5.08.2005

Q&A: What is the price of a call option that expires in 6 months...

...assuming a stock price is $80 and in the next year it will either 90 or 75. The risk free interest rate is 6%. A call option on this stock has an exercise price of $85.

CppQuant Answer

price via the binomial options model

  • Diagram Stock Price Dynamics and Option Values on "Trees"
    • Stock Price
      • Up state: $90
      • Down state: $75
    • Call Price
      • Up state: c+ = Max (0, 90 - 85) = $5.
      • Down state: c- = Max (0, 75 - 85) = $0.
  • Compute Risk Neutral Probabilities of Up and Down States
    • Up state price relative: = (S+ / S0) = 90/80 = 1.12
    • Down-state price relative: d = (S- / S0) = 75/80 = 0.94.
    • Risk-neutral "up" probability: p = ( (1 + r)^t - d) / (u - d) = ( (1.06)^0.5 0.9375) / (1.12 0.94) = 0.49
  • Compute Expected Value of Call Option: c = ( c+*p + c-*(1-p) ) / (1 + r)^t = (0.49* 5 + (1-0.49)* $0) / (1.06)^0.5= $2.39

Category: C++ Quant > Derivatives > Options > Valuation

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