Financial Risk Management of Insurance Enterprises

Valuing Interest Rate Options

Interest Rate Options• We will take a more detailed look at interest rate

options• What is fraternity row?

– Delta, gamma, theta, kappa, vega, rho

• What is the Black-Scholes formula?– What are its limitations for interest rate options?

• How do we value interest rate options using the binomial tree method?

• What is an implied volatility?

Price Sensitivity of Options

• Before moving to options on bonds, let’s digress to the “simpler” case of options on stock

• Define delta (∆) as the change in the option price for a change in the underlying stock

• Recall that at maturity c=max(0,ST-X) and p=max(0,X-ST)

– This should help make the sign of the derivatives obvious

Definition of Delta

For calls, =

For puts, =

c = call value S = stock price

p = put value

c

Sp

S

0

0

Predicting Changes in Option Value

• We can use delta to predict the change in the option value given a change in the underlying stock

• For example, if ∆= -½, what is the change in the option value if the stock price drops by $5– First, the option must be a put since ∆<0

– We know that puts increase in value as S decreases

– Change in put is (-½) x (-5) = +2.50

Similarity to Duration

• Note that ∆ is similar to duration– It predicts the change in

value based on a linear relationship

• The analog of convexity for options is called gamma (γ)– This measures the curvature

of the price curve as a function of the stock price

X

Stock price

Cal

l pri

ce

Intrinsic value Call value Delta

Other Greeks• Recall that the value of an option depends on:

– Underlying stock price (S)– Exercise price (X)– Time to maturity (T)– Volatility of stock price (σ)

– Risk free rate (rf)

• The only thing that is not changing is the exercise price

• Define “the greeks” by the partial derivatives of the option’s value with respect to each independent variable

Other Greeks (p.2)

• We’ve already seen the first and second derivative with respect to S (∆ and γ)

Theta =

Kappa =

Also known as "vega"

Rho =

c

Tc

c

rf

;

Black-Scholes

• Black and Scholes have developed an arbitrage argument for pricing calls and puts

• The general argument:– Form a hedge portfolio with 1 option and ∆ shares

of the underlying stock– Any instantaneous movement of the stock price is

exactly offset by the change in the option– Resulting portfolio is riskless and must earn risk-

free rate

The Black-Scholes Formula• After working through the argument, the

result is a partial differential equation which has the following solution

C S N d X e N d

dSX r t

t

d d t

N

r t

f

f

( ) ( )

ln( ) ( )

( )

1 2

1

2 1

2

cumulative distribution function

of a standard normal random variable

2

Some Comments about Black-Scholes

• Formula is for a European call on a non-dividend paying stock

• Based on continuous hedging argument• To value put options, use put-call parity

relationship

• It can be shown that ∆ for a call is N(d1)

– This is not as easy as it may look because S shows up in d1 and d2

Problems in Applying Black-Scholes to Bonds

• There are three issues in applying Black-Scholes to bonds

• First, the assumption of a constant risk-free rate is harmless for stock options– For bonds, the movement of interest rates is why the option

“exists”

• Second, constant volatility of stocks is a reasonable assumption– But, as bonds approach maturity, volatility decreases since at

bond maturity, it can only take on one value

Problems in Applying Black-Scholes to Bonds (p.2)

• Third, assuming that interest rates cannot be negative, there is an upper limit on bond prices that does not exist for stocks– Max price is the undiscounted value of all cash

flows

• Another potential problem is that most bonds pay coupons– Although, there are formulae which compute the

option values of dividend-paying stocks

Binomial Method

• Instead of using Black-Scholes, we can use the binomial method

• Based on the binomial tree, we can value interest rate options in a straightforward manner

• What types of options can we value?– Calls and puts on bonds– Caps and floors

Example of Binomial Method• What is the value of a 2 year call option if the

underlying bond is a 3 year, 5% annual coupon bond– The strike price is equal to the face value of $100

• Assume we have already calibrated the binomial tree so that we can price the bond at each node– Make sure our binomial model is “arbitrage free” by

replicating market values of bonds

MV=100.06

5.00%

MV=99.08

Coupon=5

5.97%

Principal =100

Coupon=5

Principal =100

Coupon=5

Principal =100

Coupon=5

MV=100.10

Coupon=5

4.89%

MV=100.96

Coupon=5

4.00%

MV=99.14

Coupon=5

5.50%

MV=100.99

Coupon=5

4.50%

Underlying Bond Values

Option Values• Start at expiration of option and work backwards

– Option value at expiration is max(0,ST-X)

• Dis count payoff to beginning of tree

MV=0.27

5.00%

MV=0.05

5.50%

MV=0.51

4.50%

Option Value = 0

Option Value = 0.10

Option Value = 0.96

Calculations

Final Payoffs B

BT

T

is Bond value at option expiration

0.05 =

max( , )

.

.

.. .

.

.. .

.

0 100

0 010

10552

051010 0 96

10452

0 270 05 051

1052

A Note About Options on Bonds

• A call option on a bond is similar to a floor– As interest rates decline, the underlying bond price

increases and the call value increases in value

– A floor also pays off when interest rates decline

• Main difference lies in payoff function– For floors, the payoff is linear in the interest rate

– For call options, the payoff has curvature because the bond price curve is convex

Implied Volatility

• Using the Black-Scholes equation or a binomial tree is useful if volatility is known– Historical volatility is frequently used

• Using the market prices of options, we can “back into” an implied market volatility– Use solver tool in spreadsheet programs or just

use trial-and-error

Use of Implied Volatility

• When creating a binomial model or similar type of tool, we should make sure that the implied market volatility is consistent with our model

• If our model has assumed a low volatility relative to the market, we are underpricing options

• This is an additional “constraint” along with arbitrage-free considerations

Next Time...

• Review of Interest Rate Swaps

• How to Value Interest Rate Swaps