Top PDF Interest rate option pricing with volatility humps

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When pricing the convexity effect in irregular interest rate deriva- tives such as, e.g., Libor-in-arrears or CMS, one often ignores the volatility smile, which is quite pronounced in the interest rate options market. This note solves the problem of convexity by replicating the irregular interest flow or option with liquidly traded options with dif- ferent strikes thereby taking into account the volatility smile. This idea is known among practitioners for pricing CMS caps. We ap- proach the problem on a more general scale and apply the result to various examples.

Modeling the term structure dynamics of (stochastic) interest rates is crucial in the research of fixed-income markets. However, problems in this area are much more com- plicated than their counterparts in equity (stock) markets. There have been several stages in the development of interest rate models. Early studies were focused on the dynamics of short rate r(t) at time t, such as Vasicek (1977), Cox, Ingersoll and Ross (1985), Hull and White (1993). Then term structure models came along, particularly the HJM framework established in the seminal work of Heath, Jarrow and Morton (1992), which governs the dynamics of forward rate f(t, T ) set at time t while becom- ing effective at T > t, or equivalently of bond price P (t, T) or yield Y (t, T ). Such an extension is interpreted as a transition from finite state models to infinite state models, because at each t, r(t) is a 1D quantity while f(t, T ) is a function (a yield curve) over different terms T ∈ [t, t + T max ]. Here T max is some fixed long term say 30 years. A

All these studies show that even these broad classes of models have been unsuccessful in accurately describing the behavior of the observed volatility smiles in the equity options markets – the empirically observed volatility smiles are typically much larger than those predicted by the theory. Clearly, some other economic phenomenon is missing from the models so far, particularly since, with frictions in the market, option-pricing models may not satisfy the martingale restriction (i.e., the price of the underlying asset implied by the option pricing model must equal its actual market value). 9 In such a situation, the no-arbitrage framework can only place bounds on option prices and hence, cannot explain the observed patterns in option prices either across strike rates or across maturities. This discussion points towards liquidity being one of the factors that might cause, or at least influence, the volatility smiles across strike rates. In this context, Constantinides (1997) concurs that, with transaction costs, the concept of the no-arbitrage price of a derivative is replaced by a range of prices, which may differ across strike rates for options. However, he distinguishes between plain-vanilla, exchange-traded derivatives (such as equity options) and customized, over-the-counter derivatives - many interest rate options fall in the latter category. From a theoretical standpoint, he argues that transaction costs are more likely to play an important role in the pricing of the customized, over-the-counter derivatives, as opposed to plain-vanilla exchange-traded contracts, since the seller has to incur higher hedging costs to cover short positions, if they are customized contracts. This issue clearly needs empirical verification and amplification.

each cell, from top to bottom are the MRE (mean relative error) and MARE (mean absolute relative error) statistics for various models as listed in Table 5. It is noted that for the SV models, all option prices are calculated using the reprojected underlying volatility and the implied risk premium of SV, while for the constant volatility models, all option prices are calculated using the implied volatility parameter. The basic conclusions we draw from the comparison are summarized as following. First, all models have substantially reduced the pricing errors of the most expensive long-term ITM options due to the use of implied volatility or volatility risk. The Black-Scholes model exhibits similar pattern of mispricing as found in other studies, namely overpricing of deep OTM options and underpricing deep ITM options. The pricing errors of long-term deep ITM options are dramatically decreased due to the larger weights put on these options in the minimization of the sum of squared option pricing errors. Second, the interest rate still only has minimal impact on option prices for both the cases of stochastic asset return volatility and constant asset return volatility. Third, all SV models outperform non-SV models due to the introduction of non-zero risk premium for conditional volatility. Compared to the Black-Scholes model, the symmetric SV models have overall lower pricing errors. Fourth, the asymmetric SV models further outperform the symmetric SV models, especially for deep OTM and deep ITM and long-term options. Finally, the asymmetric models, however, still exhibit systematic pricing errors, namely underpricing of short-term deep OTM options, overpricing of long-term deep OTM options, and underpricing of deep ITM options. This is consistent with our diagnostics of the SV model specification, i.e. the SV models fails to capture the short-term kurtosis of asset returns caused by large negative returns. These large negative returns induce a very long but thin left tail, which even SV models fail to capture. It should be noted that while percentage-wise these pricing errors appear to be large, as high as 28% for short-term OTM options, its economic implications may not be so important. For instance, for short-term deep OTM options, a 28% relative pricing errors only correspond to absolute error of roughly $ 1 = 8 on the average, which is smaller than the average bid-ask spread. Furthermore, the MARE statistics, a measure of the dispersion of the relative pricing errors, are not reduced as much as the MRE statistics, suggesting the mispricing of options by various models is less systematic.

The no-arbitrage argument assumes that the portfolio is kept globally risk free via dynamic rebalancing. The delta hedge portfolio is instantaneously risk free, but has finite risk over finite time intervals Δ t unless continuous time updating/rebalancing is accomplished to within observational error. However, an agent cannot afford to update too often (this would be quite expensive due to trading fees), and this introduces errors that in turn produce risk. This risk is recognized by traders, who do not use the risk free interest rate for r’ in (23) and (24) (where r’ determines µ ’(t) and therefore r), but use instead an expected asset return r’ that exceeds r o by a few percentage points. The reason for this choice is also theoretically clear: why bother to construct a hedge that must be dynamically balanced, very frequently updated, merely to get the same rate of return r o that a

So from 33 we may note that for the Put options will be discounted only the process S T ( ) . Now to compute the value of the option is a problem because we have stochastic interest rate so the solution is to take the default free zero coupon bond as forward measure. We assume the following process for the default free zero coupon bond:

In this paper we develop a stochastic volatility multi-factor model of the term structure of interest rates based on the Heath, Jarrow, and Morton (1992) (HJM, henceforth) framework. The model has N factors driving the forward rate curve with each factor exhibiting stochastic volatility. The model allows for hump-shaped innovations to the forward rate curve. It also allows for correlations between innovations to forward rates and stochastic volatility which implies that the model has N × 2 factors driving interest rate derivatives, except if correlations are perfect. The model has quasi-analytical zero-coupon bond option (and therefore cap) prices based on transform techniques, while coupon bond option (and therefore swaption) prices can be obtained using well known and accurate approximations. In our model the dynamics of the forward rate curve under the risk neutral measure can be described in terms of a finite number of state variables which jointly follow an affine diffusion process. This facilitates pricing of more complex interest rate derivatives by Monte Carlo simulations. We apply the flexible “extended affine” market price of risk specification developed by Cheredito, Filipovic, and Kimmel (2003). This implies that the state vector also follows an affine diffusion process under the actual measure which facilitates econometric estimation.

In this paper, we develop a tractable and flexible multifactor model of the term structure of interest rates that is consistent with these stylized facts about interest rate volatility. The model is based on the Heath, Jarrow, and Morton (1992) (HJM, henceforth) framework. In its most general form, the model has N factors, which drive the term structure, and N additional unspanned stochas- tic volatility factors, which affect only interest rate derivatives. Importantly, the model allows innovations to interest rates and their volatilities to be corre- lated. Furthermore, the model can accommodate a wide range of shocks to the term structure including hump-shaped shocks. We derive quasi-analytical zero- coupon bond option (and therefore cap) prices based on transform techniques, while coupon bond option (and therefore swaption) prices can be obtained using well-known and accurate approximations. We show that the dynamics of the term structure under the risk-neutral probability measure can be described in terms of a finite number of state variables that jointly follow an affine dif- fusion process. This facilitates pricing of complex interest rate derivatives by Monte Carlo simulations. We apply the flexible “extended affine” market price of risk specification proposed by Cheredito, Filipovic, and Kimmel (2007), which implies that the state vector also follows an affine diffusion process under the actual probability measure and facilitates the application of standard econometric techniques.

Future movement of values of risk-free interest rate and volatility are uncer- tain and as they increase, they affect call option values as depicted in the above Figure 2, Figure 3 ([5], p. 204). Sudden changes in their values may occur be- cause of economic shock. See the models suggested in [11] [12].

In this paper, we adopt the Heston–CIR hybrid model, combining the Heston stochastic volatility model and CIR (Cox– Ingersoll–Ross) stochastic interest rate model. In fact, stochastic interest rate models have already been widely adopted in pricing financial derivatives. For example, the valuation problem of European options under the correlated Heston stochastic volatility and CIR or Hull–White [ 22 ] stochastic interest rate models is considered by Grzelak & Oosterlee [ 23 ], while American option pricing under a general hybrid stochastic volatility and stochastic interest rate model is discussed in [ 24 ] with a short-maturity asymptotic expansion. Kim et al. [ 25 ] went even further and showed that adding stochastic interest rates into a stochastic volatility model could give better results compared with the constant interest rate case in any maturity. It needs to be pointed out that although Cao et al. [ 26 ] have already worked on the determination of variance swap prices under this model, their formula is not analytical since it involves solving some ODEs (ordinary differential equations) when computing the price of any variance swap, which may lead to inaccuracy problems if numerical methods are resorted to when finding solutions to these ODEs. In order to overcome this disadvantage, we present analytical pricing formulae for variance and volatility swaps, based on the derived forward characteristic function. This particular solution is actually in a series form, accompanied by a radius of convergence, ensuring the safety of its application in real markets. To demonstrate the accuracy and efficiency of the new formulae, numerical experiments are carried out to show the speed of convergence, followed by a comparison of swap prices calculated with our formulae and those obtained from Monte Carlo simulation. The influence of introducing stochastic interest rate into the Heston model is also studied.

More specifically, I computed call option prices which would obtain under three static models (mixture of distributions, compound Poisson process and Student distribution) and under thre[r]

We present a European option pricing when the underlying asset price dynamics is governed by a linear combination of the time-change Lévy process and a stochastic interest rate which follows the Vasicek proc- ess. We obtain an explicit formula for the European call option in term of the characteristic function of the tail probabilities.

the bonus policy by means of finite difference approach [12]. Participating policies with embedded surrender option have been valued within the framework of constant risk-free interest rate. Ideally, the guarantee interest rate offered by the contract is more likely to change throughout the life of the policy rather than been constant. Holders of participating policies with embedded surrender possibility might surrender their contract to take advantage of the higher yield in the financial market. Surrender options have therefore become a major concern for life insurers especially during interest rate volatility. Owing to the long maturity nature of life insurance products, if the guarantee return is not sufficiently high enough compared to other forms of investments, policy holders may terminate their existing policies early in order to go in for the higher yields offered in the capital market [5].

DOI: 10.4236/jmf.2018.81004 56 Journal of Mathematical Finance designed investment strategy can be further discussed. For example, will the in- vestment strategy during the option’s valid period influence the other aspects of stock option? Are there any differences with the option pricing under the T -forward measure and that under the risk-neutral measure? How do the option prices under the investment strategy change, if the interest rate obeys to CoxCIngersollCRoss model, BlackCDermanCToy model, BlackCKarasinski model or HullCWhite model? Moreover, option pricing with stochastic volatility may provide another story. This consideration is worth to be examined.

i=1 λ i ∆t represents the accumulative cost of borrowing the components of the underlying index or ETF. This cost, in practice, can be obtained by subtracting the average applicable “short rate” from the reference interest rate each day and accumulating it over the time period of interest. Notice, in addition to these two factors, formula (2.1.4) also shows dependence on the reference interest rate and the expense ratio of the LETF. We will show in the following paragraph, under the assumption that the price of the underlying ETF follows an Itˆ o process, formula (2.1.4) is exact, meaning ≃ can be replaced by =.

As  shown  in  Table  C1,  each  trading  day  has  a  unique  set  of  parameters.  The  training  samples  for  the  parameters on each day are the 50ETF closing prices and the O/N SHIBOR rates over the previous 500 trading days.  Therefore, the parameters for the different trading days are similar, but not the same. With regard to the parameters  of  the 50ETF closing  prices, the drift rate     denotes the  ability to achieve  long‐term stable returns. Because the  training samples are all taken from a bear market (although the 500‐day sampling period is  not long  enough  to  extend beyond the bear market), the drift parameters on the 63 trading days are barely positive. Here,     denotes  the intensity of the double exponential  jumps, and   1 ,  and   2   denote the amplitudes of the positive jump and 

The coefficients on the exogenous variables in the two equations provide important information about the common determinants of price and liquidity in this market. Higher spot rates are generally associated with higher scaled implied volatility, implying that when there are inflation concerns and expectation of rising interest rates, the dealers charge even higher prices (and wider bid-ask spreads) for away-from-the-money options. Note that the strike price effects are already controlled for, by including the LMR functions; so, this effect is incremental to the normal smile effects observed in this market. Once the effect of the spot rate is accounted for, the slope of the yield curve does not appear to have a significant effect on the ScaledIV. The impact of increasing interest rate uncertainty is similar – when swaption volatilities are higher, the scaled implied volatilities are also higher. This is indicative of a steepening of the volatility smile as options become more expensive. When there is more uncertainty in fixed income markets, dealers appear to charge even higher prices (and wider bid-ask spreads) for these options. Aggregate credit risk concerns, proxied by the default spread, do not appear to be significantly related to either price or liquidity in this market. However, equity market uncertainty does appear to be significantly associated with wider bid-ask spreads, and higher scaled implied volatilities. When there is greater uncertainty about future cash flows and discount rates in the economy, the scaled implied volatilities and bid-ask spreads of interest rate caps and floors is higher, adjusting for other effects. It appears that the revelation of information in the equity markets is one of the determinants of price and liquidity quotes posted by fixed income option dealers.

As  shown  in  Table  C1,  each  trading  day  has  a  unique  set  of  parameters.  The  training  samples  for  the  parameters on each day are the 50ETF closing prices and the O/N SHIBOR rates over the previous 500 trading days.  Therefore, the parameters for the different trading days are similar, but not the same. With regard to the parameters  of  the 50ETF closing  prices, the drift rate     denotes the  ability to achieve  long‐term stable returns. Because the  training samples are all taken from a bear market (although the 500‐day sampling period is  not long  enough  to  extend beyond the bear market), the drift parameters on the 63 trading days are barely positive. Here,     denotes  the intensity of the double exponential  jumps, and   1 ,  and   2   denote the amplitudes of the positive jump and 

The local volatility type modelling captures the surface of the implied volatilities more precisely than other approaches, (Henry-Labordere, 2009). Of importance to note is that the local volatility framework is an arbitrage free and risk neutral valuation framework. The local volatility framework is adopted to determine the European call option price with the underlying asset as the IBOR interest rate. With the introduction of derivatives market at the Nairobi Securities Exchange, an accurate pricing model for the call options on the IBOR rates will offer the best investment option for both foreign and local investors.