Derivatives 1 BFC2751 Forward and Futures Prices Investment Assets Consumption Assets Forward Pricing Value of Forward Contracts *+ Assets held by significant Assets held primarily for To avoid , %" = /"( (no income, no storage costs) *+ numbers of investors for consumption purposes - If %" > /"( , arbitrageurs can buy the asset and forward contracts on the At time !", contract value is zero asset *+ investment purposes (do not - If %" < /"( , arbitrageurs can short the asset and forward contracts on the )*+ have to be exclusively for Value of long : # = %" − ' ( asset investment (e.g. silver has Value of short forward contract: # = ' − % ()*+ Known Income ( paying known , coupon bonds) " *+ industrial uses) - %" = /" − 3 ( Forward v Futures Pricing - 3 is PV of income during contracts , bonds, gold, silver Copper, oil, corn Known (known yield rather than known income [%]) - Same when short-term interest rates are - % = / ( *)4 + Can use arbitrage arguments to determine forward/futures prices " " constant and both contracts have same maturity - 5 is average yield during contract of investment assets but not consumption assets - If strong correlation between IR and asset price, Currencies (currency trading) futures price is slightly higher *)*6 + - %" = /"( Short Selling Asset ↓(IR↓) → immediate gain (daily settlement) → gain invested at - Underlying asset: one unit of foreign currency (same as known yield) ↑ rate - /": spot price (local), %": (local), 0: local risk-free rate, 07: Short selling: selling an asset you don’t own / foreign risk-free rate Investment Commodities (can provide income, have storage costs) Possible for some, but not all investment assets *+ 1. Broker borrows securities from another client and sells them in Normal backwardation: futures price < expected future spot price - %" = /" + 9 ( for $ storage costs (inverted futures curve) - % = / ((*;<)+ for % storage costs usual way Contango: futures price > expected future spot price (pay premium to " " - Storage costs can be treated as negative yield 2. Later, you must pay securities back (to replace other client’s have delivery delayed) Consumption Assets (↑ storage,↓ income, but consumption value) securities) Summary - Suppose % > / + 9 (*+ - Investor makes profit is stock price ↓ " " - Arbitrageur can borrow /" + 9, buy and store the commodity, and short a - Must pay dividends and other benefits to owner of securities *+ - account kept with broker as insurance - Riskless profit = %" − /" + 9 ( > 0 - Relation cannot hold for long *+ Assumptions - Suppose %" < /" + 9 ( - Arbitrageur can sell the commodity, save the storage costs, invest in risk-free rate, and long a futures contract - No transaction costs *+ - Riskless profit = /" + 9 ( − %" > 0 - Same tax rate for everyone / every trade Other things to Consider - Nothing to stop relation from holding - Can borrow/lend money at same risk-free rate - %- = .-(*+ Convenience Yield (benefit derived from holding a physical asset) - Everyone takes advantage of arbitrage opportunities - .- = %-()*+ - % = / ( *;<)? + )*+ " " - ( discounts future CFs back to PV - Balances out previous inequalities Notation - Price is what you pay, value is what you get - If market is efficient: market = value (@) = interest cost – income earned + storage cost - Arbitrage occurs when market is inefficient - /": spot price today (market price) Non- paying stock @ = 0 For an investment asset: A+ - %": futures/forward price today %" = /"( - !: time to maturity Dividend paying stock @ = 0 − 5

- 0: risk-free Currency @ = 0 − 07 For a consumption asset: A)? + %" = /"( Commodity @ = 0 − 5 + B Trading Strategies Involving Options Covered Call, , Protective Put

- Covered call = long stock + short call for Hedging - Protective put = long stock + long put - If prices remain constant, call premium received ↑ return from Options alone are risky investments, however, - Obtain protection against a down market in the underlying holding stock – short term while retaining upside potential when combined with other assets (i.e. stocks) they Net payoff: ! − ! − max ! − (, 0 + , Net payoff: ! − ! + max ( − ! , 0 − 1 " $ " can reduce overall investment risk " $ " !$ is the cost of stock !$ is the cost of stock

Payoff Table !" ≤ ( !" > ( Covered Call and Protective Put Payoff Table !" ≤ ( !" > ( Long stock ! ! Long stock ! ! " " Covered Call Protective Put " " Short call 0 ( − !" - Provides premium income - Provides downside protection Long put ( − !" 0 (from short call) (from long put) Payoff !" ( - Beneficial if price stays - Beneficial if share price Payoff ( !" flat increases Net payoff !" − !$ + , ( − !$ + , - Payoff equivalent to short put - Payoff equivalent to long call Net payoff ( − !$ − 1 !" − !$ − 1 Holding stock, expect small price movement in near future Holding stock, concerned price may decrease in near future Collar

- Collar = long stock + long put + short call - Imposes floor and ceiling on value of existing investment (covered call + protective put) - Zero cost collar: premium received from SC = premium paid for LP Earn income through premium received for writing option Net payoff: !" − !$ + max (2 − !", 0 − 1 − max !" − (, 0 + , Loss on stock limited by longing Long put with K2 < !$ ! > ( ! > ( If " : gain from stock, received premium, loss on short call Short call with (5 > !$ If " : gain from stock, pay premium, put not exercised Total payoff = premium received Total payoff = gain from stock – premium paid Reduce cost of insurance, at expense of sacrificing upside gain If !" ≤ (: loss from stock, receive premium, call not exercised If !" ≤ (: loss from stock, pay premium, gain from put Total payoff = premium received – loss on stock Total payoff = premium paid

Payoff Table !" ≤ (2 (2 < !" ≤ (5 !" > (5

Long stock !" − !$ !" − !$ !" − !$

Long put on (2 (2 − !" − 1 −1 −1

Short call on (5 , , − !" − (5 + ,

Payoff (2 !" (5

Net payoff (2 − !$ − 1 + , !" − !$ − 1 + , (5 − !$ − 1 + , Black-Scholes-Merton Model THE BSM Model Applying the BSM Model Assumptions The PV of the expected (call) option value is: 1. Calculate VW and V@ (at least 4 decimal places) 01& - , = / 2 max 0, $& − ' - GBM: log-normal distribution with constant - 2. Look up =(VW) and = V@ Shift 3 on calc. gets this (VW → shift Same intuition behind the binomial model: trading is continuous → 3) - Option price = PV(expected option payoff) - Short selling of securities is permitted 3. Compute option price using the BSM equation - No taxes or transaction costs The key to option pricing is predicting the distribution of future - No dividends during the life of the 4. Check put-call parity (to verify – if necessary) share price movements - No riskless arbitrage opportunities = −VW = 1 − =(VW) - The risk-free interest rate is constant Main Assumptions of BSM Model - Options are European (non-closed forms are possible for American) Call Prices and Replication - Assumes stock price evolves according to a random process Notation named Geometric Brownian Motion (GBM) To replicate a call in general, we must: Implies two conditions: - 0: current date - Take a long in Δ\ units of the underlying, and - 1. Stock prices follow a log distribution with mean ( and constant volatility * #: maturity date (time left to maturity) - Invest ]\ at the risk-free rate - 2. Stock prices must evolve smoothly; they cannot jump (the market cannot ‘gap’) $%: current stock price - Negative ]\ indicates borrowing - The log-normal assumption: $&: stock price at time # - ': The cost of replicating this portfolio is: - The natural log of returns is normally distributed - (, *: (continuously-compounded) and volatility of - , = $%Δ^ + ]^ 9: @ stock returns (annualised) - ln ~ =((?, * #) - +: risk-free rate - , = $ Δ − ] 9; % ^ _ - - Mathematically, log-returns and continuously-compounded ,, -: prices of call and put (European) The BSM model has an identical structure: - , = $ = V − /01&'=(V ) returns represent the same thing: The Formula % W @ 9: 9: D D Therefore… - ln = B ⇔ = / ⇔ S& = $%/ 9; 9; 01& - Δ = = V , = $%= VW − '/ =(V@) ^ W 01& - ] = /01&' ∗ =(V ) Volatility * - = '/ = −V@ − $%=(−VW) _ @ =(B) is the probability under a standard normal distribution that an = VW = Δ\ Volatility in BSM model refers to standard deviation of annual log- observation is less than or equal to B The term =(V ) also has special interpretation which can be returns @ @ 1 $% * understood by considering the risk-neutral derivation of call price BSM model treats volatility as constant (assume it is known for now) VW = ln + + + # * # ' 2 - Under risk-neutral pricing, the call price is given by: Δ# (Δ# - At time : Mean = , Standard deviation = * Δ# - 01& V@ = VW − * # , = / 2 max $& − ', 0 - Volatility p. a. = Volatility p. td.∗ Trading days per annum - Where 2[B] denotes expectations under the risk-neutral probability - 252 trading days pa When $% becomes very large or when volatility approaches zero: - Then, we write - is most certain to be exercised 01& 01& - Issues with BSM model: volatility typically not constant, stock prices can sometimes ‘jump’ - , = $% − '/ , = / 2 $& − ' ∗ bc - Both VW and V@ become very large, and =(VW) and - Where =(V@) become close to 1 - Price of put option approaches zero – becomes worthless Breaking into two terms we get: 01& 01& - - = 0 (put becomes worthless) - , = / 2 $& ∗ bc − / 2 ' ∗ bc - =(−VW) and =(−V@) are both close to zero - First term = PV of what we receive from exercising the call - $%=(VW) N(V@) is just the probability of $& ≥ K - Second term = PV of what we pay on account of the 01& - i.e. the risk-neutral probability of option finishing in the money - / '=(V@)