Lecture 3

Question 1

What is an option

Answer 1

Options are financial instruments defined on specific underlying variables (e.g. assets) that give the owner the right, but not the obligation to exercise them according to certain terms. Their payoff (and value) depends on the underlying asset, as in all derivatives

Question 2

What kind of a Product is Prof. Dimitrios P. Tsomocos

Answer 2

A Yale product

Question 3

What is the main difference of an european and american option

Answer 3

European options can be exercised only at their expiration. American options can be exercised at any time before their expiration or at their expiration

Question 4

What is the upper bound of a call option

Answer 4

The stock price

Question 5

Proof that the price of a call option (c) is smaller than the share price today (S_0)

Answer 5

Proof by contradiction Assume that c > S_0 Consider the following strategy: In t=0: Sell the option and receive c. Buy the stock at t=0 and deposit c-S_0>0 at the risk-free rate r_f In T=1: If S_T >= K, then the buyer of the option will exercise it, you return the stock that you own and be left with (c-S_0)(1+r_f) + K > 0 If S_T < K , then you are left with the deposit as well as with the share. (c-S_0)(1+r_f) + S_T > 0 For a price of zero, you set up a trading strategy that gives strictly positive payoff in any case tomorrow This is strong arbitrage and it cannot hold

Question 6

What is the lower bound of a call value (c)

Answer 6

The share price today minus the discounted strike price

Question 7

Proof that the value of a call (c) is at least as high as the share price today (S_0) minus the discounted strike price K/(1+r_f)

Answer 7

Proof by contradiction Assume c < S_0 - K/(1+r_f) Consider the following trading strategy: t=0 Short sell the stock and receive S_0 Buy the option and deposit w = S_0 - c > K/(1+rf) at the risk-free rate (r_f) T=1 If S_T >= K then you exercise the option, pay K, return the stock you sold short and be left with w(1 + rf ) - K > 0 If S_T < K then you don’t exercise the option, you buy the stock in the market at ST < K and be left with w(1 + rf ) - S_T > 0 For a price of zero, you set up a trading strategy that gives strictly positive payoff in any case tomorrow This is strong arbitrage and it cannot hold

Question 8

What is the formula of the put-call parity

Answer 8

The value of a call long and the discounted strike price is the same as a put long and the share at time t=0

Question 9

Suppose there exists a payoff profile z=(z_1,…,z_s) with z_s != z_s’ A call option with strike price K matures out of the money for all states with z_s < K When can you complete markets with the payoff z and option. Especially, how many options are needed

Answer 9

If the payoff z and S-1 options with strike prices zs (except the greatest) are traded then markets are complete Remark: If the payoff z takes the same values in two states then the markets are NOT complete

Question 10

Let there be three states and the payoff of an asset be (1, 3, 6). Consider two call options with strike prices 3 and 1 respectively Are markets complete?

Which academic found that out and what is his general remark

Answer 10

Yes

Ross,1976: If the payoff z and S-1 options with strike prices z_s (except the greatest) are traded then markets are complete

Question 11

Let there be four states and the payoff of the asset (1, 3, 3, 6) Consider two call options with strike prices 3 and 1 respectively Are markets complete

Which academic found that out

Answer 11

Nope Remark: Markets are incomplete even if all options with arbitrary strike prices are traded

Ross,1976: If the payoff z and S-1 options with strike prices z_s (except the greatest) are traded then markets are complete

Question 12

An American option is worth at least as much as the corresponding European option Is that correct?

Answer 12

Yes

Question 13

An American call on a non-dividend paying stock may be worth being exercised early, i.e. its price should be equal to that of a European call Correct?

Answer 13

No An American call on a non-dividend paying stock is never worth being exercised early, i.e. its price should be equal to that of a European call

Question 14

An American put may be worth being exercised early Correct?

Answer 14

Yes

Question 15

Proof the claim that: An American call on a non-dividend paying stock is never worth being exercised early

Answer 15

Let G_n be the value of an American call option at t=n. Obviously, G_n >= V_n, where V_n is the value of the respective European call. The put-call parity for t=n gives us V_n + K/ ((1+r)^{N-n}) = U_n + S_n, where U_n is the value of the European put with the same strike price. Then, G_n >= V_n –> G_n >= Un + Sn - K/((1+r)^{N-n} >= S_n - K/((1+r)^{N-n}) > S_n - K The value of an American call at t=n is greater than what we will get from immediate exercise. Thus, an American call (on a non-dividend paying stock) should never be exercised early

Question 16

What is the Delta within Delta-Hedging

Answer 16

It is the number of stocks we have to go short (long) to hedge our position if we go long (short) on an option position, i.e. to lock on the risk-free rate

Question 17

What role plays the return of a stock if valuing an option on that stock?

Answer 17

None. Irrelevance of the stock’s return When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant

Question 18

What are assumptions in the Black Scholes model

Answer 18

The stock price follows a geometric Brownian motion with constant drift and volatility The short selling of securities with full use of proceeds is permitted There are no transactions costs or taxes. All securities are perfectly divisible There are no dividends during the life of the derivative There are no riskless arbitrage opportunities Security trading is continuous The risk-free rate of interest, r, is constant and the same for all maturities

Question 19

What do you need in the black scholes model to dynamically complete the market

Answer 19

You can dynamically complete the markets with just the stock and the bond. Thus, you can price an option on the stock using risk-neutral pricing (i.e. you can derive unique state prices that exclude arbitrage). In particular, the stock price and the option price depend on the same underlying source of uncertainty. We can form self-financing portfolios through continuous retrading, such that at the expiration date the portfolio of the bond and the stock replicate exactly the payoffs of the option. Thus, according to the LOOP the portfolio and the option should have the same price today

Question 20

Is it possible to complete the markets when at expiration date we do not have as many assets as states?

Answer 20

What we have accomplished is to dynamically complete the markets by means of self-financing portfolios, i.e. in every t we can have without any additional input or output of money a portfolio that replicates exactly the payoffs in t+dt in any state of the world. Even if markets are in general incomplete (at the expiration date we don’t have as many assets as states), we have accomplished through continuous retrading and self-financing portfolios to dynamically complete them

Question 21

What is implied volatility

Answer 21

It is the volatility implied by option prices observed in the market as if they were determined by the BS model. Using the BS formula we use the option prices that we observe as inputs and we solve for the volatility. The result is the implied volatility of the stock. Traders like to calculate implied volatilities from actively traded options on a certain asset and interpolate between them to calculate the appropriate volatility for pricing a less actively traded option on the same stock

Question 22

When the holder has the right to exercise early, is the Black Scholes or the Binominal Tree more likely to be used

Answer 22

When the holder has the right to exercise early, Binomial Trees are particularly useful for pricing

Question 23

We use the binominal tree procedure to price…

Answer 23

American calls on dividend paying stocks American puts on non-dividend and dividend paying stocks

Question 24

Show that the put call parity holds

Answer 24

Question 25

It the owner of a receiver swaption potentially paying or receiving fixed payment

Answer 25

He pays a floating rate and gets a fixed rate

Question 26

Is the owner of a payer swaption getting a fixed rate or a floating rate

Answer 26

The owner of a payer option will pay a fixed rate and receives a floating rate

Question 27

What is the Swaption Parity and how does it hold

Answer 27