While deterministic structures do trave li~nitatiorls, by iircorporatirig the volatility hump, and by yieldirig a pricing mechanism that permits allalytical solutioris to[r]

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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.

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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

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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.

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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.

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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

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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.

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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.

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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].

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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.

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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]

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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.

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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].

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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.

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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 =.

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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

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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.

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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

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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.

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