Valuation of Interest Rate Derivatives › Option-Adjusted Analysis for Mortgages
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Interest rate options appear embedded in many fixed income products. The most significant example is residential mortgages, which contain a prepayment option that borrowers can exercise when rates fall.
The Prepayment Option
A fixed-rate mortgage gives the borrower the right to prepay principal at any time. This is economically equivalent to a portfolio of:
- A fixed-rate loan to you
- A call option on the loan's value, owned by the borrower
When rates fall, borrowers refinance their mortgages, prepaying existing loans and taking out new ones at lower rates. This prepayment hurts you because you receive principal back when rates are low and must reinvest at lower yields.
Option-Adjusted Spread
The Option-Adjusted Spread (OAS) quantifies the yield premium on mortgage-backed securities after accounting for the embedded prepayment option. The calculation involves:
- Model rate paths: Generate many interest rate scenarios using a calibrated model
- Project prepayments: Apply a prepayment model that responds to rate levels
- Calculate cashflows: Determine expected cashflows on each path
- Find the spread: Solve for the spread over the benchmark curve that makes the present value of model cashflows equal to the market price
A positive OAS indicates the security offers excess yield after adjusting for option risk. You use OAS to compare MBS against corporate bonds and other fixed income alternatives on an apples-to-apples basis.
# Simplified OAS illustration
# In practice, this requires sophisticated prepayment models and Monte Carlo simulation
def simplified_oas_concept(
market_price, option_free_value, option_value, duration
):
"""
Illustrate OAS concept with simplified inputs.
OAS ≈ (Option-free yield - Market yield) adjusted for option cost
"""
# The option reduces the bond's value to the investor
theoretical_value = option_free_value - option_value
# Price difference suggests additional spread demanded
price_gap = theoretical_value - market_price
# Rough conversion: 1bp spread ≈ -duration * 1bp * price
# So spread ≈ price_gap / (duration * price * 0.0001)
implied_oas = price_gap / (duration * market_price * 0.0001)
return implied_oas
# Example: MBS trading at 98, option-free value 102, embedded option worth 3, duration 4
example_oas = simplified_oas_concept(
market_price=98, option_free_value=102, option_value=3, duration=4
)Option-Adjusted Spread Concept ================================================== Market Price: 98.00 Option-Free Value: 102.00 Prepayment Option: 3.00 Theoretical Value: 99.00 Duration: 4.0 years Approximate OAS: 26 bp Interpretation: After accounting for the prepayment option, the MBS offers ~26bp additional spread over comparable option-free securities.
