You are long a two-year 100-strike call on a 100 stock with N(d_2) = 0.45. Rates are 2% continuously compounded and rise by 100 basis points. What is the profit and loss per share from the rate move alone?

You are long a two-year 100-strike call on a 100 stock with N(d_2) = 0.45. Rates are 2% continuously compounded and rise by 100 basis points. What is the profit and loss per share from the rate move alone?

Approach: Use the closed form for the rate sensitivity of a European call, which is the discounted strike times the maturity times the exercise probability, then scale it to the size of the move.

$0.86. Rho for a European call is K*T*e^{-rT}*N(d_2) = 100*2*e^{-0.04}*0.45 = 86.47 per unit change in the rate, so a 100 basis point move is 0.01*86.47, worth $0.865 per share. The mechanism is that a higher rate raises the forward price and lowers the present value of the strike the holder will pay, so a call gains and a put loses. Rho scales with maturity, which is why it is irrelevant on a one-month option and one of the larger exposures on a two-year or five-year call. A desk running long-dated structures marks the rate curve and the dividend curve as separate risks from the volatility surface, because a parallel 100 basis point move can dominate a full point of implied volatility on that part of the book.

Follow-up: The same book is delta hedged with stock financed at the same rate. How much of that rho does the funding on the hedge cancel?

Key concepts: rho, interest rate risk, discounted strike, forward price.