Black Scholes Merton and Swaptions

Derivatives 10 min readModule 6 of 7Save

What you'll learn

LOS 29.f
Identify assumptions of the Black Scholes Merton option valuation model.
LOS 29.g
Interpret the components of the Black Scholes Merton model as applied to call options in terms of a leveraged position in the underlying.
LOS 29.h
Describe how the Black Scholes Merton model is used to value European options on equities and currencies.
LOS 29.i
Describe how the Black model is used to value European options on futures.
LOS 29.j
Describe how the Black model is used to value European interest rate options and European swaptions.

The Black Scholes Merton (BSM) option valuation model values options in continuous time, but is based on the no arbitrage condition we used in valuing options in discrete time with a binomial model. In the binomial model, the hedge portfolio is riskless over the next period, and the no arbitrage option price is the one that guarantees that the hedge portfolio will yield the risk free rate. To derive the BSM model, an “instantaneously” riskless portfolio (one that is riskless over the next instant) is used to solve for the option price.

The assumptions underlying the Black Scholes Merton model are:

1. The underlying asset price follows a geometric Brownian motion process. The continuously compounded return is normally distributed. Under this framework, change in asset price is continuous: there are no abrupt jumps.

2. The (continuously compounded) risk free rate is constant and known. Borrowing and lending are both at the risk free rate.

3. The volatility of the returns on the underlying asset is constant and known. The price of the underlying changes smoothly (i.e., does not jump abruptly).

4. Markets are “frictionless.” There are no taxes, no transactions costs, and no restrictions on short sales or the use of short sale proceeds. Continuous trading is possible, and there are no arbitrage opportunities in the marketplace.

5. The (continuously compounded) yield on the underlying asset is constant.

6. The options are European options (i.e., they can only be exercised at expiration).

The formula for valuing a European option using the BSM model is:

C_{0} = S_{0} N (d_{1}) - e^{- r T} X N (d_{2})

and

P_{0} = e^{- r T} X N (- d_{2}) - S_{0} N (- d_{1})

where

d_{1} = \frac{ln ( S _{0} / X ) + ( r + σ ^{2} /2 ) T}{σ T}, d_{2} = d_{1} - σ T

Symbol	Meaning
$C_{0}$ , $P_{0}$	Values of call and put options
$T$	Time to option expiration
$r$	Continuously compounded risk free rate
$S_{0}$	Current asset price
$X$	Exercise price
$σ$	Annual volatility of asset returns
$N (x)$	Cumulative standard normal probability
$N (- x)$	$1 - N (x)$

While the BSM model formula looks complicated, its interpretations are not:

1. The BSM value can be thought of as the present value of the expected option payoff at expiration. For a call, that means:

C_{0} = P V {S_{0} e^{r T} N (d_{1}) - X N (d_{2})}

Similarly, for a put option:

P_{0} = P V {X N (- d_{2}) - S_{0} e^{r T} N (- d_{1})}

2. Calls can be thought of as a leveraged stock investment where $N (d_{1})$ units of stock are purchased using $e^{- r T} X N (d_{2})$ of borrowed funds. A short position in bonds can also be interpreted as borrowing funds. A portfolio that replicates a put option consists of a long position in $N (- d_{2})$ bonds and a short position in $N (- d_{1})$ stocks.

3. $N (d_{2})$ is interpreted as the risk neutral probability that a call option will expire in the money. Similarly, $N (- d_{2})$ or $1 - N (d_{2})$ is the risk neutral probability that a put option will expire in the money.

Worked example: BSM model

Stock of XZ, Inc., is currently trading at $50. Suppose that the return volatility is 25% and the continuously compounded risk free rate is 3%. Calls and puts with a strike price of $45 and expiring in 6 months (T = 0.5) are trading at $7.00 and $1.00, respectively.

If N( $d_{1}$ ) = 0.779 and N( $d_{2}$ ) = 0.723, calculate the value of the replicating portfolios and any arbitrage profits on both options.

Solution

The replicating portfolio for the call can be constructed as long 0.779 shares (0.779 × $50 = $38.95), and borrow 45 × $e^{- 0.03 (0.5)}$ × (0.723) = $32.05.

net cost = $38.95 - $32.05 = $6.90

Because the market price of the call is $7.00, the profitable arbitrage transaction entails writing a call at $7.00 and buying the replicating portfolio for $6.90 to yield an arbitrage profit of $0.10 per call.

For the put option valuation, note that N(− $d_{1}$ ) = 1 − N( $d_{1}$ ) = 1 − 0.779 = 0.221 and N(− $d_{2}$ ) = 1 − 0.723 = 0.277.

The replicating portfolio for the put option can be constructed as a long bond position of 45 × $e^{- 0.03 (0.5)}$ × (0.277) = $12.28 and a short position in 0.221 shares resulting in short proceeds of $50 × 0.221 or $11.05.

net cost = $12.28 - $11.05 = $1.23

Because the market price of the put option is $1.00, arbitrage profits can be earned by selling the replicating portfolio and buying puts, for an arbitrage profit of $0.23 per put.

Options on Dividend Paying Stocks

So far we have assumed that the underlying stock does not pay dividends. If it does, we can adjust the model using a lowercase delta (δ) to represent the dividend yield, as follows:

C_{0} P_{0} d_{1} d_{2} = S_{0} e^{- δ T} N (d_{1}) - e^{- r T} X N (d_{2}) = e^{- r T} X N (- d_{2}) - S_{0} e^{- δ T} N (- d_{1}) = \frac{ln ( S _{0} / X ) + ( r - δ + σ ^{2} /2 ) T}{σ T} = d_{1} - σ T

$δ$ is the continuously compounded dividend yield.

Note that $S_{0} e^{- δ T}$ is the stock price, reduced by the present value of any dividends expected to be paid during the option’s life.

The put call parity relation must also be modified if the stock pays dividends:

P_{0} + S_{0} e^{- δ T} = C_{0} + e^{- r T} X

Options on Currencies

We can also use the Black Scholes Merton model to value foreign exchange options. Here, the underlying is the spot exchange rate instead of a stock price.

The value of an option on a currency can be thought of as being made up of two components, a bond component and a foreign exchange component. The value can be calculated as:

C_{0} = S_{0} e^{- r (B) T} N (d_{1}) - e^{- r (P) T} X N (d_{2})

and

P_{0} = e^{- r (P) T} X N (- d_{2}) - S_{0} e^{- r (B) T} N (- d_{1})

$r (P)$ is the continuously compounded price currency interest rate. $r (B)$ is the continuously compounded base currency interest rate.

For currencies, the carry benefit is not a dividend but rather interest earned on a deposit of the foreign currency. The spot exchange rate, $S_{0}$ , is discounted at the base or foreign currency interest rate, and the bond component, $e^{- r (P) T} X$ , is the exercise exchange rate discounted at the price (or domestic currency) interest rate.

The Black Model

If we ignore the mark to market feature of the futures contract, the Black model can be used to price European options on forwards and futures:

C_{0} d_{1} d_{2} = e^{- R_{f}^{c} T} [F_{T} N (d_{1}) - X N (d_{2})] = \frac{ln ( F _{T} / X ) + ( σ ^{2} /2 ) T}{σ T} = d_{1} - σ T

$σ$ is the standard deviation of returns on the futures contract. $F_{T}$ is the futures price.

Note that the Black model is just the BSM model with $e^{- R_{f}^{c} T} F_{T}$ substituted for $S_{0}$ .

Analogous to the interpretations of the BSM model, an option on futures can be thought of as follows:

The value of a call option on futures is equal to the value of a portfolio with a long futures position (the PV of the futures price multiplied by N( $d_{1}$ )) and a short bond position (the PV of the exercise price multiplied by N( $d_{2}$ )).

The value of a put option is equal to the value of a portfolio with a long bond and a short futures position. The value of a call can also be thought of as the present value of the difference between the futures price (adjusted by N( $d_{1}$ )) and the exercise price (adjusted by N( $d_{2}$ )).

Interest Rate Options

Interest rate options are options on forward rates (or options on FRAs). A call option on an FRA gains when rates rise, and a put option on an FRA gains when rates fall. Interest rates are fixed in advance (i.e., at the beginning of the loan term) and settled in arrears (i.e., paid at maturity of the loan).

While FRAs generally use a 30/360 convention, options on FRAs use an actual/365 convention.

Consider an interest rate call option on an (M×N) FRA expiring in M months that has a strike rate of X. The underlying is an (N−M) month forward rate. A call option on this FRA can be valued as:

C_{0} = (A P) e^{- r (actual /365)} [F R A_{M \times N} N (d_{1}) - X N (d_{2})] \times N P

$A P = actual /365$ is the accrual period. $N P$ is the notional principal on the FRA.

Equivalencies in Interest Rate Derivative Contracts

Combinations of interest rate options can be used to replicate other contracts, for example:

1. A long interest rate call and a short interest rate put (with exercise rate = current FRA rate) can be used to replicate a long FRA (i.e., a forward contract to receive a floating rate and pay a fixed rate).

2. Similarly, if the exercise rate = the current FRA rate, a short interest rate call and long interest rate put can be combined to replicate a short FRA position (i.e., a pay floating, receive fixed forward contract).

3. A series of interest rate call options with different maturities and the same exercise price can be combined to form an interest rate cap. Each of the call options in an interest rate cap is known as a caplet. A floating rate loan can be hedged using a long interest rate cap.

4. Similarly, an interest rate floor is a portfolio of interest rate put options, and each of these puts is known as a floorlet. Floors can be used to hedge a long position in a floating rate bond.

5. If the exercise rate on a cap and floor is same, a long cap and short floor can be used to replicate a payer swap.

6. Similarly, a short cap and long floor can replicate a receiver swap.

7. If the exercise rate on a floor and a cap are set equal to a market swap fixed rate, the value of the cap will be equal to the value of the floor.

Swaptions

A swaption is an option that gives the holder the right to enter into an interest rate swap. A payer swaption is the right to enter into a specific swap at some date in the future at a predetermined rate as the fixed rate payer. As interest rates increase, the right to take the pay fixed side of a swap (a payer swaption) becomes more valuable.

The holder of a payer swaption would exercise it and enter into the swap if the market rate is greater than the exercise rate at expiration.

A receiver swaption is the right to enter into a specific swap at some date in the future as the fixed rate receiver (i.e., the floating rate payer) at the rate specified in the swaption. As interest rates decrease, the right to enter the receive fixed side of a swap (a receiver swaption) becomes more valuable. The holder of a receiver swaption would exercise if market rates are less than the exercise rate at expiration.

A swaption is equivalent to an option on a series of cash flows (annuity), one for each settlement date of the underlying swap, equal to the difference between the exercise rate on the swaption and the market swap fixed rate.

If PVA represents the present value of such an annuity, the value of a payer swaption using the Black model can be calculated as:

pay d_{1} d_{2} = (A P) P V A [S F R N (d_{1}) - X N (d_{2})] N P = \frac{ln ( S F R / X ) + ( σ ^{2} /2 ) T}{σ T} = d_{1} - σ T

Symbol	Meaning
pay	Value of the payer swaption
AP	Reciprocal of the number of settlement periods per year in the underlying swap
SFR	Current market swap fixed rate
X	Exercise rate specified in the payer swaption
NP	Notional principal of the underlying swap

The value of a payer swaption is essentially the present value of the expected option payoff:

pay = P V (E (payoff))

Similarly, the value of a receiver swaption (which we’ll call “REC”) can be calculated as:

R E C = (A P) P V A [X N (- d_{2}) - S F R N (- d_{1})] N P

Equivalencies

A receiver swap can be replicated using a long receiver swaption and a short payer swaption with the same exercise rates. Conversely, a payer swap can be replicated using a long payer swaption and short receiver swaption with the same exercise rates. If the exercise rate is set such that the values of the payer and receiver swaptions are equal, then the exercise rate must be equal to the market forward swap fixed rate.

A long callable bond can be replicated using a long option free bond plus a short receiver swaption.

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