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Reading 49: Basics of Derivative Pricing and Valuation - Coggle Diagram
Reading 49: Basics of Derivative Pricing and Valuation
Asset Values
For equity
General form of asset valuation:
\( S_0=\frac{E(S_T)}{1+Rf+RP}\)
\(E(S_T)\): Expected asset price at time T
Rf: Opportunity cost
RP: Risk premium for uncertainty about \(S_T\)
Risk premium depends on investor risk aversion
Derivatives Values
Derivatives pricing is based on a no-arbitrage condition - law of one price
\(\frac{F_0(T)}{S_0}=(1+Rf)^T\)
\(F_0\): Forward price
\(S_0\): Asset value
This mean: Buy asset now, sell forward at \(F_0(T)\), you will earn Rf until time = T
Positions
Derivative
Long Derivative Position = Long Asset + Short Risk-free asset
Short Derivative Position = Short Asset + Long Risk-free asset
Risk-free bond
Long risk-free asset position = Long Asset + Short Derivative
Short Risk-free asset position = Short Asset + Long derivative
Asset
Long asset position = Long Derivative + Long Risk-free asset
Short asset position = Short Derivative + Short Risk-free asset
Forwards and Futures:
Value vs. Price
Price
No-arbitrage forward price:
\(F_0(T)={S_0}\times (1+Rf)^T\)
= \(F_0(T) \),a.k.a forward price in contract, a.k.a price paid at expiration
Forward Contract Value
Value = Value of a long position in the forward
Value = 0 at initiation (time 0)
Value may fluctuate, but price remain the same (exercise price at time T)
Long position
at time t (during contract life):
\(V_t(T)=S_t - \frac{F_0(T)}{(1+Rf)^(T-t)}\)
at expiration (settlement)
\(V_t(T)=S_T- F_0(T) \)
Arbitrage Strategy
If forward price is "too high" ( \(F>F_0(T)\) ): → Short Forward, borrow at Rf, buy the asset
If forward price is "too low" ( \(F < F_0(T)\) ) → Short asset, invest at Rf, long forward
Benefits and Costs
No-arbitrage forward price
with cost and benefits
\(F_0(T)=[S_0 - PV_0(ben)+PV_0(cost)] \times (1+Rf)^{T-t}\)
\(S_t\): Asset price at time t
Rf: Opportunity cost of funds
\(PV_t(cost)\): Storage, insurance to T (monetary costs)
\(PV_t(ben)\): Cash flows (monetary benefits) + Convenience yield (nonmonetary benefits)
the net cost of carry = \(Benefit - Cost = [PV_0(ben) - PV_0(cost)] \)
Higher benefit decreases forward price
Higher cost increase forward price
Effect on Forward Value
Value of forward at time t with costs and benefits
\(V_t(T)=S_t - PV_t(ben)+PV_t(cost)- \frac{F_0(T)}{(1+Rf)^(T-t)}\)
Forward Prices vs. Futures Prices
Unlike forwards, futures contracts are marked to market daily, so futures have interim cash flows
Value of futures = gain/loss since previous day; resets to zero daily at settlement
Essentially no difference for valuation
Which one is more desirable?
When Interest Rate & Futures Prices are Negatively Correlated
Forwards are more desirable
When Interest Rate & Future Prices are Positively Correlated
Futures are more desirable
When Interest Rate & Future Prices are uncorrelated
Futures Price = Forwards Price
Forward Rate Agreement (FRA)
Definition
Fixed rate = forward (contract) rate
Floating rate (e.g., LIBOR) is underlying rate
Long pays fixed rate, receives LIBOR
Receive (LIBOR - fixed rate) or
Pays (fixed rate - LIBOR)
Exchange fixed-rate payment for a floating-rate payment at a future date
Notional amount
Pricing
Swap price: PV of all net cash flows
Interest Rate Swaps
To replicate a swap with zero value at initiation, present values of off-market FRAs must sum to 0
Swap price is the fixed rate
Swap value is positive if expected short-term rates increase, negative if expected short-term rates decrease
Value
On each payment date, swap owner receives a payment based on the value of underlying at the time of each respective payment
European Options
European call option:
give you the right to buy something at price (X) on expiration date, while the market price is \( S_T\)
Exercise Value
= MAX{0, \( (S_T - X) \)}
Call is in the money if \( (S_T>X) \)
Call is out of money if \( (S_T < X) \)
European put option:
give you the right to sell something at price (X) on expiration date, while the market price is \( S_T\)
Exercise Value
= MAX{0, \( (X-S_T) \)}
Put is
out of money
if \( (S_T > X) \)
Put is
in the money
if \( (S_T < X) \)
Intrinsic or exercise value is amount in the money
Option price = Intrinsic Value + Time Value
Time value is the portion of an option's premium that is attributable to the amount of time remaining until the expiration
Factors affecting option values
Asset price
(higher asset is better for call but worse for put value)
Exercise price
(higher price is better for put but worse for call values)
Volatility of asset price
(more variable is better)
Time to expiration
(longer is better, since can be more volatile)
Risk-free rate
higher Rf increases call values but decreases put values
Cash flows from underlying asset
decrease call values but increase put values
costs from carrying the asset
increase call values but decrease put values
Deriving Put-Call parity
Protective Put
if \(S\leq X\), payoff = S + (X-S) = X
if \(S\geq X\), payoff = S + 0 = S
Protective put = long stock at price S + long a put option with exercise price (X)
Fiduciary Call
Fiduciary call: long a call option at exercise price (X) and long a pure discount bond that will pay the exercise price at expiration. Bond's PV = \( \frac{X}{(1+Rf)^T}\)
if if \(S\leq X\), payoff = 0 + X = X
if \(S\geq X\), payoff = (S-X)+X = S
Same payoffs means same values by no-arbitrage
Put-call parity:
\( S+P=C+\frac{X}{(1+Rf)^T}\)
S: Asset initial price
P: Put price (Put premium)
C: Call price (Call premium)
X: Exercise price
Rf: Risk-free rate
By rearrange the put-call parity we can create:
Synthetic Long Put: \( -P=-C+S-\frac{X}{(1+Rf)^T}\)
Long Call, Short Asset, Long Bond
Synthetic Long Call: \( -C= -P-S+\frac{X}{(1+Rf)^T}\)
Long Put, Long Asset, Short Bond
Synthetic Long Bond: \( -\frac{X}{(1+Rf)^T} = -S-P+C\)
Long Asset, Long Put, Short Call
Synthetic Long Asset: \( -S=-C-\frac{X}{(1+Rf)^T}+P\)
Long Call, Short Put, Long Bond
Put-Call-Forward Parity
We can replicate the underlying asset with a forward contract and a risk-free bond that pays the forward price at expiration
\(S_0 = \frac{F_0(T)}{(1+R_f)^T}\)
Same relationship hold
Put-call-forward parity:
\( \frac{F_0(T)}{(1+R_f)^T}+P=C+\frac{X}{(1+Rf)^T}\)
Binomial Model for Option Pricing
Example:
U = up-move factor = 1.15
D= down-move factor = 0.87
\(\pi_U\): risk-neutral probability of UP move = \(\frac{1+R_f-D}{U-D}=0.715\)
\(\pi_D\): risk-neutral probability of DOWN move = \( 1-\pi_U=0.285 \)
Rf= 7%
\(S_0=$30\)
Long a Call option at X = $30
We can do binomial Tree:
\(S_0=$30\)
28.5% chance that: after 1-year DOWN move: $30 x 0.87 = $26.1
Option will pay $0 (option out-of-money)
71.5% chance that: after 1-year UP move: $30 x 1.15 =$34.5
Payoff to call with $30 strike = $4.5
Call value = PV of Cash flows (discount at Rf): \(C_0 = \frac{4.5\times0.715+0\times0.285}{1.07} = 3 \)
American-style Options
Worth at least as much as European
American options can be exercised early (before expiration date)
Early exercise of call options is valuable if the asset pays interest or dividends
Early exercise of put options can be valuable if the are deep in the money (ex: can sell when stock price took a dive but has not recovered yet)