AppliedNumeric answerFree solution

A stock is at $100\$100 and in one year moves to either $120\$120 or $80\$80. Rates are zero. What is a $100\$100 European call worth?

Answer with a number. Fractions, powers and expressions like 23/6 or C(52,5) are read correctly in practice.

Show the answer and worked solution

Answer

1010

Build the hedge rather than guessing a probability. The call pays 2020 up and 00 down, so its delta is (20−0)/(120−80)=0.5(20-0)/(120-80) = 0.5 shares. Equivalently, solve for the risk-neutral probability that makes the stock a martingale: q=(erT−d)/(u−d)=(1−0.8)/(1.2−0.8)=0.5q = (e^{rT} - d)/(u - d) = (1 - 0.8)/(1.2 - 0.8) = 0.5, and the call is 0.5(20)+0.5(0)=$100.5(20) + 0.5(0) = \$10. The two routes agree because they are the same argument, and note that q=0.5q = 0.5 here is a coincidence of the numbers, not a general fact.

Worked solution

  1. Formula
    q=erT−du−d,C=e−rT[qfu+(1−q)fd]q = \frac{e^{rT} - d}{u - d}, \qquad C = e^{-rT}\big[q f_u + (1-q)f_d\big]
  2. Substitute
    q=1−0.81.2−0.8=0.5q = \frac{1 - 0.8}{1.2 - 0.8} = 0.5
  3. Solve
    C=0.5×20+0.5×0C = 0.5 \times 20 + 0.5 \times 0
  4. Answer
    C=$10C = \$10

Sanity check. The hedge is half a share: buy 0.5 shares for 50, borrow 40, and you have the call for 10.

Takeaway: Price by replication: the hedge sets the price, not anyone’s view.

Answer it in practice – your answer is marked and recorded.

Reported in interviews at

Optiver, SIG (Susquehanna), IMC Trading

More option pricing models questions

A stock is at $100 and in one year moves to either $120 or $80. · QuantMax