W10: Valuing Options & Options in Corporate Finance

• Factors affecting price options, Binomial trees, Black-scholes model, Application in corporate finance

Background context:

• Option contracts are financial agreements giving you the right (not obligation) to buy (call) or sell (put) an asset at a fixed price in the future.

• Calls want LOW strike prices (buy cheap), puts want HIGH strike prices (sell high)

• Strike price = when exercising, Spot price = current market price

• These are used to:

    1. Protect from financial loss through hedging

    2. Speculation (profit from calls if price rises and profit if price falls for puts), and

    3. Control large assets with less money (leverage)

• Two main styles of contract rules:

    1. American, which you can exercise anytime before expiry.

    2. European, that you can only exercise at expiry.

• American options are more flexible, thus are normally more expensive. More rights = more premiums for the same payoff at expiry if you don’t need to exercise early as many strategies only care about the payoff.

  • Especially for calls on non-dividend stocks, flexibility would be irrelevant as you wouldn’t use it anyway so the extra spent on premiums would be a waste.

  • American put will always be greater than European puts though as they can exercise early to lock in the value & time value of money

• Most models assume European options for pricing simplicity, making them more common in Index options and institutional markets.

• As opposed to directly buying/selling an asset:

  • Options give limited downside risk, better flexibility, and more leverage HOWEVER assets have no expiry (better for long term investing), full upside, and dividends, all without premiums.

Factors affecting the value of an option:

The value of an option = expected payoff (in a risk-neutral world) discounted at the risk-free rate.

* + = increase, - = decrease

Put-Call Parity (European only)

• If two portfolios give the same payoff, they must have the same value today (assuming no dividends and continuous compounding). If they’re NOT equal, an arbitrage opportunity arises (buy cheap, sell expensive)

• Formula in sheet but:

  • $c+Ke^{-rT}=p+S_0$

Example:

3 month European put with a strike price (only if exercised) of $45 is priced (price of put option aka the premium paid NOW) at $3.70. Underlying stock priced at $48 and makes no dividends during the life of the options. The risk-free rate is 5%. Calculate the price of the 3 month call option with the same strike price using the put-call parity.

c = $3.70 + $48 - $45 x e^{-0.005×0.025} = $7.26

• Put price p = 3.7

• Stock price S₀ = 48

• Strike K = 45

• Rate r = 5%= 0.05

• Time T = 0.25 (3 months)

Rearrange the formula to create synthetic instruments such as:

Why create synthetic instruments?

Because synthetic instruments are created to replicate identical payoffs, allowing arbitrage opportunities to be exploited when prices diverge. Same payoff, different combination.

Do this to:

• Arbitrage - if they have the same payoff, they should have the same price. If they don’t then one is either overpriced or under priced, thus you buy the cheap version and sell the expensive version.

  • From the example above the real call price would be $7.50, whilst the synthetic call would be $7.26. Synthetic is cheaper so sell the real one and buy the synthetic one for free profit.

• Sometimes you can’t access a certain instrument, or it may be too expensive or illiquid so you can recreate it using others.

  • i.e. if you want a call, use: Put + Stock - Bond

Synthetic instruments are how options are priced, if you can replicate a payoff then its price MUST equal the replication cost, otherwise arbitrage exists.

• Firms can use synthetics to adjust exposure and hedge risk i.e. converting equity exposure into option-like payoff

Binomial Model (American)

• Option valuation model assuming only two possibilities for stock price movement (up/down), discounting expected payoff at the risk-free rate as it assumes a risk-less portfolio

• given S0 = stock price, u = up factor, d = down factor

1. Calculate the stock price with the up or down factor

2. Calculate payoffs using

    call: $f=\max\left(S-K,0\right)$

    put: $f=\max\left(K-S,0\right)$

    * for American options, at each node do $Value=$ max(exercise now, hold value), and continuation instead of 0.

3. Calculate the risk-neutral probability

    $p=\frac{e^{rT}-d}{u-d}$

3. Price the option

    $f=e^{-rT}\left\lbrack pf_{u}+\left(1-p\right)f_{d}\right\rbrack$

4. From the final payoffs, move back using the given formula and repeat until today