**Author details**

14 Will-be-set-by-IN-TECH

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

**Figure 1.** Optimal buy (lower) and sell (upper) boundaries from negative exponential (CARA) utility maximization for a short asset-settled call with strike price *K* = 20 (solid black lines) and for the case of no option (solid red lines), with proportional transaction costs incurred at the rate of *λ* = *μ* = 0.5%, absolute risk aversion *γ* = 2.0, risk-free rate *r* = 8.5%, asset return rate *α* = 10% and asset volatility *σ* = 5%, at 1.5, 0.5, 0.25 and 0.1 period(s) from expiration *T*. For each pair of boundaries, the "buy asset"

For the so-called bounded variation follower problems, the equivalence between singular stochastic control and optimal stopping can be harnessed to provide a much simpler solution to the control problem by solving the corresponding Dynkin game. This approach can be

region is below the buy boundary and the "sell asset" region is above the sell boundary; the no-transaction region is between the two boundaries. The dashed lines correspond to the case of no

Stock Price 10 15 20 25 30 35 40 45

**0.1 period from T**

Stock Price 10 15 20 25 30 35 40 45

**0.5 period from T**

116 Stochastic Modeling and Control Singular Stochastic Control in Option Hedging

Stock Price 10 15 20 25 30 35 40 45

**0.25 period from T**

Stock Price 10 15 20 25 30 35 40 45

**1.5 periods from T**

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

transaction costs.

**4. Conclusion**

Tze Leung Lai *Department of Statistics Stanford University, U.S.A.*

Tiong Wee Lim *Department of Statistics and Applied Probability National University of Singapore, Republic of Singapore*
