example from section 10.6:
10. If the example in Section 10.6 is a pared down version of a European call, describe a similar digital option which represents a pared down put. Price it, and formulate a "digital version" of put-call parity com- paring the price of these two derivatives. Example 10.8. A digital option pays the holder $1 at some future time if the stock at that time lies above a certain threshold, and nothing otherwise. This can be considered as a simplified version of a European call with strike K, as the option pays money when the stock price is above K, and is worthless otherwise. Consider the stock whose spot price is S, and volatility is o. Let r be the risk-free rate. Compute the price of a digital option which pays $1 at time T if Sr is above the threshold K and nothing otherwise. What must our standard normal variable be for the stock price to pass the threshold? In -(r- T ST > K H Soer-T+oTZKA - =-d2 OVT So our payoff function is g(Soer-T+/T2) = 0 if 23-d2 and q(Soer- T+oT2) =1 if Z> -d2. Plugging this into the integral for the price of a general derivative, 9(Soer-o/2)T+OVTz, e-22/2 -d2 O = e-7 + e-2/2 p e -22/2 dz + 1x - V2 J-do V2 = e-T Fz(dz). Note how this is similar to the second term in the Black-Scholes formula for a European call, -KFz(d2e-T, which represents the $K the call holder pays (instead of $1 the digital option holder receives) to exercise the call should the stock price Sr pass the threshold K at time T. 10. If the example in Section 10.6 is a pared down version of a European call, describe a similar digital option which represents a pared down put. Price it, and formulate a "digital version" of put-call parity com- paring the price of these two derivatives. Example 10.8. A digital option pays the holder $1 at some future time if the stock at that time lies above a certain threshold, and nothing otherwise. This can be considered as a simplified version of a European call with strike K, as the option pays money when the stock price is above K, and is worthless otherwise. Consider the stock whose spot price is S, and volatility is o. Let r be the risk-free rate. Compute the price of a digital option which pays $1 at time T if Sr is above the threshold K and nothing otherwise. What must our standard normal variable be for the stock price to pass the threshold? In -(r- T ST > K H Soer-T+oTZKA - =-d2 OVT So our payoff function is g(Soer-T+/T2) = 0 if 23-d2 and q(Soer- T+oT2) =1 if Z> -d2. Plugging this into the integral for the price of a general derivative, 9(Soer-o/2)T+OVTz, e-22/2 -d2 O = e-7 + e-2/2 p e -22/2 dz + 1x - V2 J-do V2 = e-T Fz(dz). Note how this is similar to the second term in the Black-Scholes formula for a European call, -KFz(d2e-T, which represents the $K the call holder pays (instead of $1 the digital option holder receives) to exercise the call should the stock price Sr pass the threshold K at time T
