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## Monday, December 30, 2013

### replicate eq-fwd contract, assuming a single dividend

http://www.matthiasthul.com/joomla/attachments/article/121/ForwardReplication.pdf

Note replication portfolio is always purchased as a bundle, sometime (time t) before expiry (denoted time T).

First, let's review how to replicate a forward contract in the absence of dividends. The replication portfolio is {long 1 share, short K discount bonds}. To verify, at T the portfolio payout is exactly like long forward. By arbitrage argument, any time before expiry the portfolio value must at all times equal the fwd contract's price. I will spare you the math formula, since the real key behind the math is the replication and arbitrage.

Now, suppose there's a percentage dividend D paid out at time Td before T. In this case, let's assume the dividend rate D is announced in advance. To reduce the abstractness, let's assume D=2%, K=\$100, the stock is IBM. We are going to reinvest the dividend, not use it to offset the purchase price \$100. (This strategy helps us price options on IBM.)

The initial replication portfolio now adjusts to --{ long 0.98 IBM, short 100 discount bonds}. At T, the portfolio is exactly like long 1 forward contract. Please verify!

(In practice, dividends are declared as fixed amount like \$0.033 per share whatever the stock price, but presumably an analyst could forecast 2%.)

In simple quant models, there's a further simplification i.e. continuous dividend yield q (like 2% annually). Therefore reinvesting over a period A (like 1Y), 1 share becomes exp(qA) shares, like exp(0.02*1) = 1.0202 shares.

Q: delta of such a fwd contract's pre-maturity value? Math is simple given a good grip on fwd contract replication.
A: rep portfolio is { +1 S*exp(-qT),     -K bonds }.
A: key concept -- the number of shares (not share price) in the portfolio "multiplies" (like rabbits)  at a continuous compound rate of q. Think of q = 0.02.
A: In other words

F0 = S0*exp(-qT) - K*Z0

Differentiating wrt S0, delta = exp(-qT), which degenerates to 1 when q=0.