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# Derivatives Modeling

1. Forward Starting Option

Option price or implied volatility surfaces are available at points on a relatively sparse grid of strike and tenor pairs. Using analytical expressions to determine the local volatility function then likely yields inaccurate results due to the numerical instability from calculating first, and especially, second derivatives.

A forward starting option is an option whose strike price is not fully determined until an intermediate date before expiration.

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2. Three Factor Convertible Bond Model

The stock price process is then expressed under the bond’s coupon currency risk neutral probability measure by means of a quanto adjustment.   Under the bond’s coupon currency risk neutral probability measure, then, the short interest rate, stock price and foreign exchange rate processes respectively follow geometric Brownian motion with drift, but are driven by pair-wise correlated Brownian motions.

We next define three related random variables, which are each taken to be particular linear combinations of the original short interest rate, stock price and foreign exchange rate random variables.  Here the respective linear combinations are chosen such that the processes for the new random variables are now driven by pairwise uncorrelated Brownian motions.

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3. Hull-White Convertible Bond

Based on the Hull-White single-factor tree building approach, respective trinomial trees are constructed for the short-term interest rate and stock’s price processes.  Using the Hull-White two-factor tree building procedure, a combined tree is constructed by matching the mean, variance and correlation corresponding to each combined tree node.  The convertible bond price is given from the combined tree by backward induction.

Here the issue time refers to the coupon payment immediately prior to, or including, the valuation time; otherwise it corresponds to the bond’s issuance.  Since the valuation time is taken to be zero, the issue time must be less than or equal to zero.

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4. Exchangeable Convertible Bond

A convertible bond issuer pays periodic coupons to the convertible bond holder. The bond holder can convert the bond into the underlying stock within the period(s) of time specified by the conversion schedule. The bond issuer can call the bond and the holder can put it according to the call and put provisions. The Exchangeable feature assumes that the convertible bond and the underlying stock are issued by different parties.

Assume that the stock conversion is vulnerable. If the bond-issuer has defaulted by a time, t , then the stock price is zero. If, on the other hand, the bond-issuer has not defaulted by time t , then the stock price is given by St or 0.

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5. Extendable Swap

An extendable swap represents a forward swap agreement with an option of extending the swap for another term (swaption). The valuation model assumes the swap rates for different terms to be correlated log-normally distributed random variables and uses the Haselgrove integration method for pricing the deal.

The model estimates the swap price as a risk-neutral expectation of the difference between the bond price whose yield-to-maturity is the swap rate and the bond’s par. The swap rate is considered a log-normally distributed random variable.

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6. Callable Inverse Swap

A Callable Inverse Floating Rate Swap is a forward swap agreement with an option of canceling the swap each year starting from several years in future. The deal is priced with a two factor Black-Karasinski model.

The calibration procedure takes only an interest rate curve as input (ignoring volatility surfaces) and results in adjusting the “alpha” parameter of the model. To test the calculations over a range of parameters, we used  the “piece-wise constant parametrization” mode.

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7. Bond American Option

The model assumes the yield of an American Treasury bond to be a log-normally distributed stochastic process and uses Monte-Carlo simulation to price the deal as a European call option.

The model builds a trinomial tree for the yield process to price the deal as an American option. The time slices of the tree are evenly spaced. Node transition probabilities and the time interval between slices are determined by matching the first four moments of the underlying Brownian motion. The option is priced using the backward induction.

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8. Arrear Quanto CMS Swap

An arrear quanto constant-maturity-swap (CMS) is a swap that pays coupons in a different currency from the notional and in arrears. The underlying swap rate is computed from a forward starting CMS.

We note that the common currency unit in Europe is now taken to be the EURO.  Furthermore, the exchange rate from the EURO to an associated currency (e.g., FRF) is fixed, so there is no foreign exchange risk.  Therefore, FP London uses a common curve, EURIBOR, for discounting

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9. Variable Rate Swap

Variable rate swap is a special type of interest rate swap in which one leg of the swap corresponds to fixed rate payments while the other involves fixed rate payments for an initial period of time and a floating rate for the rest. The floating rate on that portion is defined as a minimum of two index rates.

Variable rate swap is an interest rate swap that has two legs: one fixed rate leg and a variable rate leg. The variable leg involves fixed rate payments for an initial period of time and a floating rate for the rest. The floating rate on that portion is defined as a minimum of two index rates.

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A constant maturity swap (CMS) spread option makes payments based on a bounded spread between two index rates (e.g., a GBP CMS rate and a EURO CMS rate).  The GBP CMS rate is calculated from a 15 year swap with semi-annual, upfront payments, while the EURO CMS rate is based on a 15 year swap with annual, upfront payments.

We assume that both the forward GBP and EURO CMS rates follow geometric Brownian motion under their respective -forward measures.  Here respective initial forward CMS rates are calculated.  The forward rates are then convexity adjusted from respective parallel bonds specified using

11. Early Start Swap

An early start swap is a swap that has an American style option for the counterparty of starting the swap early, within a period of three month. Otherwise, the swaps are plain vanilla fixed-for-floating swaps.

The internal rates of return of the two swaps, one starting at the beginning and the other at the end of the exercise period, are generated for the earliest exercise date, assuming that the two rate are practically perfectly correlated. Then the difference of the present values of the two swaps, if positive, is taken as the option value. This value is averaged over a number of scenarios.

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12. Digital Swap

A daily digital LIBOR swap is an interest rate swap whose reference interest rate is three-month USD Libor BBA. For each accrual period in the swap, one party receives the reference rate, and pays the reference rate plus a positive spread, but weighted by the ratio of the number of calendar days in the period that the reference rate sets below an upper level to the total number of calendar days in the period.

We assume that Libor rates follow geometric Brownian motion with no drift and constant volatility under their respective forward measures. In order to value a daily Libor-based digital payoff, the respective Libor rates at the daily setting time and at the accrual period start must then be expressed under the same forward measure.

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13. Ratchet Swap

The ratchet floating rate coupon is based on an index, e.g., 6-month EURIBOR. The rate is further subject to a minimum decrease of 0 bps and a maximum increase of a threshold, such as, 15 bps. These rates are reset two business days prior to the first day of each coupon period.

The valuation methodology is based on the Monte Carlo spot LIBOR rate model. The model generates spot rates which log-normally distributed at each reset date. These spot rates are derived from corresponding forward rates whose stochastic behavior is constructed in an arbitrage-free manner. Outcomes for the spot rate are generated for each reset date. These rates are then applied to the ratchet-type payoff structure. The ratchet instrument is then valued by discounting and averaging these payoffs.

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14. Hedge Fund Barrier Option

A hedge fund barrier call option is a note whose payoff is based on a basket of hedge funds. The deals are structured so that once the barrier (usually set at 95% of the notional) is hit, the funds in the basket are sold off, with the realized fund value depending on the redemption period of each fund..

The goal here is to estimate the market risk of the entire portfolio of such deals through analysis of a small representative sample of the portfolio and scaling up to the entire portfolio. While simulating the entire portfolio would result in a more accurate determination of the capital, the result is small enough that the dominant risk factors likely arise from sources other than market risk, and an order-of magnitude determination is likely sufficient.

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The Canada Housing Trust Swap includes variable rate mortgages and involves reinvestments made by the principal payments. The variable rate mortgages that appear in the deal are a result of these reinvestment. The model was in the context of the much more complicated problem where the notional on the mortgages was not fixed, and reinvestments were made at prevailing market prices.

The valuation of the variable rate mortgage, as it contributes to the CHT swap is very simple. It is simplified by the fact that the interest rate payable is fixed at 30 day BA, and that we are discounting using the same curve that is used to evaluate the interest received. Furthermore, the principal payments on the variable rate mortgages, including prepayments, are reinvested at par so that the principal remains constant.

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16. BMA Knockout Swap

BMA Ratio Swap with BMA Knockout is a two-legged BMA ratio swap where one leg pays a contract specified fixed rate and the other leg pays Libor times a contract specified ratio (plus a contract specified constant spread).

If we consider a deal called Libor Swap with BMA Knock-Inwhere the knockout condition is defined by a maximum level for average BMA, the coupon payments at time i S are the following:

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17. BMA Swap

In a generic fixed for floating BMA swap, the floating side is estimated using averages of the BMA Municipal Swap Index, which is published on a weekly basis. The fixed side pays interest quarterly on a 30/360 yield basis and payment dates adjusted using modified following basis. The floating side pays interest quarterly in ACT/ACT yield basis and payment dates adjusted using the modified following basis.

Three types of swap legs are available, including BMA leg as well as typical LIBOR (floating) leg and fixed leg with variable notional. BMA leg pays (or receives) weighted average of weekly BMA indices over specified periods, based on ACT/ACT day count basis (DCB). LIBOR leg receives (or pay) LIBOR rate multiplied by a fixed ratio using ACT/360 DCB.

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18. Chooser Cap

A chooser cap (floor) is different from the traditional European/Bermudan option that the owner of the chooser option has multiple chances to exercise. The rigorous definition of chooser option is given in the appendix section of this report. From the definition of the chooser option, a lower bound of the value of the chooser cap (floor) is the sum of first k maximal values of (European) caplets (floorlets). To get a good upper bound is not trivial.

From a rigorous view point, the dynamics may not be completely arbitrage-free. However, it perfectly re-produces all European caplets (floorlets) market prices automatically. Therefore, the dynamics can be considered as approximately arbitrage-free without any additional calibration. It should also be noted that volatility skewness is not considered in this dynamics.

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19. CMS Cliquet

A CMS cliquet option has two legs: One leg of this deal is based on (regular) floating rates. The other leg links to CMS swap rates. Due to the “set-in-arrear” feature in the structured leg, convexity and timing adjustments have to be considered.

Pricing the second leg of the contract is a little bit more involved. Firstly, there is an optionality which is embedded in the contract, and secondly, this leg does not incorporated a natural time lag, which implies that the convexity adjustment is needed.

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20. Arrear Forward Rate Agreement

A general FRA is a European forward rate derivative with a maturity which is not earlier than the beginning of the forward period. A vanilla FRA is the same type of security except its maturity is right at the end of the forward period. While, a set-in-arrear FRA is the one whose maturity is right at the beginning of the forward period.

Generally, convexity adjustments are required for pricing these FRAs except vanilla FRAs. Under the assumption of single factor driving force, for a FRA whose maturity is before the end of the forward period, the convexity adjustment is positive while for a FRA whose maturity is after the end of the forward period, the convexity adjustment is negative.

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