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Lesson 2: Forward Contracts I Part 2

Formalize: Forward Contracts on Investment Assets with Known Yield
(One Dividend)

How many shares of a stock do you need to buy at t = 0 such that you will end up having exactly one share of the stock at t = T, but without accumulating dividends? The stock will pay a known dividend rate of q at t  = t1 < T.

The answer to the question is buy (1 - q) shares at t = 0.

Note:

  1. The dollar amount of the dividend is: Dt1 = q · St1.
  2. The cost of obtaining a share of the stock that pays ST at t = T is: S0 (1 - q)
  3. Therefore, F0 = S0 (1 - q)(1+r)T - - - (discrete compounding) or

                            F0 = S0 (1 - q)erT - - - (continuously compounding)

Proof: If you buy one share at t = 0, you will have (ST + div), not ST, at the maturity, T. It is important to differentiate three different points in time: t = 0, t = t1, t = T

  1. At t = 0, buy (1 - q) shares
  2. At t = t1,

    ⇒
    On ex-dividend, the total number of shares =
    (1 - q) + q = 1 share. This will remain until t = T.

    • receive the dividend, qSt1 per share on (1 - q) shares: (1 - q) Dt1 = (1 - q) q St1
    • Invest dividends by buying q units of the stock
      • Ex-dividend stock price = (1 - q)St1 [b/c Spot price declines from St1 after dividend by Dt1 = q St1]
      • No. of the stock: (1 - q) q St1 / [(1 - q) St1 ] = q shares
Q.E.D. ("quod erat demonstrandum" ("that which was to be demonstrated"), is often placed at the end of a mathematical proof to indicate its completion)

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