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Digital Options Pricing and Hedging

To give some background, we first spend a little time discussing single digital options. Consider a single digital call option9 with unit payoff at time t if x* E, namely C (x*) (x* - E j. The payoff can be regarded as the limit 9 Digitals Here, There, Everywhere The discussion in this section for single digital options holds for digitals of any sort - interest rate, FX, equity, commodity, etc. The discussion also holds for barrier options that in general can have digital components. In this section, we label x as the variable, not its logarithm.

Generalized Function Approach To Fourier Pricing

From what we have seen above, a pricing system can be completely represented by a pricing kernel, which is the price of a set of digital options at each time t. We now formally define the payoff of such options, for all maturities T t. We start by denoting m (B(t, T)K) St the scaled value of the strike price, where the forward price is used as the scaling variable. This is a natural measure of moneyness of the option. Now, define k ln(m) as our key variable representing the strike. We omit the subscript t to the strike for ease of convenience, but notice that at time T, k ln(K ST). Let Xt ln(St B(t, T)). Then, the Heaviside function 0(m(Xt Xt k)), where rn 1, defines the event ST K and rn 1 refers to the complementary event. So, in what follows we will refer to the probability measure of the variable XT Xt, that is, the increment of the process between time t and time T, rather than its level at the terminal date. Anyway, since we are concerned with pricing a set of contingent claims...

Example 121 FRM Exam 1997Question 16Market Risk

B) Answer (c) is not correct since the correct market price can be set at expiration as a function of the underlying spot price. The main problem is that the delta changes very quickly close to expiration when the spot price hovers around the strike price. This high gamma feature makes it very difficult to implement dynamic hedging of options with discontinuous payoffs, such as binary options.

Approaches To Option Pricing

Where 4 (St) is the risk-neutral probability density function, T is the cumulative probability function and P(ST) is the payoff elementary statistical theory allows us to write (ST)dST d T. In various parts of the book we perform this integral explicitly for European puts and calls, binary options, knock-in options, etc.

Copula Methods In Finance A Primer

How can we recover a price consistent with market quotes The first requirement that may come to mind is to ensure that the price is consistent with the market prices for plain vanilla options on each of the two indexes. Say, for example, we can recover, using some of the models or techniques described in this chapter, the risk-neutral probability QNKY that the Nikkei index at time T will be below the level Knky. We can do the same with the S&P 500 index, recovering probability QSP. In financial terms, we are asking what is the forward price of univariate digital options with strikes Knky and KSP in statistical terms, what we are estimating from market data are the marginal risk-neutral distributions of the Nikkei and the S&P indexes. Here we give a brief preview of applications to equity-linked products, beyond the simple multivariate digital options seen above. Consider a simple case of a rainbow option, such as, for example, a call option on the minimum between two assets. These...

Exotic Options and the Smile

The second major category of instruments where the existence of the volatility smile can change pricing and hedging practices significantly is exotic options. In this section, we consider a simple knock-out call that is representative of the main ideas we want to convey. Due to the contractual equation between vanilla options, knock-out calls, and knock-in calls, our discussion immediately extends to knock-in calls as well. At the end of this section, we briefly discuss digital options and how the existence of the volatility smile affects them.

Pricing Vulnerable Options

8.4.1 Vulnerable digital options We start our analysis from the simplest products, that is digital options. This will both make the building blocks of the application clearer and open the way to more complex applications, that will follow. We remind the reader that from the previous section we denote as DCi(Ki) the default-free price of a call digital option, i.e. a contract paying one unit if and only if at the exercise date T we observe Si Ki for a given strike Ki. Assume now that a digital option is written by a counterparty A, which is subject to default risk the option will pay one unit under the joint event of the option ending in the money and survival of the counterparty A it will be worth RA, the recovery rate for maturity T, if it expires in the money and counterparty A defaults it will be worth zero otherwise. We will denote this option as V-DCi(Ki). Our task is to characterize the arbitrage-free value of such an option. We assume we are able to observe or estimate the...

Related Literature

The digital options, but that is a mere question of taste, to keep a similarity with the binomial model. Our approach blends most of the features of the literature that we have so brutally reviewed. For one thing, our attention is focused on the pricing kernel, as in the first stream of literature quoted above. For another, our focus is on disentangling the payoff of this digital option from the characteristic function in the pricing formula. Differently from Lewis (2001), we are only interested in the pricing kernel, because our task is to use the model for calibration. While this interest in calibration recalls the contribution by Carr and Madan (1999), our focus is on digital instead of European options, even though we finally obtain pricing formulas for European options that can be applied in a FFT procedure to perform calibration to market data. What we think is original with respect to the literature is that our approach is cast in the framework of generalized functions, in...

Nonsmooth payoffs

Options usually have payoffs that are not differentiable, such as all options involving the max-operator in their payoffs. Common exotic options such as digital options even possess discontinuities in their payoffs. The associated pricing problems have discontinuous final conditions or final conditions which are not differentiable. The Crank-Nicolson method, as explained in Section 3.4.3, suffers from oscillating Deltas and Gammas around the strike. This feature carries over to semidiscretizations with FE for the spatial variable and FD for time to maturity. Solving the resulting system of ODEs with the method presented in Section 3.2.9 leads to oscillating Greeks as well. Crank-Nicolson time stepping, however, is of order two, O(dt2), in contrast to explicit time stepping which is only, O(dt). Various ways to use the faster Crank-Nicolson time stepping without incurring oscillating Greeks have been explored. In a pure FD setting, only a couple of time-steps are computed with explicit...

Figure 1018

Japanese exporters last week were snapping up one- to three-month Japanese yen U.S. dollar binary options, struck within a JPY114-119 range, betting that the yen will remain bound within that range. Buyers of the options get a predetermined payout if the yen trades within the range, but forfeit a principal if it touches either barrier during the life of the option. The strategy is similar to buying a yen strangle, although the downside is capped. (Based on an article in Derivatives Week) Figure 10-18 illustrates the long binary options mentioned in the example. Looked at from the angle of yen, the binary options have similarities to selling dollar strangles.11

Figure 1014

To understand binary options, first remember the static strangle and straddle strategies. The idea was to take a long (short) volatility position, and benefit if the underlying moved more (less) than what the implied volatility suggested. Binary options form essential building blocks for similar volatility strategies, which can be implemented in a cheaper and perhaps more efficient way. Also, binary options are excellent examples of option engineering. We begin with a brief description of a European style binary option. The diagram in Figure 10-15 shows the intrinsic value of the binary where R 1. What would the time value of the binary option look like It is, in fact, easy to obtain a closed-form formula that will price binary options. Yet, we prefer to answer this question using financial engineering. More precisely, we first create a synthetic for the binary option. The value of the synthetic should then equal the value of the binary. It is fairly easy to find a replicating...

Figure 171

First there are vanilla call or put options. These were discussed in Chapters 8 and 10 and are not handled here. The second tool is touch or digital options, discussed in a later chapter, but we'll provide a brief summary below. Touch or digital options are essentially used to provide payoffs (of cash or an asset) if some levels are crossed. Most equity structured products incorporate such levels. The third tool is new it is the so-called rainbow options. These are options written on the maximum or minimum of a basket of stocks. They are useful since almost all equity structured products involve payoffs that depend on more than one stock. The fourth tool is the cliquet. These options are important prototypes and are used in buying and selling forward starting options. Note that an equity structured product would naturally span over several years. Often the investor is offered returns of an index during a future year, but the initial index value during these future years would not be...

Credit Put Option

In its simplest form, a credit option can be a binary option. With a binary credit option, the option seller will pay out a sum if and when a default event occurs with respect to a referenced credit (e.g., the underlying issuer is unable to pay its obligations as they become due). Therefore, a binary option represents two states of the world default or no default it is the clearest example of credit protection. At maturity of the option, if the referenced credit has defaulted the option holder receives a payout. If there is no default at maturity of the option, the option buyer receives nothing and forgoes the option premium. A binary credit option could also be triggered by a ratings downgrade. A European binary credit option pays out a sum only at maturity if the referenced credit is in default. An American binary option can be exercised at any time during its life. Consequently, if an American binary credit option is in the money (a default event has occurred), it will be exercised...


In financial terms, modeling the pricing kernel means recovering the forward value of a digital option, i.e. an option that pays one unit of value if some event occurs. Likewise, in our bivariate setting, recovering the pricing kernel amounts to pricing a digital option that pays one unit if two events take place. So, our problem is to find a replicating strategy for the bivariate digital option. An interesting question is whether it is possible to use univariate digital options to hedge the bivariate one. In order to focus on the bivariate feature of the pricing problem, we assume that we may replicate and price two univariate digital options with the same exercise date T written on the underlying markets S1 and S2 for strikes K1 and K2 respectively. Our problem is then to use these products to replicate a bivariate option which pays one unit if S1 K1 and S2 K2 and zero otherwise. So, a bivariate call digital option pays one unit only if both of the assets are in state H, that is, in...


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The binary call option is a European style option with payoff at expiry W(N,j) B 0 when S(N,j) K and nothing otherwise. The binary put option has payoff at expiry W(N,j) B 0 when S(N,j) K and nothing otherwise. Find the values of these options for various choices on K and B. Express the binary put option value in terms of the binary call option value.

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