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After the scenarios are generated, one has to calculate the behavior of financial instruments to movements in the underlying risk factors. Therefore two different approaches are chosen: the sensitivity approach applies relative valuation adjustments to the instrument base values proportional to the defined sensitivity. The valuation adjustments are derived from movements in value of the underlying risk factors. Secondly, in a full valuation approach, the scenario dependent input parameters are fed into a pricing function which (re)calculates the new absolute value of the instrument in each particular scenario.
The sensitivity approach is performed for all instruments which have a linear dependency on the movement of underlying risk factor values, or where not enough data or no appropriate pricing function exists for a full valuation approach.
For the risk assessment of equities, commodities or real estate instruments it is appropriate to take only the linear dependency on risk factor movements into account. Each risk factor has sensitivities to underlying risk factors. An instrument inherits the amount of shock proportional to the defined sensitivity value from the underlying risk factors.
A typical example is a developed market exchanged traded fund (ETF) with exposure to North America, Europe and Asia. The ETF will follow the price movements of these three risk factors, so the sensitivities are simply the relative exposure to the three equity markets (e.g. 0.5, 0.3 and 0.2). These sensitivities are then used to calculate the weighted shock that is applied to the actual instrument value. The same principle holds for single stock, where sensitivities to appropriate risk factors and to an idiosyncratic risk term (an uncorrelated random number) can be selected. A useful method for calibration is the multi-dimensional linear regression. The resulting betas from this regression can be taken for the sensitivities to regressed risk factors. The remaining alpha and estimation error resembles the sensitivity to the idiosyncratic risk. The exposure to the uncorrelated random number can be derived from one minus the adjusted R_square of the regression. Since the R_square gives the amount of variability that is explained by the regression model, one minus R_square is equal to the amount of uncorrelated random fluctuations which are not covered by the input parameters.
For instruments with insufficient information one can also choose the sensitivity approach.
One example are funds consisting mainly of bonds. Without look-through, one has no information about the exact cash flows of the underlying bonds. Instead, often the duration (and convexity) of the fund is known.
These two types of sensitivity (duration and convexity) can then be used to calculate the change in value to interest rate shocks. Therefore, the absolute interest rate shock at the node, which equals the Macaulay duration, is calculated as absolute difference compared to the base rate. This change in interest rate level (dIR) is directly transformed to a relative value shock (dV) by the formula
dV = duration * dIR + convexity * dIR^2
incorporating both sensitivities.
The core competency of a quantitative risk measurement project is full valuation, where the absolute value of financial instruments is calculated from raw input parameters by a special pricing function. Some input parameters to the pricing function are scenario dependent, other are inherent to the instrument. The most important input parameters have to be modeled by stochastic processes and can subsequently be fed into the pricing function, where the new, scenario dependent absolute value of the instrument is calculated. At the moment, the following full valuation pricing functions are implemented in OCTARISK:
European plain-vanilla options are priced by the Black-Scholes model (See option_bs). The Black-Scholes equation provides a best estimate of the option price. The underlying financial instrument and implied volatility are modeled as risk factors in order to calculate the new option price in each scenario. The risk free rate will be also made scenario dependent in order to capture the interest rate sensitivity of the option price. Further information is provided by numerous textbooks.
For American options a more sophisticated model has to be used for pricing. Unfortunately, binomial models (like the Cox-Rubinstein-Ross model) or finite-difference models are not feasible for a full valuation Monte-Carlo based approach, since the computation time for a large amount of time steps and MC scenarios is too high due to missing parallelization opportunities. Instead, a Willow-Tree model is implemented to price American options. Within that model, instead of using the full binomial tree with increasing number of nodes per time step, a constant number of nodes at each pricing time step is utilized to approximate the movements of the underlying price. With optimized transition probabilities, the whole model relies on a smaller amount of total nodes which significantly decreases computation time and lowers memory consumption (See option_willowtree implementation for further details).
A calibration is performed to align the model price based on provided input parameters with the observed market price. This calibration calculates an implied spread which is added to the modeled volatility as a constant offset.
The implied volatility itself is dependent on the option strike level and time to maturity (term). In order to grasp that behavior, the so called volatility smile is modeled by a moneyness vs. term volatility surface, where changes in the spot price lead to moneyness changes. Therefore, the actual implied volatility behavior (at-the-money implied volatility vs. moneyness vs. term) of the market is preserved for the pricing.
Amongst simple underlying instruments or indizes, baskets of several indizes or instrument can be used. A diversified basket volatility is calculated which serves as im plied volatility for the option derivative. Baskets itself are modeled as synthetic instruments.
Besides plain vanilla options, closed-form solutions are used for pricing of European Barrier and European Asian options. Continously geometric average asian options are priced by Kemna and Vorst model of 1990, while arithmetic average asian options are priced by Levy (1992) model.
European plain-vanilla swaptions are priced via the closed form solution of the Black-76 model or the Bachelier model. (See swaption_black76 and See swaption_bachelier for details). Again, a calibration is performed to align the model price based on provided input parameters with the observed market price. The calibrated implied volatility spread is subsequently added to the modeled volatility as a constant offset. The volatility smile for the specific term is also given by volatility cubes. The interest rate implied volatility can be set up by three axes: underlying tenor, swaption term and moneyness. For each of these combinations an implied volatility can be set. For Swaption underlyings either discount curves or fixed and floating leg swaps can be used.
Equity and Bond forwards and futures can be valuated. Therefore market indizes can be set up, which serve as underlyings for the forwards. The value of the forward or future is calculated as the payoff at maturity (underlying value minus strike) discounted back to the valuation date. For futures, net basis and accrued interest can be taken into account. (See pricing_forward for details).
Cash flow instruments are specified by the following sets of variables: cash flow dates and corresponding cash flow values. Moreover, each cash flow instrument has an actual market price and an underlying interest rate curve, which has to be provided as a separate risk factor. Before the full valuation can be carried out, the spread over yield is calculated to align the observed market price with the value given by the pricing function. The spread over yield is then assumed to be a constant offset to the scenario dependent interest rate spot curve. For each scenario, all cash flows are discounted with the appropriate interest rate and spread curve. The present value is then given by the sum of all discounted cash flows. Credit spreads are modeled as separate risk factors, thus capturing credit spread risk for cash flow instruments.
The Bond instrument class covers the full spectrum of plain vanilla bond instruments: fixed rate bonds, floating rate notes, fixed and floating swap legs, fixed rate amortizing bonds, mortage backed securities with prepayments, floating swap legs based on CMS rates, and many more. During pricing, at first the cash flows are rolled out. The forward rates for calculation of floating payments are scenario dependent. After the rollout is done, a spread over yield is calibrated in order to match the market price with the theoretical value. For calculation of the net present value of the bonds, a discount curve can be set. If the curves have attached risk factors, the pricing will be fully scenario dependent. CMS floating swaps can also have special cash flowst based on averages or capitalized CMS rates. Moreover, convexity adjustment (according to Hull or Hagan) can be taken into account.
Moreover, bonds with embedded call or put options can be modelled. Therefore a trinomial Hull-White tree is used to price these embedded European or American bond options. See (See option_bond_hw for further details.
The CapFloor class covers caps and floor instruments on discount curves. The cash flow rollout of these caps and floors also covers CMS rates and convexity adjustment. Both Black and Normal models are implemented, thus allowing for negative interest rates.
In order to model a fixed share combination of instruments, the synthetic class was introduced. The value of the synthetic instrument is calculated as the linear combination of all underlying instrument. Synthetic instruments can be used to e.g. model portfolios with full look-through or to combine fixed and floating legs to swaps. Moreover, more complex instrument including option behaviour can be modeled, if e.g. a bond and a option is combined to an instrument with embedded call / put optionality.
In order to pre-calculate cash flow values and instrument prices in other risk systems, the stochastic instrument class was introduced. Based on modelled risk factor values (e.g. correlated uniformly or normal distributed random variables), random numbers are used to determine quantile numbers and then draw cash flows or values from special curves or surfaces. These objects are used as storage containers for quantile dependent values.
Retail instruments are sold to customers in the private banking and insurance market. Typically, no second market exists for such products, making an theoretical valuation necessary. Typically, savings products and defined contribution plans have some put option behavior and can therefore be redeemed by the customer (often times to pre-agreed values and times) Moreover, savings rates can be adjusted at any point in time or extra payments could be made. Octarisk’s Retail class reflects such optionalities and allows for efficient calculation of interest and spread risk as well as sensitivities for such products allowing for a full risk assessment of all financial products in a private investors portfolio.
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