5.21.2005

Q&A: What's the arbitrage strategy given...

...the following information on put and call options on a stock: Call price: 3.1, Put price: 9, Exercise price: 60, Forward price: 55, Days to option expiration: 180 days, The continuously compounded risk-free rate: 4%.

CppQuant Answer

As the actual put is more expensive, we should sell the put and buy the synthetic put (long call, short forward and long bond).

  • bond PV = [X - F(0, T)] / (1 + r)^T = (60 - 55) / 1.04180/365 = 4.9
  • The price of a synthetic put: p0 =short forward + c0 + [X - F(0, T)]/(1 + r)^T = 0 + 3.1 + 4.9 = 8.
  • The initial up-front cash: -3.1 (long call) - 4.9 (long bond) + 0 (short forward) + 9 = 1.
  • At expiration
    • short forward = -(ST - 55)
    • long bond = 60 - 55 = 5
    • If ST < 60: the portfolio would generate 0 (long call) - (ST - 55) (short forward) - (60 - ST) (short put) + (60 - 55) (long bond) = 0.
    • If ST >= 60: the portfolio would generate (ST - 60) (long call) - (ST - 55) (short forward) + 0 (short put) + (60 - 55) (long bond) = 0.
    • The strategy would generate 1 up-front without any investment or any amount to pay back later.

Category: C++ Quant > Derivatives > Options

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