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DER 2 – Valuation of Contingent Claims

Published: 10 August 2026
Updated: 3 November 2026

Worked examples are fully visible. Check-yourself items are study aids you can reveal one at a time.

A contingent claim gives its owner a right, but never an obligation, to a payoff that depends on some underlying asset, rate or other derivative. Options are the contingent claims this reading values. The material matters well beyond the listed options market, because a very large number of ordinary investments contain options that have simply been bundled into something else: a callable bond, a convertible, an employee share scheme, a project that can be abandoned.

Every valuation model in this reading rests on one idea. The correct price of an option is the price at which nobody can extract a riskless profit from anybody else. That single condition, applied carefully, pins the value down exactly. Two families of models implement it. The binomial model works in discrete time and can handle early exercise. The Black–Scholes–Merton model, usually shortened to BSM, works in continuous time and values only options that cannot be exercised early.

Thinking like an arbitrageur

The quickest route into option valuation is to adopt the mindset of a trader hunting for free money. Such a trader prefers more wealth to less and works under two self-imposed constraints.

Constraint one: commit no capital of your own. Positions are funded by borrowing or by the proceeds of a short sale, and those proceeds stay inside the trade.
Constraint two: carry no exposure to the price of the underlying. Only market price risk on the underlying and the derivatives used is considered here. Liquidity risk and counterparty credit risk are set aside.

A position satisfying both constraints and still throwing off cash today would be a machine that prints money. Because everybody would want one, such positions cannot survive. Equivalently, if two portfolios deliver identical cash flows in every future state, they must cost the same today. That statement is the law of one price, and it is the engine behind every formula that follows.

Note that an arbitrage is always a wonderful opportunity, but not every wonderful opportunity is an arbitrage. Paying one euro today for a 99 percent chance of one million euros tomorrow and a 1 percent chance of nothing is a superb bet, yet it commits capital and carries risk, so it is not arbitrage.

The standing assumptions

Five assumptions run through the whole reading and are worth memorising as a set.

Instruments that replicate the option are identifiable and can actually be traded.
Market frictions such as transaction costs and taxes are absent.
Short selling is permitted and the seller has full use of the proceeds.
The underlying follows a known statistical distribution.
Borrowing and lending at a risk-free rate is available.

Cash flows are tracked in tables throughout, with a consistent sign convention: money leaving the arbitrageur is negative, money arriving is positive. An initial outlay of 100 euros appears as −€100, and a payoff of 1,000 yen appears as +¥1,000.

Notation and the value of an option at expiration

Let S_t be the price of the underlying observed at time t, with t measured as a fraction of a year, and let S_T be the price at expiration. A lower-case c or p denotes a European-style call or put, an upper-case C or P denotes the American-style version, and X is the exercise price, also called the strike. At initiation the subscript is dropped, so c means c₀.

At expiration there is no uncertainty left, so the option must be worth exactly what exercising it yields.

Exercise values at expiration. Any deviation is immediately arbitraged away.

Before expiration a European option cannot technically be exercised, so it has no exercise value in a legal sense. The quantity Max(0, S_t − X) is still a useful reference point, because the difference between the option price and that quantity is the time value: the market valuation of the chance that the option finishes further in the money, net of the chance that it finishes lower. Time value cannot be negative, because option payoffs are asymmetric. A call has unlimited upside and a floor of zero on the downside. Time value shrinks to zero at expiration.

Example 1 · Worked

On 15 April a 90-day European-style call option expiring on 14 July is written at a price of c = €2.50.

1. Express the option maturity as a fraction of a year, as the models require.

Solution. Divide the calendar days by 365. T = 90/365 = 0.246575 years. Every model input in this reading that involves time is expressed in years in exactly this way, including the exponents in the BSM formulas.

2. Some weeks later the contract has 35 days left. What is the elapsed time t, and what is the remaining life?

Solution. If 35 of the original 90 days remain, then 55 days have passed, so t = 55/365 = 0.150685 years. The remaining life is T − t = 35/365 = 0.095890 years. Keep the two straight: t counts time gone, while the models discount and scale volatility using the time still to run.

3. At expiration the underlying settles at a level below X. What is the call worth, and what has happened to its time value?

Solution. The call is worth Max(0, S_T − X) = zero, because S_T − X is negative and the holder simply lets the contract lapse. Time value is zero at expiration by construction, since there is no remaining chance of a better outcome. The whole €2.50 paid at initiation was time value, and all of it has decayed away.

The simplest possible model of an uncertain future allows the underlying to do exactly two things over one period. Drawn as a lattice, the starting price sits at a node, and two arcs lead to the two possible outcomes. The upper outcome is written S⁺ and the lower one S⁻. Crude as this is, everything important about option valuation is already visible in it.

The two outcomes are usually expressed as multiplicative factors on the starting price.

Up and down factors are total returns, that is one plus the rate of return.

Both factors are gross returns rather than net returns, so an up factor of 1.35 means the underlying gains 35 percent. Their size is governed by the volatility of the underlying: a more volatile underlying produces a larger u and a smaller d.

Building a hedge, then reading the price off it

Suppose a trader writes one call. A call gains when the underlying gains, so a written call loses in the up state and gains in the down state. To cancel that exposure, the trader must hold something that gains in the up state, which means buying the underlying. Let h be the number of units bought, chosen so that the position is worth the same whichever branch occurs.

Writing one call, buying h units of the underlying, and financing the difference

DER 2 – Valuation of Contingent Claims — table 1
Trade	Time step 0	Time step 1, down	Time step 1, up
Write one call	+c	−c⁻	−c⁺
Buy h units of the underlying	−hS	+hS⁻	+hS⁺
Borrow or lend	−PV(−hS⁻ + c⁻)	−hS⁻ + c⁻	−hS⁺ + c⁺
Net	+c − hS − PV(−hS⁻ + c⁻)	0	0

PV denotes discounting one period at the risk-free rate r, so PV(1) = 1/(1 + r).

Setting the two time-step-1 cash flows equal to each other, −c⁺ + hS⁺ = −c⁻ + hS⁻, and solving for h gives the hedge ratio.

Numerator and denominator follow the same up-minus-down pattern; only the instrument changes.

The mnemonic is worth carrying into the exam: option value if the underlying rises minus option value if it falls, divided by underlying value if it rises minus underlying value if it falls. Because call values move with the underlying, the call hedge ratio can never be negative.

Once the future cash flows are zero in both states, the time-zero net must also be zero, otherwise the strategy is a money machine. If the net were positive, arbitrageurs would pile in, forcing the call price down and the underlying price up until the profit vanished. If it were negative, they would run the mirror strategy, buying calls, short selling the underlying and lending, until the gap closed. Setting the time-zero net to zero and rearranging produces the valuation equation.

Equivalently, c = hS + PV(−hS⁺ + c⁺), since the hedge makes the two identical.

Read the equation in words and it says something concrete: a long call is a holding of h units of the underlying, partly paid for with a loan of PV(−hS⁻ + c⁻). A call is a geared position in the underlying, nothing more exotic.

The same machinery for puts

Puts follow identical logic with one sign flipped. Because a put loses value when the underlying rises, p⁺ is below p⁻ while S⁺ is above S⁻, so the hedge ratio comes out negative.

A negative h means the replicating trade is a short sale, not a purchase.

Trading h units when h is negative means selling short. Since −h is then positive, the term −hS is a cash inflow at time zero. So a long put is a short position in the underlying combined with lending, which is the mirror image of the call. Writing options simply reverses these trades: the writer of a call short sells the underlying and lends, while the writer of a put buys the underlying on margin.

Example 2 · Worked

Two questions on the replicating strategy inside the single-period binomial framework.

1. Which trading strategy reproduces the payoffs of a long call?

Solution. Buy h = (c⁺ − c⁻)/(S⁺ − S⁻) units of the underlying and take financing of −PV(−hS⁻ + c⁻). Laying the two side by side confirms it. Buying the call costs −c today and pays +c⁻ or +c⁺. The replicating package costs −hS − PV(−hS⁻ + c⁻) today and, by the definition of h, pays exactly +c⁻ or +c⁺. Identical payoffs in every state force identical prices. Watch the distractors: a hedge ratio built with plus signs in numerator and denominator is wrong, and short selling the underlying replicates a put, not a call.

2. Which trading strategy reproduces the payoffs of a long put?

Solution. Short sell −h = −(p⁺ − p⁻)/(S⁺ − S⁻) units of the underlying and take financing of −PV(−hS⁻ + p⁻). Since p⁺ is below p⁻ while S⁺ is above S⁻, the ratio is negative and the trade is a short sale. The financing term is positive here, so the arbitrageur is lending rather than borrowing. In short, a long put equals a short position in the underlying funded into a deposit.

The replication equations are exact, but they are fiddly to compute. A little algebra converts them into a form that is far quicker to use and that generalises cleanly to many periods. The option value equals a discounted expected payoff, where the expectation uses a particular probability that falls out of the arbitrage argument rather than out of anybody’s forecast.

The expectations form of the single-period binomial model.

FV(1) = 1 + r is the reciprocal of PV(1) = 1/(1 + r).

Written most compactly, c = PV_r[E(c₁)] and p = PV_r[E(p₁)], where the expected terminal payoffs are E(c₁) = πc⁺ + (1 − π)c⁻ and E(p₁) = πp⁺ + (1 − π)p⁻. The subscript on PV is a reminder that discounting happens at the risk-free rate.

Why this is not ordinary discounted cash flow

Two features separate this from the discounted cash flow valuation used on equities and bonds, and both are examinable.

The probability is not a forecast. π is determined objectively by u, d and r. It has nothing to do with what anybody thinks the underlying will do. It is called a risk-neutral probability precisely because no assumption about risk preferences was ever made; the number is a by-product of the arbitrage argument. Calling π a probability is a convenience, not a claim that up moves really happen that often.
The discount rate carries no risk premium. Discounting is at the risk-free rate. Both of the subjective ingredients of discounted cash flow, the expected cash flow and the risk-adjusted discount rate, have been replaced by objective quantities. That is why practitioners generally prefer this route.

Put–call parity as a shortcut

Once a call has been valued, the matching put with the same exercise price follows without further modelling work.

Two routes to the same insured payoff: hold the underlying and a put, or lend PV(X) and hold a call.

The relationship holds whatever valuation model is being used, because it is itself an arbitrage relationship. Rearranged as c = S − PV(X) + p, it says a call is the underlying, financed, plus a put.

Example 3 · Worked

A non-dividend-paying stock trades at €100. A one-year option has exercise price €100, the periodically compounded risk-free rate is 5.15 percent, and a single-period binomial model is assumed with u = 1.35 and d = 0.74.

1. Compute the optimal hedge ratio for the call.

Solution. First map out the branch values.
S⁺ = uS = 1.35(100) = 135 and S⁻ = dS = 0.74(100) = 74.
c⁺ = Max(0, 135 − 100) = 35 and c⁻ = Max(0, 74 − 100) = 0.
Then h = (35 − 0)/(135 − 74) = 35/61 = 0.573770.

2. Value the call by the expectations approach, and confirm it with the replication equation.

Solution. With FV(1) = 1 + r = 1.0515, the risk-neutral probability is
π = (1.0515 − 0.74)/(1.35 − 0.74) = 0.3115/0.61 = 0.510656.
The one-period discount factor is PV(1) = 1/1.0515 = 0.951022. So
c = 0.951022[(0.510656)35 + (1 − 0.510656)0] = 0.951022(17.87296) = €16.998.
The no-arbitrage route must agree, and it does:
c = 0.573770(100) + 0.951022[−0.573770(74) + 0] = 57.3770 − 40.3794 = €16.998.
Long 0.573770 shares costing 57.38, funded with a loan of 40.38, leaves a net outlay equal to the call price.

3. Value the matching put.

Solution. The put payoffs are p⁺ = Max(0, 100 − 135) = 0 and p⁻ = Max(0, 100 − 74) = 26. Reusing π and PV(1) from question 2,
p = 0.951022[(0.510656)0 + (1 − 0.510656)26] = 0.951022(12.72294) = €12.10.
Nothing new was estimated. Once π and the discount factor are known they serve every option on that underlying with that maturity.

If π is not a real probability, why does the answer come out right?

Because the answer was never about probabilities. The value came from a replicating portfolio whose cost is fixed by u, d, r and the payoffs. The expectations formula is that same cost rewritten. The quantity π is simply the weight that makes the rewriting work, and its resemblance to a probability is algebraic convenience. This is why two analysts who violently disagree about where the underlying is heading must still quote the same option value: their disagreement never enters the calculation.

Adding a second period changes nothing conceptually. A two-period lattice is three one-period lattices: one starting at time 0, and one starting at each of the two time-1 nodes. Solve the two later ones first, then use their answers as the payoffs for the earlier one.

Figure 1: A recombining two-period lattice

Two paths converge on the middle node at time 2, so the lattice recombines and there are three terminal values rather than four.

Keeping u and d constant throughout the lattice guarantees recombination, because udS equals duS. Take u = 1.25, d = 0.8 and S₀ = 100. An up move then a down move gives 1.25(0.8)100 = 100, and a down move then an up move gives 0.8(1.25)100 = 100. The two paths land on the same node.

This matters computationally, not just aesthetically. A recombining lattice with n time steps has exactly n + 1 terminal nodes. A lattice that fails to recombine has 2ⁿ of them, which becomes impossible to handle long before n gets interesting.

Self-financing and dynamic replication

Two properties of the replicating strategy carry names worth knowing. Dynamic replication means the option payoff is matched exactly by a trading plan laid out in advance, with the position adjusted at each node. Self-financing means those adjustments never require fresh money from outside: whatever extra units of the underlying are needed can be paid for by adjusting the borrowing. If outside money were needed, it would have to be financed externally and the arbitrage argument would break down.

Working backward

At time 2 the call is worth its exercise value at each of the three nodes: c⁺⁺ = Max(0, u²S − X), c⁺⁻ = Max(0, udS − X) and c⁻⁻ = Max(0, d²S − X). Puts are the same with the arguments reversed. At time 1 the single-period equations are applied twice, and at time 0 once more.

Example 4 · Worked

The annual rate is 3 percent, the non-dividend-paying underlying trades at 72, u = 1.356, d = 0.541, the exercise price is 75 and the call expires in two years. The completed lattice is set out below.

Two-period lattice: underlying, European call value and hedge ratio

DER 2 – Valuation of Contingent Claims — table 2
Node	Underlying	Call	Hedge ratio
Time 0	72	19.47407	0.56971
Time 1, up	97.632	33.43048	0.72124
Time 1, down	38.952	0	0
Time 2, up up	132.389	57.389
Time 2, up down	52.81891	0
Time 2, down down	21.07303	0

1. Verify the top terminal node and the time-1 hedge ratio after an up move.

Solution. Two up moves give u²S = (1.356)²(72) = 132.389, so the call is worth Max(0, 132.389 − 75) = 57.389. One up and one down gives 1.356(0.541)(72) = 52.81891, which is below the strike, so that call expires worthless. The hedge ratio at the upper time-1 node is
h⁺ = (57.389 − 0)/(132.389 − 52.81891) = 57.389/79.57009 = 0.72124.

2. Value the call at the upper time-1 node and then at time 0.

Solution. The risk-neutral probability applies at every node because u, d and r are constant:
π = (1.03 − 0.541)/(1.356 − 0.541) = 0.489/0.815 = 0.6.
At the upper time-1 node, c⁺ = (1/1.03)[(0.6)57.389 + (0.4)0] = (1/1.03)(34.4334) = 33.43048. At the lower node both time-2 payoffs are zero, so c⁻ = 0.
The time-0 hedge ratio is h = (33.43048 − 0)/(97.632 − 38.952) = 33.43048/58.68 = 0.56971, and
c = (1/1.03)[(0.6)33.43048 + (0.4)0] = 19.47.

3. Show that the replicating portfolio tracks the call and needs no fresh capital along the way.

Solution. At time 0 buy 0.56971 shares for 41.019 = 0.56971(72) and borrow 21.545, since −21.545 = (1/1.03)[−0.56971(38.952) + 0]. The net outlay is 41.019 − 21.545 = 19.47, matching the model value.
Move to time 1 after an up move. The shares are now worth 55.622 = 0.56971(97.632) and the loan has grown to 22.191 = 1.03(21.545). The portfolio is worth 55.622 − 22.191 = 33.431, equal to the call value of 33.43048 up to rounding. Dynamic replication holds.
The hedge ratio has risen from 0.56971 to 0.72124, so more shares must be bought. Required borrowing at the new node is 36.98554 = −(1/1.03)[−0.72124(52.81891) + 0]. The old loan due is 22.191, so the new borrowing covers the repayment and funds the extra shares with nothing left over and nothing missing. The strategy is self-financing.

The expectations form over two periods

Repeated substitution collapses the backward induction into a single expression. Because the middle node is reached by two paths, it carries twice the weight.

The put version is identical with p in place of c.

Here PV discounts over two periods, so with discrete rates the factor is 1/(1 + r)². Forgetting to square the discount factor is one of the most frequent errors on this calculation.

Example 5 · Worked

A non-dividend-paying stock trades at €50. The options have two years to run, the periodically compounded risk-free rate is 5 percent, the exercise price is €50, u = 1.356 and d = 0.744. Assume European-style exercise.

1. What is the risk-neutral probability of an up move?

Solution. π = [(1 + 0.05) − 0.744]/(1.356 − 0.744) = 0.306/0.612 = 0.5, or 50 percent.

2. What is the call value today?

Solution. Terminal payoffs first.
c⁺⁺ = Max[0, 1.356²(50) − 50] = 91.9368 − 50 = 41.9368.
c⁺⁻ = Max[0, 1.356(0.744)(50) − 50] = 50.4432 − 50 = 0.44320.
c⁻⁻ = Max[0, 0.744²(50) − 50] = 0.0, since 27.6768 is below the strike.
Then apply the two-period expectations formula:
c = [1/(1.05)]²[0.5²(41.9368) + 2(0.5)(0.5)(0.44320) + 0.5²(0.0)]
= 0.907029[10.4842 + 0.2216] = 0.907029(10.7058) = €9.71.

3. What is the put value today?

Solution. Use put–call parity rather than rebuilding the tree:
p = c + PV(X) − S = 9.71 + [1/(1.05)]²(50) − 50 = 9.71 + 45.35 − 50 = €5.06.
This shortcut is legitimate only for European-style options, since parity in this form assumes no early exercise.

An American-style option can be exercised at any time up to expiration. Whether that right is worth anything depends on the type of option and on whether the underlying pays anything out.

Why calls on non-dividend-paying stock are never exercised early

Take a call on a US$100 stock with an exercise price of US$10, so deep in the money that its exercise value is US$90. Suppose the call traded at exactly that US$90. An investor wanting exposure to the stock could pay US$100 for the shares, or pay US$90 for the call and hand over the remaining US$10 only at expiration, and only if the stock is still at or above US$100. The second route is strictly better, so nobody would pay only US$90 for the call. Its price must exceed the exercise value, which means selling the call always beats exercising it.

The intuition can be turned round. Exercising means you no longer believe the stock can rise, yet you are paying far more cash than you currently have committed in order to own it. The trade makes no sense. For a non-dividend-paying stock, American-style and European-style calls are therefore worth the same.

Why puts are different

A deep in-the-money put is another matter. Exercising it converts the position into cash immediately, and that cash can earn the risk-free rate. When the interest earned exceeds the remaining time value of the put, exercising early is the better decision. So American-style puts genuinely can be worth more than their European counterparts, and when that is so the expectations formula fails. Only the backward-induction, no-arbitrage route can value them, because only that route lets you test each node.

Figure 2: Backward induction with an early exercise test

Arrows run right to left because valuation starts at expiration. The test that distinguishes American-style from European-style valuation is applied at each interior node.

Consider a two-year put with a periodically compounded rate of 3 percent, a non-dividend-paying underlying at 72, exercise price 75, u = 1.356 and d = 0.541. These are the same lattice parameters used for the call above, so the underlying values are unchanged.

European-style put, then the American-style adjustment

DER 2 – Valuation of Contingent Claims — table 3
Node	Underlying	European put	European hedge ratio	American put	American hedge ratio
Time 0	72	18.16876	−0.43029	19.01710	−0.46752
Time 1, up	97.632	8.61401	−0.27876	8.61401	−0.27876
Time 1, down	38.952	33.86353	−1	36.04800	−1
Time 2, up up	132.389	0		0
Time 2, up down	52.81891	22.18109		22.18109
Time 2, down down	21.07303	53.92697		53.92697

Only the lower time-1 node changes, and that single change propagates back to time 0.

The lower time-1 node is where the action is. Its exercise value is Max(0, 75 − 38.952) = 36.048, which exceeds the European model value of 33.86353. An American-style holder would exercise there. Substituting the larger figure and rolling back gives

The early exercise premium at time 1 is 2.18447 and at time 0 it is 0.85.

The premium at the lower time-1 node is 36.048 − 33.86353 = 2.18447. At time 0 the premium is 19.02 − 18.17 = 0.85. The hedge ratio also shifts, from −0.43029 to −0.46752, because the option is now more responsive to the underlying. This is why an American-style put cannot be valued as a simple present value of expected payoffs in a multiperiod setting, although in a one-period setting it can, since there is no interior node to test.

Example 6 · Worked

Given S₀ = 26, X = 25, u = 1.466, d = 0.656, two time steps, r = 2.05 percent per period and no dividends, the European-style put lattice is as follows.

European-style put on a two-period lattice

DER 2 – Valuation of Contingent Claims — table 4
Node	Underlying	Put	Hedge ratio
Time 0	26	4.01174	−0.35345
Time 1, up	38.116	0	0
Time 1, down	17.056	7.44360	−0.99970
Time 2, up up	55.87806	0
Time 2, up down	25.00410	0
Time 2, down down	11.18874	13.81126

1. What is the early exercise premium on the American-style put?

Solution. The risk-neutral probability is π = (1.0205 − 0.656)/(1.466 − 0.656) = 0.3645/0.81 = 0.45.
At the lower time-1 node the exercise value is Max(0, 25 − 17.056) = 7.944, above the European model value of 7.44360. The premium at that node is 7.944 − 7.44360 = 0.5004.
Replace 7.44360 with 7.944 and roll back:
p = (1/1.0205)[(0.45)0 + (0.55)7.944] = (1/1.0205)(4.3692) = 4.28143.
The early exercise premium at time 0 is therefore 4.28 − 4.01 = 0.27.
The hedge ratio at time 0 also changes, from −0.35345 to −0.37721 = (0 − 7.944)/(38.116 − 17.056). Note the ordering trap: the premium is measured at time 0 against the European value, not the 0.5004 measured at the node where exercise occurs.

Dividends reduce the value of the shares on the ex-dividend date, which hurts a call holder. Most listed option contracts offer no protection against ordinary dividends, so the effect has to be modelled. The technique used here is the escrow method: assume the dividends are perfectly predictable, then split the underlying into the part that is uncertain and the part that is not.

γ is the present value of the dividends to be paid over the life of the option.

The lattice is built on the hatted quantity, the underlying stripped of its known dividends, rather than on S. At expiration the two coincide, since any dividends have already been paid by then, so the terminal node values need no adjustment. An investor who actually held the shares would finish with S_T plus the reinvested dividends. Note that exchanges may adjust the exercise price for very large special dividends, but the reading deals with regular, predictable payments.

Consider a call on a US$100 stock with exercise price US$95. The periodically compounded rate is 1.0 percent, the stock pays a US$3 dividend at time step 1, u = 1.224, d = 0.796, and the call expires in two years.

American-style call on a dividend-paying stock

DER 2 – Valuation of Contingent Claims — table 5
Node	Underlying without dividends	Call, model value	Call, exercise value	Hedge ratio
Time 0	100	12.3438	13.2497	0.6004 or 0.6445
Time 1, up	118.7644	24.9344	26.7644	0.9909
Time 1, down	77.2356	0		0
Time 2, up up	145.3676	50.3676
Time 2, up down	94.5364	0
Time 2, down down	61.4796	0

At time 0 the first hedge ratio applies to the European-style call and the second to the American-style call. The lattice involves technical details beyond the scope of the learning outcomes; the objective is to see how dividend-driven early exercise works.

The upper time-1 node is the interesting one. The present value of the US$3 dividend at time 0 is US$2.970297 = 3/1.01, so the dividend-stripped price after an up move is 118.7644 = (100 − 2.970297)(1.224). Just before the shares go ex-dividend they are worth 118.7644 + 3 = 121.7644, so exercising then yields 121.7644 − 95 = 26.7644. If the holder does not exercise, the shares drop as they go ex-dividend and the call is worth only 24.9344 at the lower, ex-dividend price.

Exercising early therefore captures the dividend that the call holder would otherwise never receive. That is the whole reason an American-style call can be worth more than a European-style one: not the passage of time, but the dividend. Here the American-style call is worth 13.2497 against 12.3438 for the European-style call, and the time-0 hedge ratio rises from 0.6004 to 0.6445.

Example 7 · Worked

Using the same inputs, verify the key figures in the lattice above.

1. Confirm the risk-neutral probability and the upper time-1 call value.

Solution. π = (1.01 − 0.796)/(1.224 − 0.796) = 0.214/0.428 = 0.5.
Two up moves take the dividend-stripped price to 118.7644(1.224) = 145.3676, so the call pays 145.3676 − 95 = 50.3676. An up then a down gives 118.7644(0.796) = 94.5364, below the strike, so that node pays zero.
The model value at the upper time-1 node is (1/1.01)[(0.5)50.3676 + (0.5)0] = 25.1838/1.01 = 24.9344.

2. Value the call at time 0 under European-style and American-style exercise, and explain the gap.

Solution. The lower time-1 node is worthless in both cases, since 77.2356(0.796) = 61.4796 and 77.2356(1.224) = 94.5364 are both below the strike.
European style: c = (1/1.01)[(0.5)24.9344 + (0.5)0] = 12.3438.
American style: replace 24.9344 with the exercise value 26.7644, giving c = (1/1.01)[(0.5)26.7644] = 13.2497.
The gap of 0.9059 is the value of the right to grab the US$3 dividend at time step 1. The hedge ratio moves in step, from (24.9344 − 0)/(118.7644 − 77.2356) = 0.6004 to (26.7644 − 0)/41.5288 = 0.6445.

The following exercise pulls together everything in the binomial sections: the expectations approach, hedge ratios at every node, the component interpretation of the model, and the difference American-style exercise makes. It is worth working through slowly, because examiners like this exact shape of question.

Example 8 · Worked

A non-dividend-paying Australian equity trades at A$7.35. Call and put options have two years to expiry, the periodically compounded risk-free rate is 4.35 percent, and the exercise price is A$8.0. The estimated factors are u = 1.445 and d = 0.715.

1. Value the European-style call and put at time step 0 and time step 1.

Solution. Start with the risk-neutral probability:
π = [(1 + 0.0435) − 0.715]/(1.445 − 0.715) = 0.3285/0.730 = 0.45.
Terminal payoffs:
c⁺⁺ = Max[0, 1.445²(7.35) − 8.0] = 7.347; c⁺⁻ = Max[0, 1.445(0.715)(7.35) − 8.0] = 0; c⁻⁻ = Max[0, 0.715²(7.35) − 8.0] = 0.
p⁺⁺ = Max[0, 8.0 − 1.445²(7.35)] = 0; p⁺⁻ = Max[0, 8.0 − 1.445(0.715)(7.35)] = 0.406; p⁻⁻ = Max[0, 8.0 − 0.715²(7.35)] = 4.24.
Expected payoffs seen from each time-1 node:
for the call after an up move, 0.45(7.347) + 0.55(0) = 3.31; after a down move, 0.45(0) + 0.55(0) = 0.0.
For the put after an up move, 0.45(0) + 0.55(0.406) = 0.2233; after a down move, 0.45(0.406) + 0.55(4.24) = 2.51.
With a one-period discount factor of 1/1.0435 = 0.958313:
c⁺ = 0.958313(3.31) = 3.17, c⁻ = 0.0, p⁺ = 0.958313(0.2233) = 0.214, p⁻ = 0.958313(2.51) = 2.41.
At time 0 the two-period expectations are
E(c₂) = 0.45²(7.347) + 2(0.45)(0.55)(0) + 0.55²(0) = 1.488, and
E(p₂) = 0.45²(0) + 2(0.45)(0.55)(0.406) + 0.55²(4.24) = 1.484.
Discounting two periods at 0.91836 gives c = 0.91836(1.488) = A$1.37 and p = 0.91836(1.484) = A$1.36.
The call and put are almost equal here by design. The spot price sits close to the present value of the exercise price, 7.35 versus 8.0/1.0435², and put–call parity written as c − p = S − PV(X) then makes the two values coincide.

2. Compute the European-style hedge ratios at both time steps and interpret the model components.

Solution. The underlying values are S⁺⁺ = 15.347, S⁺⁻ = 7.594, S⁻⁻ = 3.758, S⁺ = 10.621 and S⁻ = 5.255.
Time-1 hedge ratios:
call, up node: (7.347 − 0.0)/(15.347 − 7.594) = 0.9476; call, down node: (0.0 − 0.0)/(7.594 − 3.758) = 0.0.
put, up node: (0.0 − 0.406)/(15.347 − 7.594) = −0.05237; put, down node: (0.406 − 4.24)/(7.594 − 3.758) = −1.0, because both put payoffs below that node are in the money.
Now read the components. For the call after an up move,
c⁺ = 0.9476(10.621) + (1/1.0435)[−0.9476(7.594) + 0.0] = 10.0645 − 6.8961 = 3.1684.
That is long 0.9476 shares costing 10.0645, funded by a loan of 6.8961. The loan can also be read straight off as cost of shares minus option value, 10.0645 − 3.1684.
For the put after an up move,
p⁺ = (1/1.0435)[−(−0.05237)15.347 + 0.0] + (−0.05237)(10.621) = 0.77022 − 0.55622 = 0.2140.
That is short 0.05237 shares raising 0.55622, with 0.77022 placed on deposit. Lending equals short-sale proceeds plus the option value.
After a down move, p⁻ = (1/1.0435)[−(−1.0)7.594 + 0.406] + (−1.0)(5.255) = 7.6665 − 5.255 = 2.4115, that is short 1.0 share with 7.6665 lent.
At time 0, h_c = (3.1684 − 0)/(10.621 − 5.255) = 0.5905 and h_p = (0.2140 − 2.4115)/(10.621 − 5.255) = −0.4095. Then
c = 0.5905(7.35) + (1/1.0435)[−0.5905(5.255) + 0.0] = 4.3402 − 2.97 = 1.37, long 0.5905 shares with a 2.97 loan, and
p = (1/1.0435){−[−0.4095(10.621)] + 0.214} + (−0.4095)(7.35) = 4.37 − 3.01 = 1.36, short 0.4095 shares with 4.37 lent.

3. Now value the American-style call and put, and say what changes.

Solution. The call first. With no dividends, early exercise never pays, so the American-style call matches the European-style call: c⁺ = 3.17, c⁻ = 0.0 and c = 1.37. The exercise values confirm it, since each is at or below the model value:
Max(0, 10.621 − 8.0) = 2.621 against 3.1684; Max(0, 5.255 − 8.0) = 0.0 against 0.0; Max(0, 7.35 − 8.0) = 0.0 against 1.37.
Neither the hedge ratios nor the values move.
The put is different. At time 1 the hedge ratios are unchanged, because only one period remains, so p⁺ = 0.214 and p⁻ = 2.41 initially. The exercise values are Max(0, 8.0 − 10.621) = 0, below 0.214, but Max(0, 8.0 − 5.255) = 2.745, above 2.41. So the lower node is replaced by 2.745.
The hedge ratio becomes h_p = (0.2140 − 2.745)/(10.621 − 5.255) = −0.4717, and
p = (1/1.0435)[0.45(0.214) + 0.55(2.745)] = 1.54.
Early exercise lifts the put from 1.36 to 1.54 and steepens the hedge ratio from −0.4095 to −0.4717.

Two general lessons survive from this exercise. An option is always a position in the underlying combined with financing, and the binomial value is always a present value of expected payoffs taken under the risk-neutral measure and discounted at the risk-free rate. The binomial framework extends to other underlyings with modest surgery: currency options bring in the foreign interest rate, futures options need a lattice of futures prices. Interest rate options need something more, and that is the next topic.

When the underlying is an interest rate rather than an asset price, one convenient feature of the equity lattice disappears. The discount rate is no longer a constant sitting outside the tree; it is the very thing moving inside the tree. Valuation therefore requires a whole term structure, which is usually supplied as an arbitrage-free interest rate lattice. How such lattices are built is outside the scope here. The version used sets the risk-neutral probability of an up move at 50 percent at every node.

Two-year binomial interest rate lattice, annual compounding

DER 2 – Valuation of Contingent Claims — table 6
Node	Maturity	Rate, percent	Value of a one-period zero
Time 0	1	3.0454	0.970446
Time 1, up	1	3.9084	0.962386
Time 1, down	1	2.6034	0.974627
Time 2, up up	1	3.9706	0.961810
Time 2, up down	1	3.2542	0.968484
Time 2, down down	1	2.2593	0.977906

Maturity here means time remaining on the bond, not calendar time, so every entry is a one-period bond. The time-0 spot rate follows from the bond value: (1.0/0.970446) − 1 = 3.04540 percent.

The underlying for these options is the spot rate itself. An interest rate call is in the money when the spot rate is above the exercise rate, and an interest rate put is in the money when the spot rate is below it. That is the opposite orientation to a bond option, and confusing the two is a classic error.

The valuation procedure is the familiar expectations approach applied one period at a time. The one change is that the discount factor is taken from the node you are standing on, not from a single rate applied everywhere.

Example 9 · Worked

Using the lattice above, price a call and a put that expire in two years, are European in style, and reference the one-year spot rate on a periodically compounded basis. The notional is US$1,000,000, the exercise rate is 3.25 percent of par, the risk-neutral probability is 50 percent, and settlement is in cash at time 2 against the rate observed then.

1. Compute the terminal payoffs per US$1 of notional.

Solution. The exercise rate in decimals is 0.0325.
c⁺⁺ = Max[0, 0.039706 − 0.0325] = 0.007206.
c⁺⁻ = Max[0, 0.032542 − 0.0325] = 0.000042. Note how narrowly this node finishes in the money.
c⁻⁻ = Max[0, 0.022593 − 0.0325] = 0.0.
p⁺⁺ = Max[0, 0.0325 − 0.039706] = 0.0.
p⁺⁻ = Max[0, 0.0325 − 0.032542] = 0.0.
p⁻⁻ = Max[0, 0.0325 − 0.022593] = 0.009907.

2. Roll the payoffs back to time 1.

Solution. Use the discount factor belonging to each time-1 node.
c⁺ = 0.962386[0.5(0.007206) + 0.5(0.000042)] = 0.962386(0.003624) = 0.003488.
c⁻ = 0.974627[0.5(0.000042) + 0.5(0.0)] = 0.00002.
p⁺ = 0.962386[0.5(0.0) + 0.5(0.0)] = 0.0.
p⁻ = 0.974627[0.5(0.0) + 0.5(0.009907)] = 0.004828.
The up node discounts at 0.962386 and the down node at 0.974627. Because the underlying is the rate itself, the discount factor has to move with it.

3. Complete the valuation and scale to the notional.

Solution. Discount from time 1 to time 0 at 0.970446.
c = 0.970446[0.5(0.003488) + 0.5(0.00002)] = 0.00170216.
p = 0.970446[0.5(0.0) + 0.5(0.004828)] = 0.00234266.
Scaling by the US$1,000,000 notional gives a call value of US$1,702.16 and a put value of US$2,342.66. The whole calculation is three one-period problems chained together.

From two periods to many

Nothing stops the slicing at two. Divide the option life T into n equal steps of length T/n and the same backward induction runs on a bigger lattice. For American-style options every node gets the exercise test, taking the larger of the model value and the exercise value. As n grows the model becomes both more realistic and more laborious, and in the limit it converges on the continuous-time model in the next section.

Keep the division of labour clear. The expectations approach is quick and applies to European-style options. The no-arbitrage approach applies to both styles, handles early exercise, and is the one that explains what the option actually is.

Louis Bachelier published the first mathematically rigorous option model in 1900. Several quantitative models existed by the late 1960s, but the breakthrough came in 1973, when Fischer Black and Myron Scholes published one paper and Robert Merton another. What the BSM model adds is not a new principle but a new setting: the same no-arbitrage replication argument, run in continuous time. It is the limiting case of a binomial model as the time step shrinks toward zero, which is consistent with the statistical result that a binomial process with many steps converges toward the normal distribution. Scholes and Merton received the 1997 Nobel Prize partly for this work; Black had died in 1995, and the prize is not awarded posthumously.

Choosing a distribution

The central modelling decision is how to describe the randomness of the underlying. Bachelier proposed the normal distribution, which has real attractions: zero is attainable so bankruptcy is allowed, it is symmetric, it is easy to manipulate, and sums of normal variables stay normal. Its fatal flaw for equities is that it permits negative prices, which conflicts with limited liability. Research on stock prices through the 1950s and 1960s moved opinion toward the lognormal distribution, under which the logarithm of the return is normally distributed. Black, Scholes and Merton adopted it.

A terminal distribution is not enough, though, because replication has to be dynamic and self-financing. The model needs a description of how the price evolves at every instant, which is what a stochastic process provides. The one chosen is geometric Brownian motion.

Picture a simulation starting at S = 50 with a geometrically compounded growth rate of μ = 3 percent a year and a volatility of σ = 45 percent. Volatility here means the annualised standard deviation of the continuously compounded percentage change, that is the log return. Two features stand out. Paths that drift upward become more variable in absolute terms while paths that drift downward become less variable, because a 10 percent move on a price of 100 is 10 units but only 1 unit on a price of 10. And the price can never reach zero or go below it, which suits instruments with limited liability. Movements are erratic but never jump.

Within the model, investors are assumed to agree about everything in the distribution except the growth rate of the underlying. The standard version holds both the growth rate and the volatility constant.

The assumption list

The underlying follows geometric Brownian motion, so the continuously compounded return is normally distributed.
Prices are continuous. They move smoothly from one value to the next rather than jumping.
The underlying is liquid and can be bought and sold easily.
Trading is continuous, in the strict sense that a position can be adjusted at every instant.
Short selling of the underlying is permitted with full use of the proceeds.
There are no market frictions: no transaction costs, no regulatory constraints, no taxes.
No arbitrage opportunities exist in the market.
The options are European-style, so early exercise is ruled out.
The continuously compounded risk-free rate is known and constant, and borrowing and lending at that rate is available.
The volatility of the return on the underlying is known and constant.
Any yield on the underlying is a continuous, known and constant annualised rate.

None of these is literally true. The test of a financial model is not realism of assumptions but whether it is tractable and useful, and on that test the BSM model has done very well.

Example 10 · Worked

1. Which pair of statements is consistent with the BSM model? (A) The return on the underlying is normally distributed, and the price of the underlying can jump abruptly. (B) Brokerage costs are built into the model, and continuous trading is impossible. (C) Volatility can be predicted with certainty, and arbitrage does not exist in the marketplace.

Solution. C is correct. The model assumes volatility is known and constant, which amounts to predicting it with certainty, and it assumes no arbitrage. Every statement in A and B fails. In A the first half is right but the second is wrong, since geometric Brownian motion rules out jumps. In B both halves are wrong: the model assumes zero transaction costs, and it assumes continuous trading is possible. Note the trap in A. The model does assume a normally distributed continuously compounded return, so a careless reader accepts the pair on the strength of its first half.

Because the setting is continuous time, the interest rate r throughout this section is the annualised continuously compounded rate, and σ is annualised volatility. Start with a non-dividend-paying stock.

The BSM model for options on a non-dividend-paying stock.

N(x) is the standard normal cumulative distribution function.

N(x) returns the probability of drawing a value below x from a standard normal distribution, which has mean 0 and standard deviation 1. Read off a table or a spreadsheet, N(−1.645) = 0.05, so a d of −1.645 corresponds to 5 percent. Any negative d gives an N below 0.5, and symmetry gives the identity N(−x) = 1 − N(x), which is why the put formula can be written with negative arguments. Note that although the underlying is lognormally distributed, the N functions inside the model are standard normal.

Three ways to read the formula

As a discounted expected payoff. The model can be written c = PV_r[E(c_T)] with E(c_T) = Se^rTN(d₁) − XN(d₂), and for puts E(p_T) = XN(−d₂) − Se^rTN(−d₁). The present value factor is simply e^−rT. As with the binomial model, the expectation is taken under the risk-neutral measure and the discounting uses the risk-free rate, not a required return that reflects risk.

As two components. For a call, the stock component is SN(d₁) and the bond component is e^−rTXN(d₂); the call is the first minus the second. For a put, the stock component is SN(−d₁) and the bond component is e^−rTXN(−d₂); the put is the bond component minus the stock component.

As a dynamically managed portfolio. This is the interpretation the learning outcome asks for. The replicating portfolio costs n_SS + n_BB, where the bond is a zero-coupon bond priced at B = e^−rTX. A positive n means buying and a negative n means selling short.

Number of units in the replicating portfolio

DER 2 – Valuation of Contingent Claims — table 7
Instrument	Call	Put
Units of the underlying, n_S	N(d₁), positive	−N(−d₁), negative
Units of the zero-coupon bond, n_B	−N(d₂), negative	N(−d₂), positive

For a call, the underlying is bought and the bond is sold short, which is borrowing. So a call is a leveraged position in the underlying, exactly as the binomial model said. For a put, the underlying is sold short and the proceeds buy bonds, so a put is lending funded by a short sale. A written put is the reverse: cash is received today, the position is long the underlying and short the bond, and the borrowing exceeds the full cost of the underlying, so a short put is an over-geared long position.

The same structure in two models

DER 2 – Valuation of Contingent Claims — table 8
Model	Call: underlying	Call: financing	Put: underlying	Put: financing
Binomial	hS	PV(−hS⁻ + c⁻)	hS	PV(−hS⁻ + p⁻)
BSM	N(d₁)S	−N(d₂)e^−rTX	−N(−d₁)S	N(−d₂)e^−rTX

The parallel between the binomial hedge ratio h and N(d₁) is exact in the limit.

What N(d₂) means, and what replication costs in practice

N(d₂) has a second reading: it is the risk-neutral probability that the call finishes in the money, so N(−d₂) is the risk-neutral probability that the put does. It is emphatically not your own estimate of that probability, nor the market consensus estimate. It is the probability under the unique measure implied by the absence of arbitrage.

Because S pushes d₁ upward, N(d₁) rises as the underlying rises. Replicating a call therefore means buying more of the underlying into a rising market and selling into a falling one. Inside the theory, the accumulated losses from that buy-high, sell-low discipline add up over the option life to exactly the premium received at inception, which they must, or an arbitrage would exist. In practice transaction costs are not zero, so frequent rebalancing is expensive, and markets do jump, so hedges are imperfect. A merger announcement can move a share price discontinuously, which the model rules out by assumption. Volatility is also unknown in advance. For these reasons traded options tend to be dearer than the theory predicts, and practitioners often feed the formula a volatility a point or two above what they truly expect.

Example 11 · Worked

For options on a stock with S = 100, X = 100, r = 5 percent, T = 1.0 and σ = 30 percent, the BSM model gives PV(X) = 95.123, d₁ = 0.317, d₂ = 0.017, N(d₁) = 0.624, N(d₂) = 0.507, N(−d₁) = 0.376, N(−d₂) = 0.493, c = 14.23 and p = 9.35.

1. What is the initial replicating trade for a call buyer?

Solution. Buy n_S = N(d₁) = 0.624 shares and short sell n_B = −N(d₂) = 0.507 zero-coupon bonds priced at B = Xe^−rT = 95.123. The cost is 0.624(100) + (−0.507)(95.123) = 62.40 − 48.23 = 14.17, which is the call value of 14.23 up to rounding in the N terms. A common wrong answer uses d₁ and d₂ themselves rather than N(d₁) and N(d₂); the d values are inputs to the normal function, not quantities.

2. What is the initial replicating trade for a put buyer?

Solution. Trade n_S = −N(−d₁) = −0.376, that is short sell 0.376 shares, and buy n_B = N(−d₂) = 0.493 zero-coupon bonds. The cost is −0.376(100) + 0.493(95.123) = −37.60 + 46.90 = 9.30, against the model put value of 9.35. In one line: replicate a call by buying the underlying on margin, and replicate a put by shorting the underlying and buying bonds.

Holding the underlying instead of the derivative can produce a benefit or impose a cost. Dividends on shares, foreign interest on a currency and coupons on a bond are benefits. Storage and insurance on an agricultural commodity are costs, which enter as negative benefits. Because the model works in continuous time, these are handled as a continuous yield, written γ.

The carry-adjusted BSM model.

Carry reduces d₁ and therefore d₂ as well.

Carry-adjusted put–call parity.

Read as a discounted expectation, the expected payoffs become E(c_T) = Se^(r−γ)TN(d₁) − XN(d₂) and E(p_T) = XN(−d₂) − Se^(r−γ)TN(−d₁), while the discount factor stays e^−rT. Carry lowers the expected future value of the underlying, and the consequences are worth stating plainly: a larger carry benefit lowers the value of a call and raises the value of a put. Since carry pushes d₂ down, the risk-neutral probability that a call finishes in the money also falls.

Equity options

For stocks, γ = δ, the continuously compounded dividend yield. The replicating portfolio adapts: for calls n_S = e^−δTN(d₁), still positive, and for puts n_S = −e^−δTN(−d₁), still negative. The bond counts are unchanged at n_B = −N(d₂) for calls and n_B = N(−d₂) for puts.

The economics behind that adjustment is simple. A long position in the shares collects the dividends, and a short position has to pay them, so the burden of carrying a long position is lighter. Dividends therefore reduce the number of shares needed to replicate a call and increase the number that must be sold short to replicate a put. Higher dividends also lower d₁ and hence N(d₁), reduce the number of bonds to short for calls, and raise the number to buy for puts.

Currency options

For foreign exchange, γ = r_f, the continuously compounded foreign risk-free rate, because idle foreign currency can be placed in the foreign risk-free instrument. The quoting convention is domestic currency per unit of foreign currency, so a euro trading for 135 Japanese yen is written 135¥/€, with the euro as the foreign currency and the yen as the domestic one. This is the natural convention for a Japanese firm reporting euro holdings in yen.

A few practical points follow. The underlying is the spot exchange rate. The underlying and the exercise price must be quoted in the same units. The volatility is the volatility of the log return on the spot rate. Each contract covers a notional amount of foreign currency, and the total premium is the formula value multiplied by that notional, exactly as an equity option value is multiplied by the number of shares covered.

The two-component reading carries across: for a call the foreign exchange component is Se^−r_fTN(d₁) and the bond component is e^−rTXN(d₂), with r the domestic rate; the call is the first minus the second, and the put reverses the order using the negative arguments.

Example 12 · Worked

Three short applications of the carry-adjusted model.

1. Options on a stock have S = 60, X = 60, r = 2 percent, T = 0.5, δ = 2 percent and σ = 45 percent. Which description of the call value is right, and which of the put value?

Solution. The call is the stock price times e^−δTN(d₁) less the exercise price times e^−rTN(d₂), that is Se^−δTN(d₁) − Xe^−rTN(d₂). The put is the exercise price times e^−rTN(−d₂) less the stock price times e^−δTN(−d₁). The frequent error is attaching the dividend adjustment to the exercise price; it belongs to the underlying.
If instead δ = 0 percent, the values change, and not only because the stock component loses its adjustment. Both d₁ and d₂ depend on γ, so all three moving parts shift together.

2. Bank of China shares trade at HKD4.41 on the Hong Kong Stock Exchange. You expect the price to be steady for three months and decide to write three-month out-of-the-money calls struck at 4.60 to earn premium. Risk-free government securities pay 1.60 percent, the stock yields 0.24 percent and volatility is 28 percent. What are the six model inputs?

Solution. Underlying 4.41, exercise price 4.60, expiration 0.25 years, risk-free rate 0.0160, dividend yield 0.0024, volatility 0.28. Two traps sit in this question: the expiration must be converted from three months to 0.25 years, and the underlying and exercise price must not be swapped. The call is out of the money precisely because 4.41 is below 4.60.

3. A Japanese camera exporter receives fixed euro amounts each quarter. The spot rate is 135¥/€. A fall to 130¥/€ would mean a stronger yen and fewer yen per euro received, so the exporter considers buying an at-the-money spot euro put, which is in effect a call on the yen. The Japanese risk-free rate is 0.25 percent and the European risk-free rate is 1.00 percent. Which underlying, exercise price, risk-free rate and carry rate go into the model?

Solution. Underlying 135 and exercise price 135, since the option is at the money spot and the rate is already quoted as yen per euro. The risk-free rate is 0.25 percent, the Japanese rate, because the yen is the domestic currency under this quoting convention. The carry rate is 1.00 percent, the European rate, because the euro is the foreign currency. Inverting the quote to 1/135 is the classic wrong turn; so is using the foreign rate for discounting.

In 1976 Fischer Black adapted the BSM approach to underlyings that cost nothing to carry, of which futures and forward contracts are the leading examples. The family covered is large: equity index futures options, and, through forwards, interest rate instruments such as caps, floors and swaptions.

Assume the futures price also follows geometric Brownian motion, and set aside margin and marking to market.

Black’s model for European-style options on futures.

The risk-free rate has vanished from d₁, because a futures position costs nothing to carry.

F₀(T) is the futures price observed at time 0 for delivery at time T, and σ is the volatility of the futures price. In effect this is the BSM model with the futures price standing in for the carry-arbitrage-adjusted spot price. Put–call parity for futures options is

Both the futures price and the exercise price are settled later, so both are discounted.

The component reading survives intact. For a call, the futures component is F₀(T)e^−rTN(d₁) and the bond component is e^−rTXN(d₂), with the call being the difference. For a put the futures component uses N(−d₁), the bond component uses N(−d₂), and the bond component comes first. Equivalently, the option value is the present value of the gap between the futures price and the exercise price, with each of the two adjusted by the appropriate N term.

Example 13 · Worked

The S&P 500 Index, a spot index, stands at 1,860 and the futures contract expiring in 0.25 years trades at 1,851.65. The exercise price is 1,860, the continuously compounded risk-free rate is 0.2 percent, time to expiration is 0.25, volatility is 15 percent and the dividend yield is 2.0 percent. The model output is as follows.

Black model output for options on the futures contract

DER 2 – Valuation of Contingent Claims — table 9
Term	Call	Put
N(d₁) and N(−d₁)	0.491	0.509
N(d₂) and N(−d₂)	0.461	0.539
Value	US$51.41	US$59.76

The contract multiplier, 250 for this contract at the time of writing, is ignored. In practice the premium would be 250 times the figures shown.

1. Describe how the model produces the call value.

Solution. The call is the present value of the difference between the current futures price times 0.491 and the exercise price times 0.461, which is the formula c = e^−rT[F₀(T)N(d₁) − XN(d₂)] with numbers attached. Note the spot index of 1,860 does not enter.

2. Describe how the model produces the put value.

Solution. The put is the present value of the difference between the exercise price times 0.539 and the current futures price times 0.509, matching p = e^−rT[XN(−d₂) − F₀(T)N(−d₁)]. The pairing is what matters: the exercise price always travels with the d₂ term and the underlying with the d₁ term.

3. Which values are the underlying and the exercise price in this model?

Solution. The underlying is the futures price of 1,851.65 and the exercise price is 1,860. The spot index of 1,860 happens to equal the exercise price here, which is exactly the sort of coincidence a question uses to see whether you reach for the spot price out of habit. In the Black model the underlying is always the forward or futures price.

With interest rate options the underlying is a reference rate such as three-month MRR. An interest rate call gains when the reference rate rises and an interest rate put gains when it falls. These options are the raw material for a great deal else in the fixed income toolkit.

For a one-year interest rate call on three-month MRR, the underlying is the rate on a forward rate agreement expiring in one year. That FRA is observable today and is the rate fed into the model. The deposit underlying the FRA is a three-month deposit placed in 12 months and maturing in 15 months, and as the year passes the FRA rate normally converges on the three-month spot rate.

Advanced set, settled in arrears

Interest rates are fixed at the start of a period but paid at the end, a convention described as advanced set, settled in arrears. A bank deposit made at time t_j₋₁ has its rate fixed then, but the interest arrives at time t_j, so the deposit runs for t_m = t_j − t_j₋₁. Floating rate loans work the same way. So does the settlement of many interest rate options: if a payment of US$5,000 based on three-month MRR is determined on 15 January, the money changes hands on 15 April.

Figure 3: Advanced set, settled in arrears

The rate is fixed at option expiration but the cash arrives one accrual period later, so the discount factor runs to the later date while the volatility term uses only the time to expiration.

Rates are quoted annually while the underlying deposit is usually shorter, so an accrual period adjustment is needed. A quarterly reset FRA on a 30/360 day count has an accrual period of 0.25 = 90/360. On an actual/360 basis with 91 days in the period it becomes 0.252778 = 91/360. Typically the FRA accrual uses 30/360 while the option side uses actual days over actual days, or actual over 365.

The model

Write FRA(0, t_j₋₁, t_m) for the fixed rate available today on a forward rate agreement that expires at t_j₋₁ and whose underlying deposit matures at t_j = t_j₋₁ + t_m. For instance FRA(0, 0.25, 0.5) = 2 percent means a 2 percent fixed rate on an agreement expiring in 0.25 years with settlement in 0.75 years. Let R_X be the exercise rate and σ the annualised standard deviation of the continuously compounded change in the FRA rate.

The standard market model for an interest rate call, notional of 1.

AP is the accrual period in years.

Time to option expiration, not time to settlement, drives d₁ and d₂.

Five differences from the plain Black model deserve memorising.

The discount factor runs to the FRA maturity date t_j, written as t_j₋₁ + t_m to keep the settlement in arrears visible, and not merely to option expiration.
The underlying is a forward interest rate, not a futures price.
The exercise price is really an exercise rate.
The time to option expiration, t_j₋₁, is what enters d₁ and d₂.
The rates must be entered in decimals, so 0.02 rather than 2.0, unless the notional adjustment is divided by 100 instead.

The formulas price a notional of 1, so the actual premium is the formula value multiplied by the notional, just as a stock option value is multiplied by the number of shares covered. As with every other model here, the value is a discounted expected payoff, with the present value factor e^−rt_j running from the settlement date.

Combinations worth knowing

With the exercise rate set at the current FRA rate, long a call and short a put reproduces a receive-floating, pay-fixed FRA.
With the same exercise rate, long a put and short a call reproduces a receive-fixed, pay-floating FRA. Since FRAs are the building blocks of swaps, this is the bridge to the next section.
An interest rate cap is a strip of interest rate calls, individually called caplets, with sequential maturities. A borrower on a floating rate loan can hedge with a long cap.
An interest rate floor is a strip of interest rate puts, individually called floorlets. A holder of a floating rate bond, or any floating rate lender, can hedge with a long floor.
Long a cap and short a floor struck at the swap rate is a receive-floating, pay-fixed swap. Above the strike both the cap and the swap pay the holder; below the strike the short floor requires a payment, exactly as the swap would. Reversing both legs gives the pay-floating, receive-fixed swap.
Set the exercise rate at the swap rate and the cap and the floor must have equal value at inception. A new swap has zero value, so the cost of being long the cap and short the floor is also zero.

Example 14 · Worked

You are a speculative investor in Singapore. On 15 May you expect regulatory changes and wish to profit. On 15 June you intend to borrow 10,000,000 Singapore dollars to buy an asset you expect to resell three months later, on 15 September. Three-month SORA, the Singapore reference rate, is currently 0.55 percent. The FRA rate covering 15 June to 15 September is 0.68 percent. Worried that rates will rise, you buy an interest rate call struck at 0.60 percent.

1. Which rate is the underlying in the Black model?

Solution. The 0.68 percent FRA rate. The Black model always takes a forward or futures price as its underlying, unlike the BSM model, which takes a spot price. The current 0.55 percent spot SORA is a distractor, and the 0.60 percent is the exercise rate.

2. Over what period does the discount factor run?

Solution. From 15 May to 15 September. You are valuing on 15 May. The option expires on 15 June, when the payoff is determined by the three-month rate observed that day, but settlement in arrears means the cash arrives on 15 September. The expected payoff must therefore be discounted all the way back from 15 September. Discounting only to 15 June, the expiration date, is the standard error this question is testing.

A swaption is an option on a swap. The holder has the right, not the obligation, to enter a swap at a pre-agreed fixed rate, which serves as the exercise rate. Since swaps come in two directions, so do swaptions.

A payer swaption is an option to enter a swap paying fixed and receiving floating.
A receiver swaption is an option to enter a swap receiving fixed and paying floating.

The words call and put are usually avoided here, because it is easy to lose track of what the underlying is. The naming convention keys off the fixed leg.

Follow the buyer of a payer swaption, who profits when fixed rates rise. On exercise, that buyer enters a pay-fixed, receive-floating swap at the exercise rate R_X, then immediately enters an offsetting at-market receive-fixed, pay-floating swap at the prevailing fixed rate. The two floating legs cancel. What remains is a stream of payments equal to the difference between the current fixed swap rate and the exercise rate. That is why a swaption is valued as an annuity rather than as a single payoff.

The present value of an annuity of 1 over the swap payment dates, sometimes called the annuity discount factor.

Payer swaption, notional of 1.

Receiver swaption, notional of 1.

R_FIX is the fixed rate on the forward swap starting when the swaption expires; σ is the volatility of that forward swap rate.

Four differences from the Black model matter. There is no separate discount factor, because the annuity term already embeds the discounting over the swaption life and the payoff is a series rather than a single amount. The underlying is the fixed rate on a forward interest rate swap rather than a futures price. The exercise price is an interest rate. And the rates go in as decimals.

Component reading: for a payer swaption the swap component is (AP)PVA(R_FIX)N(d₁) and the bond component is (AP)PVA(R_X)N(d₂), with the value being the first minus the second. For a receiver swaption the same two components use the negative arguments and the bond component comes first. Written as an expectation, the swaption value is the present value of the expected payoff at expiration, where the expected payoff is the current value grossed up at the risk-free rate, since the annuity term has already done the discounting.

Equivalences

Hold a receiver swaption and write a payer swaption, both struck at one rate, and the combination behaves as a forward swap that receives fixed and pays floating.
Reverse the pair, holding the payer and writing the receiver at that same rate, and the combination behaves as a forward swap that pays fixed and receives floating.
Choose the exercise rate so that the two swaptions have equal value and that rate is the at-market forward swap rate. This is the put–call parity of the swaption market.
A long position in a callable fixed-rate bond is a long straight fixed-rate bond plus a short receiver swaption.

The last relationship rewards a moment of thought. A receiver swaption gives its buyer the right to receive fixed, so the seller must pay fixed when the buyer exercises, which happens when rates have fallen. That is precisely when a bond issuer would call the bonds. An issuer who sells a receiver swaption on matching terms has effectively converted the callable bond into a straight bond: the issuer pays the fixed rate on the underlying swap, and the floating rate received offsets the floating rate loan created by refinancing. The embedded call feature and the short receiver swaption are two descriptions of the same exposure.

Example 15 · Worked

An Australian company carries floating rate debt and has benefited from falling rates. It now fears the central bank will tighten within three months. Rather than lock in borrowing costs with a swap today, it prefers to buy a swaption expiring in three months giving the right, but not the obligation, to enter a five-year swap. The current three-month forward, five-year swap rate is 2.65 percent. The current five-year swap rate is 2.55 percent. The current three-month risk-free rate is 2.25 percent.

1. Judged against the Black model, only one of these holds. (A) What the model prices off is the forward swap rate for five years beginning in three months. (B) Discounting should be done at 2.55 percent. (C) The value of T for this swaption is five years.

Solution. A is correct. The underlying is the forward swap rate that starts when the swaption expires, here 2.65 percent. Statement B is wrong because the current five-year swap rate of 2.55 percent is not a discount rate; the discounting sits inside the annuity term. Statement C is wrong because the option expires in 0.25 years. The five years describes the tenor of the swap that would be entered, not the life of the option. Keep the two horizons separate: T in d₁ is the time to swaption expiration, while the swap tenor determines how many payments the annuity covers.

Once a valuation model exists, it can be differentiated with respect to each of its inputs. The resulting sensitivities are known collectively as the Greeks: delta, gamma, theta, vega and rho. They are static risk measures, capturing the effect of moving one factor while holding every other factor fixed. Because they come out of a model, they are model dependent, so a manager must choose a model suited to the instrument. The discussion below uses European stock options on an underlying with a continuous dividend yield δ, where δ = 0 covers non-dividend-paying stock.

Delta measures how much an instrument changes for a small change in the underlying. A long share has a delta of exactly +1.0 and a short share exactly −1.0. For options,

Both are simple functions of N(d₁).

Since N(d₁) lies between 0 and 1, call delta ranges from 0 to e^−δT and put delta from −e^−δT to 0. As the underlying rises, a call moves deeper in the money and N(d₁) heads toward 1; as it falls, N(d₁) heads toward zero. As expiration approaches, delta drifts toward 0 for an out-of-the-money option and toward 1 for one that is in the money. Delta answers how much, not how likely: it says nothing about the probability of the move it describes.

Executing a hedge

Delta hedging means taking a position in the underlying, sized by delta, so that small moves in the underlying leave the total position unchanged. For a single option, compute the delta and buy or sell that many units of the underlying. In practice a manager holds many positions at once, so delta hedging really means steering the delta of the whole portfolio. Driving it to zero produces a delta neutral portfolio.

A negative N_H means short the hedging instrument; a positive N_H means go long.

Three quick illustrations fix the mechanics. A portfolio of 100,000 shares at US$10 has a portfolio delta of 100,000, and with stock as the hedging instrument the delta of the hedge is +1, so N_H = −100,000: short 100,000 shares. A portfolio delta of 5,000 hedged with a call whose delta is 0.5 requires N_H = −5,000/0.5 = −10,000, so sell 10,000 calls. A portfolio of options with a delta of −1,500 hedged with stock requires N_H = −(−1,500)/1 = 1,500, so buy 1,500 shares.

Delta as an approximation

Delta also forecasts option price changes. Writing the new values with hats, the delta approximation is

A first-order approximation, exact only in the limit of an infinitesimal move.

Figure 4: Delta is the slope, and the gap is what gamma measures

The straight delta line touches the curved model value at the current price and falls below it on both sides, so the delta approximation understates the option value for a move in either direction.

For a small move the approximation is excellent. Push the underlying from 100 to 101 and the delta line and the model value are barely distinguishable. Push it from 100 to 150 and the model value sits well above the delta line. The error grows with the size of the move, and it is biased low in both directions, upward moves and downward moves alike, because the model value curves away from its own tangent. That is the same statement as saying delta hedging is imperfect, and it explains why continuous trading matters so much inside the theory: the hedging risk is exactly the gap between the two lines, and it is realised when the underlying moves a long way at once.

Example 16 · Worked

Apple stock trades at US$125. You write calls on 1,000 Apple shares, so you lose if Apple rises and gain if it falls. The call delta is 0.50, meaning a US$0.10 rise in Apple lifts a call on one share by US$0.05. You buy 500 Apple shares to hedge, so a US$0.10 rise costs US$50 on the written calls and earns US$50 on the shares.

1. Which of these statements is incorrect? (A) Selling Apple puts requires buying Apple stock to delta hedge. (B) Call delta is non-negative and put delta is non-positive. (C) Delta hedging neutralises exposure to the underlying.

Solution. A is the incorrect statement. A short put has positive delta, because the put itself has negative delta and selling flips the sign. Positive exposure is neutralised by short selling stock, not buying it. Statements B and C are both correct.

2. Now take S = 100, X = 100, r = 5 percent, T = 1.0, σ = 30 percent and δ = 5 percent, giving Delta_c = 0.532 and Delta_p = −0.419. You are short puts on 10,000 shares, one share per contract. What stock trade delta hedges the position?

Solution. The put delta is −0.419, so a short put has delta +0.419. The portfolio delta is 10,000(0.419) = 4,190. With stock as the hedge, Delta_H = 1.0, so
N_H = −(4,190/1) = −4,190, that is short sell 4,190 shares.

3. What if calls are used as the hedging instrument instead?

Solution. The portfolio delta is still 4,190, but now Delta_H = 0.532, so
N_H = −(4,190/0.532) = −7,875.9, that is sell 7,876 call options. Both hedges neutralise the same exposure; only the instrument differs.

4. Which description of option delta is correct? (A) It measures curvature in the option price with respect to the underlying. (B) It is the change in the option value for a small change in the underlying, everything else held constant. (C) It is the probability that the option expires in the money.

Solution. B is correct. Statement A describes gamma, not delta. Statement C describes N(d₂), which is related to but distinct from delta: delta is built from N(d₁).

Delta is a straight line drawn through a curve, so it degrades as the move gets larger. Gamma measures how fast delta itself changes when the underlying moves, and therefore how much curvature the delta line is failing to capture.

The lower-case n(d₁) is the standard normal probability density function, not the cumulative function N.

Note the change of case. Throughout the reading N denotes the cumulative distribution, while n denotes the density that generates it. Confusing them produces a badly wrong number.

A share has zero gamma, since its delta is fixed at +1 for a long position and −1 for a short one and never moves. Call gamma equals put gamma, and put–call parity shows why. Parity says c − p = S₀ − e^−rTX. Neither term on the right depends on delta, so the right-hand side has a constant delta of 1 and is insensitive to further moves in the underlying. The left-hand side must therefore have zero net gamma, which means the two gammas are equal.

Gamma is never negative and reaches its maximum near the money. Deep in the money or deep out of the money, delta barely responds to small moves, so gamma is small. Like delta, gamma changes as the underlying moves and as expiration approaches.

Gamma as a risk measure

Gamma quantifies the error left in the delta approximation, so it is a measure of non-linearity risk: the exposure that survives after a portfolio has been made delta neutral. A gamma neutral portfolio has gamma of zero.

The practical sequence is gamma first, delta second. Shares have gamma of zero, so gamma can only be changed by trading options. Once option trades have brought portfolio gamma to the desired level, the portfolio delta can be moved to its target by buying or selling shares, and because shares carry no gamma this second step leaves the gamma untouched. Doing it the other way round means the option trades needed for gamma would disturb the delta already set.

The delta-plus-gamma approximation; the put version is identical with p in place of c.

Adding the quadratic term transforms the accuracy. For small moves both approximations track the model value closely. Move the underlying from 100 to 150 and the model value is far above the delta line but now sits slightly below the delta-plus-gamma line, so the second-order estimate is much the better of the two. The direction of the residual error changes, though: the delta-plus-gamma approximation is biased low for a downward move and biased high for an upward move, whereas the plain delta approximation was biased low in both directions.

Gamma also names a real trading hazard. If the BSM assumptions held exactly, an option position could be managed without risk. Prices jump instead of gliding, and a jump leaves a delta-hedged book suddenly unhedged, because the delta that was correct a moment ago is now wrong. That exposure is what traders call gamma risk.

Example 17 · Worked

1. You hold a single option position, a short call, together with enough stock to be delta hedged. Which statement is true? (A) You are gamma neutral. (B) Buying a call will increase your overall gamma. (C) Your overall position has positive gamma, so large moves in either direction are profitable.

Solution. B is correct. Buying options, calls or puts alike, always adds gamma, because long option positions have positive gamma and the stock leg contributes none. A is wrong because a short call gives negative gamma, and holding shares does not offset it. C is wrong for the same reason and inverts the conclusion: a short gamma book loses on large moves in either direction. The only route back to gamma neutrality from a short gamma position is buying options.

Why the order of operations is not arbitrary

The instruments available split cleanly. Shares carry delta but no gamma. Options carry both. That asymmetry gives a strict ordering: fix the quantity that only options can change, then use shares to fix the quantity they can change without side effects. Reversing the order forces you to redo the first step, because every option trade you make for gamma also moves the delta you had just set. The same reasoning explains why theta cannot be managed with share trades either, since a share has no expiration date and therefore no theta.

Theta

Theta measures the change in value for a small advance in calendar time, everything else held constant. That last phrase carries a lot of weight: the theta calculation assumes literally nothing changes except the date. As calendar time advances, the time remaining to expiration falls. Shares do not expire, so a share has zero theta, and like gamma, theta cannot be adjusted by trading shares.

The profit or loss produced by the mere passage of time is called time decay. Long option positions can lose real money without the underlying moving at all, which is why managers with large option books watch theta closely. Theta is unlike delta and gamma in one important respect: there is no uncertainty about it. The underlying may or may not move, but time passes with certainty and never runs backward.

Theta is typically negative for options: as the date advances, the option loses value. Plot an at-the-money call and an at-the-money put against time to expiration and both decline toward zero, on the assumption that the underlying does not move. The decline is not linear. The nearer expiration comes, the faster value is lost.

Vega

Vega measures the change in value for a small change in volatility, everything else held constant. Vega is positive for options: more volatility means a more valuable call and a more valuable put alike. Call vega equals put vega, by the same parity argument used for gamma, since neither S₀ nor e^−rTX depends on volatility, so the vega of the right-hand side is zero and the two option vegas must offset.

Vega is different from the other Greeks in a fundamental way: the parameter it differentiates with respect to cannot be observed. Historical volatility can be computed, but the model wants future volatility, and there is no objective measure of that. Vega therefore measures sensitivity to the volatility number fed into the model rather than to any observable quantity. Of the five BSM inputs, value is most sensitive to volatility, which makes vega the most important Greek for many option books.

Vega is largest at or near the money. Volatility can generally be hedged only with other options, and volatility is itself volatile, so it is often treated as a separate asset class or risk factor. That makes it worth managing explicitly, and worth explaining to risk managers, board members and clients before, rather than after, it produces a loss.

Example 18 · Worked

Take at-the-money European options with S = 100, X = 100, r = 5 percent, T = 1 and δ = 0.

1. As volatility falls toward zero, what values do the call and the put approach, and why do they differ?

Solution. Option values converge on their lower bounds. For a European call the lower bound is the larger of zero and the underlying less the present value of the exercise price:
100 − 100e^−0.05(1) = 100 − 95.12 = 4.88.
For a European put the lower bound is the larger of zero and the present value of the exercise price less the underlying:
95.12 − 100 is negative, so the bound is 0.
The two differ by 4.88, which is exactly what put–call parity predicts, since c − p = S − PV(X). The call retains value at zero volatility because the exercise price is paid later; the put does not, because it would have the holder receiving that money later.

Rho

Rho measures the change in value for a small change in the risk-free rate, everything else held constant. Call rho is positive. Buying a call rather than the underlying avoids the financing cost of the shares, leaving money to earn interest, and the higher the rate the more that is worth. Put rho is negative. Buying a put rather than selling the underlying postpones receipt of the sale proceeds and the interest they would earn, so a higher rate makes the put less attractive.

At a zero interest rate, at-the-money calls and puts have the same value, because c − p = S₀ − e^−rTX collapses to S₀ − X, which is zero at the money. As rates rise the gap between call and put values widens. In practice the effect of rate changes on option prices is small next to the effect of changes in volatility or in the underlying, so rho is rarely the manager’s first concern. One exception matters: for interest rate options the rate is the underlying, so its influence on value is captured by delta, not rho. Rho is best understood generally as sensitivity to the rate used for discounting.

Every input to an option model is observable except one. The underlying price, the exercise price, the expiration date, the risk-free rate and the dividend yield are all known or agreed. Volatility over the life of the option, looking forward, is not. It can be estimated from history, for example by measuring the realised volatility of the past three months and using it for the next three, but history is a frail guide, and different investors will hold different views. The investor with the best forecast has the best estimate of value.

There is another way round the problem. Option prices are observable and plentiful, so the model can be inverted: hold every other input fixed and solve for the volatility that reproduces the traded price. That number is the implied volatility, and the exercise is analogous to computing yield to maturity from a bond price. There is a one-to-one relationship between implied volatility and price, so nothing is lost in the translation.

What is gained is information. Implied volatility summarises the market view of uncertainty ahead, and it also reflects demand. If buying pressure for options is not fully arbitraged away, because of transaction costs or other frictions, prices rise and implied volatility rises with them. Traders read that as a signal in its own right.

The surface is not flat

The BSM model assumes everybody agrees on a single, non-stochastic volatility. Reality disagrees. Implied volatilities differ across exercise prices, and they differ between calls and puts written on identical terms. They also differ across expirations.

Implied volatility plotted against time to expiration is the term structure of volatility.
Implied volatility plotted against exercise price is the volatility smile, or the skew when the shape is lopsided.
Plotting against both at once gives a three-dimensional volatility surface.

If the model assumptions held, the surface would be flat. It is not.

Figure 5: Implied volatility across exercise prices

The curve reaches its low point a little above the at-the-money strike and rises into both wings, with the low-strike wing on the left standing highest. A model that assumed a single volatility for every strike would draw the flat dashed line instead.

Implied volatility also moves through calendar time. A rise signals that participants are charging more for risk. If put implied volatility rises, downside protection has become dearer, so the market price of hedging has gone up. Index option markets have spawned volatility indexes that track this collective opinion, and futures and options on those indexes let managers hedge vega exposure elsewhere in a book.

The best known is the Chicago Board Options Exchange S&P 500 Volatility Index, the VIX, quoted in percentage points and designed to approximate the implied volatility of the S&P 500 over the next 30 days. It is often called the fear index, since a higher reading is read as greater investor uncertainty. Daily values over roughly two and a half decades from 2 January 1990 show long calm stretches punctuated by spikes, with an extreme reading during the 2008 global financial crisis. Remember what such a reading contains: both a higher expectation of future volatility and a stronger preference for owning rather than writing options.

Volatility as a quoting convention

Because the mapping from volatility to price is one-to-one given the other inputs, market participants quote options in volatility rather than in currency. A call trading at €14.23 might be quoted as 30.00, meaning an implied volatility of 30 percentage points. Any counterparty using the agreed model and the agreed other inputs recovers the price.

The advantage is comparability. Volatility tends to be of similar magnitude across exercise prices and expirations, so quoting in volatility strips out the effects of moneyness and time and leaves a common unit. Ask which of two calls on the same stock is more expensive when one has a longer expiration and a lower exercise price and the other has a shorter expiration and a higher exercise price, and the raw prices cannot answer. The implied volatilities can. This is also why regulators, banks, compliance teams and traders communicate about option portfolios in volatility terms: together with an agreed model it delivers a market consensus valuation, in the same way that market prices value other assets.

Example 19 · Worked

1. You run a portfolio of options on FTSE 100 futures, with the futures at 6,850. A client asks for offers on out-of-the-money puts and at-the-money puts with the same expiration. You price them at 190 and 280 futures points respectively, and the corresponding implied volatilities are 16 percent and 15.2 percent. The client wants both quotes in volatility as well as in points. Why? (A) So she can compare the two options for value and judge which is cheap. (B) So she can see where implied volatility is trading and revalue her own FTSE 100 book at current market levels. (C) Both.

Solution. C is correct. Both uses are valid. Implied volatility neutralises the effects of moneyness and time to expiration, which makes two structurally different options comparable, and it is also the natural input for marking an existing portfolio to current market levels. Note that the out-of-the-money put is the cheaper option in points, 190 against 280, yet carries the higher implied volatility, 16 percent against 15.2 percent. Price alone would have told her the opposite story about relative expensiveness.

2. A dealer offers a three-month at-the-money call on the FTSE index at 19 percent implied volatility and a one-month in-the-money put on Vodaphone at 24 percent. A trader believes FTSE volatility should be nearer 25 percent and Vodaphone volatility nearer 20 percent. What should she do? (A) Buy the FTSE call and the Vodaphone put. (B) Buy the FTSE call and sell the Vodaphone put. (C) Sell the FTSE call and sell the Vodaphone puts.

Solution. B is correct. The FTSE call is offered at 19 percent against her view of 25 percent, so it is cheap on her numbers and she expects its volatility to rise, lifting the option value: buy it. The Vodaphone put is offered at 24 percent against her view of 20 percent, so it is expensive and she expects its volatility to fall, reducing the option value: sell it. Answers A and C each get one leg right and one leg wrong.

Check yourself

1. A one-period lattice has S = 80, u = 1.25, d = 0.80, X = 80 and a per-period risk-free rate of 4 percent. Compute the hedge ratio and the risk-neutral probability, then value the call.

The branches are S⁺ = 100 and S⁻ = 64, so c⁺ = 20 and c⁻ = 0. The hedge ratio is (20 − 0)/(100 − 64) = 0.555556. The risk-neutral probability is (1.04 − 0.80)/(1.25 − 0.80) = 0.24/0.45 = 0.533333. The call value is (1/1.04)[0.533333(20)] = 10.256410, or about 10.26. Check it with replication: 0.555556(80) + (1/1.04)[−0.555556(64)] = 44.4444 − 34.1880 = 10.26.

2. Why can an American-style put on a non-dividend-paying stock be worth more than the European-style equivalent, when the same is not true of a call?

Exercising a deep in-the-money put converts it into cash that can earn the risk-free rate, and that interest can exceed the remaining time value. Exercising a call does the opposite: it hands over cash early for an asset the holder no longer expects to rise, and the call always trades above its exercise value, so selling it beats exercising it. The consequence for valuation is that an American-style put has to be valued by backward induction with an exercise test at each node, because the present value of expected payoffs formula cannot see the early exercise decision.

3. In the two-period lattice with S = 72, X = 75, u = 1.356, d = 0.541 and r = 3 percent, the European put is worth 18.16876 and the American put 19.01710. Where exactly does the difference arise, and what happens to the hedge ratio?

Only at the lower time-1 node. There the underlying is 38.952, so the exercise value is 75 − 38.952 = 36.048, against a model value of 33.86353. Replacing 33.86353 with 36.048 and rolling back one period gives 19.02, so the early exercise premium is 2.18447 at that node and 0.85 at time 0. The time-0 hedge ratio steepens from −0.43029 to −0.46752, because a larger down-state value makes the option more responsive to the underlying.

4. An interest rate call and an interest rate put are valued on the same two-period lattice. Why is the discount factor different at the two time-1 nodes, when an equity option would use the same discount factor at both?

Because the underlying is the interest rate itself. In the lattice used here the one-period bond value is 0.962386 at the up node and 0.974627 at the down node, reflecting rates of 3.9084 percent and 2.6034 percent. For an equity option the risk-free rate is an assumption held constant outside the tree; for an interest rate option it is the state variable inside the tree, so discounting has to follow it node by node. This is also why valuing interest rate options requires a whole term structure rather than a single rate.

5. State the replicating portfolio implied by the BSM model for a long call and for a long put, and say what a short put looks like.

A long call is n_S = N(d₁) units of the underlying, which is positive, plus n_B = −N(d₂) zero-coupon bonds, which is negative, so it is a leveraged long position. A long put is n_S = −N(−d₁), a short sale, plus n_B = N(−d₂) bonds, lending. A short put reverses the put, giving a positive share position and a negative bond position where the borrowing exceeds the full cost of the shares, which is an over-geared long position in the underlying.

6. An interest rate call expires on 15 June with settlement on 15 September, and is being valued on 15 May. Which period drives the discount factor, and which drives the d₁ and d₂ terms?

The discount factor runs from 15 September back to 15 May, because the option is advanced set and settled in arrears: the rate is fixed at expiry but the cash is paid one accrual period later. The d₁ and d₂ terms use only the time to option expiration, 15 May to 15 June, because that is the period over which the underlying FRA rate can still move. Mixing the two up is the standard trap in this model.

7. A trader is short a call and holds shares to be delta hedged. Explain the remaining exposure and how to remove it, in the correct order.

The book is short gamma, so any large move in the underlying, up or down, loses money, and each move leaves the delta hedge stale. Shares cannot fix this, because a share has zero gamma. The order is: first buy options to bring gamma to the target level, since only options carry gamma; then adjust the share position to restore the desired delta, which is safe because trading shares does not disturb gamma. Doing delta first would simply undo itself as soon as the gamma trades were put on.

8. Two calls on the same stock trade at different prices. One has a longer expiration and a lower exercise price; the other has a shorter expiration and a higher exercise price. Why does quoting them in implied volatility settle which is more expensive?

Their currency prices are not comparable, because moneyness and time to expiration both push the prices around for reasons that have nothing to do with how richly the option is priced. Implied volatility is the number that reproduces each price once those effects are stripped out through the model, so the two become directly comparable in a common unit. The one implying the higher volatility is the more expensive of the two, and that is the judgement the raw prices cannot deliver.

End of lesson.

Back to 07 Derivatives

DER 2 – Valuation of Contingent Claims

The no-arbitrage foundation for valuing options

Thinking like an arbitrageur

The standing assumptions

Notation and the value of an option at expiration

The one-period binomial model: hedging and replication

Building a hedge, then reading the price off it

The same machinery for puts

The expectations approach and risk-neutral probabilities

Why this is not ordinary discounted cash flow

Put–call parity as a shortcut

The two-period binomial model for call options

Self-financing and dynamic replication

Working backward

The expectations form over two periods

American-style puts and the early exercise premium

Why calls on non-dividend-paying stock are never exercised early

Why puts are different

Dividends and the early exercise of American-style calls

A comprehensive two-period valuation exercise

Valuing interest rate options in a binomial lattice

From two periods to many

Assumptions of the Black–Scholes–Merton model

Choosing a distribution

The assumption list

Components of the BSM model and the leveraged position interpretation

Three ways to read the formula

What N(d2) means, and what replication costs in practice

Carry benefits: options on dividend-paying equities and on currencies

Equity options

Currency options

The Black model and European options on futures

Interest rate options under the standard market model

Advanced set, settled in arrears

The model

Combinations worth knowing

Swaptions and their equivalence relationships

Equivalences

Delta and delta hedging

Executing a hedge

Delta as an approximation

Gamma and gamma risk

Gamma as a risk measure

Theta, vega and rho

Theta

Vega

Rho

Implied volatility and the volatility surface

The surface is not flat

Volatility as a quoting convention

Related lessons

What N(d₂) means, and what replication costs in practice