Options, Futures and Structured Products

Transcription

1 Options, Futures and Structured Products Jos van Bommel Aalto Period Options Options calls and puts are key tools of financial engineers. A call option gives the holder the right (but not the obligation) to buy an underlying security (of which the value is denoted S) for a specified price (the strike price, or exercise price (denoted X)) at maturity, denoted T. At maturity, the value of a call is thus max(0,s X) A put option gives the holder the right to sell an underlying security for a specified price before at maturity. At T the value of a put is max(0,x-s). 1

2 payoff Payoff of a Call Option at maturity, exercise price is 0 Max(0,S X) S = price of underlying X = exercise price $80 $90 0 $110 $120 $130 payoff Payoff of a Put Option at maturity, exercise price is 0 Max(0,X S) $80 $90 0 $110 $120 $130 2

3 Writing options payoff You can also sell, short or write options: This is the payoff of a Short Call with X = $110 $80 $90 0 $110 $120 $130 Of course, if you sell an option, you receive money for it. The price you receive upon selling is sometimes called the option premium writing options payoff A short put with X = $90 $80 $90 0 $110 $120 $130 If you write a put, you hope that the underlying does not fall.. 3

4 Selling or writing options What is riskier: Buying or writing options? If you write options, you need to post margin The margin for writing options tends to be higher than the premium received.. If you write options, you may get margin calls. What is riskier: writing puts, or writing calls? payoff Option Strategies What s the payoff of a long call with X = $90 a short call with X = $110? $80 $90 0 $110 $120 $130 4

5 payoff Option Strategies What s the payoff of a long put X = $110 a short put with X = $90? $80 $90 0 $110 $120 $130 payoff Option Strategies What s the payoff of a short call with X = $90 a long call with X = $110? $80 $90 0 $110 $120 $130 What s the difference with the strategy on the last slide? 5

6 Option Strategies A bond with face value! Indeed: The value of: a long put with X = $90 a short put with X = $110 Is the same as the value of: a short call with X = $90 a long call with X = $110 a bond with facevalue Otherwise you would be able to arbitrage.. Payoff Put Call parity Consider the Strategy where you buy the underlying, and go a long Put with X = $80 $90 This is called a protective put $80 $70 $60 The underlying The Put $60 $70 $80 $90 0 $110 $120 6

7 Payoff Put Call parity Now you buy a bond with PV = $80, and a Call option with X = $80 $90 $80 The bond $70 $60 The Call $60 $70 $80 $90 0 $110 $120 Put Call parity S + P = C + PV(X) Implication: Option strategies can be set up in at least two ways. 7

8 Payoff One more Strategy Buy a Call with X = $90 Buy a Put with X = $90 $40 This is called a Straddle When would you buy a straddle? The Call The put $60 $70 $80 $90 0 $110 $120 Payoff One more Strategy Of course, this straddle has a price. Assume that the Call costs $12, and the Put $8. What is the profit (or net payoff ) diagram? $0 $60 $70 $80 $90 0 $110 $120 The Call The put 8

9 Payoff Value of a CALL Option What would be the value of a Call option with X = 100, if S = 92 It s Out of The Money (OOTM) What is the value of the option if S = 0 (it s deep In The Money) Payoff Value = Intrinsic Value Max(0,X S) $70 $80 $90 0 $110 $120 Value of a CALL Option Apart from S and X, the value of an option also depends on: The volatility of the underlying, The time until maturity, T The interest rate, r Asymptote = S PV(X) How? Asymptote = $0 $70 $80 $90 0 $110 $120 9

10 Value of a CALL Option The value of a Call Option increases in r The value of a Put Option decreases in r $70 $80 $90 0 $110 $120 Value of a PUT Option The value of a Call Option increases in r The value of a Put Option decreases in r Asymptote = PV(X) S Asymptote = $0 $70 $80 $90 0 $110 $120 10

11 Determinant of Option Prices Pricing Options We are looking for a formula that gives us a value that depends on S, X, T, r,. Only as a function of S, it should look something like this: The race for a formula started when options became popular and became exchange listed in the early 1970s 11

12 The Black Scholes Merton Formula Fisher Black, Myron Scholes, and Robert Merton realised that the instantaneous return on a Call option could be replicated with a levered long position in the underlying. The leverage of the replication portfolio however changes as a function of S and T (assuming that X, and r are constant over the life of the option) The Black Scholes Merton Formula C SN rt d Xe N 1 d 2 Long position in the underlying Amount borrowed d ln 2 S X r 2 T and d T 1 d2 1 T 12

13 C SN Black Scholes Merton rt d Xe N 1 d 2 d ln 2 S X r 2 T and d T 1 d2 1 T T is the time to maturity, in years r is annual interest rate, continuously compounded. is the annual volatility. S is assumed to follow a Geometric Brownian Motion. This is a continuous process of which the returns over intervals t are independently, normally distributed with standard deviation And N(x) (0,1) is the cumulative probabability of the standard Normal distribution. In excel it is Normsdist() Black Scholes Merton Assumptions behind Black Scholes Merton There are no dividend payments. The option cannot be exercised early (they are European). The stock price follows a Geometric Brownian Motion with constant volatility. 13

14 1 st individual assignment Build your own Black Scholes option calculator, and investigate how the (Black Scholes Merton) value diagrams change as a function of r, T, and I.e. make pictures like this: Base case: X = 100 r = 5% T = 2 years = 40% Then, try: r = 2% and 10% T = 1 year, 3 years = 20%, 70% Produce three Call graphs, each one has the base case and two deviations One graph shows the effect of r, one of, one of T. Produce three Put graphs The risk of the stockmarket Historically, over many years, across many countries 6% 20% 80% 60% 40% 20% 0% 20% 40% 60% 80% 100% 6% equity premium and 20% implies that: With 68% probability, the annual return will be between r f 14% and r f + 26% With 95% probability, the annual return will be between r f 34% and r f + 46% 14

15 The risk of the average stock (e.g. Siemens, L Oréal) % % 80% 60% 40% 20% 0% 20% 40% 60% 80% 100% Why is the risk of the average stock of an index greater than the risk of the index? (while the expected return of the average stock equals the return on the index) Where do we find volatilities? From past returns: Take closing prices, and then daily returns as follows: r t = ln((p t + div t )/p t ) compute the stdev (i.e. in excel), to obtain the daily volatility multiply by the sqrt(250) (number of days in a year), presto! This gives you estimate of From option prices! We need S, X, r, T, and to find the option value. If we have the option value (the price), and S, X, r, and T, we can back out the. The sigma derived this way is called the implied volatility. 15

16 2 nd individual assignment Estimate the volatility of Facebook. Look at historic data. What was the in 2013, 2014, 2015, and 2016? (I will post a file with prices on mycourses) Back out from recent option prices Using your option calculator, and S = $140 and r = 1% Find the implied volatilities of the following three options, that mature on Jan. 19 th, On April 7 th, we had: X = $120 C = $26.20 (this is an ITM option) X = $140 C = $12.70 (this is an ATM option) X = $160 C = $4.50 (this is an OOTM option) How does the historic volatility compare with implied vol? P.s. Facebook does not pay dividends.. 16