Derivatives Flashcards
ow transaction costs are postulated as the reason for the growth
of derivatives
In broad terms, derivatives are used by funds because they
can increase fund returns and/or hedge the risk to the fund of adverse movements in the underlying such as changes in interest rates or the underlying assets
Futures contracts were initially created to
> help producers manage the market risk (hedge) of prices.
If understood and responsibly traded, derivative contracts allow traders to speculate, hedge, and earn arbitrage profit.
Net cost of carry
= Costs of carry - Benefits of carry
Benefits of carry include: Dividends, interest income and convenience yield.
Costs of carry include: Storage costs, insurance and the funding cost.
The value of a swap is equal to the present value of the:
Net cash flows from the swap
> net cash flows would consider the fixed and floating cash flows netted together
A synthetic long put can be created by combining
long call, long bond, and short the underlying
synthetic protective put
A forward contract F0(T), a risk-free bond and a put option on the underlying
binomial option pricing model steps
Step 1 Use the factors to estimate the next two possible prices of the asset
Step 2 Use the asset prices to derive the next two possible option values
Step 3 Compute the risk neutral probability
Step 4 Discount the expected value of the option at the risk-free rate
Types of Derivatives
Definition
- Financial instrument that offers a return based on the return of an underlying
asset - Cash price or spot price
- Price for immediate purchase of the underlying asset
Markets
- Exchange-traded
- Standard terms and features
- Highly regulated
Over-the-counter - Transactions created by two parties anywhere else
Types of Derivatives
Underlying assets
- Equities/Equity index
- Fixed-income instruments
- Interest rate contracts based on some market reference rate (MRR)
- Secured Overnight Financing Rate (SOFR) - overnight rate collateralized by US
Treasuries - Euro short term rate (€STR)
- Sterling Overnight Index Average (SONIA)
- Currencies
- Commodities
- Credit
- Others – e.g. weather, cryptocurrencies, longevity
Derivatives
- Contingent claims
- Forward commitments
Contingent claims
Exchange-traded
* Standard
options on
assets
* Interest rate
options
* Warrants
* Callable bonds
* Convertible
bonds
Over-the-counter
* Standard options on assets
* Interest rate options
* Callable bonds
* Convertible bonds
* Exotic options
Forward commitments
Exchange-traded (futures)
Over-the-counter
* Forward contracts
* FX forwards
* Swaps
* Credit derivatives,
e.g. CDS
Forward commitments
- Agreement to engage in a transaction at a later date at a price agreed today
- Exchange-traded
- Future contracts
- Over-the-counter
- Forward contracts and swaps
- Forward contract
- Agreement between two parties
- One party (the buyer) agrees to buy an underlying asset from the other party (the seller)
at a future date - Price is agreed at the start
- Forward market
- Private and largely unregulated with default risk
- Future
- Public, standardized transaction
- Use of clearing house removes default risk
- Swap
- Agreement between two parties to exchange a series of future cash flows
- Series of forward contracts
- At least one of the cash flows is determined by a later outcome
- Private transactions
Contingent claims
- Obligations arise only if certain conditions are met
- One party may have a choice
- Payoff is dependent on occurrence of a future event
- Options
- One party has the right, but not the obligation, to buy or sell an underlying asset at a
fixed price over a specific period of time - Pay a premium for this right to the option writer
- Call
- Right to buy
- Put
- Right to sell
Other instruments containing options (embedded derivatives)
- Convertible bonds
- Callable bonds
- Issuer holds a call option to buy back the bond before maturity
- Asset-backed securities
- Claim on a pool of securities
- Prepayment feature
- Borrowers may have the right to pay off their debts early
- Option held by the borrower
OTC
Contract terms
Liquidity
Margin
Counterparty risk
reporting
price quotes
hedging
Bespoke: tailored to meet the needs of the investor
Can be limited leading to slower execution
Historically no standardised process; regulation pushing for a more standardised process
Since 2008 global financial crisis, clearing houses are now generally used
Confidentiality
Limited; need to shop around
Specific hedging requirements can be met
Traded on exchange
Contract terms
Liquidity
Margin
Counterparty risk
reporting
price quotes
hedging
Contract specifications standardised by the exchange
Excellent on major contracts
Margin normally required
No member default risk due to clearing house
Market transparency
Highly transparent
Hedges using standardised contracts need to be actively managed
Use of Central Counterparty (CCP)
Step 1: Trade executed on an Swap Execution Facility (SEF)
Step 2: SEF trade information submitted to CCP
Step 3: CCP replaces (novates) existing trade, acting as new counterparty to both
financial intermediaries
Forwards
- Over-the-counter (OTC) futures
- Advantages over futures
- Flexibility
- Wide range of underlying assets
- Disadvantages over futures
- Historically counterparty risk
- Difficulty in closing out
- Long position makes money when market rises
- Short position makes money when market fall
- Zero sum game
Futures
- Exchange traded
- Advantages over futures
- Negligible counterparty risk
- Ease of closing out
- Disadvantages over forwards
- Less flexibility
- Smaller range of underlying assets
- Long position makes money when market rises
- Short position makes money when market fall
- Zero sum game
- Margining
- Initial margin
- Variation margining through ‘marking-to-market’
- Intra-day margin
Swaps
- Firm commitment under which two counterparties exchange a series of cash flows
in the future. - Fixed for floating swap
- The notional principal on the swap is not exchanged but is used to determine
payments - There is no initial cost since the initial value is zero
- The price of the swap (the fixed rate) is determined by solving for the constant
fixed yield that equates the present value of the expected floating payments to the
present value of the fixed payments - As market conditions change and time passes, the mark-to-market value of a
swap will deviate from zero resulting in one counterparty being in a
positive/winning position and the other being in a negative/losing position. - The swap will have many legs which can involve quarterly, semi-annual or annual
settlements - The full terms are negotiated privately between counterparties
- The swap may provide for uncollateralized exposure or terms similar to futures
margining. - An event of default usually triggers swap termination and MTM settlement
Long call (BULLISH)
Intrinsic value
Value at expiration
Profit
Maximum profit
Maximum loss
Breakeven
Max (0, ST - X)
cT
cT - c0
∞
c0
ST* = X + c0
Short call (BEARISH)
Intrinsic value
Value at expiration
Profit
Maximum profit
Maximum loss
Breakeven
Max (0, ST - X)
- cT
- cT + c0
c0
∞
ST* = X + c0
Long put (BEARISH)
Intrinsic value
Value at expiration
Profit
Maximum profit
Maximum loss
Breakeven
(0, X - ST)
pT
pT – p0
X - p0
p0
ST* = X - p0
Short put (BULLISH/ neutral)
Intrinsic value
Value at expiration
Profit
Maximum profit
Maximum loss
Breakeven
(0, X - ST)
-pT
- pT + p0
p0
X - p0
ST* = X - p0
Option premium
- The premium = Intrinsic Value + Time Value
- Time value is positive but declines to zero at expiry
- Long call/put option:
- Long call/put option: Profit = Value - Premium
Options are described as a zero sum game
which means that profit for the
long creates an equal loss for the short and vice versa
Credit default swap
Reference Asset —— Protection Buyer (Swap premium) —— Protection Seller ( Cash payment should default occur)
* Credit event
- Failure to pay (default)
- Bankruptcy
- Debt restructuring
* Reference asset
- May be an individual bond, an index of issuers or a special purpose entity with a debt portfolio, e.g. loans, bonds, mortgages
* Payout
- Cash-settled: Investor receives bond value less recovery rate
* Naked CDS
- Investor buys CDS without owning an underlying bond is seeking to gain from an
increase in the credit spread of an underlying bond
The Benefits of Derivative Markets
- Derivative instruments provide users with the opportunity to allocate, transfer
and/or manage risk without trading the underlying - Cash/Spot market prices for financial instruments and commercial goods and
services are an important source of information for the decision to buy or sell - In many instances issuers and investors face a timing difference between an
economic decision and the ability to transact in the cash market. Derivatives can
be used to bridge this gap
Examples
Issuers
- A retailer my await a shipment of goods priced in a foreign currency before selling
domestic currency to make the payment
- An issuer may want to lock in the cost of future debt in advance of the maturity of an
existing bond
Investors
- An investor may want to capitalize on a market view but not have the cash in hand
to transact in the cash market
- In anticipation of receiving a future dividend an investor may decide today how he/she will reinvest the proceeds in the future
- Derivative instruments serve as a price discovery function beyond the underlying
cash market - Future prices reveal information about the direction of cash markets in the future
- Equity market participants monitor equity index futures prices prior to the stock market opening to get an indication of the direction of cash markets in early trading
- Analysts use interest rate futures markets to understand investor expectations of a
central bank interest rate increase or decrease at a future meeting - Commodity futures prices are a gauge of supply and demand dynamics between
producers, consumers, and investors across maturities
Derivative instruments offer a number of operational advantages to cash or spot
market transactions
1. Transaction costs – commodity derivatives eliminate the need to
transport, insure, and store a physical asset in order to take a position
2. Increased liquidity – derivative markets typically have greater liquidity
as a result of the reduced capital required to create a position
3.Upfront cash requirements – initial margin on futures and premiums on
options are relatively low
4. Short positions – it is easier to take a short position since there is no
need to find a cash asset owner who is willing to lend the underlying asset
for a period of time
NOTE: derivative markets also make it less costly to exploit mispricing/arbitrage
opportunities
The Risks of Derivative Markets
Greater potential for speculative use:
High degree of implicit leverage for some derivative strategies may increase likelihood of financial stress
Lack of transparency:
Derivatives add portfolio complexity and may not be well understood by stakeholders
Basis risk:
Potential divergence between the expected value of a derivative instrument versus an underlying or hedged
transaction
Liquidity risk:
Potential divergence between the cash flow timing of a derivative instrument versus an underlying or hedged transaction
Counterparty credit risk:
Particularly when trading OTC with no central counterparty
Destabilization and systemic risk:
Excessive risk taking and use of leverage in derivative markets may contribute to market stress, as in the 2008 financial crisis
Issuer Use of Derivatives
Hedge type:
Description:
Examples:
Cash flow
Absorbs variable cash flow of floating rate asset or liability (forecasted transaction)
* Interest rate swap to a fixedrate for floating-rate debt
* FX forward to hedge forecasted sales
Issuer Use of Derivatives
Hedge type:
Description:
Examples:
Fair value
Offsets fluctuations in fair value of an asset or liability
* Interest rate swap to a floating rate for fixed-rate debt
* Commodity future to hedge inventory
Issuer Use of Derivatives
Hedge type:
Description:
Examples:
Net investment
Designated as offsetting the FX risk of the equity of a foreign operation
* Currency swap
* Currency forward
Investor Use of Derivatives
Investors use derivatives to:
* Replicate a cash market strategy
* Hedge a fund’s value against market movements
* Modify or add exposure
Examples
* Forward commitments
- Using copper forwards to create exposure to the price of copper with little initial outlay
* Contingent claims
- Using a call option to create a long exposure to an asset with very little outlay
- Using a covered call to modify the return to a long position in a stable market
The principle of arbitrage
- Arbitrage trades can be undertaken when two assets produce the same results
but trade at different prices, i.e. the ‘law of one price’ does not hold, - Buy the underpriced asset, sell the overpriced asset
- Arbitrageur receives money at the start of the holding period
- An arbitrage trade can also be executed if an asset with a known future price
does not trade at the present value of its future price - Derivatives derive their prices from underlying assets
- Can be used to hedge the underlying
- All risk is eliminated
- Position will earn the risk-free rate
- Derivatives are priced on a risk-free basis
- Called ‘risk-neutral’ pricing
Replication
- A strategy in which a derivative’s cash flow stream may be recreated using a
combination of long or short positions in the underlying asset and borrowing or
lending cash - It is typically used to mirror or offset a derivative position when the ‘law of one
price’ holds
Costs and Benefits Associated with Owning the Underlying
The relationship between spot and forward prices when there are no benefits or
costs (such as dividends or storage costs) associated with the asset is given by:
F0(T) = S0(1+r)T (1)
or
F0(T) = S0erT (2)
Where r is the opportunity cost of holding the asset
* Equation (1) Is used when S0 is the price of an asset such as a share
* Equation (2) is used when we have portfolios, such as equity portfolios, and for
FX forwards
The relationship between spot and forward prices when there are benefits or costs
F0(T) = S0 – PV0(I) + PV0(C)T (1)
or
F0(T) = S0e^(r+c-i)T (2)
Where:
* r is the opportunity cost of holding the asset
* (C, c) is the cost of ownership. Commodities, for example, have costs of
ownership such as storage, transportation, insurance
* (I, i) is the benefits of ownership such as dividends on shares or coupons on
bonds
Forward v Spot prices
Opportunity and other costs > Benefits: F0(T) > S0
Opportunity and other costs < Benefits: F0(T) < S0
Opportunity and other costs = Benefits: F0(T) = S0
Interest rate parity (IRP)
Arbitrage-free method of pricing FX forwards
continuous compounding
forward rate = S0e(r$-r£)T
NOTE: We say that USD is at a discount since we get more USD for GBP1 in 6 months
than we do today. However, we say that the FX Forward price is at a premium.
Here £ is the base currency and $ is the price currency
FX Forward versus Spot Price Relationship
Interest rate differential
Forward vs. Spot Price
Price Currency Forward
FX Forward Premium/Discount
rprice > rbase
F0(T) > S0
Discount
Premium
FX Forward versus Spot Price Relationship
Interest rate differential
Forward vs. Spot Price
Price Currency Forward
FX Forward Premium/Discount
rprice < rbase
F0(T) < S0
Premium
Discount
FX Forward versus Spot Price Relationship
Interest rate differential
Forward vs. Spot Price
Price Currency Forward
FX Forward Premium/Discount
rprice = rbase
F0(T) < S0
Neither a premium nor a discount
Neither a premium nor a discount
Convenience yield
- Represents non-monetary advantage of holding the asset
- Primarily associated with commodities
- Due to difficulty in shorting the asset or tight supplies
- Benefits increase the current price
Cost of holding the asset
- Cost of storage, insurance, potential spoilage (for soft commodities) and
opportunity cost of money invested - Reduce the current price
Costs and Benefits Associated with Owning the Underlying
Asset class
Example(s)
Benefits (I)
Costs (r, c)
Equity
Non-dividend paying stock
None
Risk-free rate
Costs and Benefits Associated with Owning the Underlying
Asset class
Example(s)
Benefits (I)
Costs (r, c)
Equity
Dividend paying stock, Equity index
Dividend, Dividend yield
Risk-free rate
Costs and Benefits Associated with Owning the Underlying
Asset class
Example(s)
Benefits (I)
Costs (r, c)
FX
Market exchange rates
None
Difference between the countries’ risk-free rates
Costs and Benefits Associated with Owning the Underlying
Asset class
Example(s)
Benefits (I)
Costs (r, c)
Commodities
Soft/hard commodities, Commodity index
Convenience yield
Risk-free rate, Storage cost
Costs and Benefits Associated with Owning the Underlying
Asset class
Example(s)
Benefits (I)
Costs (r, c)
Interest Rates
Bonds
Interest income
Risk-free rate
Costs and Benefits Associated with Owning the Underlying
Asset class
Example(s)
Benefits (I)
Costs (r, c)
Credit
Single reference entity, Credit indexes
Credit spread
Risk-free rate
Pricing and valuation for forwards
- There is no requirement for an initial cash outlay
- The contracts have an initial value of zero
- The contract value after initiation, can be positive or negative
- The contract price is a fixed price or rate for the underlying at a later date, not an
amount to be paid at the start
Pricing of a simple forward contract
- A forward contract is a delayed delivery sales contract where the seller (short) agrees to deliver the underlying asset to the buyer (long) in exchange for an amount that is fixed today and paid at the expiration of the forward contract
Price: Fixed price or interest rate agreed
Pricing: Determination of the fixed price (arbitrage theory)
Value: Amount paid/received to assume a forward contract
Valuation: Determination of value at any time
Computing the price of a forward contract
F0(T) = [S0 – PV0(I) + PV0(C)] (1+r)T
- F0(T) is the forward price agreed to at time 0 for a forward contract expiring at
time T - S0 is the spot price of the underlying asset at contract initiation
- PV0(I) is the present value of any cash flows on underlying during life of contract
- PV0(C) is the present value of any other costs incurred (e.g. storage and
insurance) other than the opportunity cost - r is the risk-free borrowing and lending rate, expressed on an EAR
- T is the time from contract initiation to expiration
Forward contract valuation
Vt(T) = [St – PVt(I) + PVt(C)] - PV[F0(T)]
Value of a forward to the buyer at any time t =
* Price of underlying at time t
* - PV at time t of remaining cash flows on the underlying
* + PV at time t of any other costs
* - PV at time t of forward contract price
* At initiation V0 = 0
During a contract’s life the Vt can be + or - since the forward price is fixed but the underlying is not
* At expiration VT = ST – F0(T)
Forward contract valuation
* Valuation during the life of a forward contract assuming no cash flows on underlying and no other costs:
Vt(T) = St - F0(T)/(1+r)(T-t)/T
FRA definition
- Agreement to pay or receive, on an agreed future date, the difference between a fixed interest rate agreed at the outset and a market reference rate (MRR) prevailing on a given future date, for a given future period (the FRA run)
- Underlying asset
- Interest payment made in a specified currency
- Payments are based on a notional amount and paid at the start of the FRA run
- No exchange of principal takes place
- NOTE: the fixed rate is the implied forward rate for the FRA run and is calculated in a similar way to the previous examples
Long FRA
* Pay fixed rate, receive floating rate
* Makes a profit when MRR
Short FRA
* Receive fixed rate, pay floating rate
* Makes a profit when MRR
MTM Valuation: Forwards versus Futures
Differences compared to forwards contracts
- Futures contracts
- Standardized terms
- Traded on a futures exchange
- More heavily regulated
- Contracts are marked to market on a daily basis (margining)
- Payment by loser to the winner
- Exchanges
- Provide a credit guarantee
- Backed by a clearing house
- Transfers the cash payments on a daily basis
- Cash flows
- Forwards – Gains/Losses paid at expiration of the contract
- MTM can be positive or negative, i.e. counterparty credit risk exists
- Futures – Receive/Pay interim cash via margining, i.e. MTM is zero
Interest Rate Futures versus Forward Contracts
- Forward Rate Agreements trade OTC
- The long agrees to pay a fixed rate and receive a MRR
- The price of the contract is the fixed rate
- The long makes money if the MRR rises
- Short term interest rate futures trade on exchanges
- The price of the contract is: 100 – (100 x MRR)
- The long makes money if the MRR goes down
- MTM is zero since the margining process transfers cash payments on a daily basis
Contract type: Interest rate futures
Gains from rising MRR: Short futures contract
Gains from falling MRR: Long futures contract
Contract type: Forward rate agreement
Gains from rising MRR: Long FRA: Fixed-rate payer
Gains from falling MRR: Short FRA: Floating rate payer
Differences compared to forwards contracts
* Correlation of futures prices and interest rates
Uncorrelated – forwards and futures prices will be the same
- Positively correlated – then a long position in a futures contract is more desirable
* When gains are made as prices and rates rise, the gain is reinvested at higher interest rates
* When losses are made as prices and rates fall, the loss is made at lower interest rates
* Futures prices > Forward prices
- Negatively correlated
* Futures prices < Forward prices
Swaps versus Series of Forwards
- Swap and A Series of Forward Contracts
- Agreement between two parties to exchange a series of future cash flows
- Series of forward contracts
- At least one of the cash flows is determined by a later outcome
- No payment upfront
- Both forwards and swaps involve counterparty credit risk
- Differences between a Swap and a Series of Forwards
Swap
Multiple settlements
Payment at end of period
Fixed rate is constant for each period
Series of forward contracts
Each forward has a single settlement
Payment at the start of the period
Fixed rate is typically different for each period
Pricing Swaps
The par swap rate is the fixed rate that equates the PV of all future expected floating rates to the PV of the fixed cash flows
If we wanted to find the three-year swap rate, we use the implied forward rates to solve for s3
The three-year swap rate identified in the previous example was 3.78%
* This implies that an investor would be indifferent to
- Paying the fixed swap rate and receiving the respective forward rates or
- Receiving the fixed swap rate and paying the respective forward rates
* Note that although the value of the swap at t = 0 is zero some individual
forward exchanges might have positive or negative values
- E.g. in one year’s time the fixed rate receiver will
* Receive the fixed swap rate of 3.78%
* Pay the initial floating rate of 2.46%
Value of fixed payments = Value of floating payments
* Receiver swap
- Treat a receive fixed swap as buying a fixed-rate bond and issuing a floating-rate bond
- Receiving the fixed swap rate and paying the respective forward rates
* Payer swap
- Treat a pay fixed swap as buying a floating-rate bond and issuing a fixed-rate bond
- Receiving the respective floating swap rates and paying the respective fixed rate
* Swaps can be used to change a company’s interest rate exposure
- A company that is borrowing at a floating rate can convert this into a fixed rate borrowing by entering into a pay fixed swap
Swap Values and Prices
- The value of a swap at inception is zero (V0(T) = 0)
- For a forward with one settlement, the value of the forward for the long is
- VT(T) = ST – F0(T) where ST is the spot price and F0(T) is the forward price
- For a swap with periodic settlements, the current MRR is the spot price and the fixed swap rate, SN, is the forward price
- The periodic settlement value = (MRR - SN) x Notional amount x Period
- The value of the swap on any settlement date equals the current settlement value as above plus the present value of all other remaining future swap settlements
- Generally the MRR is set at the beginning of each period and the fixed versus floating difference is exchanged at the end of each period
Using swaps to change the duration on a portfolio
- A bond’s duration is a measure of interest rate risk
- The higher the duration the riskier the bond or a bond portfolio
- When you are long a fixed rate bond you receive fixed coupons
- When you enter into a receive fixed swap you receive fixed payments
- Since being long a fixed-rate bond and entering into a receive fixed swap involve receiving fixed payments then doing either will increase the duration in a portfolio
- To reduce the bond portfolio duration you would short bonds or enter into a pay fixed swap
Option Exercise Value
- Options will only be exercised at maturity if the payoff is positive. That is:
- (ST – X) > 0 for a call
- (X – ST) > 0 for a put
- If not exercised the option expires worthless and the buyer’s loss equals the premium
- Before maturity (t<T) the exercise value is given by:
- Max (0, St - X/(1+r)(T-t)) for a call
- Max (0, X/(1+r)(T-t) - St) for a put
- If we assume an exercise price, X, equal to the forward price, F0(T), the exercise
value of a call option is the same as the value of a forward commitment at time t
provided that the spot price is greater than the PV(X). - If St > PV(X): St – PV(F0(T)) = Max (0, St – PV(X))
- Note that here we are ignoring the option premium
Option Time Value
- The time value of an option is equal to the difference between the current option
price and the option’s current exercise value (payoff) - ct = Max (0, St - X/(1+r)(T-t)) + Time Value
- pt = Max (0, X/(1+r)(T-t) - St) + Time Value
- Time value is generally positive but declines to zero at maturity
- The process is known as time value decay
Arbitrage and Replication
Payoff for European Options at Maturity
* cT = Max (0, ST – X)
* pT = Max (0, X – ST)
Buyer’s profit for European Options at Maturity
* Profit for call = Max (0, ST – X) – c0
* Profit for put = Max (0, X – ST) – p0
* Forward buyer
- Pays nothing up front
- Unlimited profits and loss limited to forward price
* Option buyer
- Pays premium
- Unlimited profits (for a call) and loss limited to premium
Options – Upper Bounds
- Maximum value of a European call: St
- Maximum value of a European put: X
Arbitrage and Replication
Replicating a call option at contract inception
Replicating a call option at contract inception
* Borrow X at the risk-free rate
* Use the proceeds to buy the asset at a price of S0
At maturity there are two possible outcomes
* Exercise (ST > X): sell the underlying at ST and use the proceeds to repay X
* No exercise (ST < X): No settlement is required
* A key difference between replicating forwards and options is that with options we
borrow a proportion of X based on the likelihood of exercise (hedge ratio) at time
T. The replicating transaction has to be adjusted as the likelihood of exercise
changes
Arbitrage and Replication
Replicating a put option at contract inception
Replicating a put option at contract inception
* Sell the asset short at a price of S0
* Lend the proceeds at the risk-free rate
At maturity there are two possible outcomes
* Exercise (ST < X): purchase the underlying at ST from the proceeds of the riskfree loan
* No exercise (ST > X): No settlement is required
* Again, a key difference between replicating forwards and options is that with
options we lend a proportion of X based on the likelihood of exercise (hedge ratio)
at time T. The replicating transaction has to be adjusted as the likelihood of
exercise changes
Summary of factors that impact option values
The underlying
Exercise/strike price
Time to expiration /
Risk-free rate
Volatility of underlying
Income/costs relating to underlying
Call: incr, Put: decr
Call: decr, Put: incr
Call: incr Put:incr,decr
Call: incr Put:decr
Call: incr Put:incr
Call: incr, decr Put:incr, der
Put-call parity
c0 + X/(1+rf)^T = p0 + S0
Fiduciary call
Buy call
Buy bond
Protective put
Buy put
Buy underlying asset
Synthetic call
Long put + Long underlying + Short bond p0 + S0 - X / (1 + r)T
Synthetic put
Long call + Short underlying + Long bond c0 - S0 + X / (1 + r)T
Synthetic underlying
Long call + Long bond + Short put c0 + X / (1 + r)T- p0
Synthetic bond
Long put + Long underlying + Short call p0 + S0 - c0
Covered call
Long underlying + Short Call S0 - c0 = X / (1 + r)T - p0
Put-Call Forward parity
c0 + X/(1+rf)^T = p0 + F0(T)/(1+rf)^T
Synthetic protective put
Buy Forward contract
Buy bond
Buy put
Put-Call Forward Parity and Option Applications
- Assume that at time t=0 a firm has borrowed capital in the form of a zero-coupon
bond with face value D - The market value of the firm’s assets, V0, is equal to the PV of its debt plus its
equity, E0 - V0 = E0 + PV(D)
- When the firm’s debt matures at time T, the assets are distributed amongst the
shareholders and debtholders with two possible outcomes
1. Solvency (VT > D): the firm can return capital to both its shareholders and its
debtholders
2. Insolvency (VT < D): shareholders receive nothing and debtholders receive less
than D (i.e. VT) - Unlike with Put-Call Parity here the bond is risky since debtholders only receive
D if the firm is solvent - Debtholders will demand a premium similar to a put option premium from shareholders in order to assume the risk of insolvency
Shareholders have unlimited upside and limited downside - Debtholders have limited upside whilst principal and interest payments are at risk
if the firm becomes insolvent - Payoff profiles are:
- Shareholder payoff is max(0, VT - D) which is the same payoff as a long call with strike D
- Note that a Long Call is equivalent to a Long Asset plus a Long Put
- Debtholder payoff is min(VT,D)
- Note that this equates to being long the bond with face value of D and short a put with strike D on
the firm value VT - Using S0 + p0 = c0 + PV(X) and substituting V0 for S0 and D for the risk free bond,
X, we get V0 + p0 = c0 + PV(D) - From which we can see that V0 = c0 + PV(D) - p0
- Shareholders have a position with a payoff similar to a long call
- Debtholders have a long risk-free bond plus a short put on the firm’s assets
- The short put can be interpreted as the credit spread on the firm’s debt or the premium above the risk-free rate that shareholders must pay to debtholders to assume the risk of insolvency
- From a debtholder perspective, the more valuable the sold put the more credit risk is
present in the firm’s debt
Binomial Option Pricing Model
Over a given period of time an asset’s price will go up from S0 to S1
+ or go down from S0 to S1-
* The gross return will be either
- Ru = S1
+/S0 > 1
- Rd = S1-/S0 < 1
h = (c1+ – c1-)/ (S1+ – S1-) This is known as the hedge ratio
V1+ = hS1+ – c1+
V1- = hS1- – c1-
* The return V1+ / V0 = V1- / V0 must equal (1 + risk-free rate)
- hS0 – c0 = V1+ / (1 + r)
Valuing the option using risk-neutral probabilities
u= S1+ / S0 and d=S1+/S0
pi = (1+r-d)/u-d
c0 = (pic1+ + (1-pi)c1-) / (1+r)
π and 1 - π are not the probabilities of an up and down movement but are the
probabilities that would arise if we equate the portfolio’s value with the discounted
future value using the risk-free rate as a discount factor
The Binomial Option Pricing Model
steps
Step 1 Use the factors to estimate the next two possible prices of the asset
Step 2 Use the asset prices to derive the next two possible option values
Step 3 Compute the risk neutral probability
Step 4 Discount the expected value of the option at the risk-free rate
Fra vs stir
Stir - payoff linear
Fra - profit greater than loss - convexity
Swap rate
Fixed
Detived from implied forward rates and spot rates
Swap rate
Fixed
Detived from implied forward rates and spot rates
