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程序代写案例-CHAPTER 7
时间：2022-01-09
CHAPTER 7
Arbitrage
1 Arbitrage
1.1 Arbitrage.
The following is a subset of the definition of arbitrage in the Shorter Oxford English Dictionary:
arbitrage n. & v.
A n. Commerce. Trade in bills of exchange or stocks in different markets to take advantage of their different
prices. L19.
B v.i. Commerce. Engage in arbitrage. E20.
In finance, an arbitrage opportunity means the possibility of a risk-free trading profit. This could be achieved
by making a deal which leads to an immediate profit with no risk of future loss, or making a deal which has no
risk of loss but a non-zero probability of making a future profit.
An arbitrageur is a trader who exploits arbitrage possibilities. Such possibilities do not exist for long—because
they are effectively money-making machines!
The term short-selling or shorting means selling an asset that is not owned with the intention of buying it
back later. The mechanics are as follows: suppose client A instructs a broker to short sell S. The broker
then borrows S from the account of another client and sells S and deposits the proceeds in the account of A.
Eventually A pays the broker to buy S in the market who returns it to the account he borrowed it from. (It is
possible that the broker will run out of S to borrow and then A will be short-squeezed and be forced to close
out his position prematurely.)
Example 1.1a. Suppose an investor has a portfolio which includes security A and security B. The prices at time 0
are as follows: sA = 6 and sB = 11. He assesses the prices at time 1 will be SA1 = 7 and S
B
1 = 14 if the market goes
up and SA1 = 5 and S
B
1 = 10 if the market goes down.
time 0 time 1 time 1
(market rises) (market falls)
A 6 7 5
B 11 14 10
Check there is an arbitrage opportunity.
Solution. Suppose the investor sells two A and buys one B at time 0. Hence he makes a gain of 1 at time 0. At
time 1, whether the market rises of falls, there is no change in the value of his portfolio.
Clearly, investors will sell A and buy B. This implies the current price of A will fall relative to the price of B, until
sA = sB/2.
Example 1.1b. Suppose an investor has a portfolio which includes security A and security B. The prices at time 0
are as follows: sA = 6 and sB = 6. He assesses the prices at time 1 will be SA1 = 7 and S
B
1 = 7 if the market goes up
and SA1 = 5 and S
B
1 = 4 if the market goes down.
time 0 time 1 time 1
(market rises) (market falls)
A 6 7 5
B 6 7 4
Check there is an arbitrage opportunity.
Solution. Suppose the investor buys one A and sells one B at time 0. Hence he neither gains nor loses at time 0. At
time 1, there is no change in the value of his portfolio if the market rises; he gains 1 if the market falls.
Clearly, investors will buy A and sell B. The current price of A will rise relative to the price of B, until the arbitrage
opportunity is eliminated.
ST334 Actuarial Methods cR.J. Reed Sep 27, 2016(9:51) Section 1 Page 131
Page 132 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
1.2 Existence of an arbitrage. Suppose investment A costs the amount sA and investment B costs the
amount sB at time t0. Let NPVA(t0) denote the net present value at time t0 of the payoff from investment A.
Here are three situations where an arbitrage exists:
• If NPVA(t0) = NPVB(t0) and sA 6= sB then there is an arbitrage.
• If NPVA(t0) 6= NPVB(t0) and sA = sB then there is an arbitrage.
• If sA > sB and NPVA(t0) NPVB(t0) or if sA < sB and NPVA(t0) NPVB(t0) then there is an
arbitrage.
Example (1.1a) is an example of the first formulation—apply it to two A and one B. Example (1.1b) is an
example of the second formulation. Now consider the third formulation: sA > sB andNPVA(t0) NPVB(t0).
Selling one A and buying one B at time 0 produces a cash surplus at time 0 and increases the value of the
portfolio at time t = 1.
1.3 Option pricing.
Example 1.3a. Suppose a share has price 100 at time t = 0. Suppose further that at time t = 1 its price will rise
to 150 or fall to 50 with unknown probability. Suppose the effective rate of interest is r per annum.
Suppose C denotes the price at time t = 0 of a call option for one share for the exercise price of 125 at the exercise
time of t = 1. Determine C so there is no arbitrage.
Solution. The given information is summarised in the following table:
t = 0 t = 1
probability p1 probability 1 p1
share 100 150 50
call option at t = 1 for 125 C 25 0
Consider buying x shares and y call options at time t = 0. This will cost 100x + Cy at time t = 0. At time t = 1, this
portfolio will have value 150x + 25y if the share value rises to 150, or 50x if the share value falls to 50.
Hence if y = 4x, the value of the portfolio will be 50x whether the share price rises or falls.
Suppose an investor has capital z at time 0 where z > max{100, 4C}. Consider the following 2 possible decisions:
• Investment decision A: buy 1 share, invest the rest;
• Investment decision B: buy 4 options, invest the rest.
At time t = 0, the value of the portfolio is z in both cases. For decision A, the value of the portfolio at time t = 1 is
(z 100)(1 + r) +
n 150
50
= (z 100)(1 + r) + 50 +
n 100
0
For decision B, the value of his portfolio at time t = 1 is
(z 4C)(1 + r) +
n 4⇥ 25 = 100
0
For no arbitrage, we can equate these. Hence (100 4C)(1 + r) = 50 which implies
C =
1
4
✓
100 50
1 + r
◆
We can check this is the condition for no arbitrage as follows:
Suppose C > 14
100 50/(1 + r).
At time t = 0, sell 4 options for 4C and borrow 50/(1+r). The total amount received is z = 4C+50/(1+r) > 100.
Use 100 to buy one share and the rest is the arbitrage—the rest is the amount z 100.
At time t = 1 sell the share. If the share then has price 150, repay the 50 and use the remaining 100 for paying
for the 4 options; if the share has price 50, then the options are worthless and use the 50 to repay the loan.
Suppose C < 14
100 50/(1 + r).
At time t = 0, short sell one share for 100; purchase 4 options for 4C and put the rest of the money, which is
100 4C > 50/(1 + r), in the bank.
At time t = 1, if the share has price 150, then the 4 options are worth 25 each and there will be more than 50 in
the bank; if the share has price 50, then the options are worthless but there will be more than 50 in the bank. In
both cases, the obligation of repurchasing the share which was sold short is covered.
More succinctly, compare
t = 0 t = 1: probability p1 t = 1: probability 1 p1
share 100 150 50
four call options at t = 1 for 125 4C 100 0
If no arbitrage, we must have 4C + 50/(1 + r) = 100.
7 Arbitrage Sep 27, 2016(9:51) Section 2 Page 133
2 Forward contracts
2.1 Historical background. The spot price for a commodity such as copper is the current price for immediate
delivery. This will be determined by the current supply and demand.
Suppose a manufacturer requires a supply of copper. He cannot plan his production if he does not know how
much he will have to pay in 6 months’ time for the copper he then needs. Similarly, the copper supplier does
not know how to plan his production if he does not know how much he will receive for the copper he produces
in 6 months’ time. Both supplier and consumer can reduce the uncertainty by agreeing a price today for the
“6 months’ copper”.
2.2 Forward contracts. A forward contract is a legally binding contract to buy/sell an agreed quantity of
an asset at an agreed price at an agreed time in the future. The contract is usually tailor-made between two
financial institutions or between a financial institution and a client. Thus forward contracts are over-the-counter
or OTC. Such contracts are not normally traded on an exchange.
Settlement of a forward contract occurs entirely at maturity and then the asset is normally delivered by the
seller to the buyer.
2.3 Futures contracts. A futures contract is also a legally binding contract to buy/sell an agreed quantity of
an asset at an agreed price at an agreed time in the future. Futures contracts can be traded on an exchange—this
implies that futures contracts have standard sizes, delivery dates and, in the case of commodities, quality of
the underlying asset. The underlying asset could be a financial asset (such as equities, bonds or currencies) or
commodities (such as metals, wool, live cattle).
Suppose the contract specifies that A will buy the asset S from B at the price K at the future time T . Then
B is said to hold a short forward position and A is said to hold a long forward position. Therefore, a futures
contract says that the short side must deliver to the long side the asset S for the exchange delivery settlement
price (EDSP).
On maturity, physical delivery of the underlying asset is not normally made—the short side closes the contract
with the long side by making a cash settlement .
Let ST denote the actual price of the asset at time T .
IfK > ST then the short side B makes the profitK ST .
IfK < ST then the long side A makes the profit ST K.
With futures contracts, there is daily settlement—this process is calledmark to market. At the time the contract
is made, the long side is required to place a deposit called the initial margin in a margin account. At the end
of each trading day, the balance in the margin account is calculated. If the amount falls below the maintenance
margin then the long side must top-up the account; the long side is also allowed to withdraw any excess over
the initial margin. Generally, interest is earned on the balance in the margin account. Similarly, the short side
also has a margin account.
We assume in this chapter that there is no difference between the pricing of forward contracts and futures
contracts.
Example 2.3a. Suppose the date is May 1, 2000 and the current spot price of copper is £1,000 per tonne1. This
is the current price of copper for immediate delivery. Suppose the current price of September copper is £1,100 per
tonne. Suppose further that the standard futures contract for copper is for 25 tonnes. This means that the current
price of a September copper futures contract is 25⇥ £1,100 = £27,500.
Suppose that on May 1, A buys one September futures contract for copper from B. This means that A agrees on
May 1 to pay £27,500 to B for 25 tonnes of copper on September 1 whilst B agrees to sell 25 tonnes of copper to A
on September 1 for £27,500.
1 One tonne is 1,000 kilograms. For current information, see the web site of the London Metal Exchange: www.lme.com.
现货价格
供应商和消费者今天就6个⽉期钢的价格达成⼀致可以减少不确定性
远期合约
在未来约定的时间以约定的价格买进1卖出约定放量的资产
标的资产
-乾⽅
Page 134 Section 2 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
2.4 Pricing a forward contract with no income. Suppose a forward contract specifies that A will buy the
asset S from B at the priceK at the future time T . So A is the long side and B is the short side.
Let ST denote the actual price of the asset at time T .
IfK > ST then the short side B makes the profitK ST .
IfK < ST then the long side A makes the profit ST K.
Assume that the force of interest available on a risk-free investment in (0, T ) is . By assuming the no arbitrage
assumption, we can calculate the priceK.
Method I.
Consider the following two investment portfolios:
Portfolio A. At time 0:
• buy one unit of S at the current price S0.
Value of portfolio at time 0: S0. Value of portfolio at time T: one unit of S which has value ST .
Portfolio B. At time 0:
• enter forward contract to buy one S with forward priceK at time T ;
• investKeT in a risk-free investment at force of interest .
Value of portfolio at time 0: KeT . Value of portfolio at time T: one unit of S which has value ST . (The
amountK from the risk-free investment pays for the forward contract. The size of the investment at time 0
is chosen to beKeT in order to ensure that values of the two portfolios are equal at time T .)
The no arbitrage assumption implies that
KeT = S0
and hence
K = S0eT
Method II.
We can simplify Method I as follows: equate the time 0 values. Suppose A buys one S at time 0 for the
price S0. At time T , he then has the asset S which he values atK. Equating the time 0 values gives:
S0 = KeT
Note that it has not been necessary to make any assumptions about how the price, St, of the asset varies over
(0, T ). The forward price will change over time. SupposeKt denotes the forward price for a contract taken out
at time t for maturity time T . Then Kt = Ste(Tt) where St is the price of the security at time t. As t ! T
we have Kt = Ste(Tt) ! ST . Hence as t approaches the settlement time, T , the forward price tends to the
price of the underlying asset.
Example 2.4a. The price of a stock is currently 300p. (a) What is the price of a futures contract on this stock with
a settlement day in 180 days’ time if the risk-free interest rate is 5% per annum. (Assume there are no dividends
payable over the next 180 days.) (b) After 50 days, suppose the price of the stock has risen to 308p and the risk-free
interest rate has not changed. Calculate the change in the margin account for an investor holding the futures contract
specified in (a).
Solution. (a) Equating the time 0 values gives S0 = Ket with S0 = 300, t = 180/365 and e = 1.05. Hence
K = 300 ⇥ 1.05180/365 = 307.31. (b) New price of futures contract is Kt = 308 ⇥ 1.05130/365 = 313.40. Hence,
change in margin account is the gain 313.40 307.31 = 6.09.
2.5 Pricing a forward contract with a fixed income. Now suppose the security S is a bond which pays the
coupon c at time t1 where t1 2 (0, T ).
Example 2.5a. First consider the following simplification: suppose the interest rate is 0% and hence £1 tomorrow
is worth exactly £1 today.
(a) Suppose the current price of an asset S is £10. It pays a coupon of £1 at time 1. If a forward contract is to
purchase S at time 2 for the amount £K, then how much shouldK be?
(b) If the current price of S is S0 and the coupon is c, then how much shouldK be?
Solution. (a) Here is the justification thatK should be £9.
Suppose K > 9: then an arbitrageur will buy S now for £10 and enter into a forward contract to sell the asset S
at time 2 for £K. He will receive £1 at time 1 and £K at time 2; hence he makes a risk-free profit of £(K 9) for
every 1 unit of S.
SupposeK < 9. Then an arbitrageur should sell S short now for £10 and enter into a long forward contract to buy S
at time 2 for £K. Of course he must now pay the coupon of £1 at time 1 (which goes to the person from whom S was
⽆收益远期合约的定价
7 Arbitrage Sep 27, 2016(9:51) Section 2 Page 135
borrowed.) and £K at time 2. Hence for every unit of S he makes a profit of £(10 1K) = £(9K). Similarly,
anyone owning the asset S at time 0 should make use of the same arbitrage possibility.
(b) By a similar argument,K = S0 c.
Suppose the security S is a bond which pays the coupon c at time t1 where t1 2 (0, T ). Suppose a forward
contract specifies that investor A will buy the asset S from investor B at the price K at the future time T . We
wish to decide what is the appropriate value for K under the assumption that the force of interest available on
a risk-free investment in (0, T ) is (this means that if we invest an amount b for a length of time h during the
interval (0, T ) then it grows to beh).
Method I.
Consider the following two investment portfolios:
Portfolio A. At time 0:
• buy one unit of S at the current price S0.
Value of portfolio at time 0: S0. Value of portfolio at time T: one unit of S plus amount ce(Tt1) (recall
the asset S pays coupon c at time t1 which is then invested).
Portfolio B. At time 0:
• enter forward contract to buy one S with forward priceK maturing at time T ;
• investKeT + cet1 in a risk-free investment at force of interest .
Value of portfolio at time 0: KeT + cet1 . Value of portfolio at time T: one unit of S plus amount
ce(Tt1). (The amountKeT + cet1 accumulates toK + ce(Tt1) and the amountK from the risk-free
investment pays for the forward contract. The size of the investment at time 0 is chosen to be KeT +
cet1 in order to ensure that values of the two portfolios are equal at time T .)
The no arbitrage assumption implies that
KeT + cet1 = S0
and henceK = S0eT ce(Tt1).
Method II.
Equate the time 0 values. Suppose A buys one S at time 0 for the price S0. In return he gets the coupon c
at time t1 and has the asset S at time T which he values atK. Equating the time 0 values gives:
S0 = cet1 +KeT
Note that it has not been necessary to make any assumptions about how the price, St, of the asset varies over
(0, T ).
Now suppose we have more than one coupon payment: suppose we have coupons of size ck at times tk for
k = 1, 2, . . . , n. Then the argument of equating time 0 values gives
KeT = S0
nX
k=1
cke
tk
Example 2.5b. The price of a stock is currently 250p. A dividend payment of 10p is expected in 35 days’ time
together with another dividend payment of 15p after a further 180 days. What is the price of a futures contract on
this stock with a settlement date of 250 days’ time if the risk-free interest rate is 7% per annum.
Solution. We have S0 = 250, T = 250/365 and e = 1 + i = 1.07. Equating time 0 values gives
KeT = S0
nX
k=1
cke
tk = 250 10⇥ 1.0735/365 15⇥ 1.07215/365
leading toK = 236.35.
Example 2.5c. A bond has a current price of £600. Its face value is £1,000 and it matures in exactly 5 years and
pays coupons of £30 every 6 months. The first payment is due in exactly 6 months.
The 6-month and 12-month interest rates are nominal 6% and 7% per annum compounded continuously.
By using the “no arbitrage” assumption, find the price of a forward contract to purchase the bond in exactly one year.
Solution. LetK denote the forward price. Equating time 0 prices gives: 600 = Ke0.07 + 30e0.03 + 30e0.07.
If we wish to equate time T = 1 values then we must argue as follows. We are given the force of interest rates
F0,0.5 = 0.06 and F0,1 = 0.07. Hence the forward force of interest F0.5,0.5 = 0.08 (see example(1.6a) on page 104).
This means that the forward force of interest for an investment starting in 6 months’ time is a nominal 8% per annum
compounded continuously. The coupon received in 6 months’ time can be invested and hence will grow to 30e0.04
Page 136 Section 2 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
by time 1. Hence 600e0.07 = 30e0.04 + 30 + K. This is exactly the same equation as the previous one. Both give
K = 582.28.
The price of K = 582.28 can be justified as follows. If K > 582.28, then an investor should buy S now and enter
into a forward contract to sell S forK at time 1 and a contract to invest the dividend at time 0.5 for 6 months.
His cash flow is summarised in the following table.
time 0 0.5 1
cash flow -600 30 30+K
The NPV is positive.
IfK < 582.28, then an investor with S should sell S now and enter into a forward contract to buy S forK at time 1.
His cash flow is
time 0 0.5 1
cash flow 600 -30 -30-K
The NPV is positive.
2.6 Pricing a forward contract with a known dividend yield. Now suppose the security S pays a known
dividend yield which is a nominal r% per annum compounded continuously—the value r is the average over
the life of the forward contract. This means that the dividend payment is proportional to the stock price at
the time of the dividend payment. Hence the actual size of the dividend payments over (0, T ) are unknown at
time 0 because the value of the stock price over the time interval (0, T ) is unknown at time 0. We can finesse
this problem by assuming that all dividend payments are reinvested in further units of S; hence we assume that
1 unit of the security grows to erT units of the security over the period (0, T ).
Portfolio A. At time 0:
• enter forward contract to buy one S with forward priceK maturing at time T ;
• investKeT in a risk-free investment at force of interest .
Value of portfolio at time 0: KeT . Value of portfolio at time T: one unit of S. (The amount KeT
accumulates toK which pays for the forward contract.)
Portfolio B. At time 0:
• buy erT units of S at the current price S0 (and reinvest dividend income in further units of S).
Value of portfolio at time 0: erTS0. Value of portfolio at time T: one unit of S.
The no arbitrage assumption implies that
KeT = S0erT and henceK = S0e(r)T
Example 2.6a. The price of a stock is currently 250p. It is expected to provide a dividend income of 2% of the asset
price which is received continuously during a six month period. What is the price of a forward contract on this stock
with a settlement date of 6 months’ time if the risk-free interest rate is a nominal 7% per annum with continuous
compounding.
Solution. Now S0 = 250, = 0.07 and er = 1.022. The time 0 value of one unit of stock at time T is S0erT .
Equating time 0 values givesKeT = S0erT which leads toK = 250e(0.07r)0.5 = 250e0.035/1.02 = 253.83.
2.7 Value of a forward contract at intermediate times.
Suppose a forward contract is agreed at time 0 with forward price K0 for one unit of S at time T . At time T ,
the value of the contract to the long side is ST K0, the value of the stock at time T minus the price of the
forward contract agreed at time 0. Suppose we wish to estimate Vt, the value of the forward contract to the long
side at any time t 2 [0, T ]. Then VT = ST K0, V0 = 0 and the value to the short side at any time t 2 [0, T ]
is (Vt).
Consider the following 2 portfolios:
Portfolio A. At time t
• buy the forward contract agreed at time 0 for the price Vt.
• investK0e(Tt) in a risk-free investment at force of interest for the interval (t, T ).
Value of portfolio at time t: Vt + K0e(Tt). Value of portfolio at time T : one unit of S. (Because the
amount K0e(Tt) grows to K0 which is the amount needed by the forward contract which returns one
unit of S.)
Portfolio B. At time t:
• agree to a new forward contract with priceKt for one unit of S at time T ;
7 Arbitrage Sep 27, 2016(9:51) Section 2 Page 137
• investKte(Tt) in a risk-free investment at force of interest for the interval (t, T ).
Value of portfolio at time t: Kte(Tt). Value of portfolio at time T : one unit of S.
Hence
Vt +K0e(Tt) = Kte(Tt)
and so Vt = (Kt K0)e(Tt). UsingKt = Ste(Tt) gives Vt = St S0et.
Here is a more succinct derivation: now the long side agrees at time 0 to payK0 at time T for one unit of S and
also agrees at time t to pay Kt at time T for one unit of S. Hence the value to the long side of such a forward
contract at time t is Kt K0 which is in time T values. Its value in time t values is Vt = (Kt K0)e(Tt).
UsingK0 = S0eT andKt = Ste(Tt) gives Vt = St S0et.
Example 2.7a. Suppose t0 < t1 < t2 < t3 < t4. Suppose further that the price of a stock at time t0 was P0 and
was P1 at time t1. The stock will pay coupons of C2 at time t2 and C3 at time t3. An investor entered into a long
forward contract for 1 unit of the stock at time t0 with the contract due to mature at time t4. Assuming an effective
rate of interest of 5% p.a. and no arbitrage, what is the value of the contract at time t1.
Solution. LetK0 denote the price of the original contract agreed at time t0 and let ⌫ = 1/1.05.
Equating the time t0 values gives an equation forK0: K0⌫t4 = P0 C2⌫t2 C3⌫t3 .
LetK1 denote the price of a contract agreed at time t1 for 1 unit of the stock to be delivered at time t4.
Equating the time t1 values gives an equation forK1: K1⌫t4t1 = P1 C2⌫t2t1 C3⌫t3t1 .
Hence the value of the original contract to the long side at time t1 is the difference in prices of these two contracts in
time t1 values; and that is (K1 K0)⌫t4t1 = P1 P0/⌫t1 .
2.8 Speculation, hedging, gearing and leverage.
Speculation is the process of taking a risk in the hope of making a profit.
Example 2.8a. This example is a continuation of example (2.3a).
Suppose A buys one September futures contract for copper from B. Then A will have to make a deposit, called the
margin. This may be as low as 1% for a financial futures contract. Assume it is 10% for our copper futures contract.
Hence A must deposit 10%⇥ £27,500 = £2,750. Suppose that by September, the price of copper has risen to £1,300
per tonne. Hence A makes a profit of 25 ⇥ (£13,000 £11,000) = £5,000 for an outlay of £2,750. But note that A
has taken the risk that he might lose £27,500 for an outlay of £2,750.
Suppose A believes the price of copper will fall. Hence he sells a September futures contract for 25 tonnes of copper
for £27,500. If the price of copper is only £900 per tonne on September 1, then he makes a profit of 25 ⇥ £200 =
£5,000. But suppose he is wrong and the price of copper doubles to £2,200 per tonne on September 1. Then A loses
25⇥ £1,100 = £27,500. Indeed, his potential loss is unlimited.
Gearing and leverage. These terms mean that a large gain or a large loss can be made for a small outlay.
Leverage is the term used in the USA whilst gearing is the term used in the UK.
Example 2.8b. The copper example again: examples (2.3a) and (2.8a).
Suppose A believes the price will rise. If he trades in the actual product and purchases 25 tonnes of copper today at
the current spot price of £1,000 per tonne, then this will cost him £25,000. If the price rises to £1,300 per tonne on
September 1, then he makes a profit of 25⇥ £300 = £7,500 from an investment of £25,000; this profit is 30% of his
initial investment. His largest possible loss, if the copper becomes worthless, is £25,000 which is 100% of his initial
investment.
Suppose instead of buying the actual product today, he buys a futures contract for 25 tonnes for £27,500. He then
makes a profit of £5,000 for an outlay of the deposit of £2,750; this profit is 182% of his initial investment. Of course,
he is also exposed to a potential loss of £27,500 which is 1,000% of his initial investment!
Hedging is the avoidance of risk. This implies making an investment to reduce the risk of adverse price
movements in an asset.
Example 2.8c. A UK exporter has won a contract to supply machinery to the US. The exporter will receive a
payment of $1m in six month’s time when the machinery is delivered. In order to remove the risk that an adverse
movement in the exchange rate makes the deal unprofitable, the exporter purchases a currency futures contract. This
guarantees that he can exchange the dollars which he will receive in six months’ time into sterling at a rate which is
specified today. In this way, the exporter ensures that the risk that the export deal is made unprofitable by movements
in the exchange rate is removed.
Another example: if you owned a stock, then short selling an equal amount would be a “perfect hedge”. The
term static hedge means that the hedge portfolio does not change over the period of the contract.
Page 138 Exercises 3 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
Example 2.8d. Consider again the forward contract made at time 0 to buy one S with forward priceK maturing at
time T . Find a static hedge for the short side of this simple forward contract. Suppose the risk free force of interest
is .
Solution. Recall that the short side is the party who contracts to sell one S at time T for the priceK. If the short side
just purchases one unit of S at time 0 and the price falls, then more will have been paid for S then was necessary; if
the short side just purchases one unit of S at time T to fulfil the contract and the price ST at time T is greater thenK
(the amount he receives at time T ), then a loss will be made—indeed, this potential loss is theoretically unlimited.
Suppose at time 0 the short side borrows the amount KeT = S0 at the risk-free force of interest and buys one
unit of S at the current price S0. The price of this hedge portfolio at time 0 is KeT + S0 = 0.
At time T , the short side has a debt which has grown to K which now equals the amount received under the terms
of the forward contract; in addition he owns one unit of S which must be delivered to the long side under the terms
of the forward contract. Hence this hedge portfolio means the short side is certain not to make a loss or a profit
whatever the price of the stock at time T .
3 Exercises (exs7-1.tex)
1. The price of a given share is 80p. The risk-free rate of interest is 5% per annum convertible quarterly. Assuming
no arbitrage and that the share will not pay any income, calculate the forward price for the share, for settlement in
exactly one quarter of a year. (Institute/Faculty of Actuaries Examinations, April, 2002) [2]
2. A 10-month forward contract is issued on 1 March 2002 on a stock with a price of £12 per share at that date.
Dividends of £1.50 per share are expected on 1 June, 1 September and 1 December 2002.
Calculate the forward price, assuming a risk-free rate of interest of 6% per annum convertible half-yearly and no
arbitrage. (Institute/Faculty of Actuaries Examinations, September 2002) [4]
3. (i) Explain what is meant by the “no arbitrage” assumption in financial mathematics.
(ii) A 7 month forward contract is issued on 1 January 2000 on a stock with a price of £60 per share. Dividends of
£2 per share are expected after 3 and 6 months.
Assuming a risk-free force of interest of 7% per annum and no arbitrage, calculate the forward price.
(Institute/Faculty of Actuaries Examinations, April 2000) [6]
4. An asset has a current price of £1.20. Given a risk-free rate of interest of 5% per annum effective and assuming no
arbitrage, calculate the forward price to be paid in 91 days.
(Institute/Faculty of Actuaries Examinations, September 2000) [3]
5. (i) State what is meant by a “forward contract”. Your answer should include reference to the terms “short forward
position” and “long forward position”.
(ii) A 3-month forward contract is issued on 1 February 2001 on a stock with a price of £150 per share. Dividends
are received continuously and the dividend yield is 3% per annum. In addition, it is anticipated that a special
dividend of £30 per share will be paid on 1 April 2001.
Assuming a risk-free force of interest of 5% per annum and no arbitrage, calculate the forward price per share
of the contract. (Institute/Faculty of Actuaries Examinations, April 2001) [6]
6. A one year forward contract is issued on 1 April 2003 on a share with a price of 600p at that date. Dividends of 30p
per share are expected on 30 September 2003 and 31 March 2004. The 6-month and 12-month spot risk-free interest
rates are 4% and 4.5% per annum effective respectively on 1 April 2003.
Calculate the forward price at issue, assuming no arbitrage.
(Institute/Faculty of Actuaries Examinations, September 2003) [4]
7. (i) Explain what is meant by a “forward contract”. Your answer should include reference to the terms “short
forward position” and “long forward position”.
(ii) An investor entered into a long forward contract for £100 nominal of a security seven years ago and the contract
is due to mature in 3 years’ time. The price per £100 nominal of the security was £96 seven years ago and is
now £148. The risk-free rate of interest can be assumed to be 4% per annum effective during the contract.
Calculate the value of the contract now if the security will pay a single coupon of £7 in two years’ time and this
was known from the outset. You should assume no arbitrage.
(Institute/Faculty of Actuaries Examinations, April 2003) [3+5=8]
7 Arbitrage Sep 27, 2016(9:51) Exercises 3 Page 139
8. (i) Explain what is meant by the “no arbitrage” assumption in financial mathematics.
(ii) A 3-year forward contract is to be issued on a particular company share. The current market value of the share
is £4.50 and a dividend of £0.20 per share has just been paid. The parties to the contract assume that the future
quarterly dividends will increase by 1% per quarter year compound for the first two years and by 11/2% per
quarter year compound for the final year.
Assuming a risk-free force of interest of 5% per annum and no arbitrage, calculate the forward price.
(Institute/Faculty of Actuaries Examinations, April 2004) [2+7=9]
9. The risk-free force of interest (t) at time t is given by:
(t) =
n 0.05 if 0 < t 10;
0.08 + 0.003t if t > 10.
(i) (a) Calculate the accumulation at time 15 of £100 invested at time t = 5.
(b) Calculate the accumulation at time 14 of £100 invested at time t = 5.
(c) Calculate the accumulation at time 15 of £100 invested at time t = 14.
(d) Calculate the equivalent constant force of interest from time t = 5 to time t = 15.
(ii) Calculate the present value at time t = 0 of a continuous payment stream payable at a rate of 100e0.01t from
time t = 0 to time t = 5.
(iii) A one year forward contract is issued at time t = 0 on a share with a price of 300p at that date. A dividend of
7p per share is expected at time t = 1/2. Calculate the forward price of the share, assuming no arbitrage.
(Institute/Faculty of Actuaries Examinations, September 2004) [9+4+4=17]
10. A bond is priced at £95 per £100 nominal, has a coupon of 5% per annum payable half-yearly, and has an outstanding
term of 5 years.
An investor holds a short position in a forward contract on £1 million nominal of this bond, with a delivery price of
£98 per £100 nominal and maturity in exactly one year, immediately following the coupon payment then due.
The continuously compounded risk-free rates of interest for terms of 6 months and one year are 4.6% per annum and
5.2% per annum, respectively.
Calculate the value of this forward contract to the investor assuming no arbitrage.
(Institute/Faculty of Actuaries Examinations, April 2005) [5]
11. (i) Explain what is meant by the “expectations theory” for the shape of the yield curve.
(ii) Short-term, one-year annual effective interest rates are currently 8%; they are expected to be 7% in one year’s
time, 6% in two years’ time and 5% in three years’ time.
(a) Calculate the gross redemption yields (spot rates of interest) from 1-year, 2-year, 3-year and 4-year zero
coupon bonds assuming the expectations theory explanation of the yield curve holds.
(b) The price of a coupon paying bond is calculated by discounting individual payments from the bond at the
zero coupon bond yields in (a). Calculate the gross redemption yield of a bond that is redeemed at par in
exactly 4 years and pays a coupon of 5 per annum annually in arrear.
(c) A two-year forward contract has just been issued on a share with a price of 400p. A dividend of 4p is
expected in exactly in exactly one year.
Calculate the forward price using the above spot rates of interest assuming no arbitrage.
(Institute/Faculty of Actuaries Examinations, September 2005) [2+12=14]
12. A share currently trades at £10 and will pay a dividend of 50p in one month’s time. A six-month forward contract is
available on the share for £9.70. Show that an investor can make a risk-free profit if the risk-free force of interest is
3% per annum. (Institute/Faculty of Actuaries Examinations, April 2006) [4]
13. An investor is able to purchase or sell two specially designed risk-free securities, A and B. Short sales of both
securities are possible. Security A has a market price of 20p. In the event that a particular stock market index goes
up over the next year, it will pay 25p and, in the event that the stock market index goes down, it will pay 15p.
Security B has a market price of 15p. In the event that the stock market index goes up over the next year, it will
pay 20p and, in the event that the stock market index goes down it will pay 12p.
(i) Explain what is meant by the assumption of “no arbitrage” used in the pricing of derivatives contracts.
(ii) Find the market price of B such that there are no arbitrage opportunities and assuming the price of A remains
fixed. Explain your reasoning. (Institute/Faculty of Actuaries Examinations, September 2006) [2+2=4]
14. An investor entered into a long forward contract for a security 5 years ago and the contract is due to mature in 7 years’
time. The price of the security was £95 five years ago and is now £145. The risk-free rate of interest can be assumed
to be 3% per annum throughout the 12 year period.
Assuming no arbitrage, calculate the value of the contract now if
(i) The security will pay dividends of £5 in two years’ time and £6 in four years’ time.
(ii) The security has paid and will continue to pay annually in arrear a dividend of 2% per annum of the market
price of the security at the time of payment. (Institute/Faculty of Actuaries Examinations, April 2007) [3+3=6]
Page 140 Exercises 3 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
15. A one-year forward contract is issued on 1 April 2007 on a share with a price of 900p at that date. Dividends of 50p
per share are expected on 30 September 2007 and 31 March 2008. The 6-month and 12-month spot, risk-free rates
of interest are 5% and 6% per annum effective respectively on 1 April 2007.
Calculate the forward price at issue, stating any assumptions.
(Institute/Faculty of Actuaries Examinations, September 2007) [4]
16. An 11-month forward contract is issued on 1 March 2008 on a stock with a price of £10 per share at that date.
Dividends of 50 pence per share are expected to be paid on 1 April and 1 October 2008.
Calculate the forward price at issue, assuming a risk-free rate of interest of 5% per annum effective and no arbitrage.
(Institute/Faculty of Actuaries Examinations, April 2008) [4]
17. (i) A forward contract with a settlement date at time T is issued based on an underlying asset with a current market
price of B. The annualised risk free force of interest applying over the term of the forward contract is and the
underlying asset pays no income. Show that the theoretical forward price is given by K = Bet, assuming no
arbitrage.
(ii) An asset has a current market price of 200p, and will pay an income of 10p in exactly 3 months’ time. Cal-
culate the price of a forward contract to be settled in exactly 6 months, assuming a risk free rate of interest of
8% per annum convertible quarterly. (Institute/Faculty of Actuaries Examinations, September 2008) [3+3=6]
18. (i) Explain what is meant by the “no arbitrage” assumption in financial mathematics.
An investor entered into a long forward contract for a security four years ago and the contract is due to mature in five
years’ time. The price of the security was £7.20 four years ago and is now £10.45. The risk-free rate of interest can
be assumed to be 2.5% per annum effective throughout the nine-year period.
(ii) Calculate, assuming no arbitrage, the value of the contract now if the security will pay dividends of £1.20
annually in arrear until maturity of the contract.
(iii) Calculate, assuming no arbitrage, the value of the contract now if the security has paid and will continue to pay
annually in arrear a dividend equal to 3% of the market price of the security at the time of the payment.
(Institute/Faculty of Actuaries Examinations, April 2012) [2+3+3=8]
19. A 10 month forward contract was issued on 1 September 2012 for a share with a price of £10 at that date. Dividends
of £1 per share are expected on 1 December 2012, 1 March 2013 and 1 June 2013.
(i) Calculate the forward price assuming a risk-free rate of interest of 8% per annum convertible half yearly and no
arbitrage.
(ii) Explain why it is not necessary to use the expected price of the share at the time the forward matures in the
calculation of the forward price. (Institute/Faculty of Actuaries Examinations, September 2012) [4+2=6]
20. (i) Explain the main difference:
(a) between options and futures.
(b) between call options and put options.
(ii) A one-year forward contract is issued on 1 April 2013 on a share with a price at that date of £10.50. Dividends
of £1.10 per share are expected on 30 September 2013 and 31 March 2014. On 1 April 2013, the 6-month risk-
free spot rate of interest is 4.5% per annum convertible half-yearly and the 12-month risk-free rate of interest is
5% per annum convertible half yearly.
Calculate the forward price at issue, stating any further assumptions you make.
(Institute/Faculty of Actuaries Examinations, April 2013) [4+4=8]
21. A nine-month forward contract is issued on 1 March 2012 on a share with a price of £1.80 at that date. Dividends of
10p per share are expected on 1 September 2012. Calculate the forward price at issue assuming a risk-free rate of in-
terest of 4% per annum effective and no arbitrage. (Institute/Faculty of Actuaries Examinations, September 2013) [3]
22. (i) Describe what is meant by the “no arbitrage” assumption in financial mathematics.
A 9-month forward contract is issued on 1 April 2015 on a stock with a price of £6 per share on that date. Dividends
are assumed to be received continuously and the dividend yield is 3.5% per annum.
(ii) Calculate the theoretical forward price per share of the contract, assuming no arbitrage and a risk-free force of
interest of 9% per annum.
(iii) The actual forward price per share of the contract is £6.30 and the risk-free force of interest is as in part (ii).
Outline how an investor could make an arbitrage profit.
(Institute/Faculty of Actuaries Examinations, April 2015) [2+2+2=6]
7 Arbitrage Sep 27, 2016(9:51) Exercises 3 Page 141
23. A 9 month forward contract was issued on 1 October 2015 on a share with a price at that date of £10. Dividends
of 50 pence per share are expected on 1 November 2015 and 1 May 2016. The risk-free force of interest is 5% per
annum.
(i) Calculate the forward price at issue, stating any further assumptions made and showing all workings.
(ii) Explain why the expected price of the share nine months after issue does not have to be taken into account when
Arbitrage
1 Arbitrage
1.1 Arbitrage.
The following is a subset of the definition of arbitrage in the Shorter Oxford English Dictionary:
arbitrage n. & v.
A n. Commerce. Trade in bills of exchange or stocks in different markets to take advantage of their different
prices. L19.
B v.i. Commerce. Engage in arbitrage. E20.
In finance, an arbitrage opportunity means the possibility of a risk-free trading profit. This could be achieved
by making a deal which leads to an immediate profit with no risk of future loss, or making a deal which has no
risk of loss but a non-zero probability of making a future profit.
An arbitrageur is a trader who exploits arbitrage possibilities. Such possibilities do not exist for long—because
they are effectively money-making machines!
The term short-selling or shorting means selling an asset that is not owned with the intention of buying it
back later. The mechanics are as follows: suppose client A instructs a broker to short sell S. The broker
then borrows S from the account of another client and sells S and deposits the proceeds in the account of A.
Eventually A pays the broker to buy S in the market who returns it to the account he borrowed it from. (It is
possible that the broker will run out of S to borrow and then A will be short-squeezed and be forced to close
out his position prematurely.)
Example 1.1a. Suppose an investor has a portfolio which includes security A and security B. The prices at time 0
are as follows: sA = 6 and sB = 11. He assesses the prices at time 1 will be SA1 = 7 and S
B
1 = 14 if the market goes
up and SA1 = 5 and S
B
1 = 10 if the market goes down.
time 0 time 1 time 1
(market rises) (market falls)
A 6 7 5
B 11 14 10
Check there is an arbitrage opportunity.
Solution. Suppose the investor sells two A and buys one B at time 0. Hence he makes a gain of 1 at time 0. At
time 1, whether the market rises of falls, there is no change in the value of his portfolio.
Clearly, investors will sell A and buy B. This implies the current price of A will fall relative to the price of B, until
sA = sB/2.
Example 1.1b. Suppose an investor has a portfolio which includes security A and security B. The prices at time 0
are as follows: sA = 6 and sB = 6. He assesses the prices at time 1 will be SA1 = 7 and S
B
1 = 7 if the market goes up
and SA1 = 5 and S
B
1 = 4 if the market goes down.
time 0 time 1 time 1
(market rises) (market falls)
A 6 7 5
B 6 7 4
Check there is an arbitrage opportunity.
Solution. Suppose the investor buys one A and sells one B at time 0. Hence he neither gains nor loses at time 0. At
time 1, there is no change in the value of his portfolio if the market rises; he gains 1 if the market falls.
Clearly, investors will buy A and sell B. The current price of A will rise relative to the price of B, until the arbitrage
opportunity is eliminated.
ST334 Actuarial Methods cR.J. Reed Sep 27, 2016(9:51) Section 1 Page 131
Page 132 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
1.2 Existence of an arbitrage. Suppose investment A costs the amount sA and investment B costs the
amount sB at time t0. Let NPVA(t0) denote the net present value at time t0 of the payoff from investment A.
Here are three situations where an arbitrage exists:
• If NPVA(t0) = NPVB(t0) and sA 6= sB then there is an arbitrage.
• If NPVA(t0) 6= NPVB(t0) and sA = sB then there is an arbitrage.
• If sA > sB and NPVA(t0) NPVB(t0) or if sA < sB and NPVA(t0) NPVB(t0) then there is an
arbitrage.
Example (1.1a) is an example of the first formulation—apply it to two A and one B. Example (1.1b) is an
example of the second formulation. Now consider the third formulation: sA > sB andNPVA(t0) NPVB(t0).
Selling one A and buying one B at time 0 produces a cash surplus at time 0 and increases the value of the
portfolio at time t = 1.
1.3 Option pricing.
Example 1.3a. Suppose a share has price 100 at time t = 0. Suppose further that at time t = 1 its price will rise
to 150 or fall to 50 with unknown probability. Suppose the effective rate of interest is r per annum.
Suppose C denotes the price at time t = 0 of a call option for one share for the exercise price of 125 at the exercise
time of t = 1. Determine C so there is no arbitrage.
Solution. The given information is summarised in the following table:
t = 0 t = 1
probability p1 probability 1 p1
share 100 150 50
call option at t = 1 for 125 C 25 0
Consider buying x shares and y call options at time t = 0. This will cost 100x + Cy at time t = 0. At time t = 1, this
portfolio will have value 150x + 25y if the share value rises to 150, or 50x if the share value falls to 50.
Hence if y = 4x, the value of the portfolio will be 50x whether the share price rises or falls.
Suppose an investor has capital z at time 0 where z > max{100, 4C}. Consider the following 2 possible decisions:
• Investment decision A: buy 1 share, invest the rest;
• Investment decision B: buy 4 options, invest the rest.
At time t = 0, the value of the portfolio is z in both cases. For decision A, the value of the portfolio at time t = 1 is
(z 100)(1 + r) +
n 150
50
= (z 100)(1 + r) + 50 +
n 100
0
For decision B, the value of his portfolio at time t = 1 is
(z 4C)(1 + r) +
n 4⇥ 25 = 100
0
For no arbitrage, we can equate these. Hence (100 4C)(1 + r) = 50 which implies
C =
1
4
✓
100 50
1 + r
◆
We can check this is the condition for no arbitrage as follows:
Suppose C > 14
100 50/(1 + r).
At time t = 0, sell 4 options for 4C and borrow 50/(1+r). The total amount received is z = 4C+50/(1+r) > 100.
Use 100 to buy one share and the rest is the arbitrage—the rest is the amount z 100.
At time t = 1 sell the share. If the share then has price 150, repay the 50 and use the remaining 100 for paying
for the 4 options; if the share has price 50, then the options are worthless and use the 50 to repay the loan.
Suppose C < 14
100 50/(1 + r).
At time t = 0, short sell one share for 100; purchase 4 options for 4C and put the rest of the money, which is
100 4C > 50/(1 + r), in the bank.
At time t = 1, if the share has price 150, then the 4 options are worth 25 each and there will be more than 50 in
the bank; if the share has price 50, then the options are worthless but there will be more than 50 in the bank. In
both cases, the obligation of repurchasing the share which was sold short is covered.
More succinctly, compare
t = 0 t = 1: probability p1 t = 1: probability 1 p1
share 100 150 50
four call options at t = 1 for 125 4C 100 0
If no arbitrage, we must have 4C + 50/(1 + r) = 100.
7 Arbitrage Sep 27, 2016(9:51) Section 2 Page 133
2 Forward contracts
2.1 Historical background. The spot price for a commodity such as copper is the current price for immediate
delivery. This will be determined by the current supply and demand.
Suppose a manufacturer requires a supply of copper. He cannot plan his production if he does not know how
much he will have to pay in 6 months’ time for the copper he then needs. Similarly, the copper supplier does
not know how to plan his production if he does not know how much he will receive for the copper he produces
in 6 months’ time. Both supplier and consumer can reduce the uncertainty by agreeing a price today for the
“6 months’ copper”.
2.2 Forward contracts. A forward contract is a legally binding contract to buy/sell an agreed quantity of
an asset at an agreed price at an agreed time in the future. The contract is usually tailor-made between two
financial institutions or between a financial institution and a client. Thus forward contracts are over-the-counter
or OTC. Such contracts are not normally traded on an exchange.
Settlement of a forward contract occurs entirely at maturity and then the asset is normally delivered by the
seller to the buyer.
2.3 Futures contracts. A futures contract is also a legally binding contract to buy/sell an agreed quantity of
an asset at an agreed price at an agreed time in the future. Futures contracts can be traded on an exchange—this
implies that futures contracts have standard sizes, delivery dates and, in the case of commodities, quality of
the underlying asset. The underlying asset could be a financial asset (such as equities, bonds or currencies) or
commodities (such as metals, wool, live cattle).
Suppose the contract specifies that A will buy the asset S from B at the price K at the future time T . Then
B is said to hold a short forward position and A is said to hold a long forward position. Therefore, a futures
contract says that the short side must deliver to the long side the asset S for the exchange delivery settlement
price (EDSP).
On maturity, physical delivery of the underlying asset is not normally made—the short side closes the contract
with the long side by making a cash settlement .
Let ST denote the actual price of the asset at time T .
IfK > ST then the short side B makes the profitK ST .
IfK < ST then the long side A makes the profit ST K.
With futures contracts, there is daily settlement—this process is calledmark to market. At the time the contract
is made, the long side is required to place a deposit called the initial margin in a margin account. At the end
of each trading day, the balance in the margin account is calculated. If the amount falls below the maintenance
margin then the long side must top-up the account; the long side is also allowed to withdraw any excess over
the initial margin. Generally, interest is earned on the balance in the margin account. Similarly, the short side
also has a margin account.
We assume in this chapter that there is no difference between the pricing of forward contracts and futures
contracts.
Example 2.3a. Suppose the date is May 1, 2000 and the current spot price of copper is £1,000 per tonne1. This
is the current price of copper for immediate delivery. Suppose the current price of September copper is £1,100 per
tonne. Suppose further that the standard futures contract for copper is for 25 tonnes. This means that the current
price of a September copper futures contract is 25⇥ £1,100 = £27,500.
Suppose that on May 1, A buys one September futures contract for copper from B. This means that A agrees on
May 1 to pay £27,500 to B for 25 tonnes of copper on September 1 whilst B agrees to sell 25 tonnes of copper to A
on September 1 for £27,500.
1 One tonne is 1,000 kilograms. For current information, see the web site of the London Metal Exchange: www.lme.com.
现货价格
供应商和消费者今天就6个⽉期钢的价格达成⼀致可以减少不确定性
远期合约
在未来约定的时间以约定的价格买进1卖出约定放量的资产
标的资产
-乾⽅
Page 134 Section 2 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
2.4 Pricing a forward contract with no income. Suppose a forward contract specifies that A will buy the
asset S from B at the priceK at the future time T . So A is the long side and B is the short side.
Let ST denote the actual price of the asset at time T .
IfK > ST then the short side B makes the profitK ST .
IfK < ST then the long side A makes the profit ST K.
Assume that the force of interest available on a risk-free investment in (0, T ) is . By assuming the no arbitrage
assumption, we can calculate the priceK.
Method I.
Consider the following two investment portfolios:
Portfolio A. At time 0:
• buy one unit of S at the current price S0.
Value of portfolio at time 0: S0. Value of portfolio at time T: one unit of S which has value ST .
Portfolio B. At time 0:
• enter forward contract to buy one S with forward priceK at time T ;
• investKeT in a risk-free investment at force of interest .
Value of portfolio at time 0: KeT . Value of portfolio at time T: one unit of S which has value ST . (The
amountK from the risk-free investment pays for the forward contract. The size of the investment at time 0
is chosen to beKeT in order to ensure that values of the two portfolios are equal at time T .)
The no arbitrage assumption implies that
KeT = S0
and hence
K = S0eT
Method II.
We can simplify Method I as follows: equate the time 0 values. Suppose A buys one S at time 0 for the
price S0. At time T , he then has the asset S which he values atK. Equating the time 0 values gives:
S0 = KeT
Note that it has not been necessary to make any assumptions about how the price, St, of the asset varies over
(0, T ). The forward price will change over time. SupposeKt denotes the forward price for a contract taken out
at time t for maturity time T . Then Kt = Ste(Tt) where St is the price of the security at time t. As t ! T
we have Kt = Ste(Tt) ! ST . Hence as t approaches the settlement time, T , the forward price tends to the
price of the underlying asset.
Example 2.4a. The price of a stock is currently 300p. (a) What is the price of a futures contract on this stock with
a settlement day in 180 days’ time if the risk-free interest rate is 5% per annum. (Assume there are no dividends
payable over the next 180 days.) (b) After 50 days, suppose the price of the stock has risen to 308p and the risk-free
interest rate has not changed. Calculate the change in the margin account for an investor holding the futures contract
specified in (a).
Solution. (a) Equating the time 0 values gives S0 = Ket with S0 = 300, t = 180/365 and e = 1.05. Hence
K = 300 ⇥ 1.05180/365 = 307.31. (b) New price of futures contract is Kt = 308 ⇥ 1.05130/365 = 313.40. Hence,
change in margin account is the gain 313.40 307.31 = 6.09.
2.5 Pricing a forward contract with a fixed income. Now suppose the security S is a bond which pays the
coupon c at time t1 where t1 2 (0, T ).
Example 2.5a. First consider the following simplification: suppose the interest rate is 0% and hence £1 tomorrow
is worth exactly £1 today.
(a) Suppose the current price of an asset S is £10. It pays a coupon of £1 at time 1. If a forward contract is to
purchase S at time 2 for the amount £K, then how much shouldK be?
(b) If the current price of S is S0 and the coupon is c, then how much shouldK be?
Solution. (a) Here is the justification thatK should be £9.
Suppose K > 9: then an arbitrageur will buy S now for £10 and enter into a forward contract to sell the asset S
at time 2 for £K. He will receive £1 at time 1 and £K at time 2; hence he makes a risk-free profit of £(K 9) for
every 1 unit of S.
SupposeK < 9. Then an arbitrageur should sell S short now for £10 and enter into a long forward contract to buy S
at time 2 for £K. Of course he must now pay the coupon of £1 at time 1 (which goes to the person from whom S was
⽆收益远期合约的定价
7 Arbitrage Sep 27, 2016(9:51) Section 2 Page 135
borrowed.) and £K at time 2. Hence for every unit of S he makes a profit of £(10 1K) = £(9K). Similarly,
anyone owning the asset S at time 0 should make use of the same arbitrage possibility.
(b) By a similar argument,K = S0 c.
Suppose the security S is a bond which pays the coupon c at time t1 where t1 2 (0, T ). Suppose a forward
contract specifies that investor A will buy the asset S from investor B at the price K at the future time T . We
wish to decide what is the appropriate value for K under the assumption that the force of interest available on
a risk-free investment in (0, T ) is (this means that if we invest an amount b for a length of time h during the
interval (0, T ) then it grows to beh).
Method I.
Consider the following two investment portfolios:
Portfolio A. At time 0:
• buy one unit of S at the current price S0.
Value of portfolio at time 0: S0. Value of portfolio at time T: one unit of S plus amount ce(Tt1) (recall
the asset S pays coupon c at time t1 which is then invested).
Portfolio B. At time 0:
• enter forward contract to buy one S with forward priceK maturing at time T ;
• investKeT + cet1 in a risk-free investment at force of interest .
Value of portfolio at time 0: KeT + cet1 . Value of portfolio at time T: one unit of S plus amount
ce(Tt1). (The amountKeT + cet1 accumulates toK + ce(Tt1) and the amountK from the risk-free
investment pays for the forward contract. The size of the investment at time 0 is chosen to be KeT +
cet1 in order to ensure that values of the two portfolios are equal at time T .)
The no arbitrage assumption implies that
KeT + cet1 = S0
and henceK = S0eT ce(Tt1).
Method II.
Equate the time 0 values. Suppose A buys one S at time 0 for the price S0. In return he gets the coupon c
at time t1 and has the asset S at time T which he values atK. Equating the time 0 values gives:
S0 = cet1 +KeT
Note that it has not been necessary to make any assumptions about how the price, St, of the asset varies over
(0, T ).
Now suppose we have more than one coupon payment: suppose we have coupons of size ck at times tk for
k = 1, 2, . . . , n. Then the argument of equating time 0 values gives
KeT = S0
nX
k=1
cke
tk
Example 2.5b. The price of a stock is currently 250p. A dividend payment of 10p is expected in 35 days’ time
together with another dividend payment of 15p after a further 180 days. What is the price of a futures contract on
this stock with a settlement date of 250 days’ time if the risk-free interest rate is 7% per annum.
Solution. We have S0 = 250, T = 250/365 and e = 1 + i = 1.07. Equating time 0 values gives
KeT = S0
nX
k=1
cke
tk = 250 10⇥ 1.0735/365 15⇥ 1.07215/365
leading toK = 236.35.
Example 2.5c. A bond has a current price of £600. Its face value is £1,000 and it matures in exactly 5 years and
pays coupons of £30 every 6 months. The first payment is due in exactly 6 months.
The 6-month and 12-month interest rates are nominal 6% and 7% per annum compounded continuously.
By using the “no arbitrage” assumption, find the price of a forward contract to purchase the bond in exactly one year.
Solution. LetK denote the forward price. Equating time 0 prices gives: 600 = Ke0.07 + 30e0.03 + 30e0.07.
If we wish to equate time T = 1 values then we must argue as follows. We are given the force of interest rates
F0,0.5 = 0.06 and F0,1 = 0.07. Hence the forward force of interest F0.5,0.5 = 0.08 (see example(1.6a) on page 104).
This means that the forward force of interest for an investment starting in 6 months’ time is a nominal 8% per annum
compounded continuously. The coupon received in 6 months’ time can be invested and hence will grow to 30e0.04
Page 136 Section 2 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
by time 1. Hence 600e0.07 = 30e0.04 + 30 + K. This is exactly the same equation as the previous one. Both give
K = 582.28.
The price of K = 582.28 can be justified as follows. If K > 582.28, then an investor should buy S now and enter
into a forward contract to sell S forK at time 1 and a contract to invest the dividend at time 0.5 for 6 months.
His cash flow is summarised in the following table.
time 0 0.5 1
cash flow -600 30 30+K
The NPV is positive.
IfK < 582.28, then an investor with S should sell S now and enter into a forward contract to buy S forK at time 1.
His cash flow is
time 0 0.5 1
cash flow 600 -30 -30-K
The NPV is positive.
2.6 Pricing a forward contract with a known dividend yield. Now suppose the security S pays a known
dividend yield which is a nominal r% per annum compounded continuously—the value r is the average over
the life of the forward contract. This means that the dividend payment is proportional to the stock price at
the time of the dividend payment. Hence the actual size of the dividend payments over (0, T ) are unknown at
time 0 because the value of the stock price over the time interval (0, T ) is unknown at time 0. We can finesse
this problem by assuming that all dividend payments are reinvested in further units of S; hence we assume that
1 unit of the security grows to erT units of the security over the period (0, T ).
Portfolio A. At time 0:
• enter forward contract to buy one S with forward priceK maturing at time T ;
• investKeT in a risk-free investment at force of interest .
Value of portfolio at time 0: KeT . Value of portfolio at time T: one unit of S. (The amount KeT
accumulates toK which pays for the forward contract.)
Portfolio B. At time 0:
• buy erT units of S at the current price S0 (and reinvest dividend income in further units of S).
Value of portfolio at time 0: erTS0. Value of portfolio at time T: one unit of S.
The no arbitrage assumption implies that
KeT = S0erT and henceK = S0e(r)T
Example 2.6a. The price of a stock is currently 250p. It is expected to provide a dividend income of 2% of the asset
price which is received continuously during a six month period. What is the price of a forward contract on this stock
with a settlement date of 6 months’ time if the risk-free interest rate is a nominal 7% per annum with continuous
compounding.
Solution. Now S0 = 250, = 0.07 and er = 1.022. The time 0 value of one unit of stock at time T is S0erT .
Equating time 0 values givesKeT = S0erT which leads toK = 250e(0.07r)0.5 = 250e0.035/1.02 = 253.83.
2.7 Value of a forward contract at intermediate times.
Suppose a forward contract is agreed at time 0 with forward price K0 for one unit of S at time T . At time T ,
the value of the contract to the long side is ST K0, the value of the stock at time T minus the price of the
forward contract agreed at time 0. Suppose we wish to estimate Vt, the value of the forward contract to the long
side at any time t 2 [0, T ]. Then VT = ST K0, V0 = 0 and the value to the short side at any time t 2 [0, T ]
is (Vt).
Consider the following 2 portfolios:
Portfolio A. At time t
• buy the forward contract agreed at time 0 for the price Vt.
• investK0e(Tt) in a risk-free investment at force of interest for the interval (t, T ).
Value of portfolio at time t: Vt + K0e(Tt). Value of portfolio at time T : one unit of S. (Because the
amount K0e(Tt) grows to K0 which is the amount needed by the forward contract which returns one
unit of S.)
Portfolio B. At time t:
• agree to a new forward contract with priceKt for one unit of S at time T ;
7 Arbitrage Sep 27, 2016(9:51) Section 2 Page 137
• investKte(Tt) in a risk-free investment at force of interest for the interval (t, T ).
Value of portfolio at time t: Kte(Tt). Value of portfolio at time T : one unit of S.
Hence
Vt +K0e(Tt) = Kte(Tt)
and so Vt = (Kt K0)e(Tt). UsingKt = Ste(Tt) gives Vt = St S0et.
Here is a more succinct derivation: now the long side agrees at time 0 to payK0 at time T for one unit of S and
also agrees at time t to pay Kt at time T for one unit of S. Hence the value to the long side of such a forward
contract at time t is Kt K0 which is in time T values. Its value in time t values is Vt = (Kt K0)e(Tt).
UsingK0 = S0eT andKt = Ste(Tt) gives Vt = St S0et.
Example 2.7a. Suppose t0 < t1 < t2 < t3 < t4. Suppose further that the price of a stock at time t0 was P0 and
was P1 at time t1. The stock will pay coupons of C2 at time t2 and C3 at time t3. An investor entered into a long
forward contract for 1 unit of the stock at time t0 with the contract due to mature at time t4. Assuming an effective
rate of interest of 5% p.a. and no arbitrage, what is the value of the contract at time t1.
Solution. LetK0 denote the price of the original contract agreed at time t0 and let ⌫ = 1/1.05.
Equating the time t0 values gives an equation forK0: K0⌫t4 = P0 C2⌫t2 C3⌫t3 .
LetK1 denote the price of a contract agreed at time t1 for 1 unit of the stock to be delivered at time t4.
Equating the time t1 values gives an equation forK1: K1⌫t4t1 = P1 C2⌫t2t1 C3⌫t3t1 .
Hence the value of the original contract to the long side at time t1 is the difference in prices of these two contracts in
time t1 values; and that is (K1 K0)⌫t4t1 = P1 P0/⌫t1 .
2.8 Speculation, hedging, gearing and leverage.
Speculation is the process of taking a risk in the hope of making a profit.
Example 2.8a. This example is a continuation of example (2.3a).
Suppose A buys one September futures contract for copper from B. Then A will have to make a deposit, called the
margin. This may be as low as 1% for a financial futures contract. Assume it is 10% for our copper futures contract.
Hence A must deposit 10%⇥ £27,500 = £2,750. Suppose that by September, the price of copper has risen to £1,300
per tonne. Hence A makes a profit of 25 ⇥ (£13,000 £11,000) = £5,000 for an outlay of £2,750. But note that A
has taken the risk that he might lose £27,500 for an outlay of £2,750.
Suppose A believes the price of copper will fall. Hence he sells a September futures contract for 25 tonnes of copper
for £27,500. If the price of copper is only £900 per tonne on September 1, then he makes a profit of 25 ⇥ £200 =
£5,000. But suppose he is wrong and the price of copper doubles to £2,200 per tonne on September 1. Then A loses
25⇥ £1,100 = £27,500. Indeed, his potential loss is unlimited.
Gearing and leverage. These terms mean that a large gain or a large loss can be made for a small outlay.
Leverage is the term used in the USA whilst gearing is the term used in the UK.
Example 2.8b. The copper example again: examples (2.3a) and (2.8a).
Suppose A believes the price will rise. If he trades in the actual product and purchases 25 tonnes of copper today at
the current spot price of £1,000 per tonne, then this will cost him £25,000. If the price rises to £1,300 per tonne on
September 1, then he makes a profit of 25⇥ £300 = £7,500 from an investment of £25,000; this profit is 30% of his
initial investment. His largest possible loss, if the copper becomes worthless, is £25,000 which is 100% of his initial
investment.
Suppose instead of buying the actual product today, he buys a futures contract for 25 tonnes for £27,500. He then
makes a profit of £5,000 for an outlay of the deposit of £2,750; this profit is 182% of his initial investment. Of course,
he is also exposed to a potential loss of £27,500 which is 1,000% of his initial investment!
Hedging is the avoidance of risk. This implies making an investment to reduce the risk of adverse price
movements in an asset.
Example 2.8c. A UK exporter has won a contract to supply machinery to the US. The exporter will receive a
payment of $1m in six month’s time when the machinery is delivered. In order to remove the risk that an adverse
movement in the exchange rate makes the deal unprofitable, the exporter purchases a currency futures contract. This
guarantees that he can exchange the dollars which he will receive in six months’ time into sterling at a rate which is
specified today. In this way, the exporter ensures that the risk that the export deal is made unprofitable by movements
in the exchange rate is removed.
Another example: if you owned a stock, then short selling an equal amount would be a “perfect hedge”. The
term static hedge means that the hedge portfolio does not change over the period of the contract.
Page 138 Exercises 3 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
Example 2.8d. Consider again the forward contract made at time 0 to buy one S with forward priceK maturing at
time T . Find a static hedge for the short side of this simple forward contract. Suppose the risk free force of interest
is .
Solution. Recall that the short side is the party who contracts to sell one S at time T for the priceK. If the short side
just purchases one unit of S at time 0 and the price falls, then more will have been paid for S then was necessary; if
the short side just purchases one unit of S at time T to fulfil the contract and the price ST at time T is greater thenK
(the amount he receives at time T ), then a loss will be made—indeed, this potential loss is theoretically unlimited.
Suppose at time 0 the short side borrows the amount KeT = S0 at the risk-free force of interest and buys one
unit of S at the current price S0. The price of this hedge portfolio at time 0 is KeT + S0 = 0.
At time T , the short side has a debt which has grown to K which now equals the amount received under the terms
of the forward contract; in addition he owns one unit of S which must be delivered to the long side under the terms
of the forward contract. Hence this hedge portfolio means the short side is certain not to make a loss or a profit
whatever the price of the stock at time T .
3 Exercises (exs7-1.tex)
1. The price of a given share is 80p. The risk-free rate of interest is 5% per annum convertible quarterly. Assuming
no arbitrage and that the share will not pay any income, calculate the forward price for the share, for settlement in
exactly one quarter of a year. (Institute/Faculty of Actuaries Examinations, April, 2002) [2]
2. A 10-month forward contract is issued on 1 March 2002 on a stock with a price of £12 per share at that date.
Dividends of £1.50 per share are expected on 1 June, 1 September and 1 December 2002.
Calculate the forward price, assuming a risk-free rate of interest of 6% per annum convertible half-yearly and no
arbitrage. (Institute/Faculty of Actuaries Examinations, September 2002) [4]
3. (i) Explain what is meant by the “no arbitrage” assumption in financial mathematics.
(ii) A 7 month forward contract is issued on 1 January 2000 on a stock with a price of £60 per share. Dividends of
£2 per share are expected after 3 and 6 months.
Assuming a risk-free force of interest of 7% per annum and no arbitrage, calculate the forward price.
(Institute/Faculty of Actuaries Examinations, April 2000) [6]
4. An asset has a current price of £1.20. Given a risk-free rate of interest of 5% per annum effective and assuming no
arbitrage, calculate the forward price to be paid in 91 days.
(Institute/Faculty of Actuaries Examinations, September 2000) [3]
5. (i) State what is meant by a “forward contract”. Your answer should include reference to the terms “short forward
position” and “long forward position”.
(ii) A 3-month forward contract is issued on 1 February 2001 on a stock with a price of £150 per share. Dividends
are received continuously and the dividend yield is 3% per annum. In addition, it is anticipated that a special
dividend of £30 per share will be paid on 1 April 2001.
Assuming a risk-free force of interest of 5% per annum and no arbitrage, calculate the forward price per share
of the contract. (Institute/Faculty of Actuaries Examinations, April 2001) [6]
6. A one year forward contract is issued on 1 April 2003 on a share with a price of 600p at that date. Dividends of 30p
per share are expected on 30 September 2003 and 31 March 2004. The 6-month and 12-month spot risk-free interest
rates are 4% and 4.5% per annum effective respectively on 1 April 2003.
Calculate the forward price at issue, assuming no arbitrage.
(Institute/Faculty of Actuaries Examinations, September 2003) [4]
7. (i) Explain what is meant by a “forward contract”. Your answer should include reference to the terms “short
forward position” and “long forward position”.
(ii) An investor entered into a long forward contract for £100 nominal of a security seven years ago and the contract
is due to mature in 3 years’ time. The price per £100 nominal of the security was £96 seven years ago and is
now £148. The risk-free rate of interest can be assumed to be 4% per annum effective during the contract.
Calculate the value of the contract now if the security will pay a single coupon of £7 in two years’ time and this
was known from the outset. You should assume no arbitrage.
(Institute/Faculty of Actuaries Examinations, April 2003) [3+5=8]
7 Arbitrage Sep 27, 2016(9:51) Exercises 3 Page 139
8. (i) Explain what is meant by the “no arbitrage” assumption in financial mathematics.
(ii) A 3-year forward contract is to be issued on a particular company share. The current market value of the share
is £4.50 and a dividend of £0.20 per share has just been paid. The parties to the contract assume that the future
quarterly dividends will increase by 1% per quarter year compound for the first two years and by 11/2% per
quarter year compound for the final year.
Assuming a risk-free force of interest of 5% per annum and no arbitrage, calculate the forward price.
(Institute/Faculty of Actuaries Examinations, April 2004) [2+7=9]
9. The risk-free force of interest (t) at time t is given by:
(t) =
n 0.05 if 0 < t 10;
0.08 + 0.003t if t > 10.
(i) (a) Calculate the accumulation at time 15 of £100 invested at time t = 5.
(b) Calculate the accumulation at time 14 of £100 invested at time t = 5.
(c) Calculate the accumulation at time 15 of £100 invested at time t = 14.
(d) Calculate the equivalent constant force of interest from time t = 5 to time t = 15.
(ii) Calculate the present value at time t = 0 of a continuous payment stream payable at a rate of 100e0.01t from
time t = 0 to time t = 5.
(iii) A one year forward contract is issued at time t = 0 on a share with a price of 300p at that date. A dividend of
7p per share is expected at time t = 1/2. Calculate the forward price of the share, assuming no arbitrage.
(Institute/Faculty of Actuaries Examinations, September 2004) [9+4+4=17]
10. A bond is priced at £95 per £100 nominal, has a coupon of 5% per annum payable half-yearly, and has an outstanding
term of 5 years.
An investor holds a short position in a forward contract on £1 million nominal of this bond, with a delivery price of
£98 per £100 nominal and maturity in exactly one year, immediately following the coupon payment then due.
The continuously compounded risk-free rates of interest for terms of 6 months and one year are 4.6% per annum and
5.2% per annum, respectively.
Calculate the value of this forward contract to the investor assuming no arbitrage.
(Institute/Faculty of Actuaries Examinations, April 2005) [5]
11. (i) Explain what is meant by the “expectations theory” for the shape of the yield curve.
(ii) Short-term, one-year annual effective interest rates are currently 8%; they are expected to be 7% in one year’s
time, 6% in two years’ time and 5% in three years’ time.
(a) Calculate the gross redemption yields (spot rates of interest) from 1-year, 2-year, 3-year and 4-year zero
coupon bonds assuming the expectations theory explanation of the yield curve holds.
(b) The price of a coupon paying bond is calculated by discounting individual payments from the bond at the
zero coupon bond yields in (a). Calculate the gross redemption yield of a bond that is redeemed at par in
exactly 4 years and pays a coupon of 5 per annum annually in arrear.
(c) A two-year forward contract has just been issued on a share with a price of 400p. A dividend of 4p is
expected in exactly in exactly one year.
Calculate the forward price using the above spot rates of interest assuming no arbitrage.
(Institute/Faculty of Actuaries Examinations, September 2005) [2+12=14]
12. A share currently trades at £10 and will pay a dividend of 50p in one month’s time. A six-month forward contract is
available on the share for £9.70. Show that an investor can make a risk-free profit if the risk-free force of interest is
3% per annum. (Institute/Faculty of Actuaries Examinations, April 2006) [4]
13. An investor is able to purchase or sell two specially designed risk-free securities, A and B. Short sales of both
securities are possible. Security A has a market price of 20p. In the event that a particular stock market index goes
up over the next year, it will pay 25p and, in the event that the stock market index goes down, it will pay 15p.
Security B has a market price of 15p. In the event that the stock market index goes up over the next year, it will
pay 20p and, in the event that the stock market index goes down it will pay 12p.
(i) Explain what is meant by the assumption of “no arbitrage” used in the pricing of derivatives contracts.
(ii) Find the market price of B such that there are no arbitrage opportunities and assuming the price of A remains
fixed. Explain your reasoning. (Institute/Faculty of Actuaries Examinations, September 2006) [2+2=4]
14. An investor entered into a long forward contract for a security 5 years ago and the contract is due to mature in 7 years’
time. The price of the security was £95 five years ago and is now £145. The risk-free rate of interest can be assumed
to be 3% per annum throughout the 12 year period.
Assuming no arbitrage, calculate the value of the contract now if
(i) The security will pay dividends of £5 in two years’ time and £6 in four years’ time.
(ii) The security has paid and will continue to pay annually in arrear a dividend of 2% per annum of the market
price of the security at the time of payment. (Institute/Faculty of Actuaries Examinations, April 2007) [3+3=6]
Page 140 Exercises 3 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed
15. A one-year forward contract is issued on 1 April 2007 on a share with a price of 900p at that date. Dividends of 50p
per share are expected on 30 September 2007 and 31 March 2008. The 6-month and 12-month spot, risk-free rates
of interest are 5% and 6% per annum effective respectively on 1 April 2007.
Calculate the forward price at issue, stating any assumptions.
(Institute/Faculty of Actuaries Examinations, September 2007) [4]
16. An 11-month forward contract is issued on 1 March 2008 on a stock with a price of £10 per share at that date.
Dividends of 50 pence per share are expected to be paid on 1 April and 1 October 2008.
Calculate the forward price at issue, assuming a risk-free rate of interest of 5% per annum effective and no arbitrage.
(Institute/Faculty of Actuaries Examinations, April 2008) [4]
17. (i) A forward contract with a settlement date at time T is issued based on an underlying asset with a current market
price of B. The annualised risk free force of interest applying over the term of the forward contract is and the
underlying asset pays no income. Show that the theoretical forward price is given by K = Bet, assuming no
arbitrage.
(ii) An asset has a current market price of 200p, and will pay an income of 10p in exactly 3 months’ time. Cal-
culate the price of a forward contract to be settled in exactly 6 months, assuming a risk free rate of interest of
8% per annum convertible quarterly. (Institute/Faculty of Actuaries Examinations, September 2008) [3+3=6]
18. (i) Explain what is meant by the “no arbitrage” assumption in financial mathematics.
An investor entered into a long forward contract for a security four years ago and the contract is due to mature in five
years’ time. The price of the security was £7.20 four years ago and is now £10.45. The risk-free rate of interest can
be assumed to be 2.5% per annum effective throughout the nine-year period.
(ii) Calculate, assuming no arbitrage, the value of the contract now if the security will pay dividends of £1.20
annually in arrear until maturity of the contract.
(iii) Calculate, assuming no arbitrage, the value of the contract now if the security has paid and will continue to pay
annually in arrear a dividend equal to 3% of the market price of the security at the time of the payment.
(Institute/Faculty of Actuaries Examinations, April 2012) [2+3+3=8]
19. A 10 month forward contract was issued on 1 September 2012 for a share with a price of £10 at that date. Dividends
of £1 per share are expected on 1 December 2012, 1 March 2013 and 1 June 2013.
(i) Calculate the forward price assuming a risk-free rate of interest of 8% per annum convertible half yearly and no
arbitrage.
(ii) Explain why it is not necessary to use the expected price of the share at the time the forward matures in the
calculation of the forward price. (Institute/Faculty of Actuaries Examinations, September 2012) [4+2=6]
20. (i) Explain the main difference:
(a) between options and futures.
(b) between call options and put options.
(ii) A one-year forward contract is issued on 1 April 2013 on a share with a price at that date of £10.50. Dividends
of £1.10 per share are expected on 30 September 2013 and 31 March 2014. On 1 April 2013, the 6-month risk-
free spot rate of interest is 4.5% per annum convertible half-yearly and the 12-month risk-free rate of interest is
5% per annum convertible half yearly.
Calculate the forward price at issue, stating any further assumptions you make.
(Institute/Faculty of Actuaries Examinations, April 2013) [4+4=8]
21. A nine-month forward contract is issued on 1 March 2012 on a share with a price of £1.80 at that date. Dividends of
10p per share are expected on 1 September 2012. Calculate the forward price at issue assuming a risk-free rate of in-
terest of 4% per annum effective and no arbitrage. (Institute/Faculty of Actuaries Examinations, September 2013) [3]
22. (i) Describe what is meant by the “no arbitrage” assumption in financial mathematics.
A 9-month forward contract is issued on 1 April 2015 on a stock with a price of £6 per share on that date. Dividends
are assumed to be received continuously and the dividend yield is 3.5% per annum.
(ii) Calculate the theoretical forward price per share of the contract, assuming no arbitrage and a risk-free force of
interest of 9% per annum.
(iii) The actual forward price per share of the contract is £6.30 and the risk-free force of interest is as in part (ii).
Outline how an investor could make an arbitrage profit.
(Institute/Faculty of Actuaries Examinations, April 2015) [2+2+2=6]
7 Arbitrage Sep 27, 2016(9:51) Exercises 3 Page 141
23. A 9 month forward contract was issued on 1 October 2015 on a share with a price at that date of £10. Dividends
of 50 pence per share are expected on 1 November 2015 and 1 May 2016. The risk-free force of interest is 5% per
annum.
(i) Calculate the forward price at issue, stating any further assumptions made and showing all workings.
(ii) Explain why the expected price of the share nine months after issue does not have to be taken into account when
pricing the forward. (Institute/Faculty of Actuaries Examinations, September 2015) [4+2=6]
