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  1. 009A stock trades at 100 and in one period will be either 120 or 80. Interest rates are zero. Price a call option struck at 100 by building a portfolio of shares and borrowing that copies it, and explain why the real-world probability of the up move does not appear in the price.Pricing, options and index mathsCoreOptions market makingQuant trading

    Try it first

    If you believe the stock goes up with probability 90%, what is the call worth?

    Show the worked solution

    The call is worth 10. It pays 20 if the stock goes to 120 and 0 at 80. Half a share pays 60 or 40, so half a share with a loan of 40 pays 20 or 0, exactly the call. That portfolio costs 50 - 40 = 10 today. If the call traded at any other price, you could buy the cheap one and sell the dear one for a riskless profit, so no probability is needed.

    How do you build the copy?

    Match the swing first. The call's payoff moves by 20 between the two states while the stock moves by 40, so the copy needs 20/40 = 0.5 of a share: that ratio is the option's deltaHow much an option's value changes for a one-unit change in the underlying price; here, the number of shares that copies the option.. Half a share is worth 60 or 40 at the end, which is 40 more than the call in both states. Borrow 40 today, repay 40 at the end with zero interest, and the copy pays exactly 20 or 0.

    Copy the payoff with shares and borrowing, and price the copyStock 100Call ?Stock 120Call pays 20Stock 80Call pays 0updownDelta = (20 - 0) / (120 - 80) = 0.5 shareThe copy: 0.5 share, borrow 40TodayUp (120)Down (80)0.5 share506040Loan-40-40-40Total10200Matches the call in both statesCall = cost of the copy = 10No probability of up or down was used
    A call struck at 100 on a stock that moves to 120 or 80 is copied by half a share and a loan of 40, which pays 20 or 0 exactly as the call does and costs 10 today, so the call is worth 10 with no probability used.

    Why does the chance of the up move not matter?

    Think of a shop selling a bundle of two items that you can also buy separately. The bundle's price is pinned by the parts, whatever you think about how useful the items are. The call is a bundle of half a share and a loan; the share price already reflects everyone's views about the up move, so the option inherits them and adds none of its own. A 90% view is a reason to hold the stock itself, not a reason to pay more for the call than its parts cost.

    The relationship
    Δ=Cu−CdSu−Sd=20−0120−80=0.5C0=ΔS0−B=50−40=10=q Cu+(1−q) Cd,  q=S0−SdSu−Sd=0.5\Delta = \frac{C_u - C_d}{S_u - S_d} = \frac{20-0}{120-80} = 0.5 \qquad C_0 = \Delta S_0 - B = 50 - 40 = 10 = q\,C_u + (1-q)\,C_d,\; q = \frac{S_0 - S_d}{S_u - S_d} = 0.5
    \Deltashares held in the copy
    Bthe amount borrowed, 0.5 x 80 - 0 = 40
    qthe risk-neutral weight on the up state, fixed by the prices, not by beliefs
    What it says in wordsThe copy's cost gives the price, and the same price is an average of the payoffs using weights set by today's stock price.

    What would you do if the call traded at 12?

    Sell the dear thing and buy the cheap one. Sell the call for 12, buy half a share for 50 and borrow 40, a net cash inflow of 2 today; at the end the portfolio pays exactly what you owe on the call in either state. The 2 is kept whatever happens. The weight q = 0.5 that reproduces the price is called the risk-neutral probability, but it is a pricing weight backed out of the stock price, not a forecast. Say that distinction; interviewers listen for it.

    Where candidates lose it

    The trap is pricing the call as an expected payoff under your own view: 90% of 20 is 18. That price can be arbitraged against the stock, so nobody could trade it for long, and the interviewer wants to hear that the copying portfolio pins the price.

    The second loss is getting 10 by assuming a 50% chance. The number is right by coincidence of the symmetric tree; ask yourself what happens with an up move to 130, and the risk-neutral weight changes to 1/2.5 = 0.4.

    What the interviewer asks next

    • Price the put struck at 100 and check put-call parity.
    • What changes if interest rates are 5% for the period?
    • The stock can go to 130 or 80 instead. Price the call again.