Actuarial and Financial Maths B 1 Andrew Cairns 2008/9
4 Arbitrage and Forward Contracts 2 We will now consider securities that have random (uncertain) future prices. Trading in these securities yields random future cash flows.
4.1 Arbitrage and the No-Arbitrage Assumption 3 Loosely: Arbitrage is a risk-free trading profit. (also known as a free lunch) Formally: An arbitrage opportunity exists in a financial market if either: (i) an investor can make a deal which would give an immediate profit with no risk of future loss or (ii) an investor can make a deal at zero initial cost, has no risk of future loss and a non-zero probability of future profit.
Example 4.1 Bank A charges interest at 5% p.a. effective on loans Bank B provides interest at 6% p.a. effective on deposits = Arbitrage opportunity: Borrow as much as possible from A, deposit it with B, repay A eventually and make profit. Zero initial costs, but non-zero probability (=1) of profit. 4 Short Selling Pricing of financial instruments often involves short-selling. Short-selling means selling a security that is not owned with the intention of buying it later. Short-selling yields profits when the price of the security goes down and a loss if the price goes up.
5 Example 4.2 Your broker sells on your behalf shares of a security which are owned by another client. All income of that security during the period of lending has to be given to the owner. Shares must be bought back and given back to the owner. Margins (or collateral) are required for security reasons. Example 4.3 Arbitrage opportunity involving short-selling Two assets traded over 1 time period on a stock exchange. Suppose that during this period market prices either go up or down.
Price at time 1 Asset Price now Market falls Market rises 1 7 4 8 2 9 6 12 6
Trading strategy: 7 At time 0: Sell 3 units of asset 1 short position Buy 2 units of asset 2 long position = income (at time 0): 3 7 }{{} 2 9 }{{} (short-)selling asset 1 buying asset 2 = 21 18 = 3
At time 1: 8 Buy 3 units of asset 1 ( revised holdings =0) Sell 2 units of asset 2 Income (at time 1) if market falls: Income (at time 1) if market rises: = Arbitrage 2 6 }{{} 3 4 }{{} selling asset 2 buying asset 1 2 12 }{{} 3 8 }{{} selling asset 2 buying asset 1 (instant profit at time 0; no loss at time 1) = 0 = 0
How can arbitrage be prevented? 9 If at time 0: then no arbitrage price of asset 1 = 2 3 price of asset 2
No Arbitrage Assumption 10 or There is no such thing as a free lunch. Remarks: no arbitrage opportunity can exist (i) Reasonable assumption in liquid markets. Trades of arbitrageurs will eliminate arbitrage opportunities (by force of supply and demand) (ii) No arbitrage implies: If two sets of assets have exactly the same payout (value at some future time), they must have the same value now. Example above: 3 asset 1 = 2 asset 2 at time 1. Hence at time 0: price of asset 1 = 2 3 price of asset 2. (iii) Replicating portfolio for a given set of assets = portfolio with the same value at some future time in all states of the world. No arbitrage = replicating portfolio must have the same value as the given set of assets at any intermediate time. (iv) No arbitrage and replicating portfolio are key ideas in modern financial mathematics and financial economics.
4.2 Forward Price and Hedging 11 Set-up: Asset: current price is known, future price is uncertain. Examples: Equities (shares) Fixed amount of foreign currency 1 ounce of gold Number 1, Princes Street, Edinburgh Example 4.4 A British company knows that it will receive US-$ 1 million in 3 months. Problem: Unknown future exchange rate! Company contacts bank and agrees to sell US-$ 1 million in 3 month for the now agreed exchange rate 0.6854. = At time 0 they agree that the bank will exchange 685,400 for $1M in exactly 3 months.
12 Definition 4.5 A forward contract is an agreement to buy an asset at a certain future time for a certain price. The delivery price, K, is agreed at time t = 0 The delivery price is not paid until the delivery date, T In return for K the holder of the forward contract will receive one unit of the asset. The delivery price is chosen so that the value of the forward contract is zero at time 0. No payment is made at time 0. Definition 4.6 The forward price for a particular forward contract at a particular time (t 0) is the delivery price that would make the forward contract to have zero value at that time t 0. Example 4.7 (Example 4.4 continued) 1 month later: delivery price = 0.6854, forward price will be different (in general)!
General Terminology 13 The following terminology is used in the industry. Long position in forward contract = agree to buy the asset Short position in forward contract = agree to sell the asset risk-free rate of interest = rate of interest that can be earned without assuming any risks (e.g. treasury bills, bank account) Question: How to calculate the forward price of an asset? We use the No-Arbitrage assumption.
Assumptions: 14 No arbitrage no transaction costs known risk-free force of interest δ, at which money can be deposited and borrowed Notation: T is the delivery time (in years) S t price of the asset at time t [0, T ] K forward price
4.3 Forward Contracts on Securities with no Income 15 Example 4.8 stock: S 0 = 30 risk-free force of interest δ = 0.05 p.a. No dividends payable within the next two years. What is the two-years forward price for this asset?
(i) Assume, the two-year forward price is K= 35 (delivery time T = 2) 16 An arbitrageur can adopt the following strategy: today (time t=0) in 2 years (time T = 2) 1. Borrow 30 at δ = +30 30e 2 0.05 0.05 for 2 years 2. Buy the stock -30 0 3. Enter a f.c. to sell 1 stock for 35 (=K) in 2 years 0 +35 0 35 33.16 = 1.84 = Arbitrage Conclusion: Any forward price > 33.16 would allow this arbitrage strategy.
(ii) Assume, the two-year forward price is K= 31 (delivery time T = 2) 17 An arbitrageur can adopt the following strategy: today (time t=0) in 2 years (time T = 2) 1. Short the stock +30 0 2. Lend 30 at δ = 0.05 for 2 years 3. Enter a f.c. to buy 1 stock for 31 (=K) in 2 years 30 +30e 2 0.05 0-31 0 33.16 31 = 2.16 = Arbitrage Conclusion: Any forward price < 33.16 would allow this arbitrage strategy.
Summary: 18 Any forward price > 33.16 provides an arbitrage opportunity. Any forward price < 33.16 provides an arbitrage opportunity. Therefore, the fair forward price is 33.16. This is the only price that does not provide an arbitrage opportunity. General formula for the forward price: K = S 0 exp(δt ) Example 4.9 price of stock: 134.75 risk-free force of interest: δ = 3.99% p.a. for 3-months investments = 3-months forward price: K = 134.75 exp ( 0.0399 3 ) 12 = 136.10
Hedging 19 Consider the investor who has agreed to supply the asset at time T for the forward price K = S 0 exp(δt ). Strategy 1: Do nothing before time T. Buy the asset at time T and deliver if for the price K. Receive: at time t = 0: 0 at time t = T : K S T ( could be negative) Strategy 2: Borrow K exp( δt ) = S 0 at the risk-free force of interest δ, to be repaid at time T, and buy the asset now. Receive K at time T and repay the loan. Net cashflow: at time t = 0: Ke δt S 0 = 0 at time t = T : +(K S T ) K + S T = 0 At T : Forward contract receive K pay off bank loan Long 1 unit of stock hand over to holder of forward at T
Remarks: 20 Strategy 2 = No chance of profit and no risk of loss. Strategy 2 is called a static hedge for the forward contract. In general a hedge portfolio is a portfolio which eliminates or reduces the risk associated with an investment. There is a relationship between pricing and hedging.
4.4 Forward Contracts on Securities that Provide a Known Cash Income 21 Assume, the asset will pay a known cash dividend C at a known time t [0, T ] and that the risk-free force of interest is δ. We claim that the fair forward price, K, for delivery of one unit of stock, S T, at time T is: K = ( S 0 Ce δt) e δt Proof Suppose K > ( S 0 Ce δt) e δt. An arbitrage strategy is then: At time 0: Sell one forward contract (FC) with forward price K Borrow S 0 from the bank Buy one unit of stock at S 0
At time t: 22 Receive known cash dividend of C Reduce bank loan from S 0 e δt to S 0 e δt C At time T : Receive K under terms of FC Hand over your one unit of stock under terms of FC Repay bank loan, now at ( S 0 e δt C ) δ(t t) e Net cashflow at T is: K ( S 0 Ce δt) e δt > 0 Suppose K < ( S 0 Ce δt) e δt. Adopt the opposite strategy to get an arbitrage profit at T!
Example 4.10 (example 4.9 continued) S 0 = 134.75, δ = 3.99% p.a. for 3-months investments a dividend of 10.00 is paid in one month time. 23 = 3-months forward price: K = ( 134.75 10.00 exp ( 0.0399 )) ( exp 0.0399 3 ) 12 12 = 126.03
Remarks: 24 Dividend = known cash amount payable at t < T reinvest in the bank account Dividend at t < T = known % of share price at t use dividend to buy more shares Share price just before dividend payment at t = share price just after dividend payment at t plus amount of dividend payable at t S t = S t + div t
4.5 Forward Contracts on Foreign Exchange and Securities that Provide a Known Dividend Yield 25 Example 4.11 Forward contract on currencies: At time T : receive K in domestic currency in return for 1 unit of foreign currency Notation: S t = price at t of one unit of foreign currency δ= risk-free force of interest in domestic currency δ f = risk-free force of interest in foreign currency. Claim: K = S 0 exp {(δ δ f )T } (i) Suppose K > S 0 exp {(δ δ f )T }
Strategy: At time 0: Sell one forward contract (FC) with forward price K Borrow S 0 e δ f T in the domestic currency Use the bank loan to buy e δ f T units of the foreign currency Invest e δ f T units of the foreign currency in a foreign bank account At time T : Foreign bank account now worth e δ f T e δ f T = 1 Under the terms of the FC: receive K in the domestic currency hand over your one unit of foreign currency Repay the initial bank loan, now worth 26 S 0 e δ f T e δt = S 0 e (δ δ f )T Arbitrage profit at T K S 0 e (δ δ f )T > 0
(ii) If K < S 0 e (δ δ f )T then use the reverse strategy for an arbitrage profit. 27 Therefore, K = S 0 exp((δ δ f )T ) is the only forward price that does not allow any arbitrage opportunity.
Shares paying a constant dividend yield: 28 Assume the asset provides a known dividend yield, paid continuously at rate ρ(t) = gs t at time t (g > 0) dividends are immediately reinvested in the same asset Then, starting with 1 unit at time t = 0, the reinvestment strategy will accumulate to exp(gt ) units of the asset at T. Why? N(t) units at time t, total value N(t)S t For small dt: dividend payable at t + dt is gn(t)s t+dt dt This purchases gn(t)dt extra units at t + dt So dn(t) = gn(t) N(t) = N(0)e gt
Then: K = S 0 exp {(δ g)t } 29 Example 4.12 on 9 Feb 2004 US-$ 1 = 0.5385 UK-risk-free force of interest δ = 3.99% p.a. US-risk-free force of interest δ f = g = 0.962% p.a. for 3-months investments 3-months forward price: K = 0.5385 exp { (0.0399 0.00926) 3 } 12 = 0.5426
4.6 The Value of a Forward Contract 30 Notation: f t = value of a forward contract at time t, t [0, T ] Remark: f 0 = 0 f T = S T K K S T (long position) (short position)
Long Forward Contract Consider the following two portfolios at time t. Portfolio A: Buy existing long forward contract for price f t with delivery price K l. Invest K l exp( δ(t t)) at the rate δ p.a. for (T t) years. Value of portfolio A: f t + K l exp( δ(t t)) Portfolio B: Enter a new forward contract to buy the asset for delivery price = forward price = K at time T. Invest K exp( δ(t t)) at rate δ p.a. for (T t) years. Value of portfolio B: K exp( δ(t t)) at Time T : Value of portfolio A: S T K l + K l = S T Value of portfolio B: S T K + K = S T 31
No arbitrage = 32 Value of portfolio A at time t = Value of portfolio B at time t f t + K l exp( δ(t t)) = K exp( δ(t t)) f t = (K K l ) exp( δ(t t)) Short Forward Contract Similar arguments: f t = (K l K) exp( δ(t t))
Example 4.13 3-month forward contract on US-$. Current exchange rate: 1 US-$ = 0.535 3-month UK risk-free interest rate: 4.72% = δ = log(1.0472) 3-month US risk-free interest rate: 2.4% = δ f = log(1.024) 33 Current forward price (delivery price): K l = S 0 exp {(δ δ f )T } { ( ) 3 = 0.535 exp log(1.0472) log(1.024) 12} = 0.5380049
One month later: 34 Exchange rate: 1 US-$ = 0.52 2-month UK risk-free interest rate: 4.82% = δ = log(1.0482) 2-month US risk-free interest rate: 2.56% = δ f = log(1.0256) Forward Price: Price of Short Forward Contract: K = S 0 exp {(δ δ f )T } { ( ) } 2 = 0.52 exp log(1.0482) log(1.0256) 12 = 0.5218925 f t = (K l K) exp( δ(t t)) = (0.5380049 0.5218925) exp ( log(1.0482) 2 ) 12 = 0.01598648
You should know at the end of this chapter: 35 what arbitrage opportunities are replicating portfolio, hedging the definition of a forward contract (long/short) how to calculate the forward price for securities with and without dividends how to calculate the value of a forward contract (short and long) for securities with and without dividends