Week 6 Flashcards
Derivatives:
The financial contract between two or multiple parties, where the value is derived from the future value of an underlying asset, good, or commodity.
Strategies to eliminate risk exposure (future price):
1) Buy a massive stock of resources (risk of quantity and high costs)
2) Negotiate a fixed price for a pre-specified amount with your suppliers (future contract, still exposed to quantity risk)
3) Make a reservation on future resource purchases at a price you negotiate now. (option contract)
American vs European options
American - may be exercised at any point up to expiration.
European - may be exercised only at the maturity
Call option vs put option:
Call - provides the holder with the right to purchase shares at a predetermined price. Profit if the share price > exercise price
Put - gives the holder the right to sell shares at a predetermined price. Profit if the share price < exercise price
Writing options:
Same as selling/short position. The writer loses what the buyer (long position makes), however, can only make the predetermined premium (the price of the option).
The writer has the obligation to buy/sell the shares written in the options if the holder decides to exercise.
Put-call parity:
Buying a put option and underlying asset yields the same pay-offs as buying a call option and risk-free, zero-coupon bond. Because the payoff is equal, the price must be the same as well. This is called put-call parity:
Put option + Underlying asset = Call option + PV of the exercise price (bond)
The maturity date is equal for put and call, as well as the exercise price.
Explain the lower and upper bounds of option pricing:
lower bound - the minimum price the investor will pay for the option. Equals share price - exercise price (call). This is also called the intrinsic value.
Upper bound - current share price, as it then becomes unsensible to buy option if the investor can afford to buy the underlying asset itself.
Factors determining the option value:
Contractual features:
1) Exercise price - the higher the price for a call the lower the value (opposite with put).
2) Time to expiration - the further away expiration the more valuable option.
Characteristics of the equity and the market:
1) Current share price - the higher the share price, the larger the profit for the call. The function is convex (the higher the share price the larger the increase in the value) (opposite for put).
2) Variability of the underlying.
3) Interest rate - the higher the interest rate, the more valuable the option (opposite for put).
Two-state or binomial model (replicating portfolio method in lectures):
It is possible to replicate the call option by buying a fraction of a share and borrowing to yield the same payoff.
Since the payoffs are the same the prices must be the same as well.
This approach states that there are only two possible share prices at t1. In other words, time is assumed as a discrete variable, rather than continuous.
1) Determine the delta (amount of shares to purchase)
Delta = swing of an option/swing of equity; 1$ swing in share price leads to delta change in option call.
2) Determine the amount borrowed -
Amount of borrowing = (Payoff share - payoff call)/(1+r)^t
3) Determine the value of a call:
Value of a call = share price * Delta - Amount borrowed
Risk neutrality (risk-neutral valuation):
Intuition is that risk can be eliminated by hedging. By shorting the call option it is possible to completely eliminate risk and in this case, the results will be equal to the risk-free rate.
Risk-neutral investors are indifferent to risk, therefore, they require a risk-free rate return.
r(f) = probability of rise * rise in % + (1-probability of rise) * decrease in %
Value = (p(1) * C(1) + p(2) * C(2))/(1+r)^t
Black - Scholes model:
In the real world, there are countless options for a share price at t(1); however, if we make the time period infinitesimal the share can really have two values. The formula in the Black-Scholes model can calculate the value of a call, by determining the duplicating combination at any moment.
C = S * N(d1) - E * e^(-R*t) * N(d2)
C-value of a call
S - Share price
E - exercise price
R - risk-free rate
t - time in years
N(d) - the probability that a standardized, normally distributed random variable will be less than or equal to d.
In the case of dividends S is substituted for S*e^(-qt) q represents the dividend yield.
The “Greeks”:
The Greeks measure the rate of change in the value of a call or put with respect to the major factors of underlying asset value and other factors.
Delta - measures the rate of change in the value of the call compared to the change in the underlying.
Gamma - measures the rate of change in delta with respect to change in the share price (0.05 gamma means that a 1$ increase in price leads to 0.05 increase in delta). Gamma is very small when options are deep out of/in the money, the highest when the option is at the money.
Theta - measures the change in the value of an option with the change in the time to maturity, theta is always negative.
Vega - measures the rate of change in an option’s value with respect to changes in its implied volatility.
Call option on the firm:
Equity can be viewed as a call option on the firm. Bondholders are the writers of this option, therefore, they are the owners of the company.
Shareholders receive nothing if the costs are above sales and do not execute the option (the ownership remains with bondholders). However, if the sales exceed costs (exercise price) the shareholders execute the option, repay the liabilities, and take the ownership.
Put option on the firm:
The owners are the shareholders and bondholders have purchased the put option. The shareholders own the company, but if the costs are above sales the bondholders execute the option and take the ownership.
Risky bond value:
Value of a risky bond = value of default-free bond - value of put option
