Analyst Dojo

CFA Level I Derivatives formulas

Forward and futures pricing with carry, swap values, option exercise values and bounds, put-call parity and the one-period binomial model.

102 formulas from the Derivatives lessons (6–9% of the exam), 57 to memorize. Each says how deeply to learn it and opens to a quick worked example with the keystrokes for the TI BA II Plus and the HP 12C.

How deeply to learn each part
Know the idea
Understand it in general. It frames other topics, but you will not calculate with it.
Use it
Understand it well enough to solve a question with it. Recognize when it applies.
Memorize
Learn it by heart. The exam will not give it to you.

Module 1 Derivatives and Derivative Markets

What a derivative is

Cash settlement of a forward (seller) Use it
Pshort=N (F−ST)P_{\text{short}} = N\,(F - S_T)
Quick example

Forward to sell 1,200 units at 30.00; spot at maturity is 28.50. 1,200 × (30.00 − 28.50) = 1,800 received by the seller. The buyer pays it.

On your calculator
  1. 30−28.5=×1200= 1,800
  1. 30ENTER28.5−1200× 1,800

ENTER enter (on the HP, puts the number on the stack)

Module 2 Forwards, Futures, Swaps, Options and Credit Derivatives

Forward contracts: payoff for the long and the short

Payoff to the long forward Memorize
ST−F0(T)S_T - F_0(T)
Quick example

Long 1,000 ounces of silver at USD 30.00; spot at expiration USD 33.50. (33.50 − 30.00) × 1,000 = USD 3,500.

On your calculator
  1. 33.5−30=×1000= 3500
  1. 33.5ENTER30−1000× 3500

ENTER enter (on the HP, puts the number on the stack)

Payoff to the short forward Memorize
F0(T)−STF_0(T) - S_T
Quick example

Short 8,000 bushels of wheat at USD 6.40; spot at expiration USD 5.90. (6.40 − 5.90) × 8,000 = USD 4,000.

On your calculator
  1. 6.4−5.9=×8000= 4000
  1. 6.4ENTER5.9−8000× 4000

ENTER enter (on the HP, puts the number on the stack)

Futures contracts, daily settlement and margin calls

Daily gain or loss to the long (mark to market) Use it
MTM=[ft(T)−ft−1(T)]×units×contracts\text{MTM} = [f_t(T) - f_{t-1}(T)] \times \text{units} \times \text{contracts}
Quick example

Long 4 contracts of 250 units. The settlement price rises from 80.40 to 81.10. 0.70 × 250 × 4 = +700.

On your calculator
  1. 81.1−80.4=×250×4= 700
  1. 81.1ENTER80.4−250×4× 700

ENTER enter (on the HP, puts the number on the stack)

Variation margin (margin call) Memorize
V=Minitial−Bwhen B<MmaintenanceV = M_{\text{initial}} - B \quad \text{when } B < M_{\text{maintenance}}
Quick example

Long 4 contracts of 50 units; initial 3,200 and maintenance 2,600 per contract. A 14.00 fall leaves 12,800 − 2,800 = 10,000, below 10,400. Call = 12,800 − 10,000 = 2,800.

On your calculator
  1. 4×50×14= loss 2800
  2. +|-+12800= balance 10000, below 10400
  3. 12800−10000= call 2800

+|- change sign

  1. 4ENTER50×14× loss 2800
  2. CHS12800+ balance 10000, below 10400
  3. 12800ENTER10000− call 2800

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Total gain or loss over the life of the long Use it
fT(T)−f0(T)=ST−f0(T)f_T(T) - f_0(T) = S_T - f_0(T)
Quick example

Buy 1 contract of 100 units at 52.00; the final settlement price is 55.50. (55.50 − 52.00) × 100 = +350 in total, however the daily amounts moved.

On your calculator
  1. 55.5−52=×100= 350
  1. 55.5ENTER52−100× 350

ENTER enter (on the HP, puts the number on the stack)

Interest rate swaps: fixed vs floating

Payment on one leg of a swap Use it
Payment=rate×N×1m\text{Payment} = \text{rate} \times N \times \dfrac{1}{m}
Quick example

Fixed rate 2.40% on a notional of GBP 45 million, paid semiannually: 0.024 × 45,000,000 ÷ 2 = GBP 540,000.

On your calculator
  1. 0.024×45000000÷2= 540000
  1. 0.024ENTER45000000×2÷ 540000

ENTER enter (on the HP, puts the number on the stack)

Net payment to the fixed-rate payer Memorize
P=N×(MRR−K)×1mP = N \times (\text{MRR} - K) \times \dfrac{1}{m}
Quick example

Pays fixed 5.20%, receives MRR of 4.80% on USD 80 million, quarterly. (0.048 − 0.052) × 80,000,000 ÷ 4 = −USD 80,000: it pays a net USD 80,000.

On your calculator
  1. 0.048−0.052=×80000000÷4= -80000
  1. 0.048ENTER0.052−80000000×4÷ -80000

ENTER enter (on the HP, puts the number on the stack)

Call and put options: payoff and profit at expiration

Call payoff at expiration Memorize
cT=max⁡(0, ST−X)c_T = \max(0,\, S_T - X)
Quick example

Exercise price 45, underlying at 52 at expiration. max(0, 52 − 45) = 7.00. At 41 the payoff is 0.

On your calculator
  1. 52−45= 7
  1. 52ENTER45− 7

ENTER enter (on the HP, puts the number on the stack)

Long call profit Memorize
Π=max⁡(0, ST−X)−c0\Pi = \max(0,\, S_T - X) - c_0
Quick example

X = 80, premium 3.40, S_T = 85. Payoff 5.00, profit 5.00 − 3.40 = 1.60.

On your calculator
  1. 85−80= payoff 5
  2. −3.4= 1.6
  1. 85ENTER80− payoff 5
  2. 3.4− 1.6

ENTER enter (on the HP, puts the number on the stack)

Short call profit Memorize
Π=−max⁡(0, ST−X)+c0\Pi = -\max(0,\, S_T - X) + c_0
Quick example

X = 100, premium 5.00, S_T = 108. −8.00 + 5.00 = −3.00.

On your calculator
  1. 5−108+100= -3
  1. 5ENTER108−100+ -3

ENTER enter (on the HP, puts the number on the stack)

Put payoff at expiration Memorize
pT=max⁡(0, X−ST)p_T = \max(0,\, X - S_T)
Quick example

Exercise price 120, underlying at 104. max(0, 120 − 104) = 16.00. At 131 the payoff is 0.

On your calculator
  1. 120−104= 16
  1. 120ENTER104− 16

ENTER enter (on the HP, puts the number on the stack)

Long put profit Memorize
Π=max⁡(0, X−ST)−p0\Pi = \max(0,\, X - S_T) - p_0
Quick example

X = 70, premium 2.80, S_T = 62. Payoff 8.00, profit 8.00 − 2.80 = 5.20.

On your calculator
  1. 70−62= payoff 8
  2. −2.8= 5.2
  1. 70ENTER62− payoff 8
  2. 2.8− 5.2

ENTER enter (on the HP, puts the number on the stack)

Short put profit Memorize
Π=−max⁡(0, X−ST)+p0\Pi = -\max(0,\, X - S_T) + p_0
Quick example

X = 40, premium 1.60, S_T = 35. −5.00 + 1.60 = −3.40.

On your calculator
  1. 1.6−40+35= -3.4
  1. 1.6ENTER40−35+ -3.4

ENTER enter (on the HP, puts the number on the stack)

Call breakeven price Memorize
STBE=X+c0S_T^{BE} = X + c_0
Quick example

X = 80 and premium 3.40. The call buyer breaks even at 80 + 3.40 = 83.40.

On your calculator
  1. 80+3.4= 83.4
  1. 80ENTER3.4+ 83.4

ENTER enter (on the HP, puts the number on the stack)

Put breakeven price Memorize
STBE=X−p0S_T^{BE} = X - p_0
Quick example

X = 70 and premium 2.80. The put buyer breaks even at 70 − 2.80 = 67.20.

On your calculator
  1. 70−2.8= 67.2
  1. 70ENTER2.8− 67.2

ENTER enter (on the HP, puts the number on the stack)

Credit default swaps

CDS spread payment Use it
Payment=N×s10,000×1m\text{Payment} = N \times \dfrac{s}{10{,}000} \times \dfrac{1}{m}
Quick example

EUR 12 million of protection at 85 bps a year, paid quarterly: 12,000,000 × 85 ÷ 10,000 ÷ 4 = EUR 25,500 a quarter.

On your calculator
  1. 12000000×85÷10000÷4= 25500
  1. 12000000ENTER85×10000÷4÷ 25500

ENTER enter (on the HP, puts the number on the stack)

Payment after a credit event Memorize
L=N×LGD=N×(1−RR)L = N \times \text{LGD} = N \times (1 - RR)
Quick example

USD 8 million notional, recovery 30%. 8,000,000 × (1 − 0.30) = USD 5,600,000.

On your calculator
  1. 1−0.3=×8000000= 5600000
  1. 1ENTER0.3−8000000× 5600000

ENTER enter (on the HP, puts the number on the stack)

Approximate change in CDS value Know the idea
ΔV≈Δs×N×EffDurCDS\Delta V \approx \Delta s \times N \times \text{EffDur}_{CDS}
Quick example

The spread widens by 30 bps on USD 25 million with effective duration 4.6. 0.0030 × 25,000,000 × 4.6 = USD 345,000 gain to the protection buyer.

On your calculator
  1. 30÷10000×25000000×4.6= 345000
  1. 30ENTER10000÷25000000×4.6× 345000

ENTER enter (on the HP, puts the number on the stack)

Forward commitments vs contingent claims

Long call profit when X equals the forward price Use it
Πcall=max⁡[0, ST−F0(T)]−c0\Pi_{call} = \max[0,\, S_T - F_0(T)] - c_0
Quick example

Forward price and exercise price both 250, premium 11, S_T = 262. max(0, 12) − 11 = 1.00. The long forward earns 12.00.

On your calculator
  1. 262−250−11= 1
  1. 262ENTER250−11− 1

ENTER enter (on the HP, puts the number on the stack)

Crossover: long forward equals long call Memorize
ST∗=F0(T)−c0S_T^{*} = F_0(T) - c_0
Quick example

Forward price 250 and call premium 11. Both earn the same at 250 − 11 = 239. Below it the call is better; above it the forward is.

On your calculator
  1. 250−11= 239
  1. 250ENTER11− 239

ENTER enter (on the HP, puts the number on the stack)

Crossover: long forward equals short put Use it
ST∗∗=F0(T)+p0S_T^{**} = F_0(T) + p_0
Quick example

Forward price 70 and put premium 4. Both earn the same at 70 + 4 = 74. Above it the forward is better; below it the short put is.

On your calculator
  1. 70+4= 74
  1. 70ENTER4+ 74

ENTER enter (on the HP, puts the number on the stack)

Module 3 Derivative Benefits, Risks and Uses

Why use derivatives: the four benefits

Financing cost of a position Use it
C=A×r×tC = A \times r \times t
Quick example

Financing 500,000 at 5% a year for six months costs 500,000 × 0.05 × 0.5 = 12,500. Financing only a 25,000 margin instead costs 625.

On your calculator
  1. 500000×0.05×0.5= 12500
  1. 500000ENTER0.05×0.5× 12500

ENTER enter (on the HP, puts the number on the stack)

Home-currency proceeds locked in by a forward Use it
H=N×FH = N \times F
Quick example

A firm will receive CHF 1,500,000 and sells it forward at EUR 1.04 per CHF. It locks in 1,500,000 × 1.04 = EUR 1,560,000.

On your calculator
  1. 1500000×1.04= 1560000
  1. 1500000ENTER1.04× 1560000

ENTER enter (on the HP, puts the number on the stack)

Derivative risks: leverage, basis, liquidity, counterparty and systemic

Leverage multiple Use it
L=EML = \dfrac{E}{M}
Quick example

Exposure of 90,000 held with 4,500 of margin: 90,000 ÷ 4,500 = 20×.

On your calculator
  1. 90000÷4500= 20
  1. 90000ENTER4500÷ 20

ENTER enter (on the HP, puts the number on the stack)

Return on cash committed Use it
R=EM×gR = \dfrac{E}{M} \times g
Quick example

Exposure 90,000, margin 4,500, underlying falls 1.5%. 90,000 × 0.015 ÷ 4,500 = 0.30, a loss of 30% of the margin.

On your calculator
  1. 90000×0.015÷4500= 0.3, so 30%
  1. 90000ENTER0.015×4500÷ 0.3, so 30%

ENTER enter (on the HP, puts the number on the stack)

How companies use derivatives: hedge accounting

All-in fixed rate on a swapped floating loan Use it
R=K+sR = K + s
Quick example

A loan pays the reference rate plus 0.90%. A pay-fixed swap at 4.25% receives the reference rate. All-in rate = 4.25% + 0.90% = 5.15%, whatever the reference rate does.

On your calculator
  1. 4.25+0.9= 5.15
  1. 4.25ENTER0.9+ 5.15

ENTER enter (on the HP, puts the number on the stack)

Net result of a short futures hedge on inventory Use it
N=Q [(f0−f1)−(S0−S1)]N = Q\,[(f_0 - f_1) - (S_0 - S_1)]
Quick example

1,200 ounces; futures sold at 2,034 and later at 1,990; spot falls 2,020 to 1,970. N = 1,200 × [(44) − (50)] = −7,200.

On your calculator
  1. 2034−1990−2020+1970=×1200= -7200
  1. 2034ENTER1990−2020−1970+1200× -7200

ENTER enter (on the HP, puts the number on the stack)

How investors use derivatives, compared with issuers

Covered call profit at expiration, per share Use it
P=min⁡(ST,X)−S0+c0P = \min(S_T, X) - S_0 + c_0
Quick example

Buy at 40, sell a call with X = 44 for 1.20. At S_T = 47 the share is called away at 44: 44 − 40 + 1.20 = 5.20.

On your calculator
  1. 44−40+1.2= 5.2
  1. 44ENTER40−1.2+ 5.2

ENTER enter (on the HP, puts the number on the stack)

Covered call maximum profit Use it
Pmax=X−S0+c0P_{max} = X - S_0 + c_0
Quick example

Buy at 40, sell a call with X = 44 for 1.20. The most the position can earn is 44 − 40 + 1.20 = 5.20, however high the share goes.

On your calculator
  1. 44−40+1.2= 5.2
  1. 44ENTER40−1.2+ 5.2

ENTER enter (on the HP, puts the number on the stack)

Covered call breakeven Use it
B=S0−c0B = S_0 - c_0
Quick example

Buy at 40, sell a call for 1.20. The position breaks even at 40 − 1.20 = 38.80.

On your calculator
  1. 40−1.2= 38.8
  1. 40ENTER1.2− 38.8

ENTER enter (on the HP, puts the number on the stack)

Module 4 Arbitrage, Replication and the Cost of Carry

Arbitrage and the law of one price

Spot price from a known future price Memorize
S0=F0(T) (1+r)−TS_0 = F_0(T)\,(1 + r)^{-T}
Quick example

An asset with no carry costs can be sold forward for 84.00 in nine months and r = 3%. 84.00 ÷ 1.03^0.75 = 82.16. A spot price below that is too cheap; above it, too dear.

On your calculator
  1. 1.03y^x0.75=1/x×84= 82.1583

y^x power · 1/x reciprocal

  1. 84ENTER1.03ENTER0.75y^x÷ 82.1583

ENTER enter (on the HP, puts the number on the stack) · y^x power

Arbitrage profit at expiration Use it
P=F0(T)−S0(1+r)TP = F_0(T) - S_0(1 + r)^T
Quick example

Asset at 40.00, six-month forward quoted at 41.00, r = 4%. 41.00 − 40.00 × 1.04^0.5 = 41.00 − 40.79 = 0.21 per unit (buy the asset, sell the forward). A negative P means the reverse trade earns that amount.

On your calculator
  1. 1.04y^x0.5=×40= 40.7922
  2. +|-+41= 0.2078

y^x power · +|- change sign

  1. 1.04ENTER0.5y^x40× 40.7922
  2. CHS41+ 0.2078

ENTER enter (on the HP, puts the number on the stack) · y^x power · CHS change sign

Replication and the no-arbitrage forward price

No-arbitrage forward price Memorize
F0(T)=S0 (1+r)TF_0(T) = S_0\,(1 + r)^T
Quick example

Asset at 45.00, no income or storage cost, r = 6%, delivery in nine months. 45.00 × 1.06^0.75 = 47.01.

On your calculator
  1. 1.06y^x0.75=×45= 47.0102

y^x power

  1. 1.06ENTER0.75y^x45× 47.0102

ENTER enter (on the HP, puts the number on the stack) · y^x power

No-arbitrage forward price, continuous compounding Memorize
F0(T)=S0 erTF_0(T) = S_0\,e^{rT}
Quick example

Index at 300, continuously compounded r = 2.5%, nine months. 300 × e^(0.025 × 0.75) = 305.68.

On your calculator
  1. 0.025×0.75=2NDe^x×300= 305.678

2ND second function (the shifted label above a key) · e^x e to the power x

  1. 0.025ENTER0.75×ge^x300× 305.678

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Return on an asset plus a short forward Use it
r=(F0(T)S0)1/T−1r = \left(\dfrac{F_0(T)}{S_0}\right)^{1/T} - 1
Quick example

Buy at 50.00 and sell a six-month forward at 51.00. (51 ÷ 50)^(1 ÷ 0.5) − 1 = 4.04% a year, the risk-free rate if 51.00 is the no-arbitrage price.

On your calculator
  1. 51÷50=y^x(1÷0.5)=−1= 0.0404

y^x power

  1. 51ENTER50÷0.51/xy^x1− 0.0404

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Cost of carry: income, storage and convenience yield

Spot price from the expected future price Know the idea
S0=E(ST)(1+r+λ)TS_0 = \dfrac{E(S_T)}{(1 + r + \lambda)^T}
Quick example

An asset is expected to be worth 121 in two years and investors require 10% (r plus the risk premium λ). 121 ÷ 1.10^2 = 100.00. The forward uses r only.

On your calculator
  1. 1.1y^x2=1/x×121= 100

y^x power · 1/x reciprocal

  1. 121ENTER1.1ENTER2y^x÷ 100

ENTER enter (on the HP, puts the number on the stack) · y^x power

Forward price with known income and costs Memorize
F0(T)=[S0−PV0(I)+PV0(C)] (1+r)TF_0(T) = [S_0 - PV_0(I) + PV_0(C)]\,(1 + r)^T
Quick example

Share at 120.00, present value of dividends 3.50, r = 4%, six months. (120.00 − 3.50) × 1.04^0.5 = 118.81, against 122.38 with no dividends.

On your calculator
  1. 1.04y^x0.5=×(120−3.5)= 118.8072

y^x power

  1. 120ENTER3.5−1.04ENTER0.5y^x× 118.8072

ENTER enter (on the HP, puts the number on the stack) · y^x power

Forward price with yields and costs as rates Memorize
F0(T)=S0 e(r+c−i)TF_0(T) = S_0\,e^{(r + c - i)T}
Quick example

Index at 1,200, r = 3.2%, dividend yield 2.0%, three months (continuous). 1,200 × e^((0.032 − 0.02) × 0.25) = 1,203.61.

On your calculator
  1. 0.032−0.02=×0.25=2NDe^x×1200= 1203.605

2ND second function (the shifted label above a key) · e^x e to the power x

  1. 0.032ENTER0.02−0.25×ge^x1200× 1203.605

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Currency forward prices and the interest rate differential

Currency forward rate Memorize
F0,f/d(T)=S0,f/d e(rf−rd)TF_{0,f/d}(T) = S_{0,f/d}\,e^{(r_f - r_d)T}
Quick example

NOK/GBP spot 13.20 (NOK per GBP), NOK rate 2.5%, GBP rate 4.5%, three months (continuous). 13.20 × e^((0.025 − 0.045) × 0.25) = 13.134. The higher-rate GBP trades at a forward discount.

On your calculator
  1. 0.025−0.045=×0.25=2NDe^x×13.2= 13.1342

2ND second function (the shifted label above a key) · e^x e to the power x

  1. 0.025ENTER0.045−0.25×ge^x13.2× 13.1342

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Currency forward rate, discrete compounding Use it
F0,f/d(T)=S0,f/d(1+rf1+rd)TF_{0,f/d}(T) = S_{0,f/d}\left(\dfrac{1 + r_f}{1 + r_d}\right)^T
Quick example

SEK/USD spot 10.50, SEK rate 3%, USD rate 5%, six months. 10.50 × (1.03 ÷ 1.05)^0.5 = 10.40.

On your calculator
  1. 1.03÷1.05=y^x0.5=×10.5= 10.3995

y^x power

  1. 1.03ENTER1.05÷0.5y^x10.5× 10.3995

ENTER enter (on the HP, puts the number on the stack) · y^x power

Module 5 Pricing and Valuing Forward Contracts and FRAs

Forward price vs forward value: at initiation, during the life, at expiry

Forward price, no income or costs Use it
F0(T)=S0(1+r)TF_0(T) = S_0(1 + r)^T
Quick example

A share trades at 80 and r = 3%. The nine-month forward price is 80 × 1.03^0.75 = 81.79.

On your calculator
  1. 1.03y^x0.75=×80= 81.7933

y^x power

  1. 1.03ENTER0.75y^x80× 81.7933

ENTER enter (on the HP, puts the number on the stack) · y^x power

Value at initiation Memorize
V0(T)=0V_0(T) = 0
Quick example

Entering a forward at the no-arbitrage price costs nothing and is worth zero to both sides.

Value at expiration Memorize
VT(T)=ST−F0(T)(long),F0(T)−ST(short)V_T(T) = S_T - F_0(T) \quad \text{(long)}, \qquad F_0(T) - S_T \quad \text{(short)}
Quick example

Forward agreed at 1,250 and spot ends at 1,190. Long: 1,190 − 1,250 = −60. Short: +60.

On your calculator
  1. 1190−1250= -60
  1. 1190ENTER1250− -60

ENTER enter (on the HP, puts the number on the stack)

Value during the life, long position Memorize
Vt(T)=St−F0(T)(1+r)−(T−t)V_t(T) = S_t - F_0(T)(1 + r)^{-(T-t)}
Quick example

Forward agreed at 46.00, spot now 44.50, r = 5%, 0.4 years left. PV of the forward price = 46 × 1.05^−0.4 = 45.11. Value = 44.50 − 45.11 = −0.61.

On your calculator
  1. 1.05y^x0.4+|-=×46= 45.11
  2. +|-+44.5= -0.61

y^x power · +|- change sign

  1. 1.05ENTER0.4CHSy^x46× 45.11
  2. CHS44.5+ -0.61

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Value during the life, short position Memorize
Vt(T)=F0(T)(1+r)−(T−t)−StV_t(T) = F_0(T)(1 + r)^{-(T-t)} - S_t
Quick example

Same contract from the short side: 45.11 − 44.50 = +0.61. The short gains exactly what the long loses.

On your calculator
  1. 1.05y^x0.4+|-=×46−44.5= 0.61

y^x power · +|- change sign

  1. 1.05ENTER0.4CHSy^x46×44.5− 0.61

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Valuing a forward when the asset pays income or has costs

Forward price with income and costs Use it
F0(T)=[S0−PV0(I)+PV0(C)](1+r)TF_0(T) = \left[S_0 - PV_0(I) + PV_0(C)\right](1 + r)^T
Quick example

Share at 64, PV of dividends before expiry 1.1922, r = 4%, six months. (64 − 1.1922) × 1.04^0.5 = 64.05.

On your calculator
  1. 1.04y^x0.5=×(64−1.1922)= 64.0516

y^x power

  1. 64ENTER1.1922−1.04ENTER0.5y^x× 64.0516

ENTER enter (on the HP, puts the number on the stack) · y^x power

Present value of a remaining payment Use it
PVt=D (1+r)−mPV_t = D\,(1 + r)^{-m}
Quick example

A dividend of 1.50 is due in three months and r = 6%. PV = 1.50 × 1.06^−0.25 = 1.48.

On your calculator
  1. 1.06y^x0.25+|-=×1.5= 1.48

y^x power · +|- change sign

  1. 1.06ENTER0.25CHSy^x1.5× 1.48

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Value during the life with income and costs, long Memorize
Vt(T)=[St−PVt(I)+PVt(C)]−F0(T)(1+r)−(T−t)V_t(T) = \left[S_t - PV_t(I) + PV_t(C)\right] - F_0(T)(1 + r)^{-(T-t)}
Quick example

Spot 30.00, PV of remaining dividends 0.80, forward agreed at 29.00, r = 4%, 0.5 years left. 30.00 − 0.80 − 29 × 1.04^−0.5 = 0.76.

On your calculator
  1. 1.04y^x0.5+|-=×29= 28.44
  2. +|-+30−0.8= 0.76

y^x power · +|- change sign

  1. 1.04ENTER0.5CHSy^x29× 28.44
  2. CHS30+0.8− 0.76

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Breakeven spot price Use it
St∗=PVt(I)−PVt(C)+F0(T)(1+r)−(T−t)S_t^{*} = PV_t(I) - PV_t(C) + F_0(T)(1 + r)^{-(T-t)}
Quick example

Same forward: S = 0.80 + 29 × 1.04^−0.5 = 29.24. The long breaks even when the spot price is 29.24

On your calculator
  1. 1.04y^x0.5+|-=×29+0.8= 29.24

y^x power · +|- change sign

  1. 1.04ENTER0.5CHSy^x29×0.8+ 29.24

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Currency forward value, long Use it
Vt(T)=St,f/d−F0,f/d(T) e−(rf−rd)(T−t)V_t(T) = S_{t,f/d} - F_{0,f/d}(T)\,e^{-(r_f - r_d)(T-t)}
Quick example

Forward agreed at 1.2500, spot now 1.2400, r_f = 1.5%, r_d = 2.5%, 0.25 years left. 1.2400 − 1.2500 × e^0.0025 = −0.0131 per unit of base currency.

On your calculator
  1. 0.015−0.025=×0.25=+|-2NDe^x×1.25= 1.2531
  2. +|-+1.24= -0.0131

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

  1. 0.015ENTER0.025−0.25×CHSge^x1.25× 1.2531
  2. CHS1.24+ -0.0131

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

Implied forward rates from spot rates

Discount factor Memorize
DFi=1(1+zi)iDF_i = \dfrac{1}{(1 + z_i)^i}
Quick example

The three-year spot rate is 4.5%. DF₃ = 1 ÷ 1.045³ = 0.8763. One currency unit paid in three years is worth 0.8763 today.

On your calculator
  1. 1.045y^x3=1/x 0.8763

y^x power · 1/x reciprocal

  1. 1.045ENTER3y^x1/x 0.8763

ENTER enter (on the HP, puts the number on the stack) · y^x power · 1/x reciprocal

Implied forward rate Memorize
(1+zA)A (1+IFRA,B−A)B−A=(1+zB)B(1 + z_A)^A\,(1 + IFR_{A,B-A})^{B-A} = (1 + z_B)^B
Quick example

z₂ = 2.0% and z₃ = 2.6%. The 2y1y rate satisfies 1.02² × (1 + IFR) = 1.026³, so IFR = 1.026³ ÷ 1.02² − 1 = 3.81%.

On your calculator
  1. (1.026y^x3)÷(1.02y^x2)=−1= 0.0381

y^x power

  1. 1.026ENTER3y^x1.02ENTER2y^x÷1− 0.0381

ENTER enter (on the HP, puts the number on the stack) · y^x power

Implied forward rate, solved for the forward Use it
IFRA,B−A=[(1+zB)B(1+zA)A]1/(B−A)−1IFR_{A,B-A} = \left[\dfrac{(1 + z_B)^B}{(1 + z_A)^A}\right]^{1/(B-A)} - 1
Quick example

z₁ = 2.0% and z₃ = 3.3%. The 1y2y rate is (1.033³ ÷ 1.02)^½ − 1 = 3.96% a year for the two years starting in one year.

On your calculator
  1. 1.033y^x3=÷1.02= 1.0807
  2. y^x21/x=−1= 0.0396

y^x power · 1/x reciprocal

  1. 1.033ENTER3y^x1.02÷ 1.0807
  2. 21/xy^x1− 0.0396

ENTER enter (on the HP, puts the number on the stack) · y^x power · 1/x reciprocal

Converting between compounding frequencies Use it
(1+APRmm)m=(1+APRnn)n\left(1 + \dfrac{APR_m}{m}\right)^m = \left(1 + \dfrac{APR_n}{n}\right)^n
Quick example

A 2.40% rate compounded semiannually, restated with quarterly compounding: 4 × (1.012^0.5 − 1) = 2.39%.

On your calculator
  1. 1.012y^x0.5=−1=×4= 0.0239

y^x power

  1. 1.012ENTER0.5y^x1−4× 0.0239

ENTER enter (on the HP, puts the number on the stack) · y^x power

Forward rate agreements (FRAs): settlement and uses

Fair FRA rate Memorize
FRA rate=IFRA,B−AFRA\ rate = IFR_{A,B-A}
Quick example

Three-month rate 3.10% and six-month rate 3.25% (quarterly compounding). The 3×6 FRA rate is (1.008125² ÷ 1.00775 − 1) × 4 = 3.40%.

On your calculator
  1. 1.008125y^x2=÷1.00775=−1=×4= 0.0340

y^x power

  1. 1.008125ENTER2y^x1.00775÷1−4× 0.0340

ENTER enter (on the HP, puts the number on the stack) · y^x power

FRA net payment at the end of the period Memorize
Net=(MRR−FRA rate)×N×D360Net = (MRR - FRA\ rate) \times N \times \dfrac{D}{360}
Quick example

Long FRA on USD 10 million at 5.00%, 90 days, reference rate 5.60%. (0.056 − 0.050) × 10,000,000 × 90/360 = USD 15,000 to the long.

On your calculator
  1. 0.056−0.05=×10000000×90÷360= 15000
  1. 0.056ENTER0.05−10000000×90×360÷ 15000

ENTER enter (on the HP, puts the number on the stack)

FRA cash settlement at the start of the period Memorize
Settlement=Net1+MRR×D/360Settlement = \dfrac{Net}{1 + MRR \times D/360}
Quick example

The USD 15,000 above settles at the start of the period: 15,000 ÷ (1 + 0.056 × 90/360) = USD 14,792.90.

On your calculator
  1. 15000÷(0.056×90÷360+1)= 14792.90
  1. 15000ENTER0.056ENTER90×360÷1+÷ 14792.90

ENTER enter (on the HP, puts the number on the stack)

Module 6 Pricing and Valuing Futures Contracts

Futures price at inception and how daily settlement changes value

Futures price at inception Memorize
f0(T)=[S0−PV0(I)+PV0(C)](1+r)Tf_0(T) = \left[S_0 - PV_0(I) + PV_0(C)\right](1 + r)^{T}
Quick example

A grain trades at 640 and storage costs have a present value of 8. With r = 5% and a three-month contract, f₀ = (640 + 8) × 1.05^0.25 = 655.95.

On your calculator
  1. 1.05y^x0.25=×648= 655.95

y^x power

  1. 1.05ENTER0.25y^x648× 655.95

ENTER enter (on the HP, puts the number on the stack) · y^x power

Futures price, continuous compounding Use it
f0(T)=S0 erTf_0(T) = S_0\,e^{rT}
Quick example

An index stands at 2,400 and the continuously compounded rate is 2.4%. For T = 0.75, f₀ = 2,400 × e^0.018 = 2,443.59.

On your calculator
  1. 0.024×0.75=2NDe^x×2400= 2443.59

2ND second function (the shifted label above a key) · e^x e to the power x

  1. 0.024ENTER0.75×ge^x2400× 2443.59

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Forward value during the life Use it
Vt(T)=St−F0(T)(1+r)−(T−t)V_t(T) = S_t - F_0(T)(1 + r)^{-(T-t)}
Quick example

A forward was agreed at 1,250. Three months before expiry spot is 1,275 and r = 2%. V = 1,275 − 1,250 × 1.02^−0.25 = 31.17 per unit, unpaid until expiry.

On your calculator
  1. 1.02y^x0.25+|-=×1250= 1243.83
  2. +|-+1275= 31.17

y^x power · +|- change sign

  1. 1.02ENTER0.25CHSy^x1250× 1243.83
  2. CHS1275+ 31.17

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Daily variation margin on a long futures position Use it
VMt=(ft−ft−1)×size×NVM_t = (f_t - f_{t-1}) \times \text{size} \times N
Quick example

Long 6 contracts of 40 units. The settlement price falls from 207.10 to 205.30. The long pays (205.30 − 207.10) × 40 × 6 = −432.

On your calculator
  1. 205.3−207.1=×40×6= -432
  1. 205.3ENTER207.1−40×6× -432

ENTER enter (on the HP, puts the number on the stack)

Interest rate futures: the 100 minus rate convention

Interest rate futures price Memorize
f=100−(100×MRR)f = 100 - (100 \times MRR)
Quick example

A three-month contract is quoted at 98.10. MRR = (100 − 98.10) ÷ 100 = 1.90%. A quote above 100 means a negative rate.

On your calculator
  1. 100−98.1= 1.9, so 1.90%
  1. 100ENTER98.1− 1.9, so 1.90%

ENTER enter (on the HP, puts the number on the stack)

Futures contract basis point value (BPV) Memorize
BPV=Notional×0.0001×PeriodBPV = \text{Notional} \times 0.0001 \times \text{Period}
Quick example

EUR 40 million on a three-month rate: 40,000,000 × 0.0001 × 0.25 = EUR 1,000 per basis point.

On your calculator
  1. 40000000×0.0001×0.25= 1000
  1. 40000000ENTER0.0001×0.25× 1000

ENTER enter (on the HP, puts the number on the stack)

Gain or loss on a rate futures position Use it
P&L=Δbp×BPV×NP\&L = \Delta bp \times BPV \times N
Quick example

Long 12 contracts on GBP 4 million, three-month rate. The price falls from 98.40 to 98.20, which is −20 bp. BPV = GBP 100, so P&L = −20 × 100 × 12 = −GBP 24,000.

On your calculator
  1. 98.2−98.4=×100= -20 bp
  2. ×4000000×0.0001×0.25×12= -24000
  1. 98.2ENTER98.4−100× -20 bp
  2. 4000000×0.0001×0.25×12× -24000

ENTER enter (on the HP, puts the number on the stack)

Why forward and futures prices differ

FRA cash settlement for a rate move Use it
Settlement=Δbp×BPV1+MRR×PeriodSettlement = \dfrac{\Delta bp \times BPV}{1 + MRR \times \text{Period}}
Quick example

EUR 20 million, three-month period, rate up 60 bp to 3.60%. BPV = EUR 500, so the futures pay 60 × 500 = EUR 30,000, but the FRA settles at 30,000 ÷ (1 + 0.036 × 0.25) = EUR 29,732.41.

On your calculator
  1. 20000000×60×0.0001×0.25= 30000
  2. ÷(0.036×0.25+1)= 29732.41
  1. 20000000ENTER60×0.0001×0.25× 30000
  2. 0.036ENTER0.25×1+÷ 29732.41

ENTER enter (on the HP, puts the number on the stack)

Module 7 Pricing and Valuing Interest Rate Swaps

A swap as a series of forwards: what is the same and what is different

Value of one swap period to the fixed-rate payer at inception Use it
Vi=(fi−sN)×N×DFiV_i = (f_i - s_N) \times N \times DF_i
Quick example

One-year spot 1.8% and two-year spot 2.6% give a 2-year swap rate of 2.59% and a second-year forward of 3.41%. Per 100 of notional, year 2 is worth (0.034063 − 0.025897) × 100 × 0.94996 = +0.776 to the fixed-rate payer, and year 1 is worth −0.776.

On your calculator
  1. 0.034063−0.025897=×100×0.94996= 0.7757
  1. 0.034063ENTER0.025897−100×0.94996× 0.7757

ENTER enter (on the HP, puts the number on the stack)

A swap is worth zero in total at inception Use it
∑i=1N(fi−sN)×N×DFi=0\sum_{i=1}^{N} (f_i - s_N) \times N \times DF_i = 0
Quick example

Per 100 of notional the three period values −1.056, +0.307 and +0.749 add to 0. The early loss is exactly paid for by the later gains.

Bond equivalence of a swap Memorize
Pay fixed=long FRN+short fixed-rate bond\text{Pay fixed} = \text{long FRN} + \text{short fixed-rate bond}
Quick example

A company that pays fixed and receives floating is economically long a floating-rate note and short a fixed-rate bond. It gains when rates rise. Receive fixed is the mirror image and gains when rates fall.

The swap price: solving for the par swap rate

Discount factor Use it
DFi=1(1+zi)iDF_i = \dfrac{1}{(1 + z_i)^i}
Quick example

The two-year spot rate is 2.8%. DF₂ = 1 ÷ 1.028² = 0.94627.

On your calculator
  1. 1.028y^x2=1/x 0.94627

y^x power · 1/x reciprocal

  1. 1.028ENTER2y^x1/x 0.94627

ENTER enter (on the HP, puts the number on the stack) · y^x power · 1/x reciprocal

Fixed leg equals floating leg Use it
sN∑i=1NDFi=∑i=1NIFRi DFi=1−DFNs_N \sum_{i=1}^{N} DF_i = \sum_{i=1}^{N} IFR_i \, DF_i = 1 - DF_N
Quick example

Forwards 2.20%, 3.60%, 4.10% with DFs 0.97847, 0.94443, 0.90719: the floating leg is worth 0.0928 per unit of notional, and 1 − 0.90719 = 0.0928.

On your calculator
  1. (0.022×0.97847)+(0.036048×0.94443)+(0.041047×0.90719)= 0.0928
  1. 0.022ENTER0.97847×0.036048ENTER0.94443×+0.041047ENTER0.90719×+ 0.0928

ENTER enter (on the HP, puts the number on the stack)

Par swap rate (swap price) Memorize
sN=1−DFN∑i=1NDFis_N = \dfrac{1 - DF_N}{\sum_{i=1}^{N} DF_i}
Quick example

Discount factors for years 1 to 3 are 0.96, 0.92 and 0.87. s₃ = (1 − 0.87) ÷ (0.96 + 0.92 + 0.87) = 0.13 ÷ 2.75 = 4.73%.

On your calculator
  1. 1−0.87=÷(0.96+0.92+0.87)= 0.0473
  1. 1ENTER0.87−0.96ENTER0.92+0.87+÷ 0.0473

ENTER enter (on the HP, puts the number on the stack)

Annualizing a periodic swap rate Use it
sannual=speriodic×ms_{annual} = s_{periodic} \times m
Quick example

A quarterly swap has a periodic rate of 0.45%. The annual swap rate is 0.45% × 4 = 1.80%.

On your calculator
  1. 0.0045×4= 0.018
  1. 0.0045ENTER4× 0.018

ENTER enter (on the HP, puts the number on the stack)

Swap value after inception: time and rate changes

Periodic settlement value (fixed-rate payer) Memorize
P=(MRR−sN)×N×tP = (MRR - s_N) \times N \times t
Quick example

A fund pays fixed 2.60% and receives the MRR on USD 40 million, settled semiannually. The MRR sets at 2.10%: (0.021 − 0.026) × 40,000,000 × 0.5 = −USD 100,000. The fund pays a net USD 100,000.

On your calculator
  1. 0.021−0.026=×40000000×0.5= -100000
  1. 0.021ENTER0.026−40000000×0.5× -100000

ENTER enter (on the HP, puts the number on the stack)

Swap value to the fixed-rate payer Memorize
V=∑PV(floating received)−∑PV(fixed paid)V = \sum PV(\text{floating received}) - \sum PV(\text{fixed paid})
Quick example

The remaining floating payments are worth USD 7.20 million and the remaining fixed payments USD 6.85 million. The fixed-rate payer's swap is worth 7.20 − 6.85 = +USD 0.35 million.

Swap value after the swap rate changes Use it
Vpayer=(snew−sold)×N×t×∑DFV_{payer} = (s_{new} - s_{old}) \times N \times t \times \sum DF
Quick example

A fund receives fixed 2.20% on GBP 30 million (annual). A new swap for the remaining term pays 1.70% and the remaining discount factors sum to 2.40. The receiver's value is (0.022 − 0.017) × 30,000,000 × 2.40 = GBP 360,000 (the payer's value is its negative).

On your calculator
  1. 0.022−0.017=×30000000×2.4= 360000
  1. 0.022ENTER0.017−30000000×2.4× 360000

ENTER enter (on the HP, puts the number on the stack)

Module 8 Pricing and Valuing Options

Exercise value, moneyness and time value

Exercise value of a call Memorize
EVc=max⁡(0, St−X(1+r)−(T−t))EV_c = \max\left(0,\ S_t - X(1 + r)^{-(T-t)}\right)
Quick example

Call, X = 50, S_t = 54, r = 3%, 0.5 years left. PV(X) = 50 × 1.03^−0.5 = 49.27. EV = 54 − 49.27 = 4.73.

On your calculator
  1. 1.03y^x0.5+|-=×50= 49.27
  2. +|-+54= 4.73

y^x power · +|- change sign

  1. 1.03ENTER0.5CHSy^x50× 49.27
  2. CHS54+ 4.73

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Exercise value of a put Memorize
EVp=max⁡(0, X(1+r)−(T−t)−St)EV_p = \max\left(0,\ X(1 + r)^{-(T-t)} - S_t\right)
Quick example

Put, X = 65, S_t = 60, r = 2%, 0.5 years left. PV(X) = 64.36. EV = 64.36 − 60 = 4.36.

On your calculator
  1. 1.02y^x0.5+|-=×65= 64.36
  2. −60= 4.36

y^x power · +|- change sign

  1. 1.02ENTER0.5CHSy^x65× 64.36
  2. 60− 4.36

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Exercise value at expiration Memorize
EVc=max⁡(0, ST−X),EVp=max⁡(0, X−ST)EV_c = \max(0,\ S_T - X), \qquad EV_p = \max(0,\ X - S_T)
Quick example

At expiration X = 80 and S_T = 91. Call: 11. Put: 0.

Option price = exercise value + time value Memorize
ct=EVc+TV,pt=EVp+TV,TV=price−EV≥0c_t = EV_c + TV, \qquad p_t = EV_p + TV, \qquad TV = \text{price} - EV \ge 0
Quick example

A call trades at 6.35 and its exercise value is 4.90. Time value = 6.35 − 4.90 = 1.45.

On your calculator
  1. 6.35−4.9= 1.45
  1. 6.35ENTER4.9− 1.45

ENTER enter (on the HP, puts the number on the stack)

Option price bounds and replication vs forwards

Bounds on a European call Memorize
max⁡(0, St−X(1+r)−(T−t))≤ct≤St\max\left(0,\ S_t - X(1 + r)^{-(T-t)}\right) \le c_t \le S_t
Quick example

X = 60, S_t = 64, r = 2%, one year left. PV(X) = 60 ÷ 1.02 = 58.82. Lower bound = 64 − 58.82 = 5.18; upper bound 64.

On your calculator
  1. 60÷1.02= 58.82
  2. +|-+64= 5.18

+|- change sign

  1. 60ENTER1.02÷ 58.82
  2. CHS64+ 5.18

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Bounds on a European put Memorize
max⁡(0, X(1+r)−(T−t)−St)≤pt≤X\max\left(0,\ X(1 + r)^{-(T-t)} - S_t\right) \le p_t \le X
Quick example

X = 60, S_t = 55, r = 2%, one year left. Lower bound = 58.82 − 55 = 3.82; upper bound 60.

On your calculator
  1. 60÷1.02=−55= 3.82
  1. 60ENTER1.02÷55− 3.82

ENTER enter (on the HP, puts the number on the stack)

Module 9 Put–Call Parity and Option Replication

Put–call parity: fiduciary call = protective put

Put–call parity Memorize
S0+p0=c0+X(1+r)−TS_0 + p_0 = c_0 + X(1 + r)^{-T}
Quick example

Share 25, European put 0.90, X = 22, r = 3%, T = 0.5. The call must be 25 + 0.90 − 22 × 1.03^−0.5 = 25.90 − 21.68 = 4.22.

On your calculator
  1. 1.03y^x0.5+|-=×22= 21.68
  2. +|-+25+0.9= 4.22

y^x power · +|- change sign

  1. 1.03ENTER0.5CHSy^x22× 21.68
  2. CHS25+0.9+ 4.22

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Put from parity Memorize
p0=c0+X(1+r)−T−S0p_0 = c_0 + X(1 + r)^{-T} - S_0
Quick example

Call 7.40, X = 85, S = 82, r = 3%, T = 1. PV(X) = 82.52. Put = 7.40 + 82.52 − 82 = 7.92.

On your calculator
  1. 85÷1.03= 82.52
  2. +7.4−82= 7.92
  1. 85ENTER1.03÷ 82.52
  2. 7.4+82− 7.92

ENTER enter (on the HP, puts the number on the stack)

Call from parity Memorize
c0=S0+p0−X(1+r)−Tc_0 = S_0 + p_0 - X(1 + r)^{-T}
Quick example

Share 64, put 2.85, X = 62, r = 6%, T = 1. PV(X) = 58.49. Call = 64 + 2.85 − 58.49 = 8.36.

On your calculator
  1. 62÷1.06= 58.49
  2. +|-+64+2.85= 8.36

+|- change sign

  1. 62ENTER1.06÷ 58.49
  2. CHS64+2.85+ 8.36

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Building synthetic positions with put–call parity

Synthetic share Use it
S0=c0−p0+X(1+r)−TS_0 = c_0 - p_0 + X(1 + r)^{-T}
Quick example

Call 6.80, put 4.10, X = 70, r = 2%, T = 0.5. PV(X) = 69.31. Synthetic share = 6.80 − 4.10 + 69.31 = 72.01.

On your calculator
  1. 1.02y^x0.5+|-=×70= 69.31
  2. +6.8−4.1= 72.01

y^x power · +|- change sign

  1. 1.02ENTER0.5CHSy^x70× 69.31
  2. 6.8+4.1− 72.01

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Synthetic bond Use it
X(1+r)−T=S0+p0−c0X(1 + r)^{-T} = S_0 + p_0 - c_0
Quick example

Share 36, put 2.90, call 3.40. The bond paying X is worth 36 + 2.90 − 3.40 = 35.50 today.

On your calculator
  1. 36+2.9−3.4= 35.5
  1. 36ENTER2.9+3.4− 35.5

ENTER enter (on the HP, puts the number on the stack)

Covered call equals bond minus put Memorize
S0−c0=X(1+r)−T−p0S_0 - c_0 = X(1 + r)^{-T} - p_0
Quick example

Share 70, X = 75, one year, r = 4%, put 6.50. Cost of the covered call = 75 ÷ 1.04 − 6.50 = 72.12 − 6.50 = 65.62.

On your calculator
  1. 75÷1.04= 72.12
  2. −6.5= 65.62
  1. 75ENTER1.04÷ 72.12
  2. 6.5− 65.62

ENTER enter (on the HP, puts the number on the stack)

Put–call forward parity

Share as forward plus bond Use it
S0=F0(T)(1+r)−TS_0 = F_0(T)(1 + r)^{-T}
Quick example

Forward price 2,040, r = 2%, T = 0.5, no income. A long forward plus a bond paying 2,040 costs 2,040 × 1.02^−0.5 = 2,019.90, which equals the spot price.

On your calculator
  1. 1.02y^x0.5+|-=×2040= 2019.90

y^x power · +|- change sign

  1. 1.02ENTER0.5CHSy^x2040× 2019.90

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Put–call forward parity Memorize
F0(T)(1+r)−T+p0=c0+X(1+r)−TF_0(T)(1 + r)^{-T} + p_0 = c_0 + X(1 + r)^{-T}
Quick example

F = 72, X = 68, r = 6%, T = 1, call 8.50. p = 8.50 + (68 − 72) ÷ 1.06 = 8.50 − 3.77 = 4.73.

On your calculator
  1. 68−72=÷1.06= −3.77
  2. +8.5= 4.73
  1. 68ENTER72−1.06÷ −3.77
  2. 8.5+ 4.73

ENTER enter (on the HP, puts the number on the stack)

Put minus call Memorize
p0−c0=[X−F0(T)](1+r)−Tp_0 - c_0 = \left[X - F_0(T)\right](1 + r)^{-T}
Quick example

F = 84, X = 80, r = 3%, T = 1, put 5.20. p − c = (80 − 84) ÷ 1.03 = −3.88, so the call is 5.20 + 3.88 = 9.08.

On your calculator
  1. 80−84=÷1.03= −3.88
  2. +|-+5.2= 9.08

+|- change sign

  1. 80ENTER84−1.03÷ −3.88
  2. CHS5.2+ 9.08

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Equity and debt as options on the firm

Payoffs at debt maturity Memorize
Equity=max⁡(0, VT−D),Debt=min⁡(VT, D)\text{Equity} = \max(0,\ V_T - D), \qquad \text{Debt} = \min(V_T,\ D)
Quick example

Debt face value D = 400. If V_T = 520, equity gets 120 and lenders 400. If V_T = 350, equity gets 0 and lenders 350.

Firm value from parity Memorize
V0=c0+PV(D)−p0V_0 = c_0 + PV(D) - p_0
Quick example

Equity (call) 38.37, debt D = 150 due in one year, r = 3%, put on the assets 4.00. PV(D) = 145.63. V = 38.37 + 145.63 − 4.00 = 180.00.

On your calculator
  1. 150÷1.03= 145.63
  2. +38.37−4= 180
  1. 150ENTER1.03÷ 145.63
  2. 38.37+4− 180

ENTER enter (on the HP, puts the number on the stack)

Value of risky debt Memorize
B0=PV(D)−p0B_0 = PV(D) - p_0
Quick example

D = 150, r = 3%, one year, put 4.00. Risky debt = 145.63 − 4.00 = 141.63; the put is the discount for credit risk.

On your calculator
  1. 150÷1.03=−4= 141.63
  1. 150ENTER1.03÷4− 141.63

ENTER enter (on the HP, puts the number on the stack)

Module 10 The One-Period Binomial Model

The one-period binomial tree and the hedge ratio

Share prices at the end of the period Use it
S1u=RuS0S1d=RdS0S_1^u = R_u S_0 \qquad S_1^d = R_d S_0
Quick example

Share 45, up gross return 1.20, down gross return 0.90. It ends at 45 × 1.20 = 54.00 or 45 × 0.90 = 40.50.

On your calculator
  1. 45×1.2= 54
  2. 45×0.9= 40.5
  1. 45ENTER1.2× 54
  2. 45ENTER0.9× 40.5

ENTER enter (on the HP, puts the number on the stack)

Hedge ratio Memorize
h∗=c1u−c1dS1u−S1dh^* = \dfrac{c_1^u - c_1^d}{S_1^u - S_1^d}
Quick example

Share 50 ends at 62 or 46. A call with X = 50 pays 12 or 0. h = (12 − 0) ÷ (62 − 46) = 0.75 share per call sold

On your calculator
  1. 12÷(62−46)= 0.75
  1. 12ENTER62ENTER46−÷ 0.75

ENTER enter (on the HP, puts the number on the stack)

Hedged portfolio value at expiry Use it
V1=h∗S1u−c1u=h∗S1d−c1dV_1 = h^* S_1^u - c_1^u = h^* S_1^d - c_1^d
Quick example

Same share and call, h = 0.75. Up: 0.75 × 62 − 12 = 34.50. Down: 0.75 × 46 − 0 = 34.50. The same in both states

On your calculator
  1. 12÷(62−46)=×62−12= 34.5
  1. 12ENTER62ENTER46−÷62×12− 34.5

ENTER enter (on the HP, puts the number on the stack)

Call price by replication Memorize
c0=h∗S0−V11+rc_0 = h^* S_0 - \dfrac{V_1}{1 + r}
Quick example

S₀ = 50, h = 0.75, V₁ = 34.50, r = 2%. c₀ = 0.75 × 50 − 34.50 ÷ 1.02 = 37.50 − 33.82 = 3.68*.

On your calculator
  1. 12÷(62−46)=×50−(34.5÷1.02)= 3.68
  1. 12ENTER62ENTER46−÷50×34.5ENTER1.02÷− 3.68

ENTER enter (on the HP, puts the number on the stack)

Pricing calls and puts with risk-neutral probabilities

Risk-neutral probability of an up move Memorize
π=1+r−RdRu−Rd1−π=Ru−1−rRu−Rd\pi = \dfrac{1 + r - R_d}{R_u - R_d} \qquad 1 - \pi = \dfrac{R_u - 1 - r}{R_u - R_d}
Quick example

R_u = 1.35, R_d = 0.95, r = 2%. π = (1.02 − 0.95) ÷ (1.35 − 0.95) = 0.07 ÷ 0.40 = 0.175, so 1 − π = 0.825.

On your calculator
  1. 1.02−0.95=÷(1.35−0.95)= 0.175
  1. 1.02ENTER0.95−1.35ENTER0.95−÷ 0.175

ENTER enter (on the HP, puts the number on the stack)

Call value, risk-neutral Memorize
c0=π c1u+(1−π) c1d1+rc_0 = \dfrac{\pi\, c_1^u + (1 - \pi)\, c_1^d}{1 + r}
Quick example

π = 0.175, call pays 30 or 0, r = 2%. c₀ = (0.175 × 30 + 0.825 × 0) ÷ 1.02 = 5.25 ÷ 1.02 = 5.15.

On your calculator
  1. 0.175×30÷1.02= 5.15
  1. 0.175ENTER30×1.02÷ 5.15

ENTER enter (on the HP, puts the number on the stack)

Put value, risk-neutral Memorize
p0=π p1u+(1−π) p1d1+rp_0 = \dfrac{\pi\, p_1^u + (1 - \pi)\, p_1^d}{1 + r}
Quick example

π = 0.175, put pays 0 or 12, r = 2%. p₀ = (0.175 × 0 + 0.825 × 12) ÷ 1.02 = 9.90 ÷ 1.02 = 9.71.

On your calculator
  1. 0.825×12÷1.02= 9.71
  1. 0.825ENTER12×1.02÷ 9.71

ENTER enter (on the HP, puts the number on the stack)

Risk neutrality: why real-world probabilities do not matter

Share earns the risk-free rate under π Use it
πS1u+(1−π)S1d=S0(1+r)\pi S_1^u + (1 - \pi) S_1^d = S_0 (1 + r)
Quick example

Share 50 goes to 60 or 45, r = 3%, π = 0.4333. 0.4333 × 60 + 0.5667 × 45 = 26.00 + 25.50 = 51.50 = 50 × 1.03. Risk-free growth.

On your calculator
  1. 0.433333×60+(0.566667×45)= 51.5
  2. 50×1.03= 51.5
  1. 0.433333ENTER60×0.566667ENTER45×+ 51.5
  2. 50ENTER1.03× 51.5

ENTER enter (on the HP, puts the number on the stack)

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