5.02.2005

Q&A: What should be the price of a European call...

...with an exercise price of $30 expires in 90 days, given that a European put with the same exercise price, expiration date and underlying is selling for $6. The underlying is selling for $40, and the risk free rate is 10%.

CppQuant Answer

compute the value of the synthetic call.

  • c(0) = p(0) + S(0) - X/(1 + r)^T = 6 + 40 - 30/(1 + 10%)^(90/365) = $16.7
    • time to expiration T = 90/365 = 0.2466
  • Since the synthetic call and the actual call have the same payoff, they must have the same price ($16.70) as well.

Bonus Points

  • Synthetic positions enable us to price options, because they produce the same results as options and have known prices (assuming the underlying assets make no cash payments)
  • c(0) = p(0) + S(0) - X/(1 + r)^T : the rhs is known as a synthetic call, consisting of a long put, a long position in the underlying, and a short position in the risk-free bond.
  • p(0) = c(0) + X/(1 + r)^T - S(0) : the rhs is known as a synthetic put, consisting of a long call, a short position in the underlying, and a long position in the risk-free bond.
  • p(0) + S(0) - c(0) = X/(1 + r)^T : the lhs is known as a synthetic bond.

Category: C++ Quant > Derivatives > Options

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