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

Forwards and Futures:
Value vs. Price

Price

No-arbitrage forward price:

Forward Contract Value

Value = Value of a long position in the forward

Arbitrage Strategy

This mean: Buy asset now, sell forward at $F_0(T)$, you will earn Rf until time = T

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

Value = 0 at initiation (time 0)

Long position

at time t (during contract life):

at expiration (settlement)

$V_t(T)=S_T- F_0(T) $

$V_t(T)=S_t - \frac{F_0(T)}{(1+Rf)^(T-t)}$

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 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

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

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

European Options

European call option:

European put 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) $

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

Factors affecting option values

Asset price

Exercise price

Volatility of asset price

Time to expiration

Risk-free rate

Cash flows from underlying asset

(higher asset is better for call but worse for put value)

(higher price is better for put but worse for call values)

(more variable is better)

(longer is better, since can be more volatile)

higher Rf increases call values but decreases put values

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}$

Synthetic Long Call: $ -C= -P-S+\frac{X}{(1+Rf)^T}$

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)

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

Value may fluctuate, but price remain the same (exercise price at time T)

$F_0(T)={S_0}\times (1+Rf)^T$

= $F_0(T) $,a.k.a forward price in contract, a.k.a price paid at expiration

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

Swap price: PV of all net cash flows

Value

On each payment date, swap owner receives a payment based on the value of underlying at the time of each respective payment

Time value is the portion of an option's premium that is attributable to the amount of time remaining until the expiration

Synthetic Long Bond: $ -\frac{X}{(1+Rf)^T} = -S-P+C$

Synthetic Long Asset: $ -S=-C-\frac{X}{(1+Rf)^T}+P$

Long Call, Short Asset, Long Bond

Long Put, Long Asset, Short Bond

Long Asset, Long Put, Short Call

Long Call, Short Put, Long Bond