.

Keywords:

• Martingale optimal transport
• Robust bounds
• Forward-start
• Heston

AMS Subject Classifications:

• 91G20
• 91G80
• 90C46
• 90C05
• 90C34

Acknowledgements

The authors are indebted to Claude Martini for stimulating discussions, and to the anonymous referees for helpful suggestions. AJ acknowledges financial support from the EPSRC First Grant EP/M008436/1.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the EPSRC [grant number EP/M008436/1].