CFA Level I
Derivatives
Formula Reference Sheet
Forward Contracts — Pricing & Valuation
Spot Price (No-Arbitrage)
S0 = F0(T)(1 + rf)T
Contract Value at Time t
Vt = St − PVt[F0(T)]
Value to the long position
Spot Price with Cost of Carry
S0 =
F0(T)
(1+rf)T
forward PV
+
PV(Benefits) − PV(Costs)
“Cost of Carry”
Benefits − Costs = Cost of Carry
e.g. dividends/coupons are benefits; storage costs, insurance are costs
e.g. dividends/coupons are benefits; storage costs, insurance are costs
Contract Value at Time t — With Costs & Benefits
Vt = St −
(
PVt[F0(T)] + PVt(Benefits) − PVt(Costs)
)
Put-Call Parity
Put-Call Parity (Spot Underlying)
SUnderlying Asset
+
PPut
=
CCall
+
X
(1+r)T
Risk-free Bond (PV of strike)
Forward Put-Call Parity
Fwd Price
(1+r)T
Forward Contract (PV)
+
PPut
=
CCall
+
X
(1+r)T
Risk-free Bond
Option Values
Intrinsic Value & Premium
| Concept | Call | Put |
|---|---|---|
| Intrinsic Value | max(0, Stock Price − Strike Price) | max(0, Strike Price − Stock Price) |
| Option Premium | Intrinsic Value + Time Value | |
One-Period Binomial Model
Stock Price Nodes
Up node (upper stock price)
Su1 = Ru · S0
Down node (lower stock price)
Sd1 = Rd · S0
Call prices at expiry
cu = max(0, Su − X)
cd = max(0, Sd − X)
cd = max(0, Sd − X)
Hedge Ratio & Portfolio Value
Hedge ratio (# shares needed)
h* =
(cu − cd)
(Su − Sd)
Value of hedged portfolio
V1 = h* · Su − cu
Risk Neutrality Pricing
Call Option Price Today
c0 =
π · cu + (1−π) · cd
(1+r)T
Risk-Neutral Probability (π)
π =
1 + r − Rd
Ru − Rd
π = risk-neutral probability of up move ·
Ru = up return factor ·
Rd = down return factor ·
r = risk-free rate