Free sample chapter from FRM Level I: Financial Markets and Products II.

Chapter 1: Options Markets and the Properties of Options

Chapter 1: Options Markets and the Properties of Options

The previous course ended with a hedge that worked. Ravensmoor's $800 million Core Equity Fund, with a beta of 1.15 against the Hartwell 500, was protected by selling index futures. The hedge cost nothing up front, removed 91.84% of the fund's variance, and left exactly the residual risk the manager is paid to take.

It also removed every dollar of upside. A fund that has sold futures against its whole market exposure earns the risk-free rate whatever the market does. If the index rises 10%, the futures lose what the shares gain. The investment committee wants something different: protection against a fall, and the gains from a rise kept.

That is what an option does, and this chapter prices it. It defines the contracts, then shows how far arbitrage alone constrains their prices — without any model of how the index moves. The constraints turn out to be tight. One of them, put-call parity, says that the hedge the committee wants differs from the futures hedge it already has by exactly one call option — and so it costs exactly what that call costs.

Learning Objectives

1.1 Calls, Puts and Moneyness

Call option The right, but not the obligation, to buy an underlying asset at a fixed price — the strike or exercise price — on or before a fixed date. The buyer pays a premium for the right.

Put option The right, but not the obligation, to sell an underlying asset at the strike price on or before a fixed date. The buyer of a put profits when the underlying falls.

Every option has two sides. The buyer (the holder, or the long) pays the premium and holds the right. The seller (the writer, or the short) receives the premium and takes on the obligation: to deliver the asset if a call is exercised, or to buy it if a put is. So there are four basic positions — long call, short call, long put, short put — and every strategy in Chapter 2 is built from them.

Options also differ in when they can be exercised:

• A European option can be exercised only on its expiration date.
• An American option can be exercised at any time up to and including expiration.

The labels have nothing to do with geography. Most exchange-traded options on individual shares are American, and many on broad indices are European. An American option gives the holder everything a European one does, plus a choice, so it is worth at least as much. §1.9 asks when that extra choice is worth anything.

What options are written on, and what they are for

Options are used for the same three purposes as forwards and futures, with a different shape of exposure:

• Hedging. A put sets a floor under an asset you own, and a call sets a ceiling on the cost of an asset you will need to buy. Unlike a forward, the hedge keeps the favourable outcome — at the price of the premium.
• Speculation. An option controls a large exposure for a small premium, and the most a buyer can lose is that premium. Leverage with a capped loss is what makes options attractive to speculators.
• Arbitrage. Options must trade in consistent relationships with the underlying, with each other and with interest rates. §1.7 and §1.8 derive those relationships and trade against prices that break them.

Moneyness

Moneyness Whether an option would produce a positive cash flow if it were exercised immediately. A call is in the money when the underlying is above the strike, at the money when they are equal, and out of the money when the underlying is below the strike. A put is the reverse.

Moneyness divides a premium into two parts.

Intrinsic value What an option would be worth if it were exercised now: $\max(S - K, 0)$ for a call and $\max(K - S, 0)$ for a put. It is never negative, because the holder would simply not exercise.

Time value The premium in excess of intrinsic value. It is what the market pays for the chance that the option ends up further in the money before it expires.

These are the six-month Hartwell 500 options this course works with throughout. The index stands at 4,000.00 and the multiplier is $50 per index point. The prices are quoted by the market; how such prices are produced from a model is the subject of FRM Level I: Valuation Models.

Two things in that table are worth noticing now, because the rest of the chapter explains them.

First, time value is largest at the money. The 4,000 options are pure time value — 210.48 and 190.73 points — while the in-the-money options carry less (122.62 and 106.11) despite their higher premiums. An at-the-money option's outcome is most uncertain, and uncertainty is what time value pays for.

Second, the 4,000 call costs more than the 4,000 put, although both are at the money. The difference is 19.75 points. §1.8 shows that this gap is not a matter of opinion: it is fixed by the forward price of the index, to within the rounding of the quotes.

1.2 Payoffs and Profit on an Options Position

The payoff is what a position is worth at expiration. The profit is the payoff less what was paid for it (or plus what was received). With $S_T$ the underlying's price at expiration:

The convention in this reading, and on the exam unless a question says otherwise, is to subtract the premium as paid, without interest on it for the life of the option. §1.10 carries the premium forward at the risk-free rate, because there the comparison is between two hedges with different cash flows today.

mult, strike = 50, 4000
call_prem, put_prem = 210.48, 190.73

print(f"break-evens: call {strike + call_prem:,.2f}   put {strike - put_prem:,.2f}")
for s_t in (3600, 4000, 4400):
    long_call = (max(s_t - strike, 0) - call_prem) * mult
    long_put = (max(strike - s_t, 0) - put_prem) * mult
    print(f"S_T {s_t:,}: long call {long_call:>11,.2f}   short call {-long_call:>11,.2f}"
          f"   long put {long_put:>11,.2f}   short put {-long_put:>11,.2f}")
# Output:
# break-evens: call 4,210.48   put 3,809.27
# S_T 3,600: long call  -10,524.00   short call   10,524.00   long put   10,463.50   short put  -10,463.50
# S_T 4,000: long call  -10,524.00   short call   10,524.00   long put   -9,536.50   short put    9,536.50
# S_T 4,400: long call    9,476.00   short call   -9,476.00   long put   -9,536.50   short put    9,536.50

1.3 How Exchange-Traded Options Work

Specifications

An exchange standardises every term except the price, so that options with the same terms are interchangeable and a position can be closed by trading against anyone.

• Contract size. A stock option is on 100 shares. An index option is on the index level times a dollar multiplier — $50 for the Hartwell 500 in this course; $100 is common for real broad-market indices.
• Expiration. Standard monthly options expire on the third Friday of the expiry month, on a fixed cycle of months. Exchanges also list weekly options with a few days to run, and long-term equity anticipation securities (LEAPS) with expirations up to about three years away.
• Strikes. Listed at fixed intervals around the current price — narrower for low-priced shares, wider for high-priced ones. New strikes are added when the price moves outside the existing range.
• Class and series. All options of one type (calls or puts) on one underlying form an option class. All options of a class with the same strike and expiration form an option series — the unit that actually trades.
• Position and exercise limits. The maximum number of contracts one investor may hold on one side of the market, and the maximum that may be exercised within five consecutive business days. They exist to stop one trader cornering a market through options.
• FLEX options. Exchange-traded options whose strike, expiration and exercise style the parties negotiate, retaining the exchange's clearing guarantee. They are the exchanges' answer to the over-the-counter market.

Trading characteristics

Exchange-traded options are quoted by market makers, who must post a bid and an offer on demand, within a maximum spread the exchange sets. That obligation is what makes the market liquid, and the spread is what market makers are paid for providing it.

Most option positions are closed out rather than exercised. An investor who bought a call closes it by selling a call of the same series — an offsetting order. If the counterparty to that trade is opening a new position, open interest — the number of contracts outstanding — is unchanged; if both sides are closing, it falls.

Two practical features matter for anyone trading them in size. Liquidity is concentrated in near-the-money, short-dated series, and bid-ask spreads on deep out-of-the-money options can be a large fraction of the premium. And because quoted prices are small numbers per share or per index point, a spread of a few cents is a material cost in percentage terms.

Commissions

Brokers typically charge a fixed amount per trade plus a percentage of the trade's value, with a minimum and a maximum.

Margin

Buying an option requires no margin in the usual sense: the buyer pays the premium in full, because the most the buyer can lose is the premium. (Under US rules, options with less than nine months to expiry cannot be bought on margin at all, while longer-dated options can be bought with up to 25% of the premium borrowed.)

Writing an option is different. The writer has an open-ended obligation, so the broker requires collateral. If the writer owns the underlying shares against a written call — a covered call — the shares are the collateral and no further margin is required. If not, the option is naked, and the minimum initial margin is the greater of two tests.

For a naked call:

For a naked put:

Each test is per share; multiply by the number of shares under the contracts. The 20% in test 1 becomes 15% for broad-based index options. The amount by which the option is out of the money is subtracted because that cushion must be used up before the writer loses anything. The premium received can be applied towards the requirement.

def naked_call_margin(premium, share, strike, pct=0.20):
    test_1 = premium + pct * share - max(strike - share, 0)
    test_2 = premium + 0.10 * share
    return max(test_1, test_2), test_1, test_2

def naked_put_margin(premium, share, strike, pct=0.20):
    test_1 = premium + pct * share - max(share - strike, 0)
    test_2 = premium + 0.10 * strike
    return max(test_1, test_2), test_1, test_2

m, t1, t2 = naked_call_margin(3.20, 58.00, 62.00)
print(f"Calderon 62 call:  tests {t1:.2f} / {t2:.2f} -> {m:.2f} a share, ${m * 400:,.2f} on 4 contracts")
m, t1, t2 = naked_put_margin(1.90, 58.00, 55.00)
print(f"Calderon 55 put:   tests {t1:.2f} / {t2:.2f} -> {m:.2f} a share, ${m * 500:,.2f} on 5 contracts")
m, t1, t2 = naked_put_margin(105.84, 4000.00, 3800.00, pct=0.15)
print(f"Hartwell 3,800 put: tests {t1:.2f} / {t2:.2f} -> {m:.2f} a point, ${m * 50:,.2f} a contract")
# Output:
# Calderon 62 call:  tests 10.80 / 9.00 -> 10.80 a share, $4,320.00 on 4 contracts
# Calderon 55 put:   tests 10.50 / 7.40 -> 10.50 a share, $5,250.00 on 5 contracts
# Hartwell 3,800 put: tests 505.84 / 485.84 -> 505.84 a point, $25,292.00 a contract

The margin is recalculated daily as the share price moves, and the writer receives a margin call if the account falls below the maintenance requirement — the same mechanism Chapter 4 of the previous course described for futures, applied to only one side of the trade.

Exercise and assignment

Every exchange-traded option is cleared through a clearing house — in the United States, the Options Clearing Corporation (OCC) — which becomes the buyer to every writer and the writer to every buyer, and guarantees performance. Exercise runs through it in four steps:

1. The holder instructs their broker to exercise.
2. The broker notifies the clearing firm that carries the account, which notifies the OCC.
3. The OCC assigns the exercise to a clearing member with an outstanding short position in that series, chosen at random.
4. That member assigns it to one of its customers who has written the option, by a method it has disclosed — random selection or first-in-first-out.

The writer chosen must then deliver the shares (for a call) or buy them (for a put) at the strike. A writer cannot choose whether to be assigned, which is why an American option writer carries the risk of early assignment at any time. At expiration, options that are in the money by more than a small threshold are exercised automatically unless the holder instructs otherwise. Index options are settled in cash for the difference between the index and the strike, times the multiplier.

1.4 Dividends and Stock Splits

An exchange-traded stock option is not adjusted for ordinary cash dividends. The market knows the dividends are coming, and they are already reflected in the premium: a call is worth less on a share that will pay dividends before expiry, because the share price falls on each ex-dividend date. (A cash dividend that is unusually large — commonly more than 10% of the share price — may lead the exchange to adjust the terms.)

Stock splits and stock dividends are adjusted for, because they change the number of shares without changing the value of the company. Without an adjustment, a 2-for-1 split would halve the share price and wipe out most of every call's value overnight, with no economic event to justify it.

The rule for an $n$-for-$m$ split — $n$ new shares for every $m$ old ones:

The strike times the number of shares — the total cash payable on exercise — is unchanged. A stock dividend is treated as a split: a 5% stock dividend gives 105 shares for every 100, a 105-for-100 split.

1.5 Warrants, Convertible Bonds and Employee Stock Options

Three instruments behave like call options on a company's shares but are not exchange-traded options. What separates them is who writes them: the company itself. When an exchange-traded call is exercised, existing shares change hands between investors. When one of these is exercised, the company issues new shares, and that dilutes every existing shareholder.

Warrants

Warrant A call option issued by a company on its own shares, usually attached to a bond issue to make it more attractive, and often traded separately afterwards. On exercise the company issues new shares at the strike price.

Because exercise creates shares, a warrant is worth less than an otherwise identical exchange-traded call. If a company has $N$ shares outstanding and issues $M$ warrants, each giving the right to one new share, the warrant's value is approximately

with the regular call valued on the company's share price.

Convertible bonds

Convertible bond A bond that the holder may exchange for a fixed number of the issuer's shares — the conversion ratio — at certain times during its life. It is a straight bond plus a call option on the issuer's shares, written by the issuer.

Most convertibles are also callable by the issuer. When the share price is high, the issuer calls the bond; the holder, offered the call price in cash or the larger conversion value in shares, converts. This is a forced conversion, and it means a callable convertible carries two options pointing in opposite directions: the holder's call on the shares and the issuer's call on the bond. Chapter 4 prices the issuer's call on its own.

Employee stock options

Employee stock option (ESO) A call option on a company's shares granted to an employee as compensation. It is usually granted at the money, cannot be exercised during a vesting period of several years, cannot be sold or transferred, and is typically forfeited if the employee leaves before it vests.

ESOs differ from exchange-traded options in ways that matter for their value and for the incentives they create:

• They cannot be sold. The only way an employee can turn an ESO into cash is to exercise it and sell the shares. ESOs are therefore often exercised early, which throws away time value — §1.9 shows exactly why that is costly — and means an ESO is worth less to its holder than an equivalent tradable option.
• They dilute. Like warrants, exercised ESOs are usually satisfied by issuing new shares.
• They are an expense. From 2005–06, under both international and US accounting standards, companies must expense ESOs at their fair value on the grant date. Before that, an at-the-money grant could be reported as costing nothing, which partly explains how widely they were used.
• They shape behaviour. Because an option rewards upside and does not penalise downside below the strike, ESOs can encourage executives to take on risk and to manage the share price towards vesting and exercise dates. Backdating — recording a grant as made on an earlier date when the share price was lower, so that the option was in fact granted in the money — was the subject of a wave of investigations in the mid-2000s.

1.6 The Six Factors That Move an Option's Price

Six inputs determine a stock option's price. The direction of each is examinable, and each follows from the payoff.

Price and strike. A call pays $S_T - K$, so a higher share price or a lower strike makes it more valuable; a put is the reverse.

Time. An American option with longer to run has every exercise opportunity of a shorter one, and more, so it is worth at least as much. A European option has no such guarantee: a long-dated European call on a share that pays a large dividend between two expiry dates may be worth less than a short-dated one expiring before the dividend. Hence the question marks, although in most practical cases longer European options are worth more too.

Volatility. Higher volatility makes extreme outcomes more likely in both directions. The option holder gains from the favourable extreme and is protected from the unfavourable one by the right not to exercise, so both calls and puts gain. This is the factor that distinguishes options from the underlying: a shareholder is indifferent to volatility as such, and an option holder is not.

The interest rate. A higher rate lowers the present value of the strike the call holder will pay, raising the call; and lowers the present value of the strike the put holder will receive, lowering the put. (Holding the share price fixed. In practice rates and share prices tend to move in opposite directions, and the combined effect can differ.)

Dividends. The share price falls by roughly the dividend on the ex-dividend date. That hurts a call holder and helps a put holder, since neither receives the dividend.

Here is the six-month 4,000-strike Hartwell 500 pair, with one factor moved at a time and everything else held at its base value:

1.7 Upper and Lower Bounds

Without any model of how prices move, arbitrage alone confines an option's price to a band. The notation, used for the rest of the chapter:

Upper bounds

A call gives the right to buy one share. It can never be worth more than the share itself — otherwise sell the call, buy the share, and pocket the difference:

A put gives the right to receive $K$. An American put can never be worth more than $K$. A European put pays at most $K$ at expiration, so today it can never be worth more than the present value of $K$:

Lower bound on a European call, no dividends

Compare two portfolios:

• Portfolio A: one European call plus cash of $Ke^{-rT}$, invested at the risk-free rate.
• Portfolio B: one share.

At expiration the cash has grown to $K$. If $S_T > K$, the call is exercised and A is worth $S_T$; if not, the call expires and A is worth $K$. So A is worth $\max(S_T, K)$, which is never less than B's $S_T$. A portfolio that is always worth at least as much as another must cost at least as much today:

The zero is there because an option can never have a negative price.

Lower bound on a European put, no dividends

The same logic with a put and a share (Portfolio C) against cash (Portfolio D):

With dividends

A dividend lowers the share price the option holder will face, and the holder does not receive it. Replace $S_0$ by $S_0 - D$:

For an index with a continuous dividend yield $q$, the equivalent replacement is $S_0e^{-qT}$ — in the lower bounds and in the call's upper bound, which tightens from $S_0$ to $S_0e^{-qT}$ — the number of index units you would need to hold today, reinvesting the dividends, to have exactly one unit at expiration.

from math import exp

r, t = 0.03, 0.5
share, div_pv = 58.00, 1.00 * exp(-0.03 * 0.25)

print(f"PV of K: 55 -> {55 * exp(-r * t):.4f}   62 -> {62 * exp(-r * t):.4f}   PV(div) {div_pv:.4f}")
print(f"call K55 lower bound: no dividend {share - 55 * exp(-r * t):.4f}   "
      f"with dividend {share - div_pv - 55 * exp(-r * t):.4f}")
print(f"put  K62 lower bound: no dividend {62 * exp(-r * t) - share:.4f}   "
      f"with dividend {62 * exp(-r * t) - share + div_pv:.4f}")

# The arbitrage against a call quoted at 3.00, below its 3.8188 bound
invested = share - 3.00
print(f"short share, buy call: invest {invested:.2f}, worth {invested * exp(r * t):.4f} at expiry; "
      f"locked-in minimum profit {invested * exp(r * t) - 55:.4f}")
# Output:
# PV of K: 55 -> 54.1812   62 -> 61.0769   PV(div) 0.9925
# call K55 lower bound: no dividend 3.8188   with dividend 2.8263
# put  K62 lower bound: no dividend 3.0769   with dividend 4.0695
# short share, buy call: invest 55.00, worth 55.8312 at expiry; locked-in minimum profit 0.8312

The Hartwell 500 quotes all sit comfortably inside their bounds, with the index adjusted for its dividend yield ($S_0e^{-qT} = 4,000e^{-0.01} = 3,960.1993$):

1.8 Put-Call Parity

The bounds confine one option at a time. Put-call parity links a call and a put with the same strike and expiry, and pins the difference between them exactly.

European options on a non-dividend-paying share

Return to the portfolios of §1.7:

• Portfolio A: one European call plus $Ke^{-rT}$ in cash. At expiration it is worth $\max(S_T, K)$.
• Portfolio C: one European put plus one share. At expiration: if $S_T > K$, the put lapses and C is worth $S_T$; if $S_T \le K$, the put is exercised and C is worth $K$. So C is also worth $\max(S_T, K)$.

Two portfolios with identical payoffs in every state, and no possibility of early exercise, must cost the same today:

With dividends, or a dividend yield

The share in Portfolio C now also pays dividends, which the holder of A does not receive. Adding cash equal to their present value to Portfolio A restores the equality:

For an index with a continuous yield $q$, hold $e^{-qT}$ units in Portfolio C and reinvest the dividends, so that it holds exactly one unit at expiration:

In terms of the forward price

The previous course showed that the forward price of an asset with yield $q$ is $F_0 = S_0e^{(r-q)T}$, so $S_0e^{-qT} = F_0e^{-rT}$. Substituting:

This is the version to remember, because it says what parity means. A long call plus a short put, both struck at $K$, pays $S_T - K$ at expiration whatever happens: if the index ends above $K$ you exercise the call, and if below, the put is exercised against you. Either way you buy the index at $K$. A long call and a short put together are a long forward contract with delivery price $K$ — and a forward struck at $K$ when the market forward is $F_0$ is worth $(F_0 - K)e^{-rT}$.

Three things follow immediately:

• If $K = F_0$, then $c = p$. A call and a put struck at the forward cost the same.
• If $K < F_0$, the call is dearer than the put; if $K > F_0$, the put is dearer.
• The difference does not depend on volatility, which is why §1.6 found the vega of the call and the put to be identical.

from math import exp

s0, q, r, t, mult = 4000.00, 0.02, 0.03, 0.5, 50
fwd = s0 * exp((r - q) * t)
calls = {3800: 322.62, 4000: 210.48, 4200: 128.84}
puts = {3800: 105.84, 4000: 190.73, 4200: 306.11}

print(f"F(6m) = {fwd:.4f}")
for k in calls:
    print(f"K {k:,}: C - P = {calls[k] - puts[k]:8.2f}   (F - K)e^-rT = {(fwd - k) * exp(-r * t):9.4f}")

synthetic_put = calls[4000] - (fwd - 4000) * exp(-r * t)
print(f"synthetic 4,000 put {synthetic_put:.4f} against a quote of 150.00: "
      f"{synthetic_put - 150:.4f} points, ${(synthetic_put - 150) * mult:,.2f} a contract")
# Output:
# F(6m) = 4020.0501
# K 3,800: C - P =   216.78   (F - K)e^-rT =  216.7740
# K 4,000: C - P =    19.75   (F - K)e^-rT =   19.7516
# K 4,200: C - P =  -177.27   (F - K)e^-rT = -177.2708
# synthetic 4,000 put 190.7284 against a quote of 150.00: 40.7284 points, $2,036.42 a contract

Parity for a single share

The same relationship on Calderon Securities, which pays no dividend: a six-month European call struck at $60 trades at $3.10, with the share at $58.00 and the rate at 3.00%. The put must be worth

American options: an inequality, not an equation

Parity relies on neither option being exercised early. American options can be, so for them parity weakens to a pair of bounds. With no dividends:

With dividends of present value $D$, the lower bound falls by $D$:

1.9 Early Exercise

An American option costs at least as much as its European twin, because it carries every right the European does and one more. The question is whether that extra right — to exercise early — is ever worth using.

A call on a non-dividend-paying share: never

The lower bound from §1.7 does the work. For a call on a share that pays no dividends,

The right-hand side is what you get by exercising now. The option is always worth more alive than exercised, so it should never be exercised early. Two separate reasons sit behind the strict inequality:

1. Insurance. A holder who waits keeps the right not to buy if the price falls below $K$. Exercising converts a call — which cannot lose more than its premium — into a share, which can.
2. The time value of money. Exercising early pays $K$ now rather than at expiration. Paying later is better, by the interest on $K$.

What if the holder believes the share is overpriced and wants out? Exercising is still wrong: sell the option instead, collecting its full value, intrinsic plus time. And if the holder wants to keep the option but lock in the gain, shorting the share does that without giving up the time value. An immediate consequence is that an American call on a non-dividend-paying share is worth exactly the same as a European one, $C = c$.

A call on a dividend-paying share: possibly, just before an ex-dividend date

A dividend changes the trade-off. On the ex-dividend date the share price drops by roughly the dividend, and the call holder receives none of it. Exercising just before the share goes ex-dividend captures the dividend and gives up the interest on $K$ for the remaining life of the option. So early exercise can only be optimal immediately before an ex-dividend date, and only if the dividend is larger than the interest saved by waiting:

where $t$ is the ex-dividend date. If the dividend is smaller than that, early exercise is never optimal at that date. If it is larger, early exercise may be optimal — the condition is necessary, not sufficient, because the holder also gives up the insurance value of the option.

A put: sometimes, when it is deep in the money

The interest argument runs the other way for a put. Exercising a put receives $K$ now rather than later, so early exercise earns interest — and if the put is far enough in the money, that interest is worth more than anything waiting can add.

The extreme case makes it clear. If the share price falls to zero, the put will pay exactly $K$ whenever it is exercised, and the price cannot fall further. Waiting can only lose the interest on $K$, so the holder should exercise immediately. The same is true somewhat above zero.

Summary of the rationale for early exercise

1.10 Putting It All Together: Insuring the Core Equity Fund

Ravensmoor's investment committee asked for downside protection on the $800 million Core Equity Fund, with the upside kept. The tools are now in hand.

Step 1 — how many puts. Why this step: the fund does not move one-for-one with the index; with a beta of 1.15, it moves 1.15% for each 1% move in the market. The count of index puts must therefore cover 1.15 times the fund's value in index terms, which is the beta-weighted count from the previous course:

This is the same 4,600 the previous course computed for the futures hedge before tailing. The tailing adjustment that cut the futures count to 4,589 does not apply here: it existed because a futures contract is worth its futures price and is settled daily, whereas a put pays on the index level once, at expiry.

Step 2 — what it costs. Why this step: the premium is the price of keeping the upside. With 4,600 contracts of $50 a point, each point of premium costs $230,000:

The committee chooses the 3,800 strike, which protects against index falls of more than 5%.

Step 3 — what each hedge delivers. Why this step: the two hedges differ in what they deliver, not just in what they cost, so compare fund values at expiry. Use the capital asset pricing model to translate an index level into an expected fund value, as the previous course did, and ignore the fund's residual risk — the 5.1417% of volatility that no index instrument can remove. Carry the premium forward at the risk-free rate, to $24.71 million, so that every figure is measured on the same date. Each index point moves the fund by $1.15 \times 800,000,000 \div 4,000$ dollars, which is $230,000 — exactly what each point of the put position pays.

The futures hedge turns the fund into a Treasury bill: $812.01 million whatever the market does — the risk-free return on $800 million. The puts put a floor under the fund at $736.69 million, the expected value at 3,800 less the premium, and below 3,800 the fund's losses and the puts' gains offset point for point. Above the floor, the fund keeps every point of the upside, less the premium. The puts beat the futures hedge if the index finishes above about 4,127.

Step 4 — what parity says about the difference. Why this step: §1.8 showed that a long put is a long call plus a short forward struck at the same price. So the put hedge must be the futures hedge plus a call — and the difference between the two columns of the table above must be exactly the profit on 4,600 calls struck at 3,800, bought at 322.62 and carried to expiry.

from math import exp

s0, q, r, t, mult = 4000.00, 0.02, 0.03, 0.5, 50
fund, beta = 800e6, 1.15
fwd = s0 * exp((r - q) * t)
n = beta * fund / (s0 * mult)
put_3800, call_3800 = 105.84, 322.62
premium = n * mult * put_3800

def fund_at(s_t):
    """CAPM-expected fund value in six months, residual risk ignored."""
    index_return = s_t / s0 - 1 + q * t
    return fund * (1 + r * t + beta * (index_return - r * t))

print(f"contracts {n:,.0f}; put premium ${premium:,.2f} = {premium / fund:.4%} of the fund")
for s_t in (3400, 3800, 4000, 4400):
    plain = fund_at(s_t)
    futures = plain + n * mult * (fwd - s_t)
    insured = plain + n * mult * max(3800 - s_t, 0) - premium * exp(r * t)
    calls_pnl = n * mult * (max(s_t - 3800, 0) - call_3800 * exp(r * t))
    print(f"S_T {s_t:,}: unhedged {plain / 1e6:7.2f}M  futures {futures / 1e6:7.2f}M  "
          f"puts {insured / 1e6:7.2f}M  | puts - futures {(insured - futures) / 1e6:7.2f}M"
          f"  = calls {calls_pnl / 1e6:7.2f}M")
# Output:
# contracts 4,600; put premium $24,343,200.00 = 3.0429% of the fund
# S_T 3,400: unhedged  669.40M  futures  812.01M  puts  736.69M  | puts - futures  -75.32M  = calls  -75.32M
# S_T 3,800: unhedged  761.40M  futures  812.01M  puts  736.69M  | puts - futures  -75.32M  = calls  -75.32M
# S_T 4,000: unhedged  807.40M  futures  812.01M  puts  782.69M  | puts - futures  -29.32M  = calls  -29.32M
# S_T 4,400: unhedged  899.40M  futures  812.01M  puts  874.69M  | puts - futures   62.68M  = calls   62.68M

The last two columns agree in every scenario. (They differ in the fourth decimal place, about $1,380 in total, because the quoted call and put are each rounded to the cent.)

The rest of the course is less clean. In Chapter 2 the options are bundled into strategies and structured products whose pricing is less visible. In Chapters 4 to 7 they are embedded in bonds, mortgages, futures and swaps — and in nearly every case the investor turns out to have written them.

Chapter Summary

What's Next

Chapter 2 builds on parity. Every option strategy — covered calls, protective puts, spreads, straddles, strangles — is a combination of the four positions in §1.2, possibly with the underlying and a bond, and parity prices all of them from the quotes in this chapter. The committee's hedge will be re-engineered there into a zero-cost collar, which pays for the 3,800 floor by selling away the upside above a strike that parity determines. The chapter then turns to exotic options, to products with no visible premium at all, and to the variance swap, which turns the gap between the 18% volatility implied by these quotes and the 15% the index actually realises into a cash flow.

Concept Checks

Fast recall before the exam-format questions — no calculator needed.

Exam-Style Practice

Standalone four-option questions at the real exam's pace — 2.4 minutes each.

Get the full course: FRM Level I: Financial Markets and Products II