The Encyclopedia of Economic Principals

Chapter 45

Derivatives, Options, Insurance, and Contingent Claims

Price a claim by the cost of building its payoff.

Four of the chapter's worked examples, made interactive: a Black-Scholes-Merton call, a bottomry loan forgiven if the ship is lost, a hull insurance pool with correlated losses and a put-call parity arbitrage. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Volatility and the price of a call

How much is a call worth, and how many shares hedge it, as volatility and the stock price change?

The value is the cost of a hedging strategy that replicates the call. More volatility widens the spread of terminal prices; because the payoff is convex, the upside gained outweighs the downside, which is capped at zero, so the value rises.

Equation, written in LaTeX: C=S_0N(d_1)-Ke^{-rT}N(d_2),

Equation, written in LaTeX: d_1=\frac{\ln(S_0/K)+(r+\sigma^2/2)T}{\sigma\sqrt{T}}

Equation, written in LaTeX: d_2=d_1-\sigma\sqrt{T}.

Scroll sideways for the whole equation

S0 is today's stock price, K = 85 the strike, T = 0.5 years to expiry, r = 4 percent the continuously compounded rate and sigma the annual volatility. N is the standard normal distribution function; N(d1) is the call's delta.

Predict first. Raising volatility from 25% to 35% with S0 = 80: does the call rise by more or less than 2?

Your prediction

Choose an example

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Figure: Volatility and the price of a call. Call value against the stock price for volatility 25%, lying above the expiry payoff max(S - 85, 0). At S0 = 80 the value is 4.24 and the tangent has slope 0.4438.
Volatility: 25%, Stock price today: 80
Constructed example: the chapter's hypothetical call (S0 80, K 85, T 0.5, r 4%, sigma 25% and 35%, 500 calls); volatilities 15% and 45% and stock prices 70 and 90 are added for comparison.

Calculated values

d1
-0.1414
d2
-0.3182
N(d1)
0.4438
N(d2)
0.3752
Call value C
4.24
Shares to hedge 500 short calls
221.9

With S0 = 80 and sigma = 25%, d1 = -0.1414 and d2 = -0.3182, so N(d1) = 0.4438 and N(d2) = 0.3752. The call is worth 35.5016 - 31.2578 = 4.24, and a writer of 500 calls holds 500 x 0.4438 = 221.9 shares as the opening hedge. Higher volatility lifts the curve above the expiry kink because the payoff is convex.

Worked steps

  1. ln(80/85) = -0.060625; (0.04 + 0.25^2/2)(0.5) = 0.035625
  2. Numerator = -0.060625 + 0.035625 = -0.025000
  3. d1 = -0.025000 / (0.25 x sqrt(0.5)) = -0.025000 / 0.176777 = -0.1414
  4. d2 = -0.141419 - 0.176777 = -0.318196, about -0.3182
  5. C = 80 x 0.443769 - 85 x e^(-0.02) x 0.375168 = 35.5016 - 31.2578 = 4.2438, about 4.24
  6. Hedge = 500 x 0.4438 = 221.9 shares

Use the idea

Read an option quote as a statement about volatility: with price, strike, time and rate fixed, the only free input is sigma.

Where the conclusion applies

European call, no dividends, constant volatility and rate, continuous trading without costs. Real hedges are rebalanced in steps and pay spreads.

Check your understanding: At sigma 35%, how many shares hedge 500 short calls?
N(d1) = 0.4839, so 500 x 0.4839 = 241.9 shares.

Chapter 45 source: section "Black-Scholes-Merton option pricing".

Demonstration 2 of 4

Bottomry: pricing a loan that dies with the ship

What must a merchant repay on arrival when the debt is forgiven if the ship is lost?

Bottomry bundles a loan with insurance: the lender bears the loss of the ship and is paid for it by a larger repayment in the arrival state. Dividing the required expected receipt by q gives that repayment.

Equation, written in LaTeX: qR=L(1+r)+k.

Equation, written in LaTeX: R=\frac{L(1+r)+k}{q}.

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L = 120,000 is the advance, r the lender's time return, k = 3,000 administrative cost, q the chance the ship arrives and R the repayment due only on arrival. The voyage yields 230,000 on arrival and nothing if lost.

Predict first. Does a drop in the arrival probability hurt the merchant once or twice?

Your prediction

Choose an example

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Figure: Bottomry: pricing a loan that dies with the ship. Bars for the lender and the merchant in the arrival and loss outcomes. With arrival probability 0.75 and time return 3%, the lender receives 168,800 on arrival and nothing if the ship is lost; the merchant's expected proceeds are 45,900.
Arrival probability: 0.75, Lender time return: 3%
Constructed example: the chapter's hypothetical voyage (advance 120,000, 3%, cost 3,000, q 0.75 and 0.60, proceeds 230,000); q 0.90 and time returns 0% and 6% are added for comparison.

Calculated values

Required expected receipt
126,600.00
Repayment on arrival R
168,800.00
Merchant keeps on arrival
61,200.00
Merchant expected proceeds
45,900.00

The lender needs 126,600.00 in expectation. Paid only on arrival, with probability 0.75, that requires R = 126,600.00 / 0.75 = 168,800.00. The merchant keeps 230,000 - 168,800.00 = 61,200.00 on arrival, worth 0.75 x 61,200.00 = 45,900.00 in expectation. A lower arrival chance hurts twice: revenue is less likely and the repayment owed on arrival is larger.

Worked steps

  1. Required receipt = 120,000 x 1.03 + 3,000 = 123,600.00 + 3,000 = 126,600.00
  2. R = 126,600.00 / 0.75 = 168,800.00
  3. On arrival the merchant keeps 230,000 - 168,800.00 = 61,200.00
  4. Expected proceeds = 0.75 x 61,200.00 = 45,900.00

Use the idea

Price any loan forgiven in a bad state by dividing the required expected receipt by the probability that repayment happens.

Where the conclusion applies

A risk-neutral lender, two outcomes, no recovery after a loss and a known q. Historical lenders also charged for bargaining power and uncertainty about q.

Check your understanding: At q 0.90 and 3%, what is R?
126,600 / 0.90 = 140,666.67; the merchant expects 0.90 x (230,000 - 140,666.67) = 80,400.

Chapter 45 source: section "Bottomry Loans as Options & Futures".

Demonstration 3 of 4

Frequency, correlation, and the hull premium

Which inputs move the premium, and which move the uncertainty the insurer must hold capital for?

The premium starts from the mean loss, which depends on frequency and size only. Pooling shrinks the spread of average claims when losses are independent, but correlation leaves a floor that more vessels cannot remove.

Equation, written in LaTeX: \pi=pI+a+\rho.

Equation, written in LaTeX: \operatorname{SD}(\frac{L_N}{N})=I\sqrt{p(1-p)[\frac{1}{N}+c(1-\frac{1}{N})]}.

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N = 144 vessels, I = 750,000 paid after a total loss, p the loss probability, a = 2,400 administration, rho = 1,600 loading and c the pairwise correlation of losses.

Predict first. Does correlation change the premium's expected-claims component?

Your prediction

Choose an example

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Figure: Frequency, correlation, and the hull premium. Left: premium 16,000 built from expected claim 12,000, administration 2,400 and loading 1,600. Right: SD of average claims 7,842.19 if independent and 7,842.19 at correlation 0.00.
Loss probability: 0.016, Pairwise correlation: 0
Constructed example: the chapter's hypothetical hull pool (144 vessels, 750,000, p 0.016 and 0.01, c 0 and 0.08, charges 2,400 and 1,600); p 0.025 and c 0.04 are added for comparison.

Calculated values

Expected claim pI
12,000.00
Premium
16,000.00
SD, independent
7,842.19
SD at this correlation
7,842.19
Ratio to independent
1.00

Expected claims are 0.016 x 750,000 = 12,000.00, so the premium is 16,000.00; correlation does not enter it. The SD of average claims equals 7,842.19, which is sqrt(1.00) = 1.00 times the independent level of 7,842.19. Frequency moves both the mean and the dispersion; dependence moves only the dispersion.

Worked steps

  1. pI = 0.016 x 750,000 = 12,000.00
  2. Premium = 12,000.00 + 2,400 + 1,600 = 16,000.00
  3. p(1 - p) = 0.016 x 0.984 = 0.015744
  4. 1/144 + 0.00(1 - 1/144) = (1 + 0.00 x 143)/144 = 1.00/144
  5. SD = 750,000 x sqrt(0.015744 x 1.00/144) = 7,842.19
  6. Ratio = sqrt(1.00) = 1.00

Use the idea

Price frequency into the expected claim, and price common exposure (one storm, one port) into the capital loading.

Where the conclusion applies

Equal vessels, total losses only, a single correlation for every pair and fixed nonclaim charges.

Check your understanding: With p = 0.01 and c = 0, what are the premium and the SD?
Premium 7,500 + 2,400 + 1,600 = 11,500; SD = 750,000 x sqrt(0.0099/144) = 6,218.67.

Chapter 45 source: section "Maritime Insurance & Risk Pricing".

Demonstration 4 of 4

Put-call parity arbitrage

When call, put, stock and bond quotes disagree, what trade locks in the difference?

Both packages pay max(S_T, 50) at expiry whatever happens, so they must cost the same today. Buying the cheaper and selling the dearer leaves offsetting payoffs and keeps the opening gap.

Equation, written in LaTeX: C_0+Ke^{-rT}=P_0+S_0.

Equation, written in LaTeX: 5.20+48.50=53.70.

Equation, written in LaTeX: 2.40+52=54.40.

Equation, written in LaTeX: (S_T-50)+50=S_T.

Equation, written in LaTeX: (50-S_T)+S_T=50.

Scroll sideways for the whole equation

The stock trades at 52, the call with strike 50 costs 5.20, the bond paying 50 at expiry costs 48.50 and the put price varies. S_T is the stock price at expiry.

Predict first. With the put at 1.70, which package do you buy?

Your prediction

Choose an example

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Figure: Put-call parity arbitrage. Payoffs at expiry of call plus bond and of put plus stock, which coincide at max(S_T, 50), with a net line at zero. At S_T = 60 each pays 60.00. With the put at 2.40 the opening gap is 0.70.
Put price: 2.40, Stock price at expiry: 60
Constructed example: the chapter's hypothetical quotes (stock 52, call 5.20, put 2.40, bond 48.50); puts of 1.70 and 3.10 and expiry prices 40 to 70 are added for comparison.

Calculated values

Call + bond cost
53.70
Put + stock cost
54.40
Opening receipt
0.70
Payoff of each package at S_T
60.00
Terminal net
0.00
Trade
buy the cheaper, sell the dearer

The call and bond cost 5.20 + 48.50 = 53.70; the put and stock cost 2.40 + 52 = 54.40. Buy the call and bond, short the put and the share, collecting 0.70 today. At S_T = 60 both packages pay 60.00, so the terminal net is zero and the opening difference is the whole profit before trading and financing costs.

Worked steps

  1. Call + bond = 5.20 + 48.50 = 53.70
  2. Put + stock = 2.40 + 52 = 54.40
  3. Gap = 54.40 - 53.70 = 0.70
  4. S_T = 60 > 50: call + bond pay (60 - 50) + 50 = 60.00; the put expires and the share costs 60
  5. Terminal net = 60.00 - 60.00 = 0.00

Use the idea

Check quotes against parity before trading options; a gap is profit only if it exceeds spreads, fees, borrow and collateral costs.

Where the conclusion applies

European options, no dividend before expiry, the ability to short the share and the bond quoted at the safe rate.

Check your understanding: If total trading costs are 0.85, is the 0.70 gap worth trading?
No: 0.70 - 0.85 = -0.15.

Chapter 45 source: section "Put-call parity".