DERIVATIVES 2020 Assoc. Prof. Dr. Greta KeliuotytΔ-StaniulΔnienΔ Faculty of Economics and Business Administration Finance Department 2. Basics of Derivative Pricing and Valuation 4 Pricing the Underlying Asset The four main types of underlying assets on which derivatives are based include: Equities Fixed-income securities Currencies Commodities The price of a financial asset is often determined using a present value of future cash flows approach. Determining a rate at which to discount the expected future cash flows is challenging. 5 Expectation 6 Finding current price 7 Finding current price (asset with cash flows) 8 Pricing equations for Spot assets The value of an asset today, S0, is the present value of the expected future price of an asset with no interim cash flows, E(ST), discounted at r (the risk-free rate) plus ο¬ (the risk premium) over the period from 0 to T. πΈ ππ π0 = π 1+π+λ If the asset has interim cash flows (benefits such as dividends or convenience yield) or costs (such as storage), we must adjust the pricing equation as follows: πΈ ππ π0 = π−θ+γ 1+π+λ Where γ: is the present value of any benefits and θ: is the present value of any costs 13 Pricing vs valuation Example. Long stock, short forward. S0=102, F0=100, T=1, r=0.04 PV=100/(1.04) = 96.15 V=96.15 − 102 = −5.85 14 Pricing and Valuation of Forward Contracts F0(T): The forward price established at the initiation date of contract. VT(T): The value at expiration of the forward contract. At expiration (Time = T), the value of a forward contract is VT(T) = ST – F0(T) When a forward contract is initiated (Time = 0), it is valueless to both the long and short positions. V0(T) = 0 15 Holding period of forward contract 16 Pricing and Valuation of Forward Contracts F0(T): The forward price established at the initiation date of contract. S0(T): The initial price of the underlying asset. T: The time to expiration of the forward contract. r: The risk-free rate of interest. Forward Pricing Equation: F0(T) = S0(1 + r)T 17 Forward Pricing example Assume an asset sells in the spot market for a price of $94, the risk-free rate is 4%, and the forward contract expires in six months. What is the initial value of the contract and the correct forward price? 17 Forward Pricing example Assume an asset sells in the spot market for a price of $94, the risk-free rate is 4%, and the forward contract expires in six months. What is the initial value of the contract and the correct forward price? S0(T) = $94 The value of the contract at initiation is V0(T) = 0 T = 6 months = 0.5 years 17 Forward Pricing example Assume an asset sells in the spot market for a price of $94, the risk-free rate is 4%, and the forward contract expires in six months. What is the initial value of the contract and the correct forward price? S0(T) = $94 The value of the contract at initiation is V0(T) = 0 T = 6 months = 0.5 years Forward Pricing Equation: F0(T) = S0(1 + r)T = $94(1 + 0.04)0.5 = $95.86 The forward price established at the initiation date of the contract is $95.86 18 Valuation of Forward Contracts with Costs and Benefits Forward Pricing Equation: F0(T) = (S0 – γ + θ)(1 + r)T Where: F0(T): The forward price established at the initiation date of contract S0(T): The initial price of the underlying asset T: The time to expiration of the forward contract r: The risk-free rate of interest θ: Costs to hold the spot asset: Monetary costs include storage, insurance γ: Benefits to hold the spot asset: Monetary benefits include dividends, interest Nonmonetary benefits include convenience yield 19 Forward Pricing example with Costs And Benefits Assume an asset sells in the spot market for a price of $94. The risk-free rate is 4%, and the forward contract on the asset expires in six months. If the asset pays a dividend of $2.25 and storage costs are $1 a year, what is the correct forward price? 19 Forward Pricing example with costs And Benefits Assume an asset sells in the spot market for a price of $94. The risk-free rate is 4%, and the forward contract on the asset expires in six months. If the asset pays a dividend of $2.25 and storage costs are $1 a year, what is the correct forward price? S0(T) = $94 T = 6 months = 0.5 years r = 4% θ = $1 (storage costs) γ = $2.25 (dividend) 19 Forward Pricing example with costs And Benefits Assume an asset sells in the spot market for a price of $94. The risk-free rate is 4%, and the forward contract on the asset expires in six months. If the asset pays a dividend of $2.25 and storage costs are $1 a year, what is the correct forward price? S0(T) = $94 θ = $1 T = 6 months = 0.5 years γ = $2.25 r = 4% Forward Price with carrying costs and benefits: F0(T) = (S0 – γ + θ) (1 + r)T = ($94 – $2.25+$1)(1.04)0.5 = $94.59 22 Forward vs. Futures Prices Futures contracts are marked to market daily. Accumulated futures gains and losses are built up in the margin account. Futures gains can be invested and collect interest. Futures losses must be borrowed and pay interest. Forward contracts’ gains and losses are settled at expiration. If interest rates are positively correlated with futures prices: futures prices > forward prices If interest rates are uncorrelated with futures prices: futures and forwards prices will be the same If interest rates are negatively correlated with futures prices: forward prices > futures prices F0=100, ST=103; ST-F0=3 1.f=99, loss=1 2. f=103, profit=4 23 Swap 25 Valuing a European Call Option at expiration c0 = value (price) of European call today cT = value (price) of European call at expiration C0 = value (price) of American call today CT = value (price) of American call at expiration The value of a European call at expiration is the exercise value, which is the greater of zero or the value of the underlying minus the exercise price. p0 = value (price) of European put today pT = value (price) of European put at expiration P0 = value (price) of American put today PT = value (price) of American put at expiration If the underlying asset has a value at expiration of ST : Let X be the exercise or strike price of the call. The value of the call option at expiration is: cT = Max(0,ST – X) 26 Valuing a European Call Option at expiration - example Call Example 1: ST > X. If the strike price of a call option is X = $20 and the asset price at expiration ST = $25.50, what is the value of call at expiration? Call Example 2: ST < X. If the strike price of a call option is X = $20 and the asset price at expiration ST = $18.75, what is the value of call at expiration? 26 Valuing a European Call Option at expiration - example Call Example 1: ST > X. If the strike price of a call option is X = $20 and the asset price at expiration ST = $25.50, then the call value at expiration is: cT = Max(0,25.50 – 20) = $5.50 26 Valuing a European Call Option at expiration - example Call Example 1: ST > X. If the strike price of a call option is X = $20 and the asset price at expiration ST = $25.50, then the call value at expiration is: cT = Max(0,25.50 – 20) = $5.50 Call Example 2: ST < X. If the strike price of a call option is X = $20 and the asset price at expiration ST = $18.75, then the option will have no expiration value: cT = Max(0,18.75 – 20) = $0 26 Valuing a European Call Option at expiration - example Call Example 1: ST > X. If the strike price of a call option is X = $20 and the asset price at expiration ST = $25.50, then the call value at expiration is: cT = Max(0,25.50 – 20) = $5.50 Call Example 2: ST < X. If the strike price of a call option is X = $20 and the asset price at expiration ST = $18.75, then the option will have no expiration value: cT = Max(0,18.75 – 20) = $0 27 Valuing a European Put at expiration The value of a European put at expiration is the exercise value, which is the greater of zero or the exercise price minus the value of the underlying. Let X be the exercise or strike price of the put. If the underlying asset has a value at expiration of ST: The value of the put option at expiration is: pT = Max(0,X – ST) 28 Valuing a European Put at expiration example Put Example 1: ST < X. If the strike price of a put option is X = £100 and the asset price at expiration ST = £90, what is the value of put at expiration? Put Example 2: ST > X. If the strike price of a put option is X = £35 and the asset price at expiration ST = £40, what is the value of put at expiration? 28 Valuing a European Put at expiration example Put Example 1: ST < X. If the strike price of a put option is X = £100 and the asset price at expiration ST = £90, then the put will have a value at expiration of: pT = Max(0,100 – 90) = £10 28 Valuing a European Put at expiration example Put Example 1: ST < X. If the strike price of a put option is X = £100 and the asset price at expiration ST = £90, then the put will have a value at expiration of: pT = Max(0,100 – 90) = £10 Put Example 2: ST > X. If the strike price of a put option is X = £35 and the asset price at expiration ST = £40, then the put will have no value at expiration: pT = Max(0,35 – 40) = £0 29 Moneyness Moneyness describes the relationship of the underlying price to the strike price. An option can be described as being: In-the-money: When the underlying is beyond the exercise price in the appropriate direction (higher for a call, lower for a put), the option is said to be in-the-money. At-the-money: When the underlying is precisely at the exercise price, the option is said to be at-the-money. Out-of-the-money: When the underlying has not reached the exercise price (lower for a call, higher for a put), the option is said to be out-of-themoney. 30 Factors that determine the value of a European Option The value of the underlying asset: The value of a European call option is directly related to the value of the underlying. The value of a European put option is inversely related to the value of the underlying. The exercise price: The value of a European call option is inversely related to the exercise price. The value of a European put option is directly related to the exercise price. 31 Factors that determine the value of a European Option Continued Time to expiration: The value of a European call option is directly related to the time to expiration. The value of a European put option can be either directly or inversely related to the time to expiration. The direct effect is more common. The risk-free rate of interest: The value of a European call is directly related to the risk-free interest rate. The value of a European put is inversely related to the risk-free interest rate. The volatility of the underlying asset: The values of both the European call and the European put are directly related to the volatility of the underlying. 32 Protective put S0+p0 33 Fiduciary call c0 + X/(1 + r)T 34 Protective put vs fiduciary call 35 PUT–CALL Parity equation For European options on the same underlying asset with the same strike price and expiration, the following equation must hold: πΊπ + ππ = ππ + πΏ π + π Where p0: European put price c0: European call price S0: current asset price X: strike price T: time to expiration π» 36 PUT–CALL Parity Example Consider a non-dividend-paying stock with a current price of $25/share. A 3-month European call with a strike price of $24 is selling for $3. If the risk-free rate is 2%, what is the correct price of the European put? 36 PUT–CALL Parity Example Consider a non-dividend-paying stock with a current price of $25/share. A 3-month European call with a strike price of $24 is selling for $3. If the risk-free rate is 2%, what is the correct price of the European put? π0 + π0 = π0 + π 1 + π π 36 PUT–CALL Parity Example Consider a non-dividend-paying stock with a current price of $25/share. A 3-month European call with a strike price of $24 is selling for $3. If the risk-free rate is 2%, what is the correct price of the European put? π0 + π0 = π0 + π 1 + π π $25 + π0 = 3 + $24 1 + 0.02 π0 = $3 + $23.88 − $25 π0 = $1.88 0.25 46 Arbitrage Summary The price of the underlying asset is equal to the expected future price discounted at the risk-free rate, plus a risk premium, plus the present value of any benefits, minus the present value of any costs associated with holding the asset. An arbitrage opportunity occurs when two identical assets or combinations of assets (with the same payoffs) sell at different prices, leading to the possibility of buying the cheaper asset and selling the more expensive asset to produce a risk-free return without investing any capital. In well-functioning markets, arbitrage opportunities are quickly exploited, and the resulting increased buying of underpriced assets and increased selling of overpriced assets returns prices to equivalence. 47 Derivative Pricing Summary Derivatives are priced by creating a risk-free combination of the underlying and a derivative, leading to a unique derivative price that eliminates any possibility of arbitrage. Derivative pricing through arbitrage precludes any need for determining risk premiums or the risk aversion of the party trading the option and is referred to as risk-neutral pricing. 48 Forward and Futures Pricing Summary The value of a forward contract at expiration is the value of the asset minus the forward price. The value of a forward contract prior to expiration is the value of the asset minus the present value of the forward price. The forward price, established when the contract is initiated, is the price agreed to by the two parties that produces a zero value at the start. Costs incurred and benefits received by holding the underlying affect the forward price by raising and lowering it, respectively. Futures prices can differ from forward prices because of the effect of interest rates on the interim cash flows from the daily settlement of futures. 49 Swaps Pricing Summary Swaps can be priced as an implicit series of off-market forward contracts. Each contract is priced the same, resulting in some contracts being positively valued and some being negatively valued. However, their combined values will equal zero. 50 American Vs. European Options At expiration, a European call or put is worth its exercise value, which for calls is the greater of zero or the underlying price minus the exercise price, and for puts is the greater of zero or the exercise price minus the underlying price. European calls and puts are affected by the value of the underlying, the exercise price, the risk-free rate, the time to expiration, the volatility of the underlying, and any costs incurred or benefits received while holding the underlying. Option values experience time value decay, which is the loss in value due to the passage of time (approach of expiration), plus the moneyness and the level of volatility. The minimum value of a European call is the maximum of zero or the underlying price minus the present value of the exercise price. The minimum value of a European put is the maximum of zero or the present value of the exercise price minus the price of the underlying. 51 American and European Option Pricing European put and call prices are related through put–call parity, which specifies that the put price plus the price of the underlying equals the call price plus the present value of the exercise price. European put and call prices are related through put–call–forward parity, which shows that the put price plus the present value of a risk-free bond with face value equal to the forward price equals the call price plus the present value of the exercise price. The values of European options can be obtained using the binomial model, which specifies two possible prices of the asset one period later and enables the construction of a risk-free hedge consisting of the option and the underlying. American call prices can differ from European call prices only if there are cash flows on the underlying, such as dividends or interest; capturing these cash flows are the only reason for early exercise of a call. American put prices can differ from European put prices, because the right to exercise early always has value for a put, given that there is a lower limit on the value of the underlying.