This paper focuses on a theoretical aspect of relations between the Black-Scholes implied volatility and the default probability in a general framework that the stock price is fixed at zero after default occurs. It is shown that the default probability of the company under a riskneutral measure significantly links to the implied volatility skew atWe consider European call and put options with default risk. Let C(K, T) and P(K, T) denote call and put prices with strike K and maturity T at time t = 0 respectively. Then, the call and the put prices under the assumption (1) can be written aswhere E[ · ] is an expectation operator under Q. The following lemma shows that default and survival probabilities can be expressed as the first derivative of the put and the call with respect to strike K at K = 0.Lemma 1 The partial differentials ∂P(K, T)/∂K and ∂C(K, T)/∂K exist for all K ≥ 0. Moreover, the risk-neutral default and survival probabilities of a reference company are given byand Q(τ > T) = −e rT ∂C(K, T) ∂Krespectively for any T > 0.

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Lévy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic function of the intertemporal joint distribution of the time-changed Lévy process. Using the derived representation of the characteristic function, we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader and lookback options, all of which are discretely monitored.

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Notes: We regard the prices computed by Monte Carlo with 10 7 sample paths as the exact prices denoted by EP. CEP1 and CEP2 denote the prices by the first and second order cumulant expansion respectively. CEP2* is the prices by the second order cumulant expansion with the spline interpolation. Here we define error := EP -CEP and error ratio := 100 × error/EP. E X H I B I T 3 RMBS, IO and PO Prices in the Case of w = 3, 5, 10Notes: Exhibit 4 shows RMBS, IO and PO prices, errors, and error ratios in the case of w = 20, 30, 50. E X H I B I T 4 RMBS, IO and PO Prices in the Case of w = 20, 30, 50 Notes: The solid line denotes the effective convexity of IO by CEP2*. The dashed line denotes the effective convexity of IO by MCI. The dotted line denotes the effective convexity of IO by MCF. The definition of IO effective convexity is the same as that of RMBS. E X H I B I T 1 7 Effective Convexity of IONotes: The solid line denotes the effective convexity of PO by CEP2*. The dashed line denotes the effective convexity of PO by MCI. The dotted line denotes the effective convexity of PO by MCF. The definition of PO effective convexity is the same as that of RMBS.