Calculation of One Period Binomial Option Pricing Models

ONE PERIOD BINOMIAL OPTION PRICING MODEL - CALCULATION |

| r | S0 | x (put) |

| risk-free rate | current stock price | Put option w / exercise price |

| 25% | 150 | 150 |

u | d |

stock goes down (1 + return) | stock goes up (1 - return) |

100 | 200 |

1.333 | 0.667 |

Pu = Max[0, X - uS0] | Cd = Max[0, X - dS0] |

Step 1 : Calculate Pu/Pd Pu =150-1.33(150) = 0 | Pd =$49.50 |

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Step 2 :Calculate the number of PUT options written against 1 share of stock - m |

m=S0(u-d)/(Pu-Pd) = | $150(1.333-0.667) / ($50-0) = | m = | 2.00 |

The portfolio should contain 1 share of stock long and Short 2 shares of PUT, to have final payoff equal. |

Step 3 : Portfolio Payoff in upside/downside - Value of portfolio $100.05 | $100.05 |

uS0 – mCu | dS0 – mCd |

1.33($150) – 2($50) = $100 | 0.667($150) – 2($0) = $100 |

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Step 4 : Calculate hedge riskfree portfolio : (So + -mP)(1 + rf) = uSo -mPu |

the value for P = S0[(1+rf)-u] + mPu / m(1+rf) |

p = [(1+rf) - d] / (u-d) | [(1+25%)-0.75]/(1.5-0.75) | p = | 0.875 |

(1-p) = [u - (1+rf)] / (u-d) | (1-0.667)=[1.5-(1+25%)] / (1.5-0.75) | 1-p = | 0.125 |

p = [pPu + (1-p)Pd] / (1+rf) | [(0.875*50 +0.125*0]/(1+25%) = | c = | $35.02 |

Step 5 : A Hedged Portfolio (=Initial Value of Portfolio)(=how much money you must put up) |

LONG 2 PUT and SHORT 1 share |

S0 – mP = (1 share)($150.00) – (2)($35.015) = $80 | v = | $80.04 |

Step 6 : The rate of return on the investment (at T) is: |

($80/$74.76) = 1.07 = 1 + rf.; rf = 7%. | rf = | 1.25 |

You invested $74.76 and got back $80, a 7 % return, which is the risk-free rate. |