Module 5. bsm
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The document explains how to use the Black-Scholes-Merton model to price European call and put options. It provides the formulas for the call and put prices. As an example, it calculates the call and put prices for a stock currently trading at $52 with a strike price of $50, interest rate of 3%, time to expiration of 0.5 years, and standard deviation of returns of 25%. It shows the steps of calculating the d1 and d2 values to then determine the normal cumulative distribution values to plug into the pricing formulas to get a call price of $4.5658 and put price of $1.8214.
Module 5. bsm
- 1. Black-Scholes-Merton Model Step by Step Dr. Nivine Richie
- 2. How to use the Black-Scholes-Merton model to price European calls and puts
- 3. Formula c = SN(d1) – e–rTXN(d2) p = e–rTXN(–d2) – SN(–d1) 𝑑1 = ln 𝑆 𝑋 + 𝑟 + 𝜎2 2 𝑇 𝜎 𝑇 , 𝑑2 = 𝑑1 − 𝜎 𝑇
- 4. The reading does not require you to calculate the components of the BSM model, but we will do it for this class anyway
- 5. Example (using Excel) Suppose we are given the following information on call and put options on a stock: • Stock currently trading for $52 • Strike price on calls and puts is $50 • Continuously compounded annualized rate of return is 3% • Term of the options is 0.5 year • Standard deviation of returns is 25%
- 6. Step 1. Calculate d1 𝑑1 = ln 𝑆 𝑋 + 𝑟 + 𝜎2 2 𝑇 𝜎 𝑇 𝑑1 = ln 52 50 + 0.03 + 0.212 2 0.5 0.21 0.5 = 0.4394
- 7. Step 2. Calculate d2 𝑑2 = 𝑑1 − 𝜎 𝑇 𝑑2 = 0.4394 − 0.21 0.5 = 0.2909
- 8. Step 3. Find the area under the curve at d1 and d2 In excel, use the function =NORMSDIST() to find the standard normal cumulative distribution at d1 and d2 =NORMSDIST(0.4394) gives us 0.6698 =NORMSDIST(0.2909) gives us 0.0.6144
- 9. Step 4. Plug values into BSM calculator c = SN(d1) – e–rTXN(d2) c = 52 (0.6698) – exp(-0.03*0.5) (50) (0.6614 c = 4.5658
- 10. p = e–rTXN(–d2) – SN(–d1) p = exp(-0.03*0.5) (50) (NORMSDIST (-0.2909) – 52 (NORMSDIST(-0.4394) = 1.8214 Step 4. Plug values into BSM calculator
- 11. The put value is the same value you get if you used the call and plug into put-call parity P = C – S + Xe-rT = 4.5658 – 52 + 50*exp(-0.03*0.5) = 1.8214 Step 4. OR use put-call parity to find put value