15.6 Option Pricing Models

2 different ways to value insurance company using options

  1. Equity of the firm as a call option

  2. Valuation of real options

    (Additonal source of value the firm has from its ability to engage or disengage in various projects)

15.6.1 Equity as Call Option

Merton’s method:
Owners of the firm have given assets of the firm to the debt holders, but maintain the right to purchase them at price \(D\) when the debt is due at time \(T\)

  • At \(T\), if the value of the firm \(V_T > D\) then the owners will pay back the debt

  • If \(V_T < D\), the option will not be exercised and firm goes bankrupt

\[E_T = \max[V_T - D, 0]\]

Remark.

  • Call option with strike price \(D\) and “stock” price \(V_T\)

  • Price with Black-Scholes

  • Assume debt to have one expiration date \(T\)

  • We need the volatility of the value of all assets of the firm

Application to P&C Insurers

  • Mostly theoretical

  • Moody’s KMV credit default model is based on this to estimate probabilities of default for publicly traded companies

  • Difficult to apply to P&C companies due to the additional complexity of having policyholder liabilities that look just like debt to the shareholders

    • And with policyholder liabilities it is hard to reduce it into a debt with single expiration date

15.6.2 Real Options Valuation

Not super sure how the ‘value’ of each of the options work

Definition 15.1 (Real Options) Management of a company has flexibility on when to take action on projects

Real options examples:

  • Abandonment

    (e.g. Abandoning a project so you cut off the future cash outflow)

    • Value as an American put option (You have the right to sell at a predetermined price anytime up to the expiration date)

    • \(K\) = liquidation proceeds

  • Expansion

    (e.g. similar to the abandonment but with investing)

    • Value as an American call option

    • \(K\) = additional investment

  • Contraction

    (Not total abandonment)

    • Value as American put

    • Value is the gross value of the lost capacity

    • \(K\) = cost savings (reduced investment)

  • Defer

    • Value as an American call (since it’s still an investment project)

  • Extend

    • Value as European call since it would only be exercised at the end of the project

    • Extend the life of the project by paying a fixed amount

Value is added when the firm can purchase assets below their fair value or has exclusive access to opportunities

  • e.g. flexibility to purchase assets at market price doesn’t add value

Practical Considerations, difficulty in:

  • Identifying a new business where such a real option with value exists

  • Assess the current value of these business

  • Determine if the firm has the ability to enter this business at a fixed price, or a price that significantly differs from market value

Valuation considerations, technical issues to consider:

  • Valuing the underlying cash flow

  • Time to maturity

  • Exercise type (American vs European)

  • Appropriate valuation formula (e.g. Black-Scholes assumes lognormal distribution for asset price)

Reasonableness of real option values

  • Options are more valuable when new information will be available before the expiration

  • Expansion options are only valuable if they have some exclusive right

  • Exercise price must be fixed for the option to have value