Lecture 2 Student(1)

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Derivative SecuritiesLecture 2 Futures Pricing and Trading (Chapters 3 and 5)

Nicholas ChenDr. Nicholas Chen, ICMA centre, 20111

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Review of Last LectureDifference between futures and forwardDifference between futures and optionsYour own growth option a decision to increase your salaryReal options pricing of a growth options for a firm Dr. Nicholas Chen, ICMA centre, 20112

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OutlineDetermination of Forward and Futures PricesThe value of Forward/Futures ContractsGeneral Formula of the Forward and Futures PricesReview of Capital Asset Pricing Model (CAPM)

Dr. Nicholas Chen, ICMA centre, 20113

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Forward vs. Futures PricesWhat makes the difference between them?___________in exchanges reduces the credit risk for the futures contract.We assume forward and futures contracts are the same in this module by ignoring the settlement.

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2.5Determination of Forward and Futures PriceProcedureUse no arbitrage ruleForm riskless portfolio => it should earn risk free rate (compare alternative strategies)Solve for forward priceDr. Nicholas Chen, ICMA centre, 2011

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Revisiting the Previous Arbitrage ExampleGold: An Arbitrage Opportunity?Suppose that:The spot price of gold is US$390The quoted 1-year futures price of gold is US$425The 1-year US$ interest rate is 5% per annumIs there an arbitrage opportunity? Dr. Nicholas Chen, ICMA centre, 20111.6

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Generalizing the Previous Arbitrage ExampleFor any investmentSuppose that:The spot price of an investment is S0The quoted 1-year futures price of this investment is F0The annual interest rate is rWhat is the futures price? Dr. Nicholas Chen, ICMA centre, 20111.7

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Dr. Nicholas Chen, ICMA centre, 20112.80Maturity = T|------------------------------------------------|Build a risk free portfolio: Short Forward contract and borrow to buy underlying assetTime 0Time TForward: 0 (no upfront cost)deliver (or sell ) asset at F0Cash : +S0 (borrow) - S0 (to buy) Pay back the loan -S0erTBy no arbitrage, since the strategy requires no money down, this has to hold =>___________ =0=>____________At the time 0, F0 is the delivery price K, which has to be S0erT.

Price for a forwards/futures contract

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Generalization (Cont)For any investment asset that provides no income and has no storage costs at time 0 when an contract is entered into, F0 = K = S0erT T: time until delivery date in a forward contract (in years)S0: price of asset underlying the forward contract today (spot price)K: delivery price in forward contractF0: forward price todayr: risk-free rate per annum (with continuous compounding) for an investment maturing at delivering date (in T years)

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Dr. Nicholas Chen, ICMA centre, 20112.10Prices of Forwards/Futures Change over timeAt a later date t, Ft = Ster(T t)

Why does the futures prices change over time?Because St and time t change.

When t -> T, at the maturity TFT = ST,because e-r(T t) = 1 (when t -> T)

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Convergence of forwards/futures to spot price (when t T)

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The value of a Forward/ Contract (Chap 5.7)The price of a forward contract is not its value.The value of a long forward contract, , at time t is = (Ft K )er(T-t) Similarly, the value of a short forward contract is f = (K Ft ) er(T-t) where K is delivery price in a forward contract & Ft is forward price that would apply to the contract at time t.

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Futures Value of Margin Account

Dr. Nicholas Chen, ICMA centre, 201114Taking a Long position in a futures contract does not require the upfront investment, but needs to deposit an initial margin in a futures exchange. The changes in futures prices change cause the fluctuation of the margin account balance.

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Practice Problem:Value of a ForwardA one-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.What are the forward price of today and the initial value of the forward contract? Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What are the forward price and the value of the forward contract? Dr. Nicholas Chen, ICMA centre, 201115

Practice Problem (cont)a) The forward price at , F = ______________The initial value of the forward contract, f = ___________.

b) The forward price is Ft = ___________________The delivery price in the contract is k = ______________. = ________________________




Dr. Nicholas Chen, ICMA centre, 20112.180Maturity = T|------------------------------------------------|Build a risk free portfolio: Short Forward contract and borrow to buy underlying assetTime 0Time TForward: 0 (no upfront cost)sell the asset at F0Cash : +S0 (borrow) - S0 (to buy) Pay back the loan -S0erTIncome (I) : I IerT By no arbitrage, since the strategy requires no money down, this has to hold =>________________________ = 0 =>F0= (S0-I )erT

When underlying assets pay income

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Dr. Nicholas Chen, ICMA centre, 20112.190Maturity = T|------------------------------------------------|Build a risk free portfolio: Short Forward contract and borrow to buy underlying assetTime 0Time TForward: 0 (no upfront cost)sell the asset at F0Cash : +S0 (borrow) - S0 (to buy) Pay back the loan -S0erTIncome : I IerTStorage : -U -UerTBy no arbitrage, since the strategy requires no money down, this has to hold =>_______________________ = 0 => F0= (S0- I +U)erT

When underlying assets need storage fee

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General Pricing Formula for a Futures Contract of an Investment Asset A discrete time version F0= (S0- I +U)erT A continuous time versionF0 = S0 e (r q + u)T = S0 e cT The cost of carry, c, is the interest costs, r, less the income rate, q, earned plus the proportional storage rate, u. c= r q + u

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Cost of Carry in the General Pricing Formula c= r q + u if the underlying is an investment assetNo income (non dividendpaying stock, discount bond)c = rA known cash income rate, q (stock paying known dividend, coupon bearing bond)c = r - qA known storage cost, u (commodity)c = r q + u



A foreign currency is analogous to a security providing an incomeThe holder of the currency can earn interest at the risk free interest rate, rf, prevailing in the foreign country. Futures and Forwards on Currencies

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The Cost of Carry for Futures of A Consumption AssetThe cost of carry, c, is the storage cost plus the interest costs less the income earnedFor an investment asset F0 = S0ecT For a consumption asset F0 < S0ecT because you can consume such a physical asset.Futures is less valuable than the physical asset particularly when such an asset is short of stock so that we replace c with c y, where y is the convenience yield on the consumption asset. F0 = S0 e(cy )T

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Practice Example 1The risk-free rate of interest is 7% per annum with continuous compounding, and the dividend yield on a stock index is 3.2% per annum. The current value of the index is 150. What is the six-month futures price? r = 0.07 and q = 0.032F = ___________________Dr. Nicholas Chen, ICMA centre, 201124

Practice Example 2The two-month interest rates in Switzerland and the United States are 2% and 5% per annum, respectively, with continuous compounding. The spot price of the Swiss franc is $0.8000. The futures price for a contract deliverable in two months is $0.8100. What arbitrage opportunities does this create? 1) The theoretical futures price is F = _______________2) The actual futures price is too ________. This suggests that an arbitrageur should buy Swiss francs and short Swiss francs futures. Dr. Nicholas Chen, ICMA centre, 201125

Practice Example 3The spot price of silver is $15 per ounce. The storage costs are $0.24 per ounce per year payable quarterly in advance. Assuming that interest rates are 10% per annum for all maturities, calculate the futures price of silver for delivery in nine months.Dr. Nicholas Chen, ICMA centre, 201126

Practice Example 3 (cont)The present value of storage cost0.24/4 x(1 + e-0.10x0.25 + e-0.10x0.5) = 0.176The price of the futures isF0 = (15 + 0.176) e0.1x0.75 = 16.36




CAPM IntuitionWhich investment will you pick according to our mean-standard deviation rule?No RiskE(R) = 5% Median RiskE(R) = 5%Investment AInvestment B


CAPM Intuition (Cont.)You will take the investment B only if you are rewarded with the additional premium of 7% for extra risk you are bearing

E(R) of Investment B = Risk-free rate + Risk premium = 5% + 7%No Riskr = 5%Median Riskr = 5%Risk Premium7%Dr. Nicholas Chen, ICMA centre, 201130

CAPM Intuition (Cont.)

E(R) of Investment C = Risk-free rate + Risk premium = 5% + ?

Median Riskr = 5%Risk Premium7%High Riskr = 5%Risk Premium?Investment BInvestment CDr. Nicholas Chen, ICMA centre, 201131

CAPM Intuition (cont.)We know that investment C is riskier than investment B, so the risk premium of C should be greater than the one of B by a factor of > 1, or x 7%.

E(R) of Investment B = Risk-free rate + Risk premium of B12% = 5% + 7%=> 7% = 12% - 5%E(R) of Investment C = Risk-free rate + Risk premium of C = 5% + x 7% = 5% + x (12% - 5%)


CAPM Intuition (cont.)Let investment B be the investment in the market portfolio.Its total return of 12% is the expected market return, E(Rm) Investment C be any investment, i.Its expected return is kWe know 5% is the risk free rate r

Replacing the numbers with the symbols

K of Investment C = 5% + (12% - 5%)k = r + (E(Rm) r)

That is the CAPM that won the Nobel Prize!


Your personal preference is not priced by the market


Interpretation of the CAPMThe expected, or required, for any investment can be calculated as

measures the risk of any investment relative to an investment in the market portfolio.

= 1: Portfolio returns mirror returns on market. = 1.5: Excess returns on portfolio tend to be 1.5 times the excess returns on market. = 0.5: Portfolios Excess return tend to be half of excess return on marketDr. Nicholas Chen, ICMA centre, 201135k = r + (E(Rm) r)

Lecture 2 Student(1)

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