Stochastic Models. Introduction to Derivatives. Walt Pohl. April 10, Department of Business Administration

Transcription

1 Stochastic Models Introduction to Derivatives Walt Pohl Universität Zürich Department of Business Administration April 10, 2013

2 Decision Making, The Easy Case There is one case where deciding between two projects is easy: no matter what happens, one project is always better than the other. Taking advantage of this situation is an important activity in financial markets, known as arbitrage. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

3 Example: Currency Forwards A currency forward is a contract today that locks in an exchange rate for some date in the future. The future exchange rate is known as the forward rate. (No money changes hand today.) A typical example would be a contract to trade 1 CHF for 1.05 USD in 3 months. Companies use forwards to avoid exchange rate risk changes in the exchange rate that hurt profitability. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

4 Synthesizing Currency Forwards Consider these two strategies. Enter into a currency forward to trade 1 CHF for F USD in 3 months. In 3 months you have -1 CHF, and +F USD. Suppose that you can borrow in CHF for 3 months at an interest rate of r and lend in USD at an interest rate of r f. Let today s USD-CHF exchange rate (the spot rate) be S. Then do the following: 1 Borrow 1/(1 + r) CHF for 3 months. 2 Convert it into S/(1 + r) USD. 3 Lend the USD for 3 months. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

5 Synthesizing Currency Forwards, cont d In 3 months, you have to pay back 1 CHF, but you make USD. S(1 + r f )/(1 + r) If F > S(1 + r f )/(1 + r), everyone will want forwards. If F < S(1 + r f )/(1 + r), everyone will skip the forward, and borrow CHF and lend USD instead. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

6 Arbitrage But there s more. In financial markets, you can sell the overvalued asset, and use the proceeds to buy the undervalued asset. This means you can make risk-free profits. You can frequently do this even if you don t own the underlying asset. (For example, if you can borrow stock you don t own and sell it.) Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

7 Arbitrage, Overvalued Forward If F > S(1 + r f )/(1 + r), then you can costlessly arbitrage as follows: Today Tomorrow Transaction CHF USD CHF USD Forward CHF to USD -1 F Borrow USD S/(1 + r f ) S (1+r f ) (1+r) Convert to CHF 1/(1 + r f ) S/(1 + r f ) Lend CHF -1/(1 + r f ) 1 Total F S (1+r f ) (1+r) Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

8 Arbitrage, Undervalued Forward If F < S(1 + r f )/(1 + r), then you can costlessly arbitrage as follows: Today Tomorrow Transaction CHF USD CHF USD Forward USD to CHF 1 F Borrow USD S/(1 + r f ) S (1+r f ) (1+r) Convert to CHF 1/(1 + r f ) S/(1 + r f ) Lend CHF -1/(1 + r f ) 1 Total F S (1+r f ) (1+r) Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

9 Calls A call is a contract that gives you the option to buy a stock at a future date at a fixed price X (known as the strike price) by a fixed date T (the expiration date). Let S T be the stock price on day T. If S T > X, then you exercise the option, for a net payoff of S T X. If S T < X, then you let the option expire, for a net payoff of zero. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

10 Puts A put is a contract that gives you the option to sell a stock at a future date. If S T < X, then you exercise the option, for a net payoff of X S T. If S T > X, then you let the option expire, for a net payoff of zero. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

11 American versus European Puts and calls are both known as options, since they give the option to do something later. There are some variations on when you may exercise your option: For American options, you can exercise any time up to the expiration date. European options can only be exercised on the expiration date. Insert your joke about cultural differences here. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

12 Put-Call Parity Suppose that you buy a call and sell a put with the same strike price and expiration date. If S T > X, then you will exercise the call, for a payout of S T X at expiration. If S T < X, then your customer will exercise the put, for a payout of S T X at expiration. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

13 Put-Call Parity, cont d So buying a call and writing a put are a complicated way of bringing about a payout of S T X at date T. You can do the same by buying the stock today for S 0 and borrowing X /(1 + r) until time T. Suppose that the price of the put and call are p and c. Then put-call parity should hold,. p c = X /(1 + r) S o Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

14 Extending the Idea We can take this idea further. If we are willing to postulate a model for stocks, we can (under some circumstances) use the same idea to figure out the price of the option. We do this by replicating the payoffs from the option by using: The underlying stock. A risk-free asset (like government bonds), which have a return of r. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

15 One-Step Binary Trees Suppose that tomorrow s stock price can only be one of two values: S d or S u. (d = down, u = up) Suppose we have an option that pays either P d or P u tomorrow. Then using the stock and the risk-free bond, we can replicate the payoff. P u = as u + b(1 + r) P d = as d + b(1 + r) Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

16 One-Step Binary Trees We can solve this system of equations for a and b. a = b = = P u P d S u S d r (P u as u ) r (P d as d ) The price of the option should be as + b. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

17 Delta Hedging The quantity a is known as the option s, = P u P d S u S d This method of determining the price is sometimes known as delta hedging. In general, hedging is the activity of owning an asset to cancel out another risk you face. If you sell the option, you can hedge your risk completely by buying shares. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

18 Delta Hedging, cont d The initial investment of the portfolio of 1 option, and stock is b, while the payoff is b(1 + r). The portfolio is risk-less, so has a return equal to the risk-free rate. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

19 Risk Neutrality Notice that we don t need to know the probability of an up or down move in this calculation. In fact, we can back out a market probability that the price will be up or down. Let p be the price of an option that pays 1 + r if u happens, and 0 othewise. Let q be the price of an option that pays 1 + r if d happens, and 0 othewise. If you own both options, you receive 1 + r no matter what happens, so owning both has the same payoff as the risk-free rate, and should have the same price as the risk-free bond: 1. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

20 Risk Neutrality, cont d Thus p + q = 1 satisfies the laws of probability. p and q are known as the risk-neutral probabilities of u and d, respectively. In economics, these are known as Arrow-Debreu securities. Any other option can be priced using these two numbers. An option that pays off P u and P d can be replicated by buying P u /(1 + r) of the first security and P d /(1 + r) of the second security. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

21 Risk Neutrality, cont d The price is then r (P up + P d q), which is the expected value of the payoff under this probability distribution, discounted by the risk-free rate. This is known as risk-neutral pricing. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

22 More than Two States In this set-up we have two underlying securites (the stock and bond), and two states. This generalizes to n securities and n states.this is not the direction we will go. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23

23 More Than One Step Instead, we consider multiple time steps, each allowing a single up or down move. If the time steps are small enough, the possible final states become large. More importantly, you can approximate many general stochastic processes this way. Walt Pohl (UZH QBA) Stochastic Models April 10, / 23