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Pricing Model
Equity Linked GIC Pooling Study The equity GIC product has the property that strike levels are the closing index levels on date of issue. The proposed specification for pooling employs pool idents with the same maturity, issue date, and index levels. Payoff for equity GIC requires a dynamically created basket such that the weight factors incorporate a division by the spot levels on the issue date, which converts the payoff to one based on a basket of comparative returns (rather than basket returns). To automate feeds on a daily basis we will need to create new baskets on a daily basis. The Pooling Criteria is the process of grouping like securities for the purpose of creating a pseudo-GIC to efficiently hedge the total balance. The new process will group a portion of the GICs, greater than 3 million, into pools and leave the remaining GICs in Unallocated pools. GICs exist on four source systems; IPS (non-registered), RRSP (registered), RRIF (registered) and SIT (registered and non-registered). Quanto Himalayan Option Study Analytics for pricing quanto Himalayan options on equity is discussed, where the single best return is locked in each fixing period. Specifically, we considered the impact of the quanto adjustment term on calibration and the computation of option premium and hedge ratios Himalayan options are a form of European-style, path-dependent, exotic option on a basket of equity underliers, in which intermediate returns on selected equities enter the payoff, while the equities are subsequently removed from the basket. This definition is consistent with the one in the case of domestic equities, that is, in the absence of a quanto adjustment. In the quanto case, the effect of the change in the quanto adjustment resulting from a change in the volatility is ignored. Local Volatility Model Study We review a model for computing the price, in the domestic currency, of European standard call and put options on an underlying foreign equity (stock or index) with tenor of up to 7 years. The function implements a local volatility based pricing method. Consider the calibration of the local volatility function based on market option prices or, equivalently, market Black’s implied volatilities. If there exists a smooth surface of either option price or implied volatility as a function of option strike and maturity, then this surface uniquely determines the local volatility function. Forward Starting Option Study A model is presented for pricing European exercise, forward starting options on an underlying equity. The valuation requires two interest rate curve inputs. One curve, called the “hedge curve”, is specified by a list of n term and continuously compounded zero rate pairs and is intended for use in computing forward equity values. In practice, 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. A model is used to compute option price, Delta, Gamma, Hedge Rho, Discount Rho, Vega and Theta. Three Factor Convertible Bond Study The owner of a convertible bond (CB) receives periodic coupon payments from the issuer, but can also convert the CB into the issuer’s stock. The convertible bond may also include call and put provisions, which respectively allow the issuer to buy back the convertible bond and the owner to put the convertible bond for respective preset amounts. Here the short interest rate process is assumed to be of Ho-Lee form under the bond’s coupon currency risk-neutral probability measure, and the stock price and foreign exchange rate processes are assumed to follow geometric Brownian motion with drift under their respective risk-neutral probability measures. We use a three factor, trinomial tree based model for pricing the CB. Here we construct three independent trinomial trees, which are then combined into a three-factor tree. Mutual Fund Securitization Study The purpose of the model is to determine, from a projected stream of future cashflows, whether all Commercial Paper used to fund the commissions to brokers for the sale of mutual funds will be repaid within a period. Here a broker charges the Partnership a commission on the net asset value of the mutual funds sold. The buyer of the mutual funds, however, pays nothing up front; instead, a deferred sales charge, which depends on when the mutual funds are redeemed, is assessed. The model assumes that the monthly net asset value of the mutual funds follows a deterministic process. Administration and program fees, as well as mutual fund redemptions are then based on the monthly net asset value. Issued Commercial Paper is amortized into equal monthly payments over a period of six years. Here the cashflows generated from the administration and redemption fees are paid monthly to the Partnership, to be used for the payment of outstanding Commercial Paper and associated interest. Furthermore, the model includes a test to determine whether a collateral infusion is required to aid in re-paying the Commercial Paper. The net asset value of the mutual funds follows a deterministic process, and that one party's capital contributions to fund sales commissions are amortized on a straight-line basis over a period of six years. The purpose of the model is to determine, from a projected stream of future principal and interest payments, whether all Commercial Paper will be repaid within a maximum period of thirteen years. Hull-White Convertible Bond Study A convertible bond pays the holder periodic coupon payments from the issuer, but can also convert it into the issuer’s stock. The model uses a two-factor trinomial tree for pricing the convertible bond. Here we assume that the short term interest rate process follows a stochastic differential equation (SDE) of the Hull-White form, and that the stock’s price follows geometric Brownian motion with drift. For tractability, however, the stock’s price SDE is modified so that its drift does not depend on the stochastic short-term interest rate. 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. Brownian Bridge Swap The Brownian bridge algorithm has been implemented for stress testing within the Risk Management framework. It is used for generation of multidimensional random paths whose initial and ending points are predetermined and fixed. The Brownian Bridge algorithm belongs to the family of Monte Carlo or Quasi-Monte Carlo methods with reduced variance. It generates sample paths which all start at the same initial point and end, at the same moment of time, at the same final point. In the context of stress testing this algorithm is used for efficient generation of specific scenarios subject to certain extreme and generally unlikely conditions. If paths were generated by a conventional Monte-Carlo method only a very small portion of all the paths would satisfy such conditions. Convertible Bond with Exchangeable Feature Study 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. A convertible bond with exchangeable feature, which can be converted into a stock issued by a party different from the bond issuer. 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. There are two possible cases with respect to stock conversion: Covered - the bond issuer holds the underlying stock for the life time of the convertible; Vulnerable - the bond issuer does not hold the stock prior to the stock conversion. Asset Backed Senior Note Study We consider a securitization deal, which allows the holder to purchase co-ownership interests in a revolving pool of credit card receivables. To fund the acquisition of the interests in the revolving pool, the trust issued Asset-Backed Notes, in a number of different series. A share of future collections of credit charge receivables, to which the trust is entitled, is used to pay the interest and the principal of the notes. An early amortization event is a set of circumstances specified in the offering memorandum, which if it were to occur, would effectively move the liquidation date forward to the time of its occurrence. Generally, early amortization is triggered by the failure of the pool assets to remain within a certain range of value or quality. Both the aggregate interest and the allocated collection amount depend on the remaining aggregate (senior and subordinated) principal X. Each month during the liquidation period the aggregate principal is reduced by the senior principal payment Z, unless it falls below a certain amount, which will trigger the cleanup prepurchase option, so that the entire remaining principal is paid off immediately (not shown in the formulae for simplicity). Hull White Volatility Calibration Study Hull White model is a short rate model satisfies a risk-neutral SDE of the form. We map implied Black's at the money (ATM) European swaption volatilities into corresponding Hull-White (HW) short rate volatilities. We seek to determine a HW volatility to match the market price of a certain ATM European payer swaption. In particular let Ti be a Libor rate reset point. Furthermore consider a fixed-for-floating interest rate swap of the following form, At each grid point, we compared respective Black’s and HW trinomial tree payer swaption pricing benchmarks. Specifically, using the interest rate and implied Black’s volatility. Government Bond Bootstrapping Study We discuss a method for bootstrapping a set of zero rates from an input set of US government money market securities and bonds. The government bond bootstrapping procedure requires to input a set of financial instruments, of the type below, sorted by order of increasing time to maturity: Government Bond Bootstrapping proceeds in two phases. The first phase uses short term instruments, which typically mature in one year or less. Consider, for example, a US government money market instrument with GIC Pricing Study Guaranteed investment certificate (GIC) is a financial instrument that offers a return based on a corresponding GIC rate and the performance of a basket of certain stock and bond market indices. The payoff at maturity from a GIC can be shown equal to the invested principal plus principal times the sum of the minimum guaranteed interest rate and the payoff from a European call option on the arithmetic average of the basket price, where the basket price is given by a weighted sum of the index levels. The rate of return is then given by a weighted sum of the GIC rate and the percentage change in each index level described above, but bounded below by a minimum guaranteed interest rate. At maturity, the GIC holder receives the invested principal plus the principal times this rate of return. Extendable Swap Study 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 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. Callable Inverse Swap Study 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. The Black-Karasinski class of models assumes the short term interest rates to be log-normally distributed. The spreadsheet mode used for the deal pricing has a hard-coded term structure of the mean reversion and volatility parameters, constructed as Chebyshev polynomials. Flexible GIC Study A flexible GIC is an investment with an embedded option to redeem the principal and accrued interest at any time after 30 days from the date of purchase. In other words, the holder of GIC has an option to redeem the principal and accrued interest at any time after 30 days of from the date of purchase. No interest is paid if the investment is redeemed within first 30 days from the purchase date. In the case of exercise a penalty is applied: the accrued interest is recalculated according to a lower rate than that set for the deal till maturity. The interest adjustments are typically different for the two exercise periods: the adjusted interest rate for the first exercise period is lower than that for the second one. We price the option of a flexible GIC with a one factor Hull-White model via a trinomial tree. The Hull-White model assumes a normal distribution for the rates. Our solution constructs a Hull-White tree. The calibration procedures take an interest rate curve as input (ignoring volatility surfaces) and assume volatility and mean reversion parameters as constants. American Bond Option Study A valuation model is presented for pricing an American style call option on the yield of Treasury bond. The payoff is positive if the yield exceeds a predetermined strike level. 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 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. Martingale Preserving Tree Study An important feature of the popular three factor trinomial tree is that it uses a deterministic approximation of the interest rates for constructing the stock tree. The preservation of the martingale property of the stock price is thus not guaranteed. and may potentially represent a problem. We propose a three-factor tree model that implements the Hull-White and Black-Karasinski models. The new tree model does preserve the martingale property of the stock for sufficiently long terms (with accuracy better that 10-8 for terms of at least 10 years). Given the limitations of the three-factor model that stem from the deterministic approximation of the interest rates, it performs remarkably well. Even for a bond’s term as long as 7 years the difference between the model and benchmark is within acceptable limits. The differences grow rapidly for longer terms, and the model is not recommended for terms exceeding 7 years. Arrear Quanto CMS Study 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. Assumes that, under the coupon payment currency (SEK) risk-neutral probability measure, the forward swap rate process corresponding to each swap rate fixing follows Geometric Brownian motion with drift. Each forward swap rate process is then convexity adjusted, and is furthermore expressed under the notional currency (FRF) risk neutral-probability measure by means of a quanto adjustment. The initial forward swap rate is also quanto adjusted. We note that the correlation used in the spreadsheet is between the FRF to SEK exchange rate and the SEK swap rate. Black-Karasinski Short Rate Tree Study The Black-Karasinski model is a short rate model that assumes the short-term interest rates to be log-normally distributed. The one factor Black-Karasinski model is implemented as a binomial or trinomial tree Black-Karasinski short rate tree approach can be used to price convertible bond. Convertible bond is not only a coupon paying bond but also can be converted at the discretion of the holder within the periods of time specified by the conversion schedule. Typically, the issuer has the option to buy the bond back at a predetermined strike price(s) during the callable period(s). Also, there are provisions that allow the holder to return the bond to the issuer in exchange for a predetermined cash price during certain period(s). Variable Rate Swap Study 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. The two index rates are treated as assets whose values are lognormally distributed random variable. Their pricing procedure uses discount factors retrieved from the EURIBOR curve. CMS Spread Option Study 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. 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 Early Start Swap Study 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. Early start swaps are a series of 10-year interest rate swaps. Each swap 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. We consider a swap as an exchange of a principal for a coupon bearing bond with the same principal. Indeed, from payer’s perspective, the swap’s present value is the difference between the present values of received LIBOR payments Quanto Total Return LIBOR Swap Study A quanto total return Libor Swap is a swap where one leg is a regular floating leg paying LIBOR less a constant spread and the other leg makes a single payment at the swap’s maturity equal to a leveraged non-negative return on USD-for-EURO exchange rate paid in CAD. The main focus of the valuation model is the quantoed total return on the FX rate. The payoff of the leg based on the return of the foreign exchange rate is a payoff of a European call option. Its present value is given by Black’s formula for futures with the discounting factor equal to the Canadian zero-coupon bond The payoff of the quanto-return leg is max(1.3 x (FX at maturity - initialFX)/initialFx), 0). Its present value is given by Black’s formula for futures with the discounting factor equal to the Canadian zero-coupon bond and the future price. Daily Digital LIBOR Swap Study 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. The payoff amount can be viewed as the value of the sum of a series of daily Libor digital payoffs. Here we assume that Libor rates are log-normally distributed, and derives an analytical formula for the daily digital payoff. The 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. Ratchet Swap A ratchet swap is an interest rate swap with two legs. One leg is a standard floating leg and the other leg is a ratchet leg. The ratchet leg pays a ratchet floating rate. 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. LIBOR Rate Model LIBOR Rate Model is used for pricing Libor-rate based derivative securities. The model is applied, primarily, to value instruments that settle at a Libor-rate reset point. In order to value instruments that settle at points intermediate to Libor resets, we calculate the numeraire value at the settlement time by interpolating the numeraire at bracketing Libor reset points. . Although referred to as a BGM model, the model is actually based on Jamshidian’s approach towards Libor rate modeling (i.e., where Libor rates are modeled simultaneously under the spot Libor measure). Libor rate model is very useful to price callable exotics. Many derivatives have callable features. Callable exotics are among the most challenging derivatives to price. These products are loosely defined by the provision that gives the holder or issuer the right to call the product after a lock-out period |