# N compounding periods in each year rate of interest r

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n compounding periods in each year rate of interest r A invested for t years yields A (1 + r/n) t*n Continuous compounding corresponds to the situation where the length of the compounding period goes to zero. Therefore, an amount A invested for t years is worth lim n→∞ A(1 + r/n) t*n = A e rt at maturity

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Compounding period and the value of €1 with 100% interest per year Compounding Value Annually €1.00×2 1 = €2 Semi-annually €1.00×1.5² = €2.25 Quarterly \$1.00×1.25 4 = \$2.4414 Monthly \$1.00×(1.0833) 12 = \$2.613035 Weekly \$1.00×(1.00192) 52 = \$2.692597 Daily \$1.00×(1.0002739) 365 = \$2.714567 Continuous \$2.7182818
Continuous Compounding You have to deal correctly with compounding to take proper account of the time value of money. The calculations get a little messy when the number of holding periods in a year is not an integer. A simplification, which is used throughout derivatives mathematics, is to use continuously compounded rates.

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1. An Arbitrage Opportunity? Suppose that: The spot price of a non-dividend paying stock is \$40 The 3-month forward price is \$43 The 3-month US\$ interest rate is 5% per annum Is there an arbitrage opportunity? 10
2. Another Arbitrage Opportunity? Suppose that: The spot price of non-dividend paying stock is \$40 The 3-month forward price is \$39 The 3-month US\$ interest rate is 5% per annum Is there an arbitrage opportunity? 11

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Notation for Valuing Futures and Forward Contracts S 0 : Spot price today F 0 : Futures or forward price today 12 T : Time until delivery date r : Risk-free interest rate for maturity T
The Forward Price If the spot price of an investment asset is S 0 and the futures price for a contract deliverable in T years is F 0 , then F 0 = S 0 e rT where r is the 1-year risk-free rate of interest. In our examples, S 0 =40, T =0.25, and r =0.05 so that F 0 = 40 e 0.05×0.25 = 40.50 13

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If Short Sales Are Not Possible.. Formula still works for an investment asset because investors who hold the asset will sell it and buy forward contracts when the forward price is too low 14
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