Lecture 5. Trading With Portfolios. 5.1 Portfolio. How Can I Sell Something I Don t Own?

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Everett Barker
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1 Lecture 5 Trading With Portfolios How Can I Sell Something I Don t Own? Often market participants will wish to take negative positions in the stock price, that is to say they will look to profit when the stock goes down. This practice is called short selling. But how can you sell something if you don t own it? Well, traders are able to do this by exchanging an IOU (I owe you) contract with a stock holder to gain ownership of the stock for a period of time. They then sell the stock on the market and invest the cash, waiting for the stock price to drop so they are able to buy it back for a lower price. Once the trader has bought it back they are able to hand the stock back to the holder cancelling out their IOU. There are laws surrounding short selling as it can be used to artificially raise supply (the number of stocks for sale) above the actual number of holders who want to sell, and since an increase in supply normally results in a drop in price so there is obviously an incentive to do this. It has been blamed for several market crashes and is normally restricted or even completely banned under certain circumstances. 5.1 Portfolio A portfolio is range of investments held by an individual. We assume that portfolios may contain both positive and negative positions in stocks, bonds, call and put options. Mathematically, we tend to denote portfolios as Π, and your position in each asset will be positive if you are holding it (long) and negative if you are selling it (short). Example 5.1. Starting from zero, create a portfolio that is long or short one share by borrowing or investing money. 42

2 Solution

3 Therefore the following portfolio Π = S + 2C P describes a situation where the investor is long a stock, long two call options and short one put option. Example 5.2. Draw the payoff diagrams for going both long and short on both a put and call. Solution 5.2. See presentation slides for Lecture Trading With Portfolios There are a variety of reasons why an investor may choose to buy a combination of options, stocks and bonds rather than just one or the other. We call the combination of financial contracts that an investor holds the portfolio, and in this lecture we discuss some of the common combinations that might be held by investors. We look at the payoff of those portfolios, and how to construct the payoff diagrams for the resulting portfolio. It is common for students to ask why an investor would take a certain position, particularly if it looks as though the payoff at the end could be negative. To understand this you need to remember that the price of taking a certain position will be related to the likely payoffs there is no easy way to make money on the stock market. If there is a large probability that there will be a negative payoff at the end of the contract (you pay out money rather than receiving it) then it is likely that the portfolio will be very cheap (or even negative in price, so you receive money at the start) to set up. The investor has to balance up the risk of winning and losing against 44

4 their view on the market. Setting up different portfolios will allow them to maximise returns given what they think will happen in the future. There are two main concerns for an investor that is trading in the market, that is profit and risk. In Lecture 6 we expand on the definition of risk, which is related to the variance of the payoffs. When an investor buys or sells a contract they either wish to increase their profit or hedge their risks. If the investor wishes to make a profit, they will normally design a portfolio with large positive payoffs that increases the risk. If the investor wishes to hedge or reduce risk, they design or adapt their existing portfolio so that both potential profit or losses are lower. Hedge is something an investor does to reduce their risk. This usually means that they are protecting themselves against a large financial loss. However in Financial Mathematics a reduction in risk might mean that both large losses and profits are avoided, resulting in a less risky portfolio Straddle One of the most common portfolios is the straddle. This involves buying both a call option and a put option with the same strike price and maturity date at the same time. The value of this portfolio is given by Π(S, t) = C(S, t; E) + P (S, t; E) (5.1) and therefore the payoff at maturity (t = T ) is Π(S, T ) = { E S if S < E (you exercise the put) S E if S E (you exercise the call). (5.2) So this has large positive winnings if the stock price has a large downward movement or if the stock has a large upward movement. Given that the investor wins both ways, it follows that the cost of setting up such a portfolio will be relatively high. Example 5.3. Draw the payoff diagram for going long on a straddle and short on a straddle. Solution 5.3. Fill in Figure 5.1 and Figure 5.2. The opposite position of a short straddle is what you get if you sell those same two options. We write the value of this portfolio as Π(S, t) = C(S, t; E) P (S, t; E). (5.3) 45

5 Figure 5.1: Payoff diagram for a long straddle. 46

6 Figure 5.2: Payoff diagram for a short straddle. 47

7 At maturity we have Π(S, T ) = { (E S) if S < E (the buyer exercises the put) (S E) if S E (the buyer exercises the call). (5.4) Now this contract has negative payoff in all scenarios, so why would you do it? Well if the payoff is always negative then the person you sell it to you will have to give you a lot of money at the start. This means that you make money by hoping that there isn t a large movement up or down. As long as movements in stock price are small you are able to charge enough at the start to still make a profit after you have paid out any winnings at the end. This can be risky as you tend to win small amounts very often but there is always a small chance you might lose big and potentially go bankrupt making this a very risky strategy! Example 5.4. An example of large profits. Assume that: S 0 = 40, E = 40, C 0 = 2, P 0 = 2. Can you find the investors expected return if they believe that the stock price at T is modelled by the following tree? S 0 = 40 p = 3 4 p = Solution

8 5.2.2 Bull Spread A bull spread is a slightly cheaper way to bet on an upward movement in the stock price than just buying the call option. We create it by buying a call then selling one with a slightly higher exercise price, this gives Π(S, t) = C(S, t; E 1 ) C(S, t; E 2 ) (5.5) where E 2 > E 1. So why would you invest like this? Well if you expect only a small movement in price this is a cheaper way to exploit this than buying the call option. This can be considered as a hedge on the initial investment of buying the call, even though it is limiting potential winnings (as opposed to losses as you might expect). Example 5.5. Sketch the payoff diagram for a bull spread. Solution

9 Fill in Figure 5.3. You may hear the term bull to describe market conditions, as a bull market is one in which confidence is high and stocks are growing. The term bear indicates the opposite, and it refers to a situation where stocks are losing value, so a bear spread is used exploit down movements in price. We create a bear spread by buying and selling a put option, or Π(S, t) = P (S, t; E 1 ) P (S, t; E 2 ) (5.6) where E 1 > E 2. Example 5.6. Sketch the payoff diagram for a bear spread. Solution 5.6. Left as an exercise. 5.3 Risk Free Investments When dealing with contracts that make payments in the future there is always a risk that you won t get paid, even banks sometimes run out of money! In economics, we like to imagine that there does exist an investment in the economy that is completely risk-free, and we can then use this investment to compare against our options, portfolios etc. Even before we begin applying our model to the stock price there are things we can say something about the price of financial contract, for instance the upper and lower bounds, that must be true under any situation. The risk free investment is so important because the true market price (the price everyone agrees on) is known, since the only thing that makes a value subjective is risk. We talk about this more in the next lecture, but the main focus of this course is describing a strategy in which you can take risky financial derivatives and make them completely risk free. This ability to effectively remove risk is what transformed the world of investment banking. 50

10 Figure 5.3: Payoff diagram for bull spread. 51

11 5.3.1 Bond A Bond is a contract that yields a known amount F, called the face value, on a known time T, called the maturity date. The authorised issuer (for example, government) owes the holder a debt and is obliged to repay the face value at maturity and may also make interest payments (the coupon). We will assume in this course that the only bonds traded are those issued by governments that can be assumed to be risk free. The term B(t) will denote a risk free investment in government bonds. Face Value/Principle The final payment amount as described in the contract, usually denoted F. The holder of the bond can collect the payment of F on the maturity date. Maturity The maturity date is the date written in the contract on which the holder with receive the face value payment. Interest Rate This is the growth rate of the bond, or how much value is added for the holder over a period of time. If I deposit money into a restricted access savings account in a bank then the bank is effectively selling me a bond. Imagine I pay them say 1000 now to receive 1100 in five years time. I have bought a bond with face value F = 1100 and maturity date T = 5, the interest rate could be worked out from e r 5 = 1.1. The bank now owes me money and I am hoping that they will have the money in five years time to pay me! Coupon These are payments from the seller of the bond to the buyer. Usually if the bond has a very long maturity date (they can be up to 50 years or more!) the value of the bond would be very low without these payments. Coupons enable the seller of the bond to receive more money from the initial sale. Zero Coupon Bond Simply a bond that doesn t pay coupons. Example 5.7. Given that the return on a risk free bond can be defined as db B = rdt, (5.7) where r is the risk-free interest rate. Calculate the value of the bond if B(t = T ) = F and r is constant. 52

12 Solution 5.7. No-Arbitrage Principle In the next lecture we introduce the no-arbitrage principle which is the bedrock of mathematical finance. It allows us to draw equivalence between different types of financial contracts and construct arguments on how to price financial contracts. 53

Mathematics of Financial Derivatives

Mathematics of Financial Derivatives Lecture 8 Solesne Bourguin bourguin@math.bu.edu Boston University Department of Mathematics and Statistics Table of contents 1. The Greek letters (continued) 2. Volatility