Derivatives Slides part 6

Option pricing with Black&Scholes

Eero Kasanen Derivatives Ch 2

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N-PERIOD BINOMIAL TREE – BLACK&SCHOLES

Putting in periods, and solving backwards, we can solve n-period tree. This leads into a generalization of the previous results

c = e-rT

n! pj(1-p)n-jmax(S0ujdn-j – K,0) j =0 (n - j )! j !

n

Dividing time T into smaller time intervals we get more accurate results. If we divide the time T into shorter and shorter periods so that the number of periods n is getting larger and larger, in the limit we get the Black&Scholes formula for European call options c.

Eero Kasanen Derivatives Ch 2

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BLACK&SCHOLES FORMULA

c = S0N(d1) – Ke-rTN(d2)

S0 s 2 ln ÷ + r + ÷ T èK è 2 d1 = s T

S0 s 2 ÷ + r - ÷T = d - s T d2 = ln èK è 1 2

s T

where c = price of a European call option S0 = current stock price (known) r = risk-free rate (roughly known) T = time to expiration (known) K = exercise price (known) σ = volatility(standard deviation) of stock return pa (estimated) N(x) = value of cumulative Normal distribution at x

Eero Kasanen Derivatives Ch 2 3

IMPLIED VOLATILITY

All other Black&Scholes parameters (S0,r,T,K) can be readily observed but volatility σ is not directly observable. Historical volatilities can be calculated but B&S formula really needs forward looking volatility. Market participants like to talk about implied volatility. If you observe the market price of an option c and other parameters S,r,T,K, you can back out what the parameter σ must have been to make Black&Scholes formula to hold. This σ is the implied volatility. VIX index measures implied volatility of 30-day options on S&P 500 index. It estimates market′s view of future uncertainty (volatility). Example: if c = 4.76, S0 = 42, K = 40, r = 0.1, T = 0.5 then implied volatility σ = 0.2 (check with your own B&S formula)

Eero Kasanen Derivatives Ch 2 4

ESTIMATING HISTORICAL VOLATILITY

1.

1 n (u (i ) u ) 2 n 1 i 1

Eero Kasanen Derivatives Ch 2

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REAL WORLD EXAMPLE OF BLACK&SCHOLES

Apple stock 14.3.13 Stock price $428.35, historical volatility 3 month 38.64%, Exercise price $400, T = 100 days (≈3 months to Jun 21), LIBOR USD 3 month rate 0.28% pa (= (360/100)ln(1+0.0028(100/360)) =0. 279% pa continuous compounding). Calculating call option value

d1 = [ln(428.35/400)+(0.00279+0.38642/2)(100/360)]/0.3864x 100 / 360 = 0.4419 d2 = 0.4419 – 0.3864 100 / 360= 0.2382 N(d1) = 0.6707 (use NORM.DIST function in Excel) N(d2) = 0.5941 (use NORM.DIST function in Excel) c = S0N(d1) – Ke-rTN(d2) = 428.35x0.6707 – 400e-0.00279x(100/360)0.5941 = 49.82 ******** The real call option on Apple traded at $42.55 mid-price. In the market 14.3.13 implied volatility of Apple stock is 29.8%, lower than historical vola 38.64%. Also the option market is using roughly 13 week T bill rates at 0.001% as relevant interest rates. Putting in new estimates we get c = 42.59

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