Q&A: What is the forward price given the following...

...The options and a forward contract expire in 90 days. The continuously compounded risk-free rate is 5%, and the exercise price is 95. The call price is 10, and the put price is 3.9.

CppQuant Answer

According to the put-call-forward parity, c0 + [X - F(0, T)]/(1 + r)^T = p0 (where [X - F(0, T)]/(1 + r)^T is the bond PV)

  • F(0, T) = X - (p0-c0)*(1 + r)^T = 95-(3.9-10)*1.05^(90/365) = $101.17
  • A synthetic forward contract is a combination of a long call, a short put and a zero-coupon bond with face value (X - F(0, T)).
  • we may either short or long this bond, depending on whether the exercise price of these options is lower (ie. short if X < F(0, T)) or higher than the forward price.

Bonus Points

  • To see the put-call-forward parity: Consider two portfolios
    • Portfolio A consists of a long call and a long position in a zero-coupon bond with face value of X - F(0, T)
    • Portfolio B consists of a long put and a long forward.
    • At initiation, the value of Porfolio A = c0 + [X-F(0, T)]/(1+r)^T, Porfolio B = p0.
    • At expiration the value of the portfolios are
      • if S(t) <= X: Porfolio A = X-F(0,T) = Porfolio B
      • if S(t) > X: Porfolio A = S(t)-F(0,T) = Porfolio B
    • As two portfolios have exactly the same payoff, their initial investments should be the same as well. That is: c0 + [X - F(0, T)]/(1 + r)^T = p0
  • For a synthetic forward contract: consider a portfolio consisting of a long call, short put and a long position in a zero-coupon bond with face value of X - F(0, T). At expiration the value of the portfolio is:
    • if ST <= X: 0 (value of long call) + [-(X - ST)] (value of short put) + [X - F(0, T)] (value of long bond) = ST - F(0, T)
    • if ST > X. : [ST - X] (value of long call) + 0 (value of short put) + [X - F(0, T)] (value of long bond) = ST - F(0, T)
    • As a forward contract's payoff at expiration is also ST - F(0, T), the portfolio's initial value must be equal to the initial value of the forward contract (which is 0): c0 - p0 + [X - F(0, T)]/(1 + r)^T = 0

Category: C++ Quant > Derivatives > Options

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