Derivative Securities Law of One Price Payoff Diagrams for common derivatives Valuation of Derivatives Put-Call Parity

Derivative Securities The Law of One Price Two assets with identical cash flows in the future will have the same current price in an arbitrage-free market Sometimes the “assets” may be a portfolio, or combinations of smaller assets in order to generate the similar cash flows

Payoff Diagram for Forward Contracts Suppose you know that on September 30 th you will want to sell a bushel of corn. You have some alternatives: 1) Wait until 9/30 and take the then current price 2) Agree with another party on a price you both will honor on 9/30 – this would take away some of the risk of the price dropping drastically between now and 9/30

Payoff Diagram for Forward Contracts You enter into a forward contract with a cereal maker to sell one bushel of corn to him at $15.30 What will be the gain or loss to you on 9/30? Clearly dependent upon the price of a bushel of corn on 9/30…

Payoff Diagram for Forward Contracts Gain to you will be cash you receive for the bushel of corn less the market value of the bushel of corn on 9/30 Example: If corn is selling for $15.40 on 9/30, then you will have received $15.30 from the cereal maker and given away an asset worth $15.40 Your loss is $0.10 – Same as the cereal maker’s gain

Payoff Diagram for Forward Contracts Gain to you will be cash you receive for the bushel of corn less the market value of the bushel of corn on 9/30 Example: If corn is selling for $15.00 on 9/30, then you will have received $15.30 from the cereal maker and given away an asset worth $15.00 Your gain is $0.30 – Same as the cereal maker’s loss

Payoff Diagram for Forward Contracts Gain for agreeing to sell an asset through a forward contract is (often called going “short” forward): Cash Received – Current Value of Asset Sold Gain for agreeing to buy an asset through a forward contract is (often called going “long” forward): Current Value of Asset Sold - Cash Paid

Notation In notational terms, we can define: S t = Market Value of the underlying asset at time t T = Time of maturity of futures F(t, T) = Strike price of futures Long Futures Payoff = S T - F(t, T) Note that it is dependent on the Market Value at T

Forward Prices We can use the Law of One Price to get at what the theoretical strike price is for a forward contract….. Consider Portfolio 1: Purchase the asset in the spot market At time t, it is worth S t As we get closer to time T, there is a net cash benefit of owning the asset, such as dividends received, less storage costs, etc. Let these net benefits be D. At time T, the portfolio is worthS T + D

Forward Prices We can use the Law of One Price to get at what the theoretical strike price is for a forward contract….. Consider Portfolio 2: Agree to pay F at time T, and also want to have a total of D at time T At time t, deposit (F+D) e -r(T-t) in an account to earn a risk free rate, r As we get closer to time T, interest is earned on the account At time T: The account has grown to (F+D) We pay F to acquire the asset Our holdings are now worth S T + D

Forward Prices We can use the Law of One Price to get at what the theoretical strike price is for a forward contract….. Portfolio 1 and Portfolio 2 are both have a payoff of S T + D at time T so under the Law of One Price the two portfolios must be equal at time t Implies S t = (F+D) e -r(T-t) F = S t e r(T-t) – D So the “correct theoretical” forward settlement price is the current spot increased at risk free interest less the net benefit of owning the asset. Market supply and demand, however, will assist in adjusting the theoretical price to an actual price. Sometime D is positive, sometimes negative, sometime negligible For some assets, D is expressed as a rate of interest so that F = S t e (r-d)(T-t)

Payoff Diagram for Option Contracts Call Option: The right to purchase an asset from another party at a pre-determined price (K) on or before a point in time (T) Typically a price involved to purchase this right and is called the call “premium” – denote this as c. If market price of the asset at T is larger than K, then you will exercise the call. You could buy the share for K, immediately sell it for S T. Your payoff is S T – K - c

Payoff Diagram for Option Contracts Put Option: The right to sell an asset to another party at a pre-determined price (K) on or before a point in time (T) Typically a price involved to purchase this right and is called the call “premium” – denote this as p. If market price of the asset at T is less than K, then you will exercise the put. You could buy the share for S T, immediately sell it for K. Your payoff is K – S T - p

Payoff Diagram for Option Contracts Back to the Law of One Price….. What is the payoff diagram for owning an asset? Is there some way to combine puts, calls and asset to get some similar pictures?

Payoff Diagram for Option Contracts We can use these ideas to create an equation called put-call parity. If we know the prices of three of the following four variables, we can solve for the remaining one. 1. p = Price of Put 2. c = Price of Call 3. S = Price of underlying asset 4. r = risk free rate of return c – p = S – Ke -r(T-t)

Payoff Diagram for Option Contracts For put-call parity to hold, the following assumptions must be made: No dividends payable on underlying asset Borrowing and investing available at risk-free rate, r No transaction fees or taxes Short selling and borrowing allowed in any fractional quantities No arbitrage opportunities

Option Valuation European versus American Options: European options can be exercised only on the expiration date – no interim right to exercise American options can be exercised at any time up to and including the expiration date Most option valuation routines are derived assuming a European option as an initial basis - Easier to think about the defining event occurring at one point in time - Since American options have at least as many rights as European options, the value of a European option can be thought to be a minimum value for the value of an American option

Option Valuation Mathematics of Option Valuation – probably one of the most important areas of modern finance Robert C. Merton and Myron S. Scholes won the 1997 Nobel Prize in Economics for their work in this field Unfortunately, Fischer Black died in 1995 before he could be properly recognized We’ll look a little bit of the math and then describe how you can move from simple models to rigorous development

Option Valuation Fundamental underlying concept is that the market value of a European-style derivative contract based on an underlying asset is equal to a combination of the market values of the asset and a default-free bond Essentially, values of derivatives can be replicated by forming a portfolio with market value nS – B; the key is how much n and how much B – that is what we’ll try to solve for

Option Valuation Perhaps a first easy example: We’ll create a function defining the payoff of the derivative as f(s). So the cash flow from the derivative would be f(S T ) at maturity Let f(S T ) = S T This is a derivative is the most simplest form. To replicate the payoff of the derivatives, how much n and B do we need? Clearly n = 1 and B = 0. Note that n and B can be negative if we need it to be, implying selling the asset short or loaning risk-free amounts instead of borrowing.

Option Valuation In our simple model, let’s consider what would happen if one of two ending states occur. Either… The asset can appreciate at rate u and grow to S t e u(T – t) Or, the asset can depreciate at rate d and grow to S t e d(T – t) This is kind of a one-period Bernoulli trial It has two possible outcomes and we could place probabilities on the likelihood of each outcome

Option Valuation Now let’s see what happens when we consider the two states in the “derivative cash flow” function, f(s) Let f u = f(S t e u(T – t) ) = cash flow in “up” state Let f d = f(S t e d(T – t) ) = cash flow in “down” state Then using our replicating portfolio, we can create two equations with two unknowns: f u = n(S t e u(T – t) ) – Be r(T-t) asset will appreciate and loan will grow f d = n(S t e d(T – t) ) – Be r(T-t) asset will depreciate and loan will grow

Option Valuation Solving for n and B: nS t = (f u – f d ) / (e u(T – t) - e d(T – t) ) B = e r(T – t) [(f u e d(T – t) - f d e u(T – t) ) / (e u(T – t) - e d(T – t) )]

Option Valuation Solving for n and B: So….. If we use the most simple derivative payoff function, f(S T ) = S T Then nS t = (f u – f d ) / (e u(T – t) - e d(T – t) ) = [f(S t e u(T – t) ) – f(S t e d(T – t) )] / (e u(T – t) - e d(T – t) ) = (S t e u(T – t) – S t e d(T – t) ) / (e u(T – t) - e d(T – t) ) = S t (e u(T – t) – e d(T – t) ) / (e u(T – t) - e d(T – t) ) = S t This implies n = 1.

Option Valuation Solving for n and B: So….. If we use the most simple derivative payoff function, f(S T ) = S T Then B = e r(T – t) [(f u e d(T – t) - f d e u(T – t) ) / (e u(T – t) - e d(T – t) )] = e r(T – t) [f(S t e u(T – t) ) e d(T – t) - f(S t e d(T – t) ) e u(T – t) ] / (e u(T – t) - e d(T – t) ) = e r(T – t) [S t e u(T – t) e d(T – t) - S t e d(T – t) e u(T – t) ] / (e u(T – t) - e d(T – t) ) = 0

Option Valuation The General Result: When you know the payoff function of the derivative, you can generate a replicating portfolio for the derivative in the general form of nS – B If you write out “ns – B”, the value can generally be thought of as a “discounted probability-weighted payoff” nS – B = e r(T – t) [ q f u + (1-q) f d ] where q = (e r(T – t) - e d(T – t) ) / (e u(T – t) - e d(T – t) )

Option Valuation So for a derivative contract we can imply that the value of the derivative is e r(T – t) [ q f u + (1-q) f d ] For a specific derivative, say a call option, we can then factor in its specific payoff function. Here, f(s) = (S – K) if S is high enough, or f(s) = 0 if S is below the strike. We know that the strike, K, will have to either fall below Se d(T – t), between Se d(T – t) and Se u(T – t), or above Se u(T – t) SituationPayoff K > Se u(T – t) 0 Se d(T – t) < K < Se u(T – t) e r(T – t) [ q f u ] K < Se d(T – t) e r(T – t) [ q f u + (1-q) f d ]

Option Valuation Rewriting the payoffs: SituationPayoff K > Se u(T – t) 0 Se d(T – t) < K < Se u(T – t) e r(T – t) q (Se u(T – t) – K) K < Se d(T – t) e r(T – t) [ q (Se u(T – t) – K) + (1-q) (Se d(T – t) – K) ] The last payoff can also be written as: e r(T – t) [ q (Se u(T – t) – K) + (1-q) (Se d(T – t) – K) ] e r(T – t) [ q Se u(T – t) – qK + Se d(T – t) – K - q Se d(T – t) + qK] e r(T – t) [ q Se u(T – t) + Se d(T – t) – K - q Se d(T – t) ] e r(T – t) [ q Se u(T – t) + (1-q) Se d(T – t) – K]

Option Valuation Rewriting the payoffs: SituationPayoff K > Se u(T – t) 0 Se d(T – t) < K < Se u(T – t) e r(T – t) q (Se u(T – t) – K) K < Se d(T – t) e r(T – t) [ q Se u(T – t) + (1-q) Se d(T– t) – K] From this, we can make some general conclusions for this “simplistic” value of a call option: The value is going to be dependent on the value of the underlying asset and some sort of probability distribution of how that asset appreciates in value between now and maturity The value is going to be dependent on the present value of the strike price and some sort of probability distribution of how the asset appreciates in value between now and maturity

Option Valuation Black and Scholes made some additional assumptions around this model and came up with this valuation of the call option: c = S N(d 1 ) - K e -r(T-t) N(d 2 ) Where d 1 = [ ln (S/K) + (r + σ 2 / 2) (T – t) ] / (σ ) d 2 = d 1 - (σ ) N(x) denotes the cumulative standard normal distribution function