# What can we say about digital puts and calls with different strike prices?

I am a noob to the field of quantitative finance. I am reading this book by Mark S. Joshi. Can you help me make sense of one of the exercise questions? Here is the question (from page 40 of the book):

Exercise 2.3 Let $$P$$ be a digital put struck at $$K_1$$ and $$C$$ be a digital call struck at $$K_2$$. (A digital put pays 1 if spot is below the strike at expiry, and a digital call pays 1 if the spot is above the strike.) What can we say about the prices of $$C$$ and $$P$$ in each of the following cases?

1. $$K_1=K_2$$;
2. $$K_1 < K_2$$;
3. $$K_1 > K_2$$.

Here are the solutions (from page 474):

Exercise 2.3

1. Precisely one of the derivatives pays off so the value of the two together at expiry will be equal to 1. Therefore $$\text{DC}(K_1) + \text{DP}(K_1) = \text{ZCB}.$$
2. At most one of the derivatives pays off so the value of the two together at expiry will be 1 or 0. Therefore $$\text{DC}(K_1) + \text{DP}(K_1) < \text{ZCB}.$$
3. At least one of the derivatives pays off so the value of the two together at expiry will be 1 or 2. Therefore $$\text{DC}(K_1) + \text{DP}(K_1) > \text{ZCB}.$$

I have not seen this notation before. Do you know what the functions DC, DP and ZCB mean? I'm guessing DC denotes the pay-off for the digital call, DP denotes the pay-off for the digital put. What is ZCB? Is it 1, if so, why did the author not just use 1?

• ZCB = Zero coupon bond? 1. basically is a zero coupon bond that pays a 1 at expiration. Dec 14, 2020 at 17:30
• Yes, I think that is probably it. Thank you! Dec 14, 2020 at 17:31
• The function DC() denotes the value of the Digital Call and not they payoff at expiration. I have encountered these functions in the Black and Scholes model for barrier options. For instance, a down-and-out call option is often referred to as DOC() and and down-and-in call DIC(), and so on. Dec 14, 2020 at 23:19