Question

Case: Today is April 30th, 2020, and a trader sold 100,000 European call options on the SPY ETF with an exercise price of $310, expiring on June 30, 2020. The trader was able to write these options for $3.80. Additional Information: Suppose that the SPY obeys to a GBM process under risk neutral probability, such that (1) Sx = Sex exp ((- - -)+0B4) where Sc is the SPY price at time t, r is the risk-free rate, and o is the volatility of the index. Assume that r = 0%. Implied Volatility (5 Points) Given the market price of the option, what is the implied volatility of the SPY ETF? Hint: This is a one-to-one matching problem. You have an option pricing formula, i.e. Black- Scholes Model (BSM), and a market price. You need to find the implied o* that equates the BSM price to the market price.

The price of the underlying asset is $290.48

Answer 1

Solution:

With the use of the Black Scholes formula, we can calculate the call and put option premium. Using goal seek function we can calculate the implied volatility for a given call or put premium.

Time = 2/12 years = 0.17

Implied volatility = 22.28%

2 A B C Black Scholes Model D E F G H I J K L M N O P Q Black scholes formula d1 -0.7557195 <-=(LN(C4/C3)+C6*(C5+C72)/(2)))/(SQRT(C6)*07) -0.8373691 <-=(LN(C4/C3)+C6*(C5-C742)/(2))/(SQRT(C6) *C7) 3 Strike Price 4 Spot Price 5 Interest rate 6 Time 7 Volatility 310 290.48 0% 0.17 20% c= S, N(d)- K e "N(d)) p= Ke N(-d)-S, N(-d) N(di) Nd2) 0.2249087 <-=NORM.DIST(F3,0,1,TRUE) 0.2011926 <=NORM.DIST(F4,0,1,TRUE) where d. - In(S, /K)+(r+02/237 OT Ni-d1) N(-d2) 0.78 <-1-F6 0.80 <=1-F7 d. = Incs, /K)+(r–02 1217 – d, -ONT 12 Goal Seek ? X Call Option Put Option 2.96 <=F6*C4 - C3*EXP(-C6*C5)*F7 22.48 <=C3*EXP(-C5*C6)*F10 -C4*F9 13 Set cell: SF5121 15 To value: 3.80 16 By changing cell: SCS7|| OK Cancel 19

н к L M N O P Q Black Scholes Model Black scholes formula d1 d2 -0.6694151 <-=(LN(C4/C3)+C6*(C5+(C7^2)/(2))/(SQRT(C6) *C7) -0.7603892 <-=(LN(C4/C3)+C6*(C5-(C7^2)/(2)))/(SQRT(C6) *C7) 3 Strike Price 4 Spot Price 5 Interest rate 6 Time 7 Volatility 310 290.48 0% 0.17 22.28% c=S, Nd)-K e "N(d)) p= Ke" N(-d,)-S, N(-d) N(dl) N(D2) 0.2516154 <-=NORM.DIST(F3,0,1,TRUE) 0.223511 <=NORM.DIST(F4,0,1,TRUE) where d. - In(S, /K)+(r +62/217 Goal Seek Status ? X Ni-d1) N(-d2) 0.75 <-1-F6 0.78 <=1-F7 G/T In( SK)+(1-0-/2)T_dat Goal Seeking with Cell F12 found a solution. Step Pause Call Option Put Option 3.80 <=F6*C4 - C3*EXP(-C6*C5)*F7 23.32 <=C3*EXP(-C5*C6)*F10 -C4*F9 Target value: 3.8 Current value: 3.80 D OK Cancel 005