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EC372 Bond and Derivatives Markets
Topic #7: Options III: Applications
R. E. Bailey
Department of Economics
University of Essex
Outline
Contents
1
Stock Index Options
1
2
Options on Futures
2
2.1
3
Synthetic futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Interest rate options
4
4
Options and Portfolio Risks
6
5
Portfolio insurance
7
Reading: Economics of Financial Markets, chapter 20
1
Stock Index Options
Stock index options
• Main difference from single company options: ‘exercise’ requires cash settlement
Example: FT-SE 100 index option
– European style
– Quotation: Index points, each point worth £10.
– Tick size: 0.5 point, worth £5.
– Delivery dates: end of March, June, September, December + four nearest months
• Example:
– Suppose that c = 150, June expiry, exercise price of 5000
– Buy 1 option for £1500 = 10 × 150.
– End June, suppose that FTSE-100 index = 5300.
– Exercise option for £3000 = (5300 − 5000) × 1 × 10
– Net gain: £1500 = 3000 − 1500
– Prior to end June, option could have been sold.
1
2
Options on Futures
Options on futures
• Main features:
– Underlying asset: a futures contract
– Expiry date: shortly before futures contract delivery date
– Typical style: American
– Call: option to buy a futures contract (long position)
– Put: option to sell a futures contract (short position)
• Example 1: American call option on oil futures
– Buy one option with X = $90 on futures with December delivery
– Each futures contract is for 1000 barrels of oil
– Suppose that: f (September, December) = $94
– Suppose option is exercised in September:
1. Option payoff (gross of premium): $4000 = 1000 × (94 − 90)
2. One futures contract is received from option writer
– In frictionless market: futures contract could be sold without loss
– In a frictionless market: payoff from selling option ≥ exercise
Why? For American call options, C ≥ f − X, where the futures contract, with price
f , plays the role of the underlying asset. Hence, sale of the option for C yields more
than exercise, f − X. If this were not the case, so that C < f − X, there would be a
instant arbitrage opportunity: buy an option and exercise it immediately. As always the
existence of frictions (e.g. transaction costs) could upset the argument: the issue is the
magnitude of the frictions.
Options on futures, continued
• Example 2: American put option on gold futures
– Buy one option with X = $1800 on futures with December delivery
– Each futures contract is for 100 troy ounces of refined gold
– Suppose that: f (October, December) = $1720
– Suppose option is exercised in October:
1. Option payoff (gross of premium): $8000 = 100 × (1800 − 1720)
2. Option writer sells one futures contract on behalf of option holder
– In frictionless market: could buy futures contract without loss
– In a frictionless market: payoff from selling option ≥ exercise
Why? Check the reasoning for call options, above. For American put options, P ≥
X − f , otherwise there is an arbitrage opportunity. Hence, rather than exercising the
option, its sale in the market will always yield at least as much as exercising it.
• With market frictions (transaction costs), it may be profitable to exercise the option, and to
retain the futures contract (to offset later or deliver)
2
More details for options on futures:
A call option on a futures contract confers the right to take a long position in the futures contract. A
put option confers the right to take a short position in the futures contract. The relationship between
options and the assigned futures position can be summarised as follows:
Long futures
Short futures
Call options
long call (holder)
short call (writer)
Put options
short put (writer)
long put (holder)
The writer of options on futures receives the option premium in return for an obligation to take a
futures position opposite to the buyer in the event that the option is exercised.
2.1
Synthetic futures
Synthetic futures
• Payoffs at expiry, T : call option: cT = max[0, fT − X]
pT = max[0, X − fT ]
• Payoffs on options can replicate payoff on futures:
– Long futures = long call + short put
– Short futures = short call + long put
• Synthetic futures: option pairs that replicate futures payoffs
• Put-call parity for European options: c +
• For European options on futures: c +
X
=p+S
R
X
f
=p+
R
R
– Notice presence of f /R, not f
– Why? Because f is paid only at T (delivery date)
Option Payoffs:
Hence, if the options are exercised at expiry, T , the payoffs cT and pT are
Call option: cT = max[0, fT − X];
Put option: pT = max[0, X − fT ]
where X denotes the exercise price.
Recall that the payoff for a long futures position held to maturity is given by fT − f per contract,
i.e. the contract is purchased at price f and offset by selling it for fT at maturity.
Suppose that the futures price today, f , equals the options’ exercise price, X, f = X. Consider
a portfolio comprising the purchase of one call option and the writing of one put option — sometimes
called a ‘synthetic futures contract’. If at maturity fT > X, the call option is worth fT −X = fT −f ,
exactly equal to the payoff on the futures contract. (The put option is not exercised if fT > X.)
Alternatively, if fT < X, the put option is exercised with a (negative) payoff fT − X = fT − f ,
exactly equal to the payoff on the futures contract. (The call option is not exercised if fT < X.)
Hence, the payoffs for the portfolio of one long call and one short put are exactly the same at
expiry as for a long futures contract, when the futures contract is purchased at the option exercise
price.
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The payoff at maturity for a short futures position is exactly the same as for a portfolio comprising one long put option and one short call option, again assuming that f = X.
Suppose that f 6= X. In this case, the put-call parity relationship is helpful. With a futures
contract as the underlying asset, the put-call parity relationship for European options is
c+
X
f
=p+
R
R
(1)
where R is the abbreviated form of R(t, T ), the interest factor.
Notice that the put-call parity relationship implies that f = X + R(c − p) — the difference in
the option prices compensates for any difference between f and X.
Example Suppose that f = $122, X = $100 and R = 1.10. The put-call parity relationship implies
that c − p = $20. A long futures position has a payoff of fT − 122 per contract. Compare this
with the payoff from buying one call and writing one put (i.e. a synthetic futures contract). The
payoff at expiry on the portfolio of options is fT − 100. (If fT > 100 the call is exercised; if
fT < 100 the put is exercised.) However, the net cost of purchasing a call and writing a put equals
c − p = $20, an amount that must be borrowed at the outset, and repaid with interest at date T :
R(c − p) = 1.10(20) = $22. Thus the net payoff from the options portfolio equals fT − 100 − 22,
precisely the same as for the futures contract.
3
Interest rate options
Interest rate options
• Interest rate option: underlying asset is an interest bearing security
• Example: option on 3-month sterling futures contract (ICE NYSE)
– Unit of trading: one futures contract
– Option premium: quoted in ticks, £6.25, each tick 0.005%
– Premiums are not paid at the outset but via margin account
• Example: put option
– Suppose: P = 0.24 (48 ticks), X = 95.00, T = end June
– Buy one option, cost £300 = 24 × 12.50
– At expiry, suppose EDSP = 94.00.
– Option is automatically exercised (it is profitable to do so)
– Short futures position assigned and immediately cash settled
∗ Sell at X = 95, buy at EDSP = 94
– Gain (gross of premium): (95 − 94) × 100 × 12.50 = £1250
– Net gain: £1250 − 300 = £950
Interest rate options, continued
• Applications:
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– Interest rate cap – for a borrower
– Interest rate floor – for a lender
• Example: interest rate cap
– Purpose: protect against future interest rate rise
– Buy put options (option to sell futures contract at X)
– If at T interest rate is high, EDSP is low
– If EDSP is below X: exercise option – futures gain compensates for high interest rate
– If at T interest rate is low, EDSP is high
– If EDSP is above X: exercise dies unexercised – borrow at the low market interest rate
– But the cap is not free: – option premium must be paid, whether or not they are exercised
Effective rate
Rate cap
Effective rate
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Rate floor
6
5.24%
4.74%
-
5.00%
Market rate
-
5.00%
Market rate
Figure 1: Interest rate caps and floors.
An interest rate cap (left panel) places a ceiling on the rate at which funds can be borrowed. Notice that the cap is not free: the effective interest rate (i.e. cost of funds)
exceeds the market rate at low market interest rates. An interest rate floor (right panel)
places a lower bound on the rate at which funds can be lent. Again, notice that the floor
is not free: the effective interest rate (i.e. return on funds) is less than the market rate for
high market interest rates.
Example: interest rate cap
Assume that a company plans, in March, to borrow £500,000 for three months from July 1st
at whatever market interest rate then rules. Suppose that a put option with exercise price of 95.00,
expiring at the end of June, currently trades at a premium of 0.24, i.e. 48 ticks, where each tick equals
0.005 percentage points, worth £6.25 per tick.
One put option is purchased for £300 = 24 × 12.50. Now suppose that, at expiry, the futures
price (i.e. the EDSP) is 94.00, calculated from a market interest rate of 6.00% (94.00 = 100 − 6.00).
The option is automatically exercised and the investor is assigned a short futures position.
The futures position is then cash-settled with the sale of one contract for 95.00 at a date when the
market price equals 94.00 (the EDSP), thus yielding a gain of 100 ticks, i.e. £1, 250 = 100 × 12.50.
(For the futures contract a tick is worth £12.50.) Consequently, there is a net gain of £950, equal to
0.76% of £500,000 for three months. Hence, if £500,000 is borrowed at 6%, the effective borrowing
rate is capped at 5.24%.
Effectively, a futures contract has been sold at 95.00 and repurchased at 94.00. The purchase
of the option ensures that no loss is incurred if the futures price is greater than 95.00 at the maturity
date. This benefit is not free. It’s cost is reflected in the option premium of 48 ticks (remember
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that an option tick for this contract is 0.005%, as distinct from the futures contract where the tick is
0.01% – it’s just a feature of the contract specification).
Each one-point increment in the interest rate is matched by a different payoff on the option
contract as shown in the following table.
Interest
rate∗
9.00%
8.00%
7.00%
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
Futures
Gain (+) % gain (+) Effective
price∗
or loss (−) or loss (−) borrowing rate∗
91.00
+4700 +3.76%
9.00 − 3.76 = 5.24%
92.00
+3450 +2.76%
8.00 − 2.76 = 5.24%
93.00
+2200 +1.76%
7.00 − 1.76 = 5.24%
94.00
+950 +0.76%
6.00 − 0.76 = 5.24%
95.00
−300 −0.24%
5.00 + 0.24 = 5.24%
96.00
−300 −0.24%
4.00 + 0.24 = 4.24%
97.00
−300 −0.24%
3.00 + 0.24 = 3.24%
98.00
−300 −0.24%
2.00 + 0.24 = 2.24%
99.00
−300 −0.24%
1.00 + 0.24 = 1.24%
(∗ Rates and prices realized at the option expiry date.)
If the interest rate (on the expiry date) is at or below 5%, the option is allowed to die, unexercised. The company then benefits from lower interest rates but pays 0.24% above the market rate,
the extra reflecting the option premium. But if the interest rate (as of the option expiry date) exceeds
5%, the company caps its borrowing cost at 5.24%.
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Options and Portfolio Risks
Hedging with options
• Hedging using options as the hedge instrument
– Portfolio value: Wt = N St + M ct
– Hence: ∆W = N ∆S + M ∆c
∆c
– Or: ∆W = N + M
∆S
∆S
– Remember: c = f (S, X, τ, R, σ)
– Hedging objective: choose M to make ∆W = 0
– From the Black-Scholes model it can be shown that:
∆c/∆S =≈ ∂c/∂S = Φ(x1 )
– Hence, hedge ratio: M/N = −∆S/∆c ≈ −1/Φ(x1 )
• Complications:
1. Hedge ratio depends on everything that affects c.
2. Option hedge-ratios are ‘dynamic’: vary over time
3. c is a non-linear function of S – hedge ratio varies as S varies
Hedging according to the above principles is often known as Delta hedging.
Some Greek letters
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The partial derivative of f (S, X, τ, R, σ) with respect to each of its arguments is crucial in
designing strategies involving options. Collectively, the partial derivatives are known as ‘the Greeks’
— gamma, theta, rho and even ‘vega’ appear, as well as delta.
The meaning of each term is as follows: gamma equals ∂ 2 f /∂S 2 , i.e. the rate of change of delta
with respect to S; theta equals the rate of change of the option premium with respect to time, t, or
−∂f /∂τ , because τ ≡ T − t; rho equals the rate of change of the option premium with respect to
the risk-free interest rate, i.e. ∂f /∂r (where r denotes the risk-free interest rate, assumed constant,
so that R = er(T −t) ); and vega equals ∂f /∂σ, the rate of change in the premium with respect to
volatility.
The presence of gamma, ∂ 2 f /∂S 2 6= 0, serves as a reminder that option prices are nonlinear functions of their arguments (at least in the context of the Black-Scholes model). That is, the
magnitude of delta varies with the value of S.
One implication of the nonlinearity is that the gamma should be taken into account to improve
the accuracy in approximating ∆c/∆S. This becomes of significance the larger is ∆S.
Another implication of the nonlinearity is that effective hedging strategies employing options
are ‘dynamic’ in the sense that the number of options held changes as the price of the underlying
asset changes. Changes in option holdings should, in principle, be made continuously via a process
sometimes called ‘rebalancing’. Such a process can be expensive in transactions costs, consequently
reducing the attractiveness of options as hedge instruments.
5
Portfolio insurance
Terminology: the contracts resulting from the strategies studied below are now commonly known
as structured products, typically sold by banks to their clients (large companies with specific needs).
The principles are exactly the same as for the older term ‘portfolio insurance’.
Portfolio insurance
• Place a floor under a portfolio value, without restricting gains
• Variant 1: Stop-loss selling
– Sell assets when prices hit a lower ‘trigger’, and purchase when prices rise to a higher
‘trigger’ level
– Stop-loss selling relies on asset prices changing ‘smoothly’
– May be expensive in transaction costs
• Variant 2: Purchase put options
– Buy put options that can be exercised if asset prices fall too low
– Option premiums are positive (insurance premium)
– Portfolios typically comprise many assets (lots of options needed)
– Use stock index options – but may not reflect the portfolio composition.
– Traded options tend to have only a short time to expiry
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Payoff 6
(a)
X −p
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@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
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@
@
@
0
@
@
X
@
@
@
@
@
@
@
- S
T
A
Payoff 6
B
(b)
X
X −p
0
- S
T
X
Figure 2: Portfolio Insurance with a put option
Panel (a) shows the payoff from a put option as a function of the underlying portfolio’s
market value at the expiry date, T . (The portfolio is assumed to comprise a single asset,
with price ST at T .) The dashed line in panel (b) shows the value of an uninsured
portfolio, i.e. the payoff equals ST . Purchase of the put option provides a lower bound
of X − p for the payoff when ST < X. In the event that ST > X, the payoff increases
one-for-one with the market value of the underlying portfolio, though its level is lower
by the amount of the option premium, p.
Portfolio insurance, 2
• Variant 3: Portfolio insurance with call options
– Invest in risk-free bonds and purchase call options
– If asset prices rise, exercise the options
– Shortcomings: as for portfolio insurance using put options
• Variant 4: create synthetic put options
– The Black-Scholes analysis suggests that portfolios of risky assets and bonds can replicate (‘mimic’) option payoffs
– ⇒ choose risky asset and bond portfolios to create same payoffs as options
– Complications:
1. The premium, though implicit, is positive
2. Involves dynamic replication – continual purchase and sale of assets
3. Effectiveness relies on same assumptions as Black-Scholes
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4. Requires purchase of assets when prices are increasing, and sales when falling –
may be destabilizing
Portfolio Insurance: summary
The objective of portfolio insurance is to place a floor under the value of a portfolio without
restricting increases in its value. Costs of portfolio insurance: (a) a premium has to be paid (either,
explicitly for traded options, or, implicitly for synthetic options); and (b) continual rebalancing of
the portfolio may incur high transaction costs. Active portfolio management in the form of portfolio
insurance may yield benefits but it is not free.
*****
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