Derivatives V2 Flashcards
[Equity Forwards and Futures: Pricing and No-Arbitrage Valuation]
1- Pricing Framework
– Equity forwards and futures are priced using the cost of carry model, which reflects the relationship between the spot price of the equity and its future value, adjusted for costs and benefits of holding the asset over time.
– The fair value assumes no arbitrage, meaning that identical positions in the spot and derivative markets should yield the same return over time.
2- No-Arbitrage Pricing Formula (Without Dividends)
– Formula:
F₀ = S₀ × e^(r × T)
Where:
— F₀ = Forward or futures price
— S₀ = Spot price of the equity
— r = Risk-free rate (continuously compounded)
— T = Time to maturity (in years)
– Interpretation:
The forward price equals the spot price grown at the risk-free rate over time. Arbitrage opportunities arise if the actual forward price deviates from this level.
3- No-Arbitrage Pricing Formula (With Known Discrete Dividends)
– Formula:
F₀ = (S₀ - PV(div)) × e^(r × T)
Where:
— PV(div) = Present value of expected dividends over the life of the contract
– Interpretation:
The expected dividends are subtracted because a forward/futures holder does not receive them. The adjusted spot price is then compounded forward at the risk-free rate.
4- No-Arbitrage Pricing Formula (With Continuous Dividend Yield q)
– Formula:
F₀ = S₀ × e^((r - q) × T)
Where:
— q = Continuous dividend yield
– Interpretation:
The continuous dividend yield reduces the effective growth rate of the spot price, resulting in a lower forward price compared to the no-dividend case.
5- Arbitrage and Valuation Logic
– If F_actual > F₀, an arbitrageur can short the forward and buy the stock, financing it at the risk-free rate.
– If F_actual < F₀, the arbitrageur can short the stock and enter a long forward position.
– In both cases, arbitrage ensures that the actual forward price converges to the theoretical no-arbitrage price.
6- Difference Between Futures and Forwards
– Futures are marked-to-market daily and have no credit risk, while forwards are private agreements settled at maturity and carry counterparty risk.
– If interest rates are constant and there are no cash flows, futures and forwards have the same price.
– With stochastic interest rates or correlation between rates and the asset, futures prices may deviate due to daily margining.
[Reverse Carry Arbitrage: Theory and Application]
1- Theoretical Concept
– Reverse carry arbitrage occurs when an investor sells the underlying asset short and simultaneously enters a long forward contract to buy it in the future.
– This strategy is used to exploit situations where the actual forward price is too low relative to the theoretical no-arbitrage price.
– However, if the forward is correctly priced (i.e., consistent with the carry arbitrage model), no arbitrage opportunity exists, and the strategy yields zero net profit regardless of future spot price movements.
– In the absence of cash flows (like dividends), the forward price is determined solely by the spot price and the cost of carry (financing rate).
2- Logic of the Arbitrage Setup
– Today:
— Sell the stock short at the current spot price (receive S₀).
— Invest the proceeds at the risk-free rate.
— Enter into a long forward contract to buy the stock at F₀ in the future.
– At maturity (T):
— Use proceeds from the risk-free investment to settle the forward.
— Use the forward contract to repurchase the stock and close the short position.
– If the forward price is equal to F₀ = S₀ × e^(r × T), the gain/loss on the forward exactly offsets the cost of repurchasing the stock, resulting in no arbitrage profit.
3- Numerical Example (Consistent with the Image)
– Given:
— Spot price S₀ = 75
— Risk-free rate r = 4% (annually compounded)
— Time T = 1 year
– Theoretical forward price:
F₀ = 75 × (1.04) = 78
Scenario A: Spot rises to 80
– Long forward gains: 80 - 78 = +2
– Short stock repurchase cost: -80
– Return from risk-free investment: +78
– Net cash flow = 0
Scenario B: Spot falls to 71
– Long forward loses: 71 - 78 = -7
– Short stock repurchase cost: -71
– Return from risk-free investment: +78
– Net cash flow = 0
4- Conclusion
– When the forward price reflects the correct no-arbitrage value, reverse carry arbitrage results in zero profit in all outcomes.
– This confirms that forward prices embed all financing costs, and arbitrage only exists if the forward deviates from its theoretical level.
[Carry Arbitrage Model: Without vs With Underlying Cash Flows]
1- Without Underlying Cash Flows
– Applies to assets like non-dividend-paying stocks or zero-coupon instruments.
– The forward price reflects only the financing cost of holding the asset.
– No adjustments are needed since the asset generates no intermediate value.
– Result: Forward price grows purely at the risk-free rate from the spot price.
2- With Underlying Cash Flows
– Applies when the asset provides value during the holding period (e.g., dividends, coupons, yields).
– The spot price must be adjusted downward by the present value of those cash flows.
– With discrete cash flows: subtract PV of known payments.
– With continuous yield: reduce the carry rate by the yield.
– Result: Forward price reflects both financing cost and the opportunity cost of missed cash flows.
[Forward Contract Valuation (No Cash Flows)]
1- Setup
– An investor entered a 9-month forward contract at F₀ = 105.
– After 6 months, the spot price is Sₜ = 101, and the risk-free annual rate is 5% (compounded annually).
– The remaining time to maturity is 3 months = 0.25 years.
2- Valuation Formula (Long Position)
Vₜ = Sₜ − F₀ / (1 + r)^(T−t)
– Applying the values:
Vₜ = 101 − 105 / (1.05)^0.25 = 101 − 103.73 = −2.73
– Interpretation:
The long forward is worth −$2.73, indicating a loss to the long party, as the spot price is lower than the present value of the forward price.
3- Alternative Method Using Offset Forward Price
– Calculate a 3-month forward price based on current spot:
Fₜ = 101 × (1.05)^0.25 = 102.24
– Then use:
Vₜ = PV(Fₜ − F₀) = (102.24 − 105) / (1.05)^0.25 = −2.73
– Same result confirms the no-arbitrage valuation principle holds using either spot-price or offsetting-forward approach.
[Forward Contract Pricing with Underlying Cash Flows]
1- Concept
– When the underlying asset generates cash flows (e.g., dividends or benefits) or incurs costs (e.g., storage), the arbitrage-free forward price must reflect these.
– The carry arbitrage model adjusts the spot price by the future value of carry costs (CC₀) and carry benefits (CB₀) over the contract term.
2- Investor A: Stock with Dividend
– Spot price = 50
– Dividend = 2 in 2 months (present value benefit)
– Contract term = 6 months
– r = 5% annually compounded
F₀ = FV(S₀) − FV(Dividend)
F₀ = 50(1.05)^(6/12) − 2(1.05)^(4/12) = 49.20
– Interpretation:
The forward price is lower than the spot price because the dividend is a benefit the forward buyer will not receive. Its present value is subtracted, and its future value is discounted from the forward price.
3- Investor B: Productive Asset with Costs and Benefits
– Spot price = 500
– Present value of benefits (CB₀) = 40
– Present value of costs (CC₀) = 15
– Term = 12 months
– r = 5%
F₀ = FV[S₀ + CC₀ − CB₀] = (500 + 15 − 40)(1.05) = 498.75
– Interpretation:
The forward price adjusts the spot price upward for maintenance costs and downward for future benefits. The net is grown at the risk-free rate to obtain the fair forward price.
[Pricing and Valuation of Forwards and Futures: General Carry Model]
1- Forward Price with Carry Costs and Benefits
– The general pricing model accounts for all net costs and benefits of holding the underlying asset until maturity.
– Carry costs (CC) include expenses like storage, insurance, and spoilage—these increase the forward price.
– Carry benefits (CB) include income from the asset such as dividends or coupons—these reduce the forward price.
2- General Formula (Discrete Framework)
F₀(T) = FV₀,T(S₀ + CC₀ − CB₀)
– The forward price is the future value of the spot price, adjusted for any upfront costs or benefits associated with carrying the asset.
3- Continuous Compounding Version
F₀(T) = S₀ × e^[(r_c + CC − CB) × T]
– This reflects net carrying cost applied to the spot price over time.
– If no carry benefits, set CB = 0.
– If no carry costs, set CC = 0.
[Principles of Arbitrage-Free Pricing]
To ensure that arbitrage-free pricing holds in theoretical models, several key assumptions are made:
1- Replicating Instruments Are Available
– Any derivative or forward contract can be exactly replicated using spot assets and financing, allowing for fair value pricing through replication.
2- No Market Frictions
– There are no transaction costs, bid-ask spreads, taxes, or other barriers to trading, ensuring that arbitrage strategies can be executed freely.
3- Short Selling Is Allowed
– Investors can sell securities they do not own, which is essential for strategies like reverse carry arbitrage and other replication arguments.
4- Borrowing and Lending at the Risk-Free Rate
– All investors have equal access to the risk-free rate for both borrowing and lending, which standardizes the cost of carry and ensures consistent discounting.
[Forward Contract Valuation (With Underlying Cash Flows)]
1- Setup
– Investor A entered a 6-month forward at F₀ = 49.20 one month ago.
– Current spot price Sₜ = 52, and the stock will pay a $2 dividend in 1 month.
– Time remaining until contract expiration = 5 months
– Risk-free rate = 5% annually compounded
2- Step 1: Price of Equivalent 5-Month Forward
Fₜ = FV(Sₜ) − FV(Dividend)**
Fₜ = 52(1.05)^(5/12) − 2(1.05)^(4/12) = 51.04
3- Step 2: Value of Existing Forward
Vₜ = PV(Fₜ − F₀) = (51.04 − 49.20) ÷ (1.05)^(5/12) = 1.80
4- Alternative Approach
Vₜ = Sₜ − [F₀ ÷ (1 + r)^(T − t)] + PV(CC) − PV(CB)
Vₜ = 52 − 49.20 ÷ (1.05)^(5/12) − 2 ÷ (1.05)^(1/12) = 1.80
Interpretation:
The forward has gained value (+1.80) for the long party since the contract was initiated.
This value accounts for both the change in spot price and the fact that the dividend (a carry benefit) will not be received by the long forward holder.
[Valuation of Forward and Futures Contracts]
1- Forward Contract Valuation
– Value is based on the difference between the current forward price and the original contract price, discounted to present value.
– For a long position:
V_t(T) = PV_t,T[F_t(T) − F₀(T)]
– For a short position:
V_t(T) = PV_t,T[F₀(T) − F_t(T)]
– At maturity:
— Long: V_T = S_T − F₀(T)
— Short: V_T = F₀(T) − S_T
2- Futures Contract Valuation
– Futures are marked-to-market daily, so the value resets to zero after each settlement.
– Right before marking to market:
— Long: v_t(T) = f_t(T) − f_{t−}(T)
— Short: v_t(T) = f_{t−}(T) − f_t(T)
– After daily settlement:
— Contract value = 0
Simple Explanation of Forward Rate Agreements (FRAs)
A Forward Rate Agreement (FRA) is like a bet on future interest rates. Two parties agree to a specific interest rate for a loan or deposit that will start in the future. Here’s how it works:
Two Roles in an FRA:
Fixed-rate payer: This party “locks in” an interest rate and will pay it no matter what happens in the market.
Floating-rate payer: This party agrees to pay whatever the actual interest rate is at the future time.
Why Do FRAs Exist?
FRAs are used to manage the risk of future interest rate changes. For example, a company that knows it will borrow money in six months can use an FRA to “lock in” the borrowing cost today.
How the Settlement Works:
On the agreement’s end date (called the expiration date), the actual interest rate in the market is compared to the fixed rate that was agreed upon.
If the market rate is higher than the agreed fixed rate, the fixed-rate payer gains money because they locked in a lower rate.
If the market rate is lower, the floating-rate payer benefits because they pay less interest than expected.
Key Points to Remember:
FRAs are settled in cash, not by actually borrowing or lending money.
Settlement is based on the difference between the agreed fixed rate and the actual floating rate.
Difference Between FRAs and Swaps
While FRAs and swaps both involve agreements about future interest rates, they work differently:
Timeframe:
FRA: A one-time agreement about a single period in the future. For example, you agree on the interest rate for a six-month loan starting three months from now.
Swap: A longer-term agreement that involves multiple periods. For example, you might exchange interest rate payments every six months for five years.
Structure:
FRA: Only one settlement happens (at the expiration date of the agreement).
Swap: Many settlements happen, one for each payment period over the life of the swap.
Use:
FRA: Typically used to manage risk for a specific short-term future loan or deposit.
Swap: Used for longer-term risk management or to change the structure of ongoing debt (e.g., from fixed to floating interest payments or vice versa).
[Pricing and Valuation of Interest Rate Forwards and Futures]
1- Forward Rate Agreement (FRA) Pricing
– FRAs are over-the-counter contracts where one party receives a fixed interest rate and the other receives a floating rate (e.g., Libor).
– The FRA fixed rate (FRA₀)** is determined at initiation using the no-arbitrage forward rateformula, derived from two discount bonds:
FRA₀ = [ (1 + Lₜrₜᵣ) / (1 + Lₜhₜₙ) − 1 ] × (1 / tₘ)
Where:
— Lₜrₜᵣ = Spot rate from today to the end of the loan period
— Lₜhₜₙ = Spot rate from today to the FRA settlement date
— tₘ = Day-count fraction for the loan period
– This ensures the FRA rate aligns with market-implied forward rates, eliminating arbitrage between borrowing/lending across different maturities.
2- FRA Settlement and No-Arbitrage Value
– At expiration, the floating rate is observed, and the difference between it and FRA₀ determines the settlement cash flow, discounted back to time h (FRA expiry).
Settlement Amount (Floating Receiver):
= NA × (Lₘ − FRA₀) × tₘ / (1 + Dₘ × tₘ)
Where:
— Lₘ = Actual floating rate set at time h
— NA = Notional amount
— Dₘ = Discount factor based on the floating rate term
– This formula reflects the present value of the interest rate difference over the loan period.
3- FRA Value Before Expiry
– The mark-to-market value of an existing FRA is based on entering an offsetting FRA at the current market rate **FRAg:
Vg = [FRAg − FRA₀] × tₘ / (1 + Dₜ₋g × tᵣ₋g)
– This value represents the present value of the difference between the agreed fixed rate and current market forward rate, discounted over the remaining life of the FRA.
4- Interest Rate Futures
– Interest rate futures (e.g., Eurodollar futures) are standardized, exchange-traded contracts priced using the same principles.
– However, they are marked-to-market daily, which causes their value to reset to zero after each session.
– Because of daily settlement and convexity bias, futures prices slightly differ from FRA prices, especially under volatile interest rate conditions.
Summary:
– FRA pricing uses no-arbitrage forward rates implied by the yield curve.
– Settlement and valuation depend on the difference between actual and contracted rates, adjusted for time value.
– Futures follow similar pricing logic but differ in treatment due to daily margining and market structure.
[Example: FRA Payment Valuation]
1- Setup
– Notional amount (NA) = £10,000,000
– FRA type = 1×6 receive-fixed
– FRA rate (FRA₀) = 2.1%
– Observed floating rate (Lₘ) = 2.6%
– Discount rate for settlement (Dₘ) = 2.3%
– Loan period (tₘ) = 5 months = 5/12
2- FRA Settlement Formula (Fixed Receiver / Short FRA)
Settlement = − [NA × (Lₘ − FRA₀) × tₘ] ÷ [1 + Dₘ × tₘ]
Settlement = − [10,000,000 × (0.026 − 0.021) × (5/12)] ÷ [1 + 0.023 × (5/12)]
= − [10,000,000 × 0.005 × 0.4167] ÷ 1.0096
= −20,636
3- Interpretation
– Since the company is the fixed receiver, the negative result indicates a loss of £20,636.
– The floating receiver gains the same amount (+£20,636), consistent with FRA zero-sum logic.
– The valuation accounts for the difference in rates over the period and discounts the settlement to present value.
[Example: FRA Fixed Rate Calculation]
1- Setup
– 90-day Euribor = 5.6%
– 270-day Euribor = 6.1%
– FRA type = 3 × 9 → Loan starts in 3 months (90 days) and lasts for 6 months (180 days)
– h = 90, m = 180, T = 270
2- FRA Fixed Rate Formula
FRA₀ = [ (1 + L_T × (T/360)) / (1 + L_h × (h/360)) − 1 ] × (360 / m)
Plugging in values:
FRA₀ = [ (1 + 0.061 × (270/360)) / (1 + 0.056 × (90/360)) − 1 ] × (360 / 180)
FRA₀ = 6.26%
3- Interpretation
– The calculated FRA rate (6.26%) is the no-arbitrage fixed rate for a 180-day loan starting in 90 days.
– It is implied by the current term structure and ensures no arbitrage between lending over 270 days vs. lending 90 days then rolling into a 180-day FRA.
[Example: FRA Valuation]
1- Setup
– We are now 25 days into the 3 × 9 FRA.
– Remaining days to FRA start (h) = 65
– FRA maturity = T = 245 days → loan length (m) = 180 days
– 65-day Euribor = 5.9%
– 245-day Euribor = 6.5%
– FRA₀ = 6.26% (from previous calculation)
2- Step 1: Recalculate FRA Rate with Updated Inputs
FRA₂₅ = [ (1 + 0.065 × (245/360)) / (1 + 0.059 × (65/360)) − 1 ] × (360/180) = 6.65%
3- Step 2: Value of the FRA (Floating Receiver / Long FRA)
V₂₅ = [ (FRA₂₅ − FRA₀) × (180/360) ] ÷ [1 + 0.065 × (245/360)]
V₂₅ = [0.0665 − 0.0626] × 0.5 ÷ [1 + 0.065 × (245/360)] = 0.0019
4- Interpretation
– The FRA has gained value for the long (floating receiver) because the market interest rate rose above the originally locked rate.
– The value 0.0019 refers to a gain per unit of notional, which would be multiplied by the notional amount to get the monetary value.
Understanding a 3 × 9 Forward Rate Agreement (FRA)
1- Overview of a 3 × 9 FRA:
– A 3 × 9 FRA refers to a forward rate agreement that expires in 3 months, with a payoff based on the 6-month Libor rate that starts at that time.
– The “3 × 9” format means:
— “3” is the number of months until the FRA expires (deferral period).
— “9” is the total number of months until the underlying deposit matures (3 months deferral + 6 months deposit).
2- Structure of a Short Position in a 3 × 9 FRA:
– A short position involves:
– 1- Going short on a 9-month Libor deposit to avoid locking in a lower rate in the future.
– 2- Going long on a 3-month Libor deposit to hedge against rate increases in the short term.
3- Key Payment Mechanics:
– When the FRA expires in 3 months:
— The floating interest rate for a 6-month period (Libor) will determine the payoff.
— The FRA’s settlement is made based on the difference between the agreed fixed rate and the floating rate.
— Payments are netted at this point.
[Settlement and Valuation of FRAs: Timeline and Application]
1- FRA Timeline Structure
– Time 0: FRA is initiated and priced.
– Time h: FRA expires; floating rate (e.g., LIBOR) is observed.
– Period m: The term of the underlying loan (or deposit) starts at h and matures at h + m = T.
– FRA payoffs are settled at time h, but they reflect the interest rate difference over the m-day period beginning at h.
2- Payoff Mechanism and Roles
– **Floating receiver (long position) profits if the observed floating rate Lₕ(m) is above the agreed FRA rate.
– Fixed receiver (short position) profits if the floating rate is below the FRA rate.
– The cash difference is discounted to the FRA expiration date to reflect early settlement.
3- Settlement Formula (Floating Receiver / Long FRA)
Settlement = [ NA × (Lₕ(m) − FRA₀) × tₘ ] / [1 + Dₕ(m) × tₘ]
Where:
— NA = Notional amount
— Lₕ(m) = Observed floating rate at time h
— FRA₀ = FRA rate fixed at initiation
— tₘ = Year fraction of loan period
— Dₕ(m) = Discount factor for the loan term at time h
Note: Use the same formula with a negative sign to compute the short (fixed receiver) payoff.
4- Example Recap
– 1×6 FRA (fixed-receiver) on £10,000,000
– FRA₀ = 2.1%
– Lₕ(150-day LIBOR) = 2.6%
– tₘ = 150 ÷ 360 = 0.4167
– Dₕ = 2.3%
Settlement Amount = [10,000,000 × (0.026 − 0.021) × 0.4167] ÷ [1 + 0.023 × 0.4167] ≈ £20,294
Since this is a fixed-receiver FRA, the value is negative and represents a loss to the short party.
Key Insight:
– Settlement reflects the interest differential over the loan period, discounted back to time h.
– Always interpret FRA settlement relative to the party’s position and direction of the rate movement.
Unique Issues Affecting Fixed-Income Forward and Futures Contracts
1- Accrued Interest in Bond Pricing
– Bonds can be quoted in two ways:
— 1- Clean price: Excludes accrued interest.
— 2- Dirty price: Includes accrued interest.
– For forward and futures contracts, adjustments may need to be made to account for accrued interest, especially when determining the actual settlement amount.
2- Multiple Bonds Deliverable by the Seller
– Certain contracts allow sellers to choose from a set of eligible bonds for delivery.
– This flexibility can introduce pricing variations and potential uncertainty for the buyer.
3- Cheapest-to-Deliver Bonds
– When multiple bonds are eligible for delivery, the seller often selects the cheapest-to-deliver bond, minimizing their cost of fulfilling the contract.
– The cheapest-to-deliver bond is determined by comparing the cost of the bond to the delivery price specified in the contract.
[Pricing and No-Arbitrage Valuation of Fixed-Income Forwards and Futures]
1- Key Pricing Principle
– Fixed-income forwards and futures are priced using the carry arbitrage model, accounting for coupon payments (carry benefits) and whether accrued interest is included in the spot bond price.
– The no-arbitrage forward price ensures that holding the bond or replicating its payoff via forward contracts yields the same value.
2- Accrued Interest Calculation
AI = (NAD / NTD) × (C / n)
Where:
— NAD = Days since last coupon
— NTD = Days in the coupon period
— C = Annual coupon
— n = Number of coupon payments per year
3- Pricing When Accrued Interest Is Included
– Let S₀ be the full bond price (dirty price):
F₀ = FV₀,T(S₀ − CB₀)
– CB₀ = Present value of coupon payments (carry benefits), denoted PVCI₀,T
– CC₀ = 0 (no carry costs for bonds)
4- Pricing When Accrued Interest Is Not Included
– Let B₀ be the clean price (excluding accrued interest):
F₀ = FV₀,T(B₀ + AI₀ − PVCI₀,T)
– Adjusts the clean price upward by current accrued interest and subtracts the present value of future coupons.
5- Bond Futures with Multiple Deliverables
– When multiple bonds are deliverable, the quoted futures price is:
F₀ = Q₀ × CF
Where:
— Q₀ = Quoted futures price
— CF = Conversion factor for the specific deliverable bond
6- Interpretation
– If forward/futures prices deviate from these theoretical values, arbitrage is possible by exploiting the mispricing between holding the bond and entering into the forward.
– These formulas ensure forward prices fully reflect the time value of money and coupon income, maintaining arbitrage-free alignment between spot and forward markets.
[Example: Bond Futures Price Calculation]
1- Setup
– Bond par value = $1,000
– Coupon rate = 4% annually → $20 semiannual coupon
– Current bond price (including accrued interest) = $990
– Next coupon in 80 days
– Futures contract expires in 210 days
– Risk-free rate = 5.10% annually
2- Step 1: Compute Future Value of the Bond Price
F₀ = FV[B₀ + AI₀] − AI_T − FVCI
Where:
— B₀ + AI₀ = 990 (dirty price)
— AI_T = Accrued interest at delivery (not applicable here as price is dirty)
— FVCI = Future value of coupon interest that will be received before futures expiry
FVCI = 20 × (210 − 80)/180 × (1.051)^[(210−80)/360] = 20(130/180)(1.051)^(130/360)
3- Plug Into Formula
F₀ = 990(1.051)^(210/360) − 20(130/180)(1.051)^(130/360) = 984.35
4- Interpretation
– The futures price is $984.35, which reflects the bond’s full price grown at the risk-free rate minus the future value of the coupon received before delivery.
– This ensures no arbitrage between buying the bond and entering into a futures contract.
[Quiz - Equilibrium Quoted Futures Price for 10-Year Treasury Note]
1- Overview of the Concept
– The equilibrium quoted futures price is derived using the carry arbitrage model.
– It represents the fair price of a futures contract accounting for accrued interest, financing cost, and the time value of money.
– This calculation uses the quoted dirty price of the bond, adjusted for accrued interest at expiration and the present value of any coupon interest during the life of the contract.
2- Formula Used
– Carry arbitrage formula for equilibrium quoted futures price:
— “Q0 = (1 ÷ CF) × [ FV(B0 + AI0) - AIT - FVCI ]”
— Where:
—- Q0: Equilibrium quoted futures contract price.
—- CF: Conversion factor = 0.7025.
—- B0: Dirty price of the bond = 104.00.
—- AI0: Accrued interest at initiation = 0.17.
—- AIT: Accrued interest at expiration = (120 ÷ 180 × 0.02 ÷ 2) = 0.67.
—- FVCI: Present value of any coupon interest to be received during contract = 0.
—- FV(…): Future value at expiration, compounded at annualized rate.
3- Step-by-Step Calculation
– Step 1: Combine dirty price and accrued interest at initiation
— B0 + AI0 = 104.00 + 0.17 = 104.17
– Step 2: Compound this value forward 90 days (i.e., 3 months or 3 ÷ 12)
— Annualized risk-free rate = 1.65%
— FV = 104.17 × (1 + 0.0165)^(3 ÷ 12) = 104.17 × 1.0041 ≈ 104.597
– Step 3: Subtract accrued interest at expiration
— AIT = 0.67 (calculated using 120 days till next coupon, 180-day convention, 2% annual coupon)
— Adjusted value = 104.597 - 0.67 = 103.927
– Step 4: Divide by the conversion factor to get quoted price
— Q0 = 103.927 ÷ 0.7025 ≈ 147.94
4- Final Answer
– Equilibrium quoted futures price = 147.94
Accrued Interest (AI)
1- Overview of the Concept
– Accrued interest is the portion of the bond’s coupon payment that has been earned but not yet paid since the last coupon payment date.
– Bonds can be quoted with accrued interest (dirty price) or without it (clean price).
2- Formula for Accrued Interest
– Formula: AI = (NAD ÷ NTD) × (C ÷ n).
3- Explanation of Variables
– AI: Accrued interest since the last coupon payment.
– NAD: Number of accrued days since the last coupon payment.
– NTD: Total number of days in the coupon payment period.
– C: Stated annual coupon amount.
– n: Number of coupon payments per year.
4- Calculation Example
– If an investor earns $10 per month on a bond with an annual coupon of $120, the accrued interest after two months would be:
— NAD = 2 months (out of 12 total months).
— C = $120, and n = 12 (monthly payments).
— AI = (2 ÷ 12) × (120 ÷ 12) = $20.
[Valuing Fixed-Income Forwards and Futures]
1- Bond Futures Valuation
– The value of a bond futures contract is simply the daily price change from the previous day’s settlement due to marking to market.
– After daily settlement, the contract’s value resets to zero.
2- Bond Forward Valuation
– The value of a bond forward contract at any point before maturity is the present value of the difference between the current forward price and the original forward price:
V_t = PV_t,T[F_t(T) − F₀(T)]
– This reflects the gain or loss relative to the agreed forward terms.
A receive-floating, pay-fixed swap is equivalent to being long a floating-rate bond and short a fixed-rate bond. The investor is borrowing at a fixed rate and investing in a floating rate. For the swap to have zero initial value, the price of the floating-rate bond and fixed-rate bond should be the same. This is usually done by assuming both are selling at par.
Types of Payment Structures: “Advanced Set” and “Settled in Arrears”
1- Advanced Set, Settled in Arrears (Used in Swaps and FRAs)
– Advanced Set:
— The interest rate for a future period is determined at the start of that period.
— Example: If the interest payment covers a 6-month period starting today, the rate is “set” at the beginning of the 6 months (today).
– Settled in Arrears:
— The payment is made at the end of the period, after the 6 months have passed.
— Example in Plain Words:
—- Suppose you agree on an interest rate today for the next 6 months. At the end of those 6 months, the payment is calculated using the “set” rate and paid then.
2- Advanced Set, Advanced Settled (Used in FRAs)
– Advanced Set:
— Similar to the above, the interest rate is determined at the start of the period.
– Advanced Settled:
— The payment is made at the beginning of the period, instead of at the end.
— To ensure fairness, the payment is discounted back to today’s value since it is made earlier than normal.
– Example in Plain Words:
— Suppose today you agree on an interest rate for a 6-month loan starting 3 months from now. When the loan period starts in 3 months, the rate is already set (advanced set), and you make the payment immediately at the start of the period (advanced settled), discounted to present value.
Clarification on FRA Properties
1- Property 1: FRAs are standardized contracts traded on exchanges.
– This property is incorrect. FRAs are over-the-counter (OTC) contracts, not standardized contracts traded on exchanges. They are privately negotiated agreements between counterparties.
– The underlying for an FRA is the interest rate, not a financial instrument like stocks or bonds, which is common in many derivative contracts.
2- Property 2: In an FRA, cash is exchanged at initiation and when interest is paid on each leg of the agreement.
– This property is also incorrect. Cash is not exchanged at initiation under an FRA. Instead, cash is exchanged only when the FRA is settled.
– FRAs are typically advanced set (the rate is determined at the beginning of the period) and advanced settled (the cash payment is made at the start of the settlement period after discounting for time value).
[Pricing and Valuing Swap Contracts]
1- Swap Pricing Concept
– A plain vanilla interest rate swap involves exchanging fixed-rate payments for floating-rate payments.
– The fixed rate (r_FIX) is set so that the present value of fixed and floating legs is equal at initiation (i.e., value of the swap is zero).
– The fixed rate is derived using the discounted present value of future cash flows.
2- Swap Pricing Equation
r_FIX = [1.0 − PV₀,tₙ(1)] ÷ ∑_{j=1}^{n} PV₀,tⱼ(1)
Where:
— PV₀,tⱼ(1) = Present value factor for each payment date using discount factors from the yield curve
— tₙ = Final swap payment date
3- Example
– Assume a 3-year annual-pay interest rate swap, and the discount factors for years 1 to 3 are:
— PV₀,t₁ = 0.95
— PV₀,t₂ = 0.91
— PV₀,t₃ = 0.87
– The denominator is the sum of PVs of the fixed payments:
∑ PV₀,tⱼ = 0.95 + 0.91 + 0.87 = 2.73
– PV of notional at maturity (for floating leg):
PV₀,tₙ = 0.87
– Now plug into the formula:
r_FIX = (1.0 − 0.87) ÷ 2.73 = 0.13 ÷ 2.73 ≈ 0.0476 or 4.76%
Interpretation:
– The fair fixed rate on this 3-year swap is 4.76%, which equates the value of fixed and floating legs at initiation.
Fixed Rate for the Fixed Leg of a Libor-Based Interest Rate Swap
1- Overview of the Concept
– To determine the fixed rate (rFIX) for the fixed leg of a Libor-based interest rate swap, the present value of all cash flows must equal the notional principal at the swap’s initiation. This involves using spot rates and adjusting them for Libor spreads.
2- Formula
– Fixed rate formula:
“rFIX = [1 - PV0,tn] ÷ T∑_t=1 PV0,t(1)”
— Where:
—- rFIX: Fixed rate of the swap.
—- PV0,tn: Present value of one unit of currency at the swap’s maturity.
—- T∑_t=1 PV0,t(1): Summation of all present value factors over the term of the swap.
3- Explanation of Variables
– PV0,tn: Present value factor for the final maturity term, calculated as “PV0,tn = 1 ÷ [1 + Libor rate × t]”.
– T∑_t=1 PV0,t(1): Sum of the present value factors for all periods in the swap’s term.
4- Steps to Solve
– Add the Libor spread (45 basis points) to each zero-coupon Treasury spot rate to derive Libor rates:
— Year 1: 4.30% + 0.45% = 4.75%.
— Year 2: 4.70% + 0.45% = 5.15%.
— Year 3: 5.00% + 0.45% = 5.45%.
— Year 4: 5.15% + 0.45% = 5.60%.
— Year 5: 5.25% + 0.45% = 5.70%.
– Calculate PV factors using these Libor rates:
— Year 1: PV factor = 1 ÷ [1 + 0.0475] = 0.954654.
— Year 2: PV factor = 1 ÷ [1 + 0.0515]^2 = 0.904444.
— Year 3: PV factor = 1 ÷ [1 + 0.0545]^3 = 0.852826.
— Year 4: PV factor = 1 ÷ [1 + 0.0560]^4 = 0.804163.
— Year 5: PV factor = 1 ÷ [1 + 0.0570]^5 = 0.757923.
– Sum the PV factors: T∑_t=1 PV0,t(1) = 4.27401.
– Use the formula to calculate rFIX:
“rFIX = [1 - 0.757923] ÷ 4.27401 = 0.0566 (or 5.66%).”
Valuing AMC’s Interest Rate Swap One Year After Initiation
1- Overview of the Concept
– The value of AMC’s interest rate swap is determined by comparing the fixed rate the company is paying (from the original five-year swap) to the current fixed rate for a four-year swap, which reflects market conditions after one year.
2- Formula for the Swap’s Value
– Value formula:
“V = (FSt - FS0) × T∑_t=1 PV0,t(1) × Notional amount”
3- Explanation of Variables
– V: Value of the swap.
– FSt: Fixed rate of the swap at time t, which is 5.57%.
– FS0: Fixed rate of the swap at initiation, which is 5.66%.
– T∑_t=1 PV0,t(1): Sum of present value factors for the remaining four years of the swap.
– Notional amount: Principal amount used to calculate cash flows, $10,000,000.
4- Steps to Calculate
Step 1: Compute the new fixed rate (FSt) of 5.57% for a four-year swap.
– Formula to calculate the fixed rate for the swap:
“FSt = (1 - PV0,tn) ÷ T∑_t=1 PV0,t(1)”
– Sub-step A: Calculate the PV factor for the last payment (PV0,tn).
— Using the zero-coupon spot rate for year 4 (5.60%):
“PV0,tn = 1 ÷ (1 + 0.0560)^4 = 0.804163.”
– Sub-step B: Compute the sum of PV factors for the remaining four years (T∑_t=1 PV0,t(1)):
— Year 1: Libor spot rate = 4.75%, PV factor = 1 ÷ (1 + 0.0475)^1 = 0.954654.
— Year 2: Libor spot rate = 5.15%, PV factor = 1 ÷ (1 + 0.0515)^2 = 0.904444.
— Year 3: Libor spot rate = 5.45%, PV factor = 1 ÷ (1 + 0.0545)^3 = 0.852826.
— Year 4: Libor spot rate = 5.60%, PV factor = 1 ÷ (1 + 0.0560)^4 = 0.804163.
“T∑_t=1 PV0,t(1) = 0.954654 + 0.904444 + 0.852826 + 0.804163 = 3.516087.”
– Sub-step C: Calculate the new fixed rate (FSt):
“FSt = (1 - 0.804163) ÷ 3.516087 = 0.0557 or 5.57%.”
Step 2: Calculate the PV factors for the remaining term.
– The PV factors calculated above are the same as in Step 1, with the sum already determined as 3.516087.
Step 3: Calculate the value of the swap (V).
– Use the original fixed rate (FS0 = 5.66%) and the new fixed rate (FSt = 5.57%):
“V = (FSt - FS0) × T∑_t=1 PV0,t(1) × Notional amount.”
“V = (0.0557 - 0.0566) × 3.516087 × 10,000,000.”
“V = -0.0009 × 3.516087 × 10,000,000.”
“V = -33,122.35.”
5- Final Swap Value
– The value of the swap from AMC’s perspective is approximately -$33,100. This negative value indicates a loss for AMC if the swap were to be unwound at this point.
[Pricing and No-Arbitrage Valuation of Interest Rate Swaps]
1- Pricing Concept
– Interest rate swaps involve exchanging fixed-rate payments for floating-rate payments on a notional amount.
– At initiation, the swap has zero value, meaning the present value of fixed payments equals the present value of expected floating payments.
– The fixed rate (r_FIX) is set using the no-arbitrage condition, ensuring neither party gains at initiation.
2- Swap Pricing Formula
r_FIX = [1 − PV₀,tₙ] ÷ ∑_{j=1}^{n} PV₀,tⱼ
Where:
— PV₀,tⱼ = Discount factor for each payment date
— PV₀,tₙ = Discount factor for the notional repayment at maturity
— n = Number of payment periods
3- No-Arbitrage Valuation After Initiation
– If market rates change, the swap gains or loses value depending on how the new fixed rate compares to the original rate.
– The value of the swap to the fixed-rate receiver is:
V_swap = (New r_FIX − Old r_FIX) × ∑ PV₀,tⱼ × NA
– If the new fixed rate rises above the original, the fixed payer loses and the fixed receiver gains, and vice versa.
4- Interpretation
– The swap pricing formula guarantees no arbitrage at initiation by aligning the present value of both legs.
– After initiation, swap value reflects changes in the interest rate environment, similar to a portfolio of forward rate agreements (FRAs).
Currency Swaps
1- Overview of Currency Swaps
– A currency swap exchanges cash flows in different currencies, typically involving payments that are either fixed or floating interest rates.
– There are four types of currency swaps:
— Fixed-for-fixed.
— Floating-for-fixed.
— Fixed-for-floating.
— Floating-for-floating.
2- Key Features of Currency Swaps
– Unlike other swaps, currency swaps often involve the exchange of notional amounts at both initiation and expiry of the swap.
– The exchange rate used for the notional amount exchange is fixed at the time the swap is initiated.
— This ensures that the exchange rate remains consistent for both initiation and settlement.
– Subsequent periodic payments are not netted because they occur in different currencies, requiring separate payment streams.
Valuation of a Currency Swap at a Later Date
1- Overview
– The value of a currency swap at a later date (t) is the difference between the current values of the two underlying bonds, adjusted for the exchange rate at time t.
– This method assumes the investor receives fixed payments in currency a and makes fixed payments in currency b.
2- Formula for Valuation
– Formula: VCS = Va - St * Vb.
3- Explanation of Variables
– VCS: Value of the currency swap at time t.
– Va: Current value of the bond denominated in currency a.
– Vb: Current value of the bond denominated in currency b.
– St: Exchange rate at time t (units of currency a per unit of currency b).
[Pricing and No-Arbitrage Valuation of Currency Swaps]
1- Concept
– A currency swap involves exchanging principal and interest payments in two different currencies.
– One party pays interest in one currency while receiving interest in another, often using different fixed or floating rates.
– At inception, the present value of both legs must be equal, ensuring no arbitrage and zero net value for both parties.
2- Pricing Approach
– Each leg of the swap is valued as a separate fixed- or floating-rate bond denominated in its respective currency.
– Use the appropriate domestic discount curve for each currency to calculate present values.
– At initiation:
PV(foreign cash flows converted to domestic) = PV(domestic cash flows)
3- Swap Fixed Rate (Domestic Leg)
– To price the fixed rate on the domestic leg (assuming fixed-for-fixed structure):
r_FIX_domestic = [1 − PV(notional repayment)] ÷ ∑ PV(coupon dates)
– Where PVs are discounted using the domestic yield curve.
4- No-Arbitrage Value After Initiation
– If market rates change, the swap may gain or lose value depending on which side is now more favorable.
– The value of the swap is:
V_swap = PV(received leg in domestic currency) − PV(paid leg in domestic currency)
– Foreign leg is converted at the current spot FX rate.
5- Example
– Assume a USD/EUR swap:
— Notional: $10 million exchanged for €9 million
— Both pay annual fixed coupons: USD at 4%, EUR at 3.5%
— Discount USD cash flows using USD curve, EUR flows using EUR curve
— Convert EUR leg to USD using spot rate (e.g., 1.11 USD/EUR)
– If PV(USD leg) = $9.8 million and PV(EUR leg in USD) = $9.9 million →
V_swap = 9.9 − 9.8 = +$0.1 million
– The USD payer has gained value since the EUR payments are now more valuable.
6- Interpretation
– Currency swaps allow hedging or exposure to different currencies with tailored interest structures.
– No-arbitrage pricing ensures fair terms at inception, and any valuation change reflects shifts in interest rates or exchange rates.
[Example: Currency Swap Pricing with Spot Rates]
1- Setup
– British company borrows €10 million for 2 years and enters a currency swap to pay in GBP.
– Exchange rate: €1 = £1.14 → GBP notional = £11.4 million
– Swap has semiannual reset → 4 periods
– Spot rate curves provided for GBP and EUR (used to calculate discount factors)
– Objective: Compute the fixed swap payments in GBP and EUR
2- Step 1: Compute Present Value Factors
– For each 6-month period, discount factors are calculated using semiannual compounding:
PV = 1 / [1 + (spot rate / 2)]^n, where n is the number of periods
– Summing PVs:
— GBP PV sum = 3.8664
— EUR PV sum = 3.9751
— Final notional PV (2-year point):
GBP = 0.9403, EUR = 0.9881
3- Step 2: Compute Fixed Swap Rates
– Formula:
r_FIX = [1 − PV_notional] ÷ [∑ PV_coupons] × (1/AP)
Where AP = 0.5 (semiannual frequency)
– GBP Leg:
r_FIX_GBP = (1 − 0.9403) ÷ 3.8664 × (1 ÷ 0.5) = 3.088%
– EUR Leg:
r_FIX_EUR = (1 − 0.9881) ÷ 3.9751 × (1 ÷ 0.5) = 0.598%
4- Step 3: Calculate Fixed Payments
– Notional amounts:
— EUR leg = €10 million → payment = 10,000,000 × 0.00299 = €29,900
— GBP leg = £11.4 million → payment = 11,400,000 × 0.01544 = £176,016
5- Interpretation
– These fixed payments reflect no-arbitrage values based on current spot rates and FX conversion.
– Each party pays interest in its respective currency using the fixed rates derived from the discount curves.
– Notional principal is exchanged at initiation and maturity at the fixed exchange rate.
[Fixed Rate for a Currency Swap Contract]
1- Overview of the Concept
– The fixed rate (rFIX) for a currency swap contract ensures the present value of fixed payments equals the notional principal at the start of the swap. This makes the swap’s initial value zero.
2- Formula
– Fixed rate formula:
“rFIX = [1 - PV0,tn] ÷ T∑_t=1 PV0,t(1)”
— Where:
—- rFIX: Fixed rate of the swap.
—- PV0,tn: Present value of one unit of currency at the swap’s maturity.
—- T∑_t=1 PV0,t(1): Summation of all present value factors for the swap’s term.
3- Explanation of Variables
– PV0,tn: Calculated using the formula “PV0,tn = 1 ÷ [1 + spot rate × (days to maturity ÷ 360)]” for the final term.
– T∑_t=1 PV0,t(1): Sum of the present value factors for each term within the swap’s maturity.
4- Steps to Solve
– Calculate PV factors for all maturities using spot rates:
— For 90 days: PV factor = 1 ÷ [1 + 0.028 × (90 ÷ 360)] = 0.993049.
— For 180 days: PV factor = 1 ÷ [1 + 0.031 × (180 ÷ 360)] = 0.984737.
— For 270 days: PV factor = 1 ÷ [1 + 0.033 × (270 ÷ 360)] = 0.975848.
— For 360 days: PV factor = 1 ÷ [1 + 0.034 × (360 ÷ 360)] = 0.967118.
– Sum these PV factors: T∑_t=1 PV0,t(1) = 3.920751.
– Calculate PV0,tn for the final maturity term (360 days): PV0,tn = 0.967118.
5- Final Calculation
– Substitute values into the formula:
“rFIX = [1 - 0.967118] ÷ 3.920751 = 0.008387 (quarterly rate).”
– Convert to annualized rate:
“Annual rate = 0.008387 × 4 = 3.35%.”
Equity Swaps Overview
1- Definition of Equity Swaps
– An equity swap is a financial contract in which at least one payment is determined by an equity return.
– The other payment is typically determined by a fixed interest rate, floating interest rate, or another equity return.
– Equity swaps are used to modify the equity exposure of a portfolio.
2- Types of Equity Swaps
– 1- Receive-equity return, pay-fixed: The investor receives the return on an equity asset while paying a fixed interest rate.
– 2- Receive-equity return, pay-floating: The investor receives the return on an equity asset while paying a floating interest rate.
– 3- Receive-equity return, pay-another equity return:
— This involves swapping the returns of two different equity assets.
— It can be synthetically created by combining two “receive-equity return, pay-fixed” swaps.
3- Characteristics of the Equity Portion
– The equity return can be based on:
— An individual stock.
— A stock index.
— A custom portfolio.
– Returns can include or exclude dividends.
– The equity return may be positive or negative.
[Pricing and No-Arbitrage Valuation of Equity Swaps]
1- Concept
– An equity swap is a derivative contract where one party pays (or receives) the return on an equity asset (e.g., a stock or equity index), while the other party pays (or receives) a fixed or floating interest rate.
– Common structures:
— Equity total return vs. fixed rate
— Equity total return vs. floating rate (e.g., LIBOR + spread)
2- Pricing at Initiation
– At initiation, the value of the swap is zero: the present value of expected cash flows from both legs must be equal.
– The fixed or floating rate leg is determined using current market rates.
– The equity leg reflects the notional return on the equity over the payment period (including price change and any dividends).
3- No-Arbitrage Fixed Rate
– The fixed rate is set such that:
PV(Fixed Payments) = PV(Expected Equity Return)
– If payments are made periodically, this is similar to a series of forward contracts on the equity, reset at each period.
4- Mark-to-Market Valuation After Initiation
– As equity values and interest rates change, the swap gains or loses value.
– Swap Value to Equity Receiver = Current Equity Value − PV of Fixed/Floating Payments Remaining
– Swap Value to Equity Payer = PV of Fixed/Floating Payments Remaining − Current Equity Value
5- Interpretation
– If equity performance exceeds the fixed or floating leg, the equity receiver benefits.
– If the equity underperforms, the equity payer benefits.
– No-arbitrage pricing ensures that at inception, neither party can secure a riskless profit, and future gains or losses depend on actual equity performance versus the contracted interest leg.
[Example: Equity Swap Valuation]
1- Setup
– A receive-equity, pay-fixed equity swap was entered 3 months ago.
– Fixed rate = 2.5%, equity was at Sₜ₋ = 1,200, and the term structure is flat at 2.0%.
– The goal is to find the stock price Sₜ that makes the value of the swap = 0 today.
2- Step 1: Value the Fixed Leg
– The fixed leg is treated like a bond, and its value is calculated using discounted cash flows of fixed payments and notional repayment:
V_FIX(C₀) = (0.025 / 1.02^0.75) + (0.025 / 1.02^1.75) + (1.025 / 1.02^2.75) = 1.0195
3- Step 2: Value of the Equity Leg
– The equity leg is based on the return of the equity relative to the initial value:
V_EQ,ₜ = V_FIX(C₀) − (Sₜ / Sₜ₋)
– Solve for Sₜ that sets V_EQ,ₜ = 0:
0 = 1.0195 − (Sₜ / 1,200) → Sₜ = 1.223 × 1,200 = 1,223
4- Interpretation
– The swap is fairly valued when the equity has appreciated to 1,223.
– At this level, the value of receiving the equity return equals the present value of the fixed leg.
– This ensures no gain or loss for either counterparty if the contract were to be marked-to-market.
[Quiz - Valuation of Pay-Fixed Receive-Equity Swap]
1- Overview of the Swap Valuation
– The swap is a pay-fixed, receive-equity total return swap.
– The notional principal is SAR 100 million with quarterly payments.
– The valuation is performed 90 days after initiation using updated market data from Exhibit 2.
2- Formula Used
– Value of pay-fixed receive-equity swap:
— “V = - [ FBt(C0) - (St / St-1) × Par ]”
— Where:
—- FBt(C0): Present value of fixed payments (quarterly fixed coupon + principal).
—- St / St-1: Ratio of equity index value at t and t-1 (JSI at 90 days ÷ JSI at initiation).
—- Par: Notional principal = SAR 100 million.
3- Step-by-Step Calculation of Receive-Fixed Leg
– Fixed rate at initiation: 1.39285% quarterly.
– Coupon payment: 100 million × 1.39285% = SAR 1,392,850.
– Present values for fixed leg:
— Q2: 0.989364 × 1,392,850 = 1,378,036
— Q3: 0.976563 × 1,392,850 = 1,360,206
— Q4: 0.962464 × 1,392,850 = 1,340,568
— Principal: 0.962464 × 100,000,000 = 96,246,400
– Total PV of fixed leg: SAR 100,325,210
4- Calculation of Equity Return Component
– JSI_t = 3,745 (from Exhibit 2)
– JSI_t-1 = 3,658 (from Exhibit 1)
– Ratio (St / St-1) = 3,745 / 3,658 = 1.0238
– Equity leg value: 1.0238 × 100 million = SAR 102,380,000
5- Final Valuation
– Swap value from Grant’s perspective:
— V = - [100,325,210 - 102,380,000]
— V = SAR 2,053,139
[Quiz - Valuation of Pay-Floating Receive-Equity Swap]
1- Overview of the Swap Structure
– Grant considered a pay-floating, receive-equity total return swap.
– The notional principal is SAR 100 million.
– The contract terms are advanced set and settled in arrears.
– The floating rate is based on the 90-day JIBAR at initiation (4.5%).
– The valuation pertains to the first quarterly cash flow received after 90 days.
2- Formula Used
– Net payment received = [ (Equity return) - (Floating rate) ] × Notional
— Equity return = (JSI_t ÷ JSI_t-1) − 1
— Floating rate = JIBAR ÷ 4 (quarterly rate)
3- Calculation of Floating Rate Component
– JIBAR at initiation = 4.5%
– Quarterly floating rate = 4.5% ÷ 4 = 1.125%
4- Calculation of Equity Return Component
– JSI_t = 3,745 (from Exhibit 2)
– JSI_t-1 = 3,658 (from Exhibit 1)
– Equity return = (3,745 ÷ 3,658) − 1 = 0.023783 = 2.3783%
5- Final Valuation of Net Payment
– Net payment = (2.3783% − 1.125%) × 100,000,000
– Net payment = 1.2533% × 100,000,000 = SAR 1,253,349
Assessment of Reynolds’ Comments on Fixed-Income Futures Contracts
1- Comment 1: “To value a bond from a country that only lists the clean price, we need to add accrued interest to the price to get the full price.”
– This comment is correct. The clean price of a bond reflects the present value of future cash flows without considering accrued interest.
– To obtain the dirty price (or full price), the accrued interest must be added.
– Accrued interest is determined by multiplying the coupon amount by the fraction of the accrual period that has elapsed since the last coupon payment.
2- Comment 2: “The conversion factor ensures that the seller of a futures contract will be indifferent with respect to which bond to deliver at maturity.”
– This comment is incorrect. While the conversion factor is designed to equalize the values of eligible bonds for delivery, it is not perfect.
– In practice, sellers will opt to deliver the cheapest-to-deliver bond, which may not align with the theoretical neutrality implied by the conversion factor.
[Binomial Option Valuation Model and Its Component Terms]
1- Concept
– The binomial option valuation model is a discrete-time model that calculates the value of options using a tree of possible future prices.
– The option is valued by modeling up (S⁺) and down (S⁻) movements in the underlying asset price over a specified period, then discounting expected payoffs back to present using risk-neutral probabilities.
– It is especially suited for American-style options, where early exercise must be evaluated at each node.
2- Option Exercise Value
– At the final node (expiration), option value is based on intrinsic value:
— Call option payoff: c_T = Max(0, S_T − X)
— Put option payoff: p_T = Max(0, X − S_T)
Where:
— S_T = asset price at maturity
— X = strike price
3- Hedge Ratio (h)
– The hedge ratio defines how many units of the underlying asset are needed to replicate the option payoff using a hedging portfolio:
— For calls: h = (c⁺ − c⁻) ÷ (S⁺ − S⁻) ≥ 0
— For puts: h = (p⁺ − p⁻) ÷ (S⁺ − S⁻) ≤ 0
Where:
— c⁺, c⁻ = option values in up and down states
— S⁺, S⁻ = underlying asset prices in up and down states
– This replicating portfolio consists of h shares of the asset and (1 − hS₀) in lending or borrowing.
4- Economic Interpretation
– A long call mimics borrowing to buy the asset → potential for leveraged upside.
– A long put mimics short-selling the asset and lending the proceeds → protection against decline.
– These replication strategies demonstrate no-arbitrage pricing by showing that the option price must equal the cost of a portfolio that replicates its payoffs.
5- Conclusion
– The binomial model combines flexibility, step-by-step logic, and arbitrage-free valuation.
– It breaks complex pricing into simple, intuitive steps, showing how option payoffs can be matched by dynamic trading in the underlying and cash.
If a stock does not pay dividends, it will never be optimal to exercise a call option early. The value of the call option will always exceed its exercise value. But for a put option that is deep in the money, it may be optimal to exercise early. The proceeds could be immediately reinvested at the risk-free rate to earn more than the current value of the put option.
It may be optimal to exercise an American call option early if the underlying stock pays dividends. The stock price will drop by the amount of the dividend right after a dividend is paid, which reduces the value of the call option. If a call is in the money based on the price of the underlying stock before a dividend is paid, there may be value to be captured from early exercise that would be lost if the option could not be exercised until maturity.
[Present Value Analysis of a European Option’s Expected Payoff]
1- Concept
– The value of a European option can be interpreted as the present value of its expected payoff at expiration, discounted at the risk-free rate under a risk-neutral probability framework.
– Since European options can only be exercised at maturity, their value depends solely on their terminal payoff and the probability of different price outcomes.
2- Expected Payoff Framework
– Under risk-neutral valuation, we assume the expected return of the underlying asset is the risk-free rate (r).
– The option’s expected payoff is calculated using risk-neutral probabilities (p) of up/down movements and is then discounted to present value:
— Call value: c₀ = e^(−rT) × E_Q[Max(0, S_T − X)]
— Put value: p₀ = e^(−rT) × E_Q[Max(0, X − S_T)]
Where:
— E_Q = expectation under risk-neutral probabilities
— S_T = stock price at expiration
— X = strike price
— T = time to expiration
3- Application in the Binomial Model
– In a single-period binomial model:
— Let c⁺ and c⁻ be the call payoffs in the up and down states, respectively.
— Then:
c₀ = e^(−rΔt) × [p × c⁺ + (1 − p) × c⁻]
4- Interpretation
– This structure shows that an option is worth the discounted average payoff it would generate in a risk-neutral world.
– It avoids the need to estimate actual probabilities or expected returns, aligning with no-arbitrage pricing principles.
– The approach reinforces the idea that option value is driven by time value, volatility, and probability-weighted outcomes rather than intrinsic value alone.
[Valuing European and American Options Using the Expectations Approach]
1- Expectations Approach for European Options
– The value of a European option is the present value (PV) of its expected payoff at expiration, calculated using risk-neutral probability (π):
— Call option:
c = PV[π × c⁺ + (1 − π) × c⁻]
— Put option:
p = PV[π × p⁺ + (1 − π) × p⁻]
– Where:
— π = (1 + r − d) ÷ (u − d)
— r = risk-free rate per period
— u, d = up/down factors
— c⁺, c⁻ and p⁺, p⁻ = option values in up/down states
2- Key Insight
– This method assumes a risk-neutral world, meaning the underlying asset grows at the risk-free rate, not its actual expected return.
– The resulting value reflects the discounted expected payoff, ensuring no-arbitrage pricing.
3- American-Style Options
– For American options, use the same backward induction method, but at each node compare:
— The calculated value (from expectations) vs.
— The immediate exercise value
– Formula:
Option value = max[Expected PV, Intrinsic Value]
– This allows for the possibility of early exercise, which is especially important for deep-in-the-money puts or when dividends are expected.
4- Summary
– European option values are based purely on expected payoffs discounted at the risk-free rate.
– American options require a node-by-node check for optimal exercise, making them more complex but more flexible.
[Arbitrage Opportunity Involving Options]
1- Concept
– An arbitrage opportunity exists when a trader can construct a position involving options and/or the underlying asset that generates a riskless profit with no initial net investment.
– These strategies exploit mispricing between related instruments, violating the no-arbitrage condition.
2- Example: Put-Call Parity Violation
– Put-call parity defines a fundamental relationship between European call and put options on the same underlying asset:
C₀ + X × e^(−rT) = P₀ + S₀
– If this relationship doesn’t hold, arbitrage is possible.
3- Arbitrage Scenario
Assume:
– C₀ + X × e^(−rT) < P₀ + S₀ → Call side is underpriced
Arbitrage Strategy:
– Buy the underpriced side: Buy the call and risk-free bond (X × e^(−rT))
– Sell the overpriced side: Short the put and sell the underlying asset
– Net investment = negative (cash inflow)
– Payoff at expiration is riskless and zero-net → risk-free profit is locked in today
4- Interpretation
– This arbitrage earns a guaranteed profit without market risk or capital at risk.
– Such mispricings are usually short-lived as arbitrageurs trade to eliminate the price gap, restoring the parity condition.
5- Summary
– Arbitrage in options typically arises from violations in pricing relationships like put-call parity, conversion/reversal trades, or misaligned synthetic positions.
– These trades enforce pricing consistency across derivative markets, supporting market efficiency.
[Example: Single-Period Binomial Call Valuation]
1- Setup
– Non-dividend paying stock: S₀ = 40
– Strike price: X = 40 (at-the-money)
– Risk-free rate: r = 3%
– Up factor: u = 1.25 → S⁺ = 40 × 1.25 = 50
– Down factor: d = 0.80 → S⁻ = 40 × 0.80 = 32
– Call payoffs:
— c⁺ = Max(0, 50 − 40) = 10
— c⁻ = Max(0, 32 − 40) = 0
2- Hedge Ratio Approach
– Hedge ratio:
h = (c⁺ − c⁻) ÷ (S⁺ − S⁻) = (10 − 0) ÷ (50 − 32) = 5 ÷ 9
– Call value:
c = hS − [(hS⁺ − c⁺) ÷ (1 + r)]
c = (5/9)(40) − [(5/9)(50) − 10] ÷ 1.03 = 4.96
3- Expectations Approach
– Risk-neutral probability:
π = (1 + r − d) ÷ (u − d) = (1.03 − 0.80) ÷ (1.25 − 0.80) = 0.511
– Call value:
c = [π × c⁺ + (1 − π) × c⁻] ÷ (1 + r) = [0.511 × 10 + 0 × 0.489] ÷ 1.03 = 4.96
4- Interpretation
– Both approaches confirm that the value of the at-the-money call option is $4.96.
– This value reflects the present value of expected payoff in a risk-neutral world and can be replicated using a hedged portfolio of stock and borrowing.
– The binomial model validates no-arbitrage pricing and provides flexibility for more complex, multi-period scenarios.
[Example: Two-Period Binomial Model Call Valuation]
1- Setup
– Underlying stock: S₀ = 40
– Strike price: X = 42
– u = 1.10 → up 10% each period
– d = 0.92 → down 8% each period
– r = 3% per period (risk-free)
– European call expiring in two periods
2- Stock Price Tree
– S⁺⁺ = 40 × u² = 48.40
– S⁺⁻ = S⁻⁺ = 40 × u × d = 40.48
– S⁻⁻ = 40 × d² = 33.86
3- Terminal Call Payoffs
– c⁺⁺ = Max(0, 48.40 − 42) = 6.40
– c⁺⁻ = c⁻⁺ = Max(0, 40.48 − 42) = 0
– c⁻⁻ = Max(0, 33.86 − 42) = 0
4- Risk-Neutral Probability
π = (1 + r − d) ÷ (u − d) = (1.03 − 0.92) ÷ (1.10 − 0.92) = 0.611
5- Option Value at t₀
– Use expectation approach for c₀:
c₀ = [π² × c⁺⁺ + 2π(1−π) × 0 + (1−π)² × 0] ÷ (1.03)²
c₀ = (0.611)² × 6.40 ÷ (1.0609) = 2.25
6- Interpretation
– The value of the two-period European call is $2.25.
– Only the up-up path results in a non-zero payoff; other paths expire out-of-the-money.
– The binomial model captures this with probability weighting and risk-free discounting, reflecting no-arbitrage pricing.
[Example: Two-Period Binomial American-Style Put Option Valuation]
1- Setup
– Underlying stock: S₀ = 40
– Strike price: X = 42
– u = 1.40 → up 40% per period
– d = 0.60 → down 40% per period
– r = 3% per period (risk-free)
– American put option (can be exercised early)
2- Stock Price Tree
– S⁺⁺ = 40 × u² = 78.40
– S⁺⁻ = S⁻⁺ = 40 × u × d = 33.60
– S⁻⁻ = 40 × d² = 14.40
– S⁺ = 40 × u = 56
– S⁻ = 40 × d = 24
3- Payoffs at Final Nodes (t = 2)
– p⁺⁺ = Max(0, 42 − 78.40) = 0
– p⁺⁻ = Max(0, 42 − 33.60) = 8.40
– p⁻⁻ = Max(0, 42 − 14.40) = 27.60
4- Risk-Neutral Probability
π = (1 + r − d) ÷ (u − d) = (1.03 − 0.60) ÷ (1.40 − 0.60) = 0.5375
5- Valuation at t = 1
– p⁻ = [π × p⁺⁻ + (1 − π) × p⁻⁻] ÷ (1 + r)
= [0.5375 × 8.40 + (1 − 0.5375) × 27.60] ÷ 1.03 = 16.78
– But: Intrinsic value at S = 24 is 42 − 24 = 18 > 16.78 → early exercise optimal
→ Use 18 instead of 16.78 for p⁻
– p⁺ = [π × p⁺⁺ + (1 − π) × p⁺⁻] ÷ (1 + r)
= [0.5375 × 0 + (1 − 0.5375) × 8.40] ÷ 1.03 = 3.77
– No early exercise since intrinsic value at S = 56 is 0 < 3.77
6- Valuation at t = 0
– p₀ = [π × p⁺ + (1 − π) × p⁻] ÷ (1 + r)
= [0.5375 × 3.77 + (1 − 0.5375) × 18] ÷ 1.03 = 10.05
7- Interpretation
– The American-style put is worth $10.05, higher than its European counterpart due to the possibility of early exercise at t = 1 when S = 24.
– This example highlights how American options must be evaluated at each node for potential exercise, using backward induction and comparing intrinsic vs. continuation value.
[Example: Valuation of a European Put Option on Interest Rates Using a Binomial Model]
1- Setup
– Type: 2-year European-style put option on a 1-year spot interest rate
– Strike rate (X): 3.25%
– Notional amount: 1,000,000
– Current rate: 3.0454%, modeled with a binomial tree
– Risk-neutral probability: π = 0.5
2- Terminal Payoffs (Year 2)
– If rate = 3.9706% or 3.2542% → Out of the money → p = 0
– If rate = 2.2593% → Payoff = (3.25% − 2.2593%) × 1,000,000 = 9,907
3- Intermediate Node Valuations (Year 1)
– Node at 2.6034%:
p⁻ = [π × 0 + (1 − π) × 9,907] ÷ (1 + 0.026034) = 4,828
– Node at 3.9084%:
p⁺ = 0, since both future payoffs are 0
4- Root Node Valuation (Year 0)
– Use discounted expected value:
p₀ = [π × 0 + (1 − π) × 4,828] ÷ (1 + 0.030454) = 2,343
5- Interpretation
– The present value of the put option is $2,343
– Only the down-down path leads to a non-zero payoff
– Pricing uses risk-neutral probabilities and backward induction consistent with no-arbitrage logic in option pricing on interest rates.
[Binomial Replication and Borrowing for Long Call Option]
1- Problem Setup
– The goal is to replicate a long position in a European call option using a portfolio of stock and borrowing.
– Underlying: FTC stock trading at 34 with no dividends.
– Call option: 1-year maturity, strike price = 33.
– Binomial model:
— Up factor (u) = 1.15 → Su = 34 × 1.15 = 39.10
— Down factor (d) = 0.90 → Sd = 34 × 0.90 = 30.60
– Risk-free rate = 4%.
2- Step 1: Determine Option Payoffs
– Call payoffs at maturity:
— Up scenario: max(39.10 − 33, 0) = 6.10
— Down scenario: max(30.60 − 33, 0) = 0
3- Step 2: Compute Hedge Ratio (h)
– h = (Cu − Cd) ÷ (Su − Sd)
– h = (6.10 − 0) ÷ (39.10 − 30.60) = 6.10 ÷ 8.50 = 0.717647
4- Step 3: Construct Replicating Portfolio
– Shares bought: 34 × 0.717647 = 24.40
– Borrowed amount = Present value of 30.60 × 0.717647
— Borrowed = (30.60 × 0.717647) ÷ 1.04 ≈ 21.12
5- Step 4: Interpret Replicating Strategy
– The replicating portfolio uses $24.40 to buy shares.
– $21.12 is borrowed, resulting in net outlay = −3.28 (cost of the option).
– In one year, borrowed amount grows to 21.12 × 1.04 = 21.96.
– Portfolio value in up move = 39.10 × 0.717647 − 21.96 = 6.10
– In down move = 30.60 × 0.717647 − 21.96 = 0
Conclusion:
– The amount borrowed to replicate the long call position is approximately 21.12 per share.
– This ensures a payoff identical to the call’s and satisfies the no-arbitrage condition.
[Assumptions of the Black–Scholes–Merton Option Valuation Model]
1- Underlying Asset Dynamics
– The price of the underlying asset follows a geometric Brownian motion with continuous compounding.
– Returns on the asset are lognormally distributed and independently and identically distributed (i.i.d.).
2- Market Conditions
– Markets are frictionless, meaning no transaction costs, taxes, or bid-ask spreads.
– Trading of the asset and option occurs continuously.
– Investors can borrow and lend at the risk-free rate, which is constant and known.
– Short selling is allowed with full use of proceeds.
3- Option and Asset Characteristics
– The underlying asset pays no dividends (for the basic BSM model).
– The option is European-style, meaning it can only be exercised at maturity.
– The volatility σ of the underlying asset is constant and known over the life of the option.
4- No Arbitrage
– The model is built on the principle of no arbitrage, meaning there are no riskless profit opportunities in the market.
[Black–Scholes–Merton Model for Non-Dividend-Paying Stocks]
1- Call Option Valuation
– Formula: c = S × N(d1) − X × e^(-rT) × N(d2)
2- Put Option Valuation
– Formula: p = X × e^(-rT) × N(−d2) − S × N(−d1)
3- Definitions of d1 and d2
– d1 = [ln(S ÷ X) + (r + σ² ÷ 2) × T] ÷ (σ × √T)
– d2 = d1 − σ × √T
4- Interpretation
– N(d2): Probability that the call option finishes in the money.
– 1 − N(d2): Probability that the put option finishes in the money.
These expressions price European options under the assumptions of lognormal stock returns, constant volatility, and no dividends.
[Black–Scholes–Merton (BSM) Model: Call Option Valuation Example]
1- Problem Setup
– A European call option is written on a non-dividend-paying stock.
– Current stock price (S) = 75
– Strike price (X) = 80
– Time to maturity (T) = 200 ÷ 365 = 0.5479 years
– Continuously compounded risk-free rate (r) = 4%
– Continuously compounded dividend yield (δ) = 2.7%
– Annual volatility (σ) = 30%
Objectives:
– 1- Compute the current value of the call option.
– 2- Determine the number of shares in the replicating portfolio.
– 3- Find the risk-neutral probability that the call ends up in the money.
2- Step 1: Adjusted Stock Price for Dividend Yield
– Because the underlying pays continuous dividends, we must discount the spot price using the dividend yield.
– Adjusted spot:
— S × e^(-δT) = 75 × e^(-0.027 × 0.5479) ≈ 73.90
3- Step 2: Calculate d1 and d2
– d1 = [ln(S ÷ X) + (r - δ + 0.5σ²)T] ÷ (σ√T)
– d1 = [ln(73.90 ÷ 80) + (0.04 - 0.027 + 0.5 × 0.09)(0.5479)] ÷ (0.3 × √0.5479)
– d1 ≈ -0.15
– d2 = d1 − σ√T
– d2 ≈ -0.15 − 0.3 × √0.5479 ≈ -0.37
4- Step 3: Lookup Cumulative Normal Values
– N(d1) = N(-0.15) = 0.4404
– N(d2) = N(-0.37) = 0.3557
5- Step 4: BSM Call Option Formula
– Formula:
— c = S × e^(-δT) × N(d1) − X × e^(-rT) × N(d2)
– Substituting values:
— c = 73.90 × 0.4404 − 80 × e^(-0.04 × 0.5479) × 0.3557
— c ≈ 32.53 − 27.82 = 4.71
6- Step 5: Number of Shares in Replicating Portfolio
– Formula: N(d1) × e^(-δT)
– = 0.4404 × e^(-0.027 × 0.5479) ≈ 0.4339
– Interpretation: To replicate the option’s payoff, hold 0.4339 shares of the underlying per option contract.
7- Step 6: Risk-Neutral Probability of Expiring In-The-Money
– The probability that the call ends up in the money under the risk-neutral measure is N(d2).
– Therefore:
— Risk-neutral probability = 0.3557
[Assumptions of the Black–Scholes–Merton Option Valuation Model]
1- Underlying Asset Dynamics
– The price of the underlying asset follows a geometric Brownian motion process with constant drift and volatility.
– Log returns of the asset are normally distributed and independent over time.
2- No Arbitrage and Market Efficiency
– Markets are frictionless: no transaction costs, taxes, or bid–ask spreads.
– There are no arbitrage opportunities.
– Securities are perfectly divisible and liquid.
3- Trading and Borrowing
– Continuous trading of the underlying asset is possible.
– Investors can borrow and lend unlimited amounts at a known, constant risk-free rate.
– Short selling of the underlying asset is permitted without restrictions.
4- Option Characteristics
– The model applies to European-style options only, which can only be exercised at maturity.
– No early exercise is permitted (as in American options).
5- Dividends
– The basic BSM model assumes the underlying asset pays no dividends during the life of the option.
– A modified version adjusts for continuously paid dividend yield.
6- Risk-Neutral Valuation
– The model uses risk-neutral pricing: expected returns are replaced by the risk-free rate under the risk-neutral measure.
[Interpretation of the Black–Scholes–Merton Model Components for Call Options]
1- Replication View
– A European call option can be interpreted as a leveraged portfolio combining a long position in the underlying with borrowed funds.
– This reflects the option buyer’s exposure to the underlying asset without paying the full price upfront.
2- Call Option Formula
– Formula:
c = S × e^(-δT) × N(d1) − X × e^(-rT) × N(d2)
Where:
— S × e^(-δT) × N(d1) = Present value of the exposure to the underlying (adjusted for dividends).
— X × e^(-rT) × N(d2) = Present value of the exercise price, i.e., the amount to be borrowed and repaid if the option ends up in the money.
3- N(d1) and the Hedge Ratio
– N(d1) represents the delta of the call option, or the number of shares needed to replicate the call option payoff.
– This quantity shows how much exposure to the underlying is embedded in the option.
4- N(d2) and Risk-Neutral Probability
– N(d2) reflects the risk-neutral probability that the option finishes in the money, i.e., that the call will be exercised.
5- Leveraged Position Interpretation
– Buying a call is equivalent to:
— Long N(d1) shares of the stock, financed by
— Borrowing the present value of the strike price X, scaled by N(d2).
– This structure allows for amplified exposure to the underlying with limited downside risk, mimicking a leveraged investment strategy.
[Black–Scholes–Merton (BSM) Valuation for European Options on Equities and Currencies]
1- European Options on Equities
– The BSM model values European call and put options on non-dividend-paying or dividend-paying stocks.
– For stocks with continuous dividend yield (δ), the call option formula becomes:
— Call: c = S × e^(-δT) × N(d1) − X × e^(-rT) × N(d2)
— Put: p = X × e^(-rT) × N(−d2) − S × e^(-δT) × N(−d1)
– Here:
— S = Current stock price
— X = Strike price
— r = Risk-free rate
— T = Time to maturity
— σ = Volatility
— δ = Dividend yield
— d1 = [ln(S/X) + (r − δ + 0.5σ²)T] ÷ (σ√T)
— d2 = d1 − σ√T
2- European Options on Currencies
– For options on foreign currencies, the BSM model is adapted to reflect foreign interest rates.
– The foreign risk-free rate acts as a dividend yield, and the formulas become:
— Call: c = S × e^(-rf T) × N(d1) − X × e^(-rd T) × N(d2)
— Put: p = X × e^(-rd T) × N(−d2) − S × e^(-rf T) × N(−d1)
– Where:
— S = Spot exchange rate (domestic/foreign)
— rf = Foreign interest rate (acts like dividend yield)
— rd = Domestic interest rate
— d1 = [ln(S/X) + (rd − rf + 0.5σ²)T] ÷ (σ√T)
— d2 = d1 − σ√T
3- Interpretation and Usage
– The BSM model provides a theoretical fair value under no-arbitrage and risk-neutral assumptions.
– It is used for:
— Pricing listed equity options and currency options.
— Delta-hedging strategies based on N(d1) (hedge ratio).
— Assessing risk-neutral probabilities of expiring in-the-money via N(d2).
[Valuation of European Options on Futures Using the Black Model]
1- Overview of the Black Model
– The Black model is a variation of the Black–Scholes–Merton framework adapted for pricing European options on futures contracts.
– Since futures require no upfront cost, the model discounts the expected option payoff at the risk-free rate.
2- Pricing Formulas
– Call option on futures:
— c = F0(T) × e^(−rT) × N(d1) − X × e^(−rT) × N(d2)
– Put option on futures:
— p = X × e^(−rT) × N(−d2) − F0(T) × e^(−rT) × N(−d1)
3- Components and Variables
– F0(T): Current futures price with maturity T
– X: Strike price of the option
– T: Time to maturity (in years)
– r: Risk-free rate (continuously compounded)
– σ: Volatility of the futures price
– d1 = [ln(F0(T) ÷ X) + 0.5σ²T] ÷ (σ√T)
– d2 = d1 − σ√T
4- Interpretation
– Since futures require no initial investment, F0(T) is used directly as the underlying value.
– Both call and put options are discounted back using e^(−rT) to account for the time value of money.
– The model assumes that futures prices are lognormally distributed and that markets are frictionless and arbitrage-free.
5- Practical Application
– Commonly used in pricing options on:
— Interest rate futures
— Commodity futures
— Equity index futures
– Provides a closed-form solution suitable for hedging and trading strategies involving derivative contracts on futures.
[Black Model for Valuing European Interest Rate Options]
1- Overview of Interest Rate Options
– European interest rate options are commonly structured as caps or floors.
– A cap is a portfolio of call options on interest rates (caplets), providing protection against rising rates.
– A floor is a portfolio of put options on interest rates (floorlets), protecting against falling rates.
– Each caplet or floorlet has a sequential maturity and a common strike rate.
2- Pricing with the Black Model
– The Black model values each caplet or floorlet individually using a modified Black–Scholes approach, where the underlying is a forward rate agreement (FRA).
– Caplet value:
— Caplet = Notional × (Δt) × [F × N(d1) − X × N(d2)] × e^(−rT)
– Floorlet value:
— Floorlet = Notional × (Δt) × [X × N(−d2) − F × N(−d1)] × e^(−rT)
3- Variables Used
– F: Forward interest rate for the period (from FRA curve)
– X: Strike rate (cap or floor rate)
– T: Time to the payment date of the caplet/floorlet
– Δt: Year fraction for the period (e.g., 0.5 for semiannual)
– r: Discount rate used for present value (usually the zero rate)
– σ: Volatility of the forward rate
– d1 = [ln(F ÷ X) + 0.5σ²T] ÷ (σ√T)
– d2 = d1 − σ√T
4- Interpretation
– Each caplet/floorlet provides a payoff only if the reference forward rate exceeds (or falls below) the strike rate.
– The total value of a cap or floor is the sum of the present values of its individual caplets or floorlets.
– This method allows the pricing of structured protection against rate movements over multiple periods.
5- Applications
– Used by borrowers or lenders to hedge interest rate risk on floating-rate debt.
– Common in structured products, swaps, and corporate debt management.
[Black Model for Valuing European Swaptions]
1- Overview of Swaptions
– A swaption is a European-style option granting the right, but not the obligation, to enter into an interest rate swap on a future date.
– The key types are:
— Payer swaption: Right to pay fixed and receive floating.
— Receiver swaption: Right to receive fixed and pay floating.
– Exercise of the swaption results in the initiation of a swap at the strike fixed rate.
2- Black Model Application
– The Black model treats the underlying swap as a forward contract on a fixed rate.
– The value of the swaption is based on the difference between the market-implied forward swap rate and the strike rate, discounted over the option maturity.
– Payer swaption value:
— Value = PV_Annuity × [F_swap × N(d1) − X × N(d2)]
– Receiver swaption value:
— Value = PV_Annuity × [X × N(−d2) − F_swap × N(−d1)]
3- Variables Used
– F_swap: Forward swap rate (market rate for the swap starting at option maturity)
– X: Strike rate of the swaption (fixed rate specified in the contract)
– PV_Annuity: Present value of the swap’s fixed leg annuity (acts as notional)
– T: Time to expiration of the swaption
– σ: Volatility of the forward swap rate
– d1 = [ln(F_swap ÷ X) + 0.5σ²T] ÷ (σ√T)
– d2 = d1 − σ√T
4- Interpretation
– A payer swaption gains value when forward swap rates rise above the strike, enabling the holder to lock in paying below-market rates.
– A receiver swaption benefits from falling rates, enabling the holder to receive above-market fixed payments.
– The Black model assumes European-style exercise (only at expiration), and uses the risk-neutral valuation framework to compute fair value.
5- Practical Use
– Commonly used in managing interest rate risk and structuring callable or putable bond features.
– Provides flexibility to lock in or hedge future borrowing or lending costs via optional swap entry.
[Interpretation of Option Greeks]
1- Delta
– Measures the sensitivity of the option’s price to changes in the price of the underlying asset.
– Call options: Delta ranges from 0 to 1.
– Put options: Delta ranges from -1 to 0.
– Interpretation: A delta of 0.60 means the option price will increase by approximately 0.60 units if the underlying rises by 1 unit.
2- Gamma
– Measures the rate of change of delta with respect to changes in the underlying price (i.e., the curvature).
– High gamma indicates greater sensitivity of delta to price changes.
– Most relevant for at-the-money options with short time to expiration.
– Interpretation: A higher gamma implies the option’s delta can shift rapidly, increasing risk for delta hedgers.
3- Vega
– Measures the sensitivity of the option’s price to changes in implied volatility.
– Both calls and puts have positive vega: their values rise with increasing volatility.
– Interpretation: A vega of 0.12 means that if implied volatility rises by 1%, the option’s value increases by 0.12.
4- Theta
– Measures the sensitivity of the option’s price to the passage of time (i.e., time decay).
– Theta is usually negative for long option positions: value erodes as expiration approaches.
– Interpretation: A theta of -0.05 means the option’s value will decrease by 0.05 per day, assuming all else remains constant.
5- Rho
– Measures the sensitivity of the option’s price to changes in the risk-free interest rate.
– Call options: Rho is positive (value increases with rates).
– Put options: Rho is negative (value decreases with rates).
– Interpretation: A rho of 0.25 means a 1% increase in interest rates raises the option value by 0.25.
Key Takeaways
– Delta: Directional risk.
– Gamma: Convexity and hedging sensitivity.
– Vega: Volatility risk.
– Theta: Time decay.
– Rho: Interest rate exposure.
[Delta Hedging Execution]
1- Objective
– Delta hedging aims to neutralize the directional risk of an option position by offsetting changes in the underlying asset’s price.
– It is primarily used by dealers and institutional investors to maintain a market-neutral position.
2- Key Concept
– The delta (Δ) of an option estimates how much the option’s price will move for a 1-unit change in the underlying asset’s price.
– A delta hedge involves holding Δ shares of the underlying asset per option contract to offset the option’s exposure.
3- Execution Steps
– 1- Determine option delta: Use the Black-Scholes-Merton model or binomial models to calculate the current delta of the option.
– 2- Calculate hedge ratio: Multiply delta by the number of options held to get the number of underlying shares to buy/sell.
– 3- Take opposite position in the underlying:
— For a long call option (positive delta), sell Δ shares of the underlying.
— For a long put option (negative delta), buy |Δ| shares of the underlying.
4- Example
– Suppose an investor holds 100 European call options on a stock with a delta of 0.60.
– Total delta = 100 × 0.60 = 60
– To hedge: sell 60 shares of the underlying stock.
– This creates a delta-neutral position where gains/losses from the option are offset by losses/gains in the stock.
5- Rebalancing
– Delta is not constant; it changes as the underlying price, time to maturity, and volatility change.
– Delta hedges require frequent rebalancing, especially as expiration nears or market conditions shift.
[Gamma Risk in Options Trading]
1- Definition of Gamma
– Gamma (Γ) measures the rate of change of delta with respect to changes in the price of the underlying asset.
– It reflects the curvature or non-linearity of the option’s price sensitivity to the underlying asset.
2- Interpretation
– High gamma means delta changes rapidly, especially near the strike price and close to expiration.
– Gamma is highest for at-the-money options and decreases for deep in- or out-of-the-money options.
3- Role in Hedging
– Gamma indicates the stability of a delta hedge:
— A position with low gamma maintains a stable delta, requiring less frequent rebalancing.
— A position with high gamma will see delta shift quickly, making hedging more difficult and costly.
4- Risk Implications
– Gamma risk refers to the unpredictable changes in delta that can cause large and sudden losses in hedged portfolios.
– Traders with large short option positions face adverse gamma risk, especially when volatility rises or the asset moves sharply.
5- Gamma and Theta Trade-Off
– Gamma risk is closely tied to theta (time decay):
— A long gamma position (e.g., long options) tends to lose value over time (negative theta).
— A short gamma position (e.g., short options) benefits from time decay but suffers when the market moves sharply.
[Implied Volatility in Options Trading]
1- Definition of Implied Volatility (IV)
– Implied volatility is the market’s forecast of the future volatility of the price of the underlying asset, derived from the market price of an option.
– It is the value of volatility that, when input into an option pricing model (like Black–Scholes–Merton), makes the model price equal to the market price of the option.
2- Interpretation
– IV reflects expected future uncertainty; it does not predict direction, only magnitude of price movement.
– A higher IV implies greater expected movement in the underlying, while a lower IV implies relative price stability.
3- Use in Options Trading
– Traders use IV to assess whether options are cheap or expensive relative to historical or realized volatility.
– Comparing IV to historical volatility can guide strategies like:
— Buying options when IV is low (expecting a future rise).
— Selling options when IV is high (expecting a future decline or mean reversion).
4- Volatility Surface
– IV is not constant across all options on the same asset; it varies by strike price and maturity, forming a volatility surface.
– Common shapes include the volatility smile and volatility skew, reflecting market biases.
5- Role in Strategy and Risk Management
– IV is a key input in option pricing, hedging, and risk metrics (like Vega).
– Traders monitor IV changes for volatility arbitrage and to anticipate market sentiment shifts.
Because the ending value is known in advance (i.e., it is the same in both the up and down scenarios), the hedged portfolio is risk-free. Therefore, it is appropriate to discount at the risk-free rate” : means that regardless of whether the underlying asset’s price goes up or down, the combined value of the asset and the derivative will reach a predetermined value. Because the portfolio’s value is certain, this justifies the use of the risk-free rate for the purpose of discounting, as the portfolio’s value does not depend on market movements and is essentially risk-free.
An investor can still be exposed to risk even with a Delta-Gamma hedge position because of Gamma Risk.
This occurs if a stock price jumps rather than move continuously. It leaves the position suddenly unhedged
