5.11.2005

Q&A: What is the price of a call option that expires in 2 years...

Suppose the price of a stock is $100. It can either rise or fall by 10% in any 12 months. The risk-free rate is 6%. The exercise price of a call option on the stock is $100.

CppQuant Answer

  • Risk-neutral "up" probability: p = ( (1 + r)^t - d) / (u - d) = (1.06 - 0.9) / (1.1 - 0.9) = 0.8
  • Stock in 2 years (at expiration) (forward traversing the tree )
    • S++ = 100 x 1.1 x 1.1 = $121
    • S+- = S-+ = 100 x 1.1 x 0.9 = $99.
    • S-- = 100 x 0.9 x 0.9 = $81. * Call Prices in 2 years (at expiration)
    • c++ = Max (0, S++ - 100) = $21
    • c+- = c-+ = Max (0, S+- - 100) = $0.
    • c-- = Max (0, S-- - 100) = $0.
  • Call Prices in 1 year (traverse the tree backwards)
    • c+ = ( c++*p + c+-*(1-p) ) / (1 + r) = (0.8 x 21 + 0.2 x 0) / 1.06 = $15.85.
    • c- = ( c+-*p + c--*(1-p) ) / (1 + r) = 0
  • c = ( c+*p + c-*(1-p) ) / (1 + r) = (0.8 x 15.85 + 0.2 x 0) / 1.06 = $11.96.

Bonus Points

  • Different hedge ratios at each time point:
    • n+ = (c++ - c+-) / (S++ - S+-)
    • n- = (c-+ - c--) / (S-+ - S--)
    • n = (c+ - c-) / (S+ - S-)

Category: C++ Quant > Derivatives > Valuation

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