## Pricing Capped Calls with Automatic Exercise

A capped call with automatic exercise is an option where the payoff is capped at a given level L. The auto exercise feature further means that whenever the option’s intrinsic value touches or exceeds L, the option will automatically be exercised and the holder will be paid L. The following adjustments will be made in the basic option pricing procedure:

**Step 2**: The terminal payoff will be min{L, max[(S_{T}-K),0]}.

**Step 3**: The following condition will be added to the formula at non-terminal nodes. If the underlying asset’s price at the node, S, is greater than or equal to K+L then the option will be exercised immediately and the option value at that node, C, will be equal to L. If S is less than K+L then the option will not be exercised and the option value at that node, C, will be the expected present value of the values at the successive nodes as given by formula (2). For time step, n=2 this is given below:

Column | A | B | C |

Row | S_{T} |
C_{T} |
T-1?t |

1 |
=u^{2}S_{0} |
min[L, max{(A1-K),0}] | =if[A1>=K+L, L, exp(-r?t)*{p*B1+(1-p)*B2}] |

2 |
=A1*d | min[L, max{(A2-K),0}] | = if[A2>=K+L, L, exp(-r?t)*{p*B1+(1-p)*B3}] |

3 |
=A2*d | min[L, max{(A3-K) ,0}] | = if[A3>=K+L, L, exp(-r?t)*{p*B2+(1-p)*B4}] |

4 |
=A3*d | min[L, max{(A4-K) ,0}] | = if[A4>=K+L, L, exp(-r?t)*{p*B3+(1-p)*B5}] |

5 |
=A4*d | min[L, max{(A5-K),0}] | = if[A5>=K+L, L, exp(-r?t)*{p*B4+(1-p)*B5}] |

Figure 90: Formulas for spreadsheet implementation of binomial tree for capped calls with automatic exercise

The adjusted formulas in this column will then be copied to D and later columns depending on the number of steps employed.

For the following parameters the numerical example is given below:

n | 4 | T | 0.1 | x | 0.9988 | L | 5 | u | 1.0486 |

r | 0.05 | ?t | 0.025 | K | 45 | S_{0} |
47 | ||

? | 0.3 | d | 0.9537 | p | 0.5013 | q | 0.4987 |

Column | A | B | C | D | E | F |

Row | S_{T} |
C_{T} |
T-1?t | T-2?t | T-3?t | T-4?t |

1 |
56.820 | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 |

2 |
54.187 | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 |

3 |
51.677 | 5.000 | 5.000 | 5.000 | 5.000 | 5.000 |

4 |
49.283 | 4.283 | 3.500 | 3.572 | 3.625 | 3.661 |

5 |
47.000 | 2.000 | 2.145 | 2.251 | 2.323 | 2.501 |

6 |
44.823 | – | 1.001 | 1.074 | 1.377 | 1.431 |

7 |
42.746 | – | – | 0.501 | 0.538 | 0.814 |

8 |
40.766 | – | – | – | 0.251 | 0.269 |

9 |
38.877 | – | – | – | – | 0.126 |

Figure 91: Spreadsheet for capped calls with automatic exercise example

The price of the option is 2.501.

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