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Options Valuations (Derivatives - part 2) Summary
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Detailed and well summarised study notes of the "Derivatives - Options Valuations" section taught in Finance IIB at UCT (FTX3045S).
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DERIVATIVES - OPTION VALUATIONINTRODUCTION
- Traded away price risk by locking in for better or worse, a fixed price.
- Right but not the obligation
- Contingent obligation
- “IN THE MONEY” – exercising option would give you a positive payoff
- Worst case scenario: Option at expiration has a payoff of 0.
- Options are bets on volatility
- What is the value of an option that is currently “OUT OF THE MONEY” but has yet to
expire?
• Still probability something may happen in market to make it in the money
Intrinsic value
- Value the option would have if it were about to expire
- = to S0 - X: in-the-money call option and X-S0: in-the-money put option
- Gives the payoff that could be obtained by immediate exercise
- Set equal to zero for out-of-the-money or at-the-money options
Time value
- Difference between premium price and intrinsic value
- Greater the time to expiration, greater the chance of volatility, greater chance of
finishing in the money.
- Part of the option’s value that may be attributed to its positive time to expiration.
- Option’s time value: volatility value
- Volatility value lies in the value of the right not to exercise if out-of-the-money
- As stock price increases (call) or decreases (put), volatility value minimal
Adjusted intrinsic value of the option
- Present value of your obligation is the present value of X.
- Net value of the call option is S0 - PV(X) → Xe-rT or (x/(1+r)T
- Net value of put option is PV(X) – S0
- Assuming no dividends paid until after expiration
- If stock pays dividends before expiration: entitled to interim dividend
- S0 - PV(X) – PV(D): call
- PV(X) – PV(D) – S0: put
,The call option valuation function
- Can’t be negative – flat in beginning
- Greatest time value = greatest volatility
Determinants of option prices
Call formulas:
- (ST – X)
- S0 – PV (X)
- S0 – PV (X) – PV (D)
- PV(X): X/(1+rf)
Put formulas:
- (X – ST)
- PV (X) – S0
- PV (X) – S0 – PV(D)
- PV(X): X/(1+rf)
, Determinants:
- Stock price: The higher S, the more likely call option will expire in-the-money and the
more profitable call option will be.
- Exercise price: Higher X reduces call option payoff and increases put option payoff
- Volatility: Volatility in the underlying asset increases expected payoff to the option –
enhancing its value.
• Expected cash flow from the option depends on the volatility of the stock –
option’s expected payoff increases along with the volatility of the underlying
asset.
- Time to expiration: For more distant expiration dates, there is more time for
unpredictable future events to affect the volatility of the underlying asset and prices.
- Interest rates: Higher interest rates reduce the present value of the exercise price.
- Dividend pay-outs: dividend payout policy of the firm affects option values.
• A high dividend payout policy reduces the rate of growth of the stock price
• Decreases potential payoff from the call option and increases potential payoff
from the put option
RESTRICTIONS ON OPTION VALUES
- Must have a positive value, payoff is 0 at worst
- Value of call option cannot exceed the value of the underlying stock: Upper bound
on option price
- Value of an option cannot be negative
- Additional lower bound on a call option: value of levered equity
Compare two portfolios:
1. Call option on one share of stock
2. Leveraged equity position: One share and borrowing of PV(X) and PV(D)
Costs of establishing leverage equity position:
S0 – (X+D)/(1+rf)T
Options payoff is always > leveraged equity position
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