Question #21 - Reading 31 Valuation of Contingent Claims

Reading: Reading 31 Valuation of Contingent Claims

PDF File: Reading 31 Valuation of Contingent Claims.pdf

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Shared Context

of 111 Regarding options on a stock without dividends, it is: A) sometimes worthwhile to exercise calls early but not puts. B) sometimes worthwhile to exercise puts early but not calls. C) never worthwhile to exercise puts or calls early. Lowell Wood is using the binomial option-pricing model to price interest rate options. She has obtained the following 2-year, annual rate tree (based on an assumed volatility of interest rates of 25%). Exhibit 1 Wood has been asked help a colleague with the valuation of an interest rate put. The interest rate put option has 2 years to maturity and a strike price of 4.5% and is based on 360 day MRR. The option has a notional principal of $10m. Wood has discovered that the Black model may be used to price options on interest rates by viewing the interest rate option as an option on a FRA. She is currently writing a research note for her team and makes the following three notes regarding the Black model: Note 1: "When using the Black model care needs to be taken to ensure that the payoff is discounted from the end of the borrowing and lending period (i.e., the maturity of the rate underlying the FRA), rather than the exercise date of the option." Note 2: "Given an interest rate option is an option on a FRA, call options will gain in value when interest rates rise and put options will fall in value." Note 3: "The accrual period needs to be factored in when valuing the option. This is because quoted rates are annual rates but in reality, the time between the FRA expiration and the maturity of borrowing and lending may not be one year. The accrual period can be viewed as a fraction of a year." Wood asks for information about interest rate caps and floors. Newman makes the following comments: Comment 1: "A long FRA can be viewed as equivalent to a long interest rate call and a short interest rate put with the same strike and time to expiration." Comment 2: "Given a cap is a series of interest rate call options with identical strike prices and a floor is a series of interest rate put options also with identical strike prices, a short cap and long floor with identical strike prices would create a pay fixed receive floating interest rate swap."

Question

Using the information about the interest rate put and the spot and forward rates in Exhibit 1, which of the following is closest to the value of the put? Assume that the option cash settle at time 2.

Answer Choices:

A. $44,250

B. $64,250

C. $84,250

Explanation

Step 1: At the expiry of the option at T2 consider whether the option will be exercised in each of the forward scenarios. Remember an interest rate put option allows the holder the right but not obligation to pay floating and receive fixed. The strike price (in this case 4.5%) is the fixed rate in an interest rate option. The put will be exercised in the forward rate is less than the fixed rate. Forward rate 6.8% – option lapses Forward rate 4.12% – option exercised Forward rate 2.5% – option exercised Step 2: We now calculate the pay off on the option at the options expiry given each interest rate scenario. This assumes the payoff on the option is at T2, which is technically incorrect as in reality the payoff is at the end of the borrowing and lending period (T3) rather than the expiry of the option (T2). If we exercise the interest rate option we then enter pay floating receive fixed until the end of the borrowing and lending period. Calculate the payoff on the put at T2 in each scenario: (interest rate received – interest rate paid) × days / 360 × notional principal forward rate 6.8% – option lapses payoff zero forward rate 4.12% = (0.045 – 0.0412) × 360 / 360 × $10m = $38,000 forward rate 2.5% = (0.045 – 0.025) × 360 / 360 × $10m = $200,000 Step 3: Discount the payoffs back through the binomial interest rate tree to T0. Note that in a binomial interest rate tree we always have a 50% chance of an up move and a 50% chance of a down move. Discount the probability weighted amounts from T2 back to T1 at the relevant forward rate. Value at T1 upper node: [($0 + $38,000) ½] / 1.0495 = $18,103.86 Value at T1 lower node: [($38,000 + $200,000) ½] / 1.03 = $115,533.98 Discount the probability weighted amounts back from T1 to T0 at the 1 period spot rate to arrive that the price (premium) on the interest rate put. [($18,103.86 + $115,533.98) ½] / 1.04 = $64,248.96 Diagrammatically:

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