Question

Derivatives & Risk Management - Options Market Pricing Models)

Be able to explain and apply the binomial options pricing model (BOPM):

o This applies to both the arbitrage pricing version (i.e., the 5 steps version) and the risk neutral pricing version.

o What are the probabilities of the up and down state-of-nature (S.O.N.) for any given option? How does this relate to the expected moneyness of an option?

o What is the “economic story” behind the BOPM. Explain the logic and how it leads to an equilibrium price that can be enforced with the Law of One Price (i.e., how the call premium can be arbitrage enforced). There are lots of parts to this story, but remember that the foundation of this model is the concept of a replicating portfolio (aka the “partially covered call”).

o Explain why the no-arbitrage condition must exist for a call. In other words, explain what it means for a call to be over-priced or under-priced relative to the BOPM.

EVEN PARTIAL ANSWERS APPRECIATED if unable to provide an entire answer

Answer 1

1 . The binomial option pricing model is an options valuation method developed in 1979. The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the option's expiration date. The model reduces possibilities of price changes, and removes the possibility for arbitrage.
The binomial option pricing model assumes a perfectly efficient market. Under this assumption, it is able to provide a mathematical valuation of an option at each point in the timeframe specified. The binomial model takes a risk-neutral approach to valuation and assumes that

underlying security prices can only either increase or decrease with time until the option expires worthless.
Binomial Pricing Example
A simplified example of a binomial tree has only one time step. Assume there is a stock that is priced at $100 per share. In one month, the price of this stock will go up by $10 or go down by $10, creating this situation:

Stock Price = $100

Stock Price (up state) = $110

Stock Price (down state) = $90

Next, assume there is a call option available on this stock that expires in one month and has a strike price of $100. In the up state, this call option is worth $10, and in the down state, it is worth $0. The binomial model can calculate what the price of the call option should be today. For simplification purposes, assume that an investor purchases one-half share of stock and writes, or sells, one call option. The total investment today is the price of half a share less the price of the option, and the possible payoffs at the end of the month are:

Cost today = $50 - option price

Portfolio value (up state) = $55 - max ($110 - $100, 0) = $45

Portfolio value (down state) = $45 - max($90 - $100, 0) = $45

The portfolio payoff is equal no matter how the stock price moves. Given this outcome, assuming no arbitrage opportunities, an investor should earn the risk-free rate over the course of the month. The cost today must be equal to the payoff discounted at the risk-free rate for one month. The equation to solve is thus:

Option price = $50 - $45 x e ^ (-risk-free rate x T), where e is the mathematical constant 2.7183

Assuming the risk-free rate is 3% per year, and T equals 0.0833 (one divided by 12), then the price of the call option today is $5.11.

Due to its simple and iterative structure, the binomial option pricing model presents certain unique advantages. For example, since it provides a stream of valuations for a derivative for each node in a span of time, it is useful for valuing derivatives such as American options. It is also much simpler than other pricing models such as the Black-Scholes model.

The value of an option is determined by six variables relating to the underlying asset and financial markets.
1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.

This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.

6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved.This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts.

Pricing model (no arbitrage) :

The general approach to option pricing is first to assume that prices
do not provide arbitrage opportunities.
Then, the derivation of the option prices (or pricing bounds) is
obtained by replicating the payoffs provided by the option using
the underlying asset (stock) and risk-free borrowing/lending.
Illustration with a Call Option
Consider a call option on a stock with exercise price X.
(Assume that the stock pays no dividends.)
At time 0 (today):
Intrinsic Value = Max[S-X, 0],
The intrinsic value sets a lower bound for the call value:
C > Max[S-X, 0]
In fact, considering the payoff at time T, Max[ST-X, 0] we can
make a stronger statement:
C > Max[S-PV(X), 0] ? Max[S-X, 0]
where PV(X) is the present value of X (computed using a
borrowing rate).
If the above price restriction is violated we can arbitrage.