Fin management materials / 11 P4AFM-Session12_j08

SESSION 12 – OPTIONS

3.4The Grabbe Model

¾In 1984 Orlin Grabbe produced a modified version of the Black-Scholes model which can be used to value currency options.

¾This is referred to as the Grabbe (1984) model or the FOREX modified Black-Scholes options pricing model. The formulae are published in the exam:

c = e-rt F0N(d1)–XN(d2) Or

p = e-rt XN(-d2)–F0N(-d1) Where:

d1 = 1n (F0/X)+s2T/2 s T

d2 = d1 – s T

c= price of call option

p= price of a put option

e= the exponential constant

r= annual risk free interest rate

T= time to expiry (in years)

X= exercise price/strike price

s= annual standard deviation of the exchange rate

1211

SESSION 12 – OPTIONS

Example 6

Euro/Sterling spot rate = 0.69 (indirect quote i.e. €1 = £0.69)

3 month Euro LIBOR = 2.7656%

3 month Sterling LIBOR = 4.6231%

Monthly volatility of Euro against Sterling = 6.35%

Required:

Calculate the price of Euro/Sterling at the money call options where each option has a contract size of 100,000 Euros and 3 months to expiry. Assume that deposits earn a rate of return equal to sterling LIBOR.

Solution

1212

SESSION 12 – OPTIONS

3.5Valuation of Real Options

¾Real Options Pricing Theory suggests that the true value of a project = Traditional NPV + value of “embedded options”

¾A common embedded option is an abandonment or exit option. For example consider a pharmaceutical company that has an option to sell the patent rights to a drug rather than continue production itself.

¾In many ways an abandonment option resembles an US style put option. The Black Scholes model can be used to make a conservative estimate of the value of such an option.

¾The Black Scholes model may however underestimate the value of the option as it is designed for European style options whereas many options within company projects may be American style.

¾More accurate valuation can be performed using binomial models.

Example 7

A company has just commenced a project with the following data:

Conventional NPV $10m

Capital Expenditure $90m

10 year life

Volatility of project cash flows = 45%

Risk free rate = 5%

The company has the option to abandon the project at any time and sell the technology for an estimated $40m.

Required:

Use the Black Scholes model to estimate the minimum value of the abandonment option.

Solution

1213

SESSION 12 – OPTIONS

3.6The Merton Model

¾In 1974 Robert Merton developed a model for estimating credit spreads and the probability of default on corporate debt.

¾Merton based his model on Black and Scholes view that equity can be regarded as a European call option over the company’s net assets.

¾Merton used the Black Scholes model to value a firm’s equity under the assumption that the firm’s debt is a discounted bond with zero coupon and redemption at face value at time t - the time to expiry of the equity option.

¾Having valued the equity Merton then uses Modigliani and Miller’s assertion that the market value of equity plus the market value of debt equals the total value of the firm’s assets i.e.

Value of debt = value of assets – value of equity

¾By comparing the theoretical market value of the debt to its face value the yield can be easily implied.

¾The yield can then be compared to the risk free rate to estimate the credit spread on the bond and hence the firm’s default risk.

3.7Credit derivatives

The Merton Model (perhaps together with the Z-score or Zeta models) may reveal significant default risk on a bond, or at least risk of the spread increasing. An investor in such bonds can hedge against such risks using various credit derivatives:

¾Credit forward contracts – the holder of a credit forward can use it to lock a particular credit spread. If the actual spread becomes wider than the contracted spread the buyer of the contract receives compensation from the seller, and vice versa. No premium is paid but the holder does not benefit from falling spreads.

¾Credit swaps – can be used to exchange risky returns for guaranteed flows. The investor pays the counterparty the total returns on a bond and receives a payment based upon LIBOR. Can be referred to as the swap of a “dirty” cash flow for a “clean” cash flow.

¾Credit options – common types include:

Credit spread options – the buyer of the option receives a payout if the actual spread on a bond exceeds the strike spread of the option.

Predetermined payout option – which pays the holder a fixed sum if a specified event occurs e.g. default or downgrade in credit rating.

1214

SESSION 12 – OPTIONS

¾ Assume an investor holds shares and that the delta of call options on this share = 0.5 Delta hedging would take place as follows:

Number of call options to sell = Number of shares bought

Nd(1)

So if holding 100 shares we would need to sell (100/0.5) = 200 calls to construct a delta hedge.

If the share price falls by 10 cents there will be a $10 loss on the shareholding. However the price of each option will have fallen by 5 cents. We can then buy back each option at 5 cents lower price than we previously sold them at. This gives a gain on options of 0.05 × 200 = $10.

Buying shares and then selling calls (“long share/short calls”) in the hedge ratio therefore ensures a delta neutral portfolio where gains and losses cancel each other out.

¾However, the problem with delta hedging is that delta is not a constant. It changes as the share price changes. Note from the graph above that there is not a linear relationship between share prices and option prices.

¾The portfolio will therefore need to be rebalanced as delta changes.

¾The frequency of rebalancing depends on the rate of change of delta - measured by gamma

4.3Gamma

Definition

Gamma measures the rate of change of delta (the sensitivity of delta) as the

¾If an option is deep “out of the money” then its delta will be low and will stay low – hence gamma will also be low.

¾If an option is deep “in the money” its delta will be high and will stay high – hence gamma will be low.

¾If an option is near to or “at the money” then delta changes rapidly – and gamma will be high i.e. a delta hedge would need constant rebalancing.

1216

SESSION 12 – OPTIONS

4.4Theta

Definition

Theta measures how much value is lost over time. The theta is usually expressed as the amount of loss per day.

Option

value

Passing of time

¾Time decay reduces the value of both call options and put options.

4.5Vega

Definition

Vega measures how much option prices change with changes in share price volatility.

¾Vega is always positive and therefore as price volatility increases the option premium (for puts or calls) will also increase.

¾Vega can also be referred to as Kappa or Lamba.

4.6Rho

Definition

Measures how much option prices change with changes in the interest rates.

¾If interest rates rise the price of call options will also rise

¾If interest rates rise the price of put options will fall.

1217

SESSION 12 – OPTIONS

4.7Summary of the Greeks

Key points

The holder/buyer of an option has the right, but not the obligation, to buy (if calls) or sell (if puts) the underlying asset at a fixed price (the strike/exercise price) on (if European style) or before (if American style) an agreed date (the expiry date).

The value of an option is reflected in its premium i.e. price and has two main components – intrinsic value and time value.

Black and Scholes produced the most famous model for valuing options – designed for European call options but easily converted to puts using putcall parity. The value of American options is usually close to their European equivalent.

Grabbe later applied the Black Scholes model to valuing currency options, and Merton applied it to valuing corporate debt. The model can also be used to value options embedded within company projects for Real Options Pricing Theory.

“The Greeks” analyse how the price of an option varies with key factors. The most famous Greek is delta – popular with fund managers for setting up perfectly hedged portfolios to protect against short term volatility.

1218

SESSION 12 – OPTIONS

FOCUS

You should now be able to:

¾describe the main features of options;

¾differentiate between traded options and OTC options;

¾discuss the determinants of option pricing;

¾perform calculations involving the Black-Scholes model, including put-call parity and delta hedging;

¾perform calculations involving the Grabbe variant of the Black Scholes model;

¾apply the Black Scholes model to valuing real options embedded within investment projects;

¾explain the principles of the Merton Model for valuing corporate debt;

¾Discuss “the Greeks”.

1219

SESSION 12 – OPTIONS

EXAMPLE SOLUTIONS

Solution 1

(a)July call option @ 500 pence − out of the money as the exercise price is higher than the current market price

(b)April put option @ 460 pence − out of the money as the exercise price is below the current market price

(c)October call option @ 460 pence − in the money as the exercise price is below the current market price

Solution 2

(a) (i) July call option @ 460 pence

This would be exercised as the shares would be bought @ 460 pence and they have a current market value of 490 pence

(ii)July call option @ 500 pence

Currently this would not be exercised as you would not buy shares for 500 pence that are only currently worth 490 pence

Intrinsic value = zero

(b) (i) October put option @ 460 pence

This would not be exercised as this gives the right to sell shares worth 490 pence currently for just 460 pence.

(ii) October put option @ 500 pence

This would be exercised as it gives a right to sell share worth 490 pence for 500 pence.

1220