Course O · Options and Derivatives

Formula sheet

O1 Forwards and futures: the price of waiting

The forward price: today's price, carried

F = S × (1 + r)T
F = S × erT   (interest added continuously)
F
the forward price, agreed today, for delivery at time T
S
the underlying's price today
r
the interest rate per year, as a decimal (5% is 0.05), the same for borrowing and lending
T
the years until delivery (half a year is 0.5)
(1 + r)T
(1 + r) multiplied by itself T times (T can be a fraction)
e
a fixed number, about 2.718

O1 Forwards and futures: the price of waiting

Income and storage shift the carry

F = (S − I + U) × (1 + r)T
F
the forward price, agreed today, for delivery at time T
S
the underlying's price today
I
the present value today of income paid to the holder before delivery, such as dividends (0 if none): I = each payment ÷ (1 + r)(years until paid), added up
U
the present value today of storage costs paid up to delivery (0 if none), worked out the same way
r
the interest rate per year, as a decimal, the same for borrowing and lending
T
the years until delivery

O2 What no-arbitrage forbids: option bounds and put–call parity

The bounds on an option's price

max(S − K ÷ (1 + r)T, 0) ≤ C ≤ S
max(K ÷ (1 + r)T − S, 0) ≤ P ≤ K ÷ (1 + r)T
time value = price − intrinsic value
C, P
the European call's and put's prices today, same strike and expiry
S
the underlying's price today
K
the strike
r
the interest rate per year, as a decimal (5% is 0.05), the same for borrowing and lending
T
the years until expiry
K ÷ (1 + r)T
the strike's present value: what lent today grows to K at expiry

O2 What no-arbitrage forbids: option bounds and put–call parity

Put–call parity

C + K ÷ (1 + r)T = P + S
C − P = S − K ÷ (1 + r)T
C, P
the European call's and put's prices today, same strike and expiry
S
the underlying's price today
K
the strike
r
the interest rate per year, as a decimal, the same for borrowing and lending
T
the years until expiry

O3 Pricing without a crystal ball: the binomial tree

Pricing by copying: the one-step binomial model

Δ = (Cu − Cd) ÷ (S·u − S·d)
B = (Δ·S·d − Cd) ÷ (1 + r)  ·  C = Δ·S − B
q = ((1 + r) − d) ÷ (u − d)
C = [q·Cu + (1 − q)·Cd] ÷ (1 + r)
S
the share's price at the start of the step
u, d
the up and down factors: the share ends the step at S·u or S·d
r
the interest rate for one step, for borrowing and lending
Cu, Cd
the contract's payoff (in a longer tree, its value) at the end of the step if up / if down
Δ
delta: the shares in the copy
B
what the copy borrows at the start (negative: lends)
C
the contract's no-arbitrage price at the start of the step
q
the risk-neutral probability: a weight from u, d and r, not the real chance of up

O4 Black–Scholes, from the tree

The Black–Scholes formula

C = S·N(d1) − PV(K)·N(d2)
d1 = [ln(S ÷ PV(K)) + ½σ²T] ÷ (σ√T)  ·  d2 = d1 − σ√T
P = C − S + PV(K)  ·  PV(K) = K ÷ (1 + r)T
S
the share's price today
K
the strike
T
the years left until the last day
r
the interest rate a year, for borrowing and lending
σ
the volatility: the standard deviation of the share's yearly log return, ln(end price ÷ start price)
N(x)
the share of the bell curve's area left of x standard deviations from its middle
N(d2)
the q-chance that the call ends above its strike
N(d1)
the call's delta (the binomial tree lesson's word): the shares held today in the copy, the shares and loan that reproduce the call's payoff
C, P
the European call's and put's prices today, before costs

O5 The Greeks

The Greeks of a European call

Δ = N(d1)  ·  Γ = φ(d1) ÷ (Sσ√T)  ·  vega = Sφ(d1)√T
Θ = −Sφ(d1)σ ÷ (2√T) − rc·PV(K)·N(d2)
change in price ≈ Δ·m + ½Γ·m² + Θ·t + vega·s
Δ, Γ, Θ
delta, gamma and theta of a European call (one that can be used only on its last day): delta per euro of share price, gamma per euro, theta per year (÷ 365 for a day)
vega
per 1.00 of volatility (÷ 100 for a percentage point)
φ(x)
the bell curve's height at x: e−x²/2 ÷ √(2π)
rc
the yearly rate r written as a continuous rate, ln(1 + r): 0.0488 for 5%
m, t, s
small changes: the share's move in euros, the years that pass (t in years), the change in volatility (s as a decimal: 0.02 for two points)
S, K, T, σ
the share's price today, the strike, the years left, the volatility
d1, d2, N, PV(K)
as in the Black–Scholes formula: d1 = [ln(S ÷ PV(K)) + ½σ²T] ÷ (σ√T), d2 = d1 − σ√T, PV(K) = K ÷ (1 + r)T

O6 Implied volatility and the smile

Implied volatility, found by bisection

CBS(S, K, T, r, σimp) = C
σmid = (σlo + σhi) ÷ 2; if CBS(σmid) > C then σhi = σmid, else σlo = σmid
width after n halvings = (σhi − σlo) ÷ 2n
C
the option's price: a quote, or here a simulated average
CBS(…, σ)
the Black–Scholes price with volatility σ (for a put, CBS − S + PV(K))
σimp
the implied volatility
σlo, σhi
the ends of the range that holds σimp; σmid its middle
n
the number of halvings

O7 Testing an options idea honestly

Win rate is not the average

E = p × W − (1 − p) × L
E
the expectancy: the average result per trade
p
the win rate, as a decimal
W
the average win
L
the average loss, as a positive number

O7 Testing an options idea honestly

A short sample can miss the rare event

chance of no event in n years = (1 − q)n
q
the chance that a year has the event, as a decimal
n
the number of years in the sample

O8 Capstone: hedge an option and explain the P&L

Gamma P&L: where a hedged option's P&L comes from

P&L ≈ Σdays ½ Γt St² (σimp² − σreal²) Δt
average P&L = (1 + r)T × [C(σimp) − C(σreal)]
Σdays
the sum over the year's days
Γt, St
gamma (worked out at σimp) and the share's price on day t
σimp
the volatility the price was charged at
σreal
the volatility the share moves with
Δt
one day in years, 1 ÷ 365
interest
the simulation also grows each day's part with interest to the last day, at most × 1.05; the first line leaves this out
C(σ)
the Black–Scholes price of the call at volatility σ
r, T
the interest rate a year and the years to the last day
P&L
the seller's cash on the last day, hedged with delta at σimp, before costs