Question

Problem 4: 

You are a trader who trades both puts and calls on SleazeCo. Information about current market conditions is displayed below.

$$ \begin{array}{lclcc} \text { Stock Price } & \text { Exercise Price } & \text { Expiration Date } & \text { Call Price } & \text { Put Price } \\ \hline 88 & 90 & 1 / 12^{\text {th }} \text { of a year } & 2.8546 & 4.6032 \\ 88 & 95 & 1 / 12^{\text {th }} \text { of a year } & 1.2978 & 7.8240 \end{array} $$

The annualized continuously-compounded risk-free rate is .06 (6%). 

1. Given the information above, are there any arbitrage opportunities?

2. 2. If no, explain why. If yes, describe one set of trades you could make now to exploit the arbitrage opportunity. Show that this strategy generates an arbitrage profit.

3. 3. The volatility of the return on SleazeCo. stock is 35 percent per year (i.e., σ= .35). What are the Black-Scholes values for the calls and puts above?

   
   

Answer 1

stock price (SP) - 88 Exercise price (x) = 90 period – th of year ie, month Call premiom - 2.8546 put premium - 4.6032 calculation of fair value of call option Vealu = [sp - present value of x] + Uput Veall = (88 – 900 + 4.6032 90 88 - +4.6032 0.005 e. = 1.00501257 - Check in working Notes for 0.005 Vcal = (88 - 89.5511) + 4.6032 Fair value of each – 3.0521 But actual premium is 2.8546 . Call option is underpriced. It means put option is over priced. Arbitrage is possible.

Action for Arbitrage (1) Buy & call option (Holder) (ii) sell pat option (iii) short sell thare today. Steps for arbitrage & Identify net cash flow today = – 2.8546+ 4.6032+86 = 89.7586 m How 0.005 7.56 2 Deposit I 89.7586 @ 61. For month (3) withdraw deposit @ I mos = 89.75664 e = 69.7566% 1.00571257 - 900065 - 20. 2085 At I month time, we have to gettle call option & put option & also buy share in market out flow - Stripe price = 90 & Arbitrage gun - step ₂ - step = 90.2085 – 90 = 0.2085

stoepe price(sp)= BE Exercise police (x) > 9095 period - 1 month Call premium = 1.2978 pul premium - 7.8240 calculation of fair value of coll option V = (SP- pr of x') + put Call 95 Fairvalue Veall = 88 0.06x + 7.8240 e + =/88 0.005 708240 check wooking lote. + 7.8240 ( 88- 1.00501257 Fair value of La = 1.2978 190987 Actual call premium - 1.2978 . call option is correctly priced so Arbitrage u not possible (Reasons gunen in next page)

P-4 put-call y using Case (1) 8/1080 Sased on geen information in the problem, encoblem, We can calculate the arbitrage existance by using put party theory Situations of Arbitrage possibility If antenat premiom > pair coll premium call is orcupriced && we can observe that put is Underpriced. :. Arbitrage is possible. 2) If actreal call premium u & pair call premium call is underpriced & we can observe that put is our priced so. Arbitrage is possible & If Actual call premiom u fair call premium call & correctly priced & we can obserue that put is also correctly priced - Arbitrage å not possible

BLACH sellores MODEL for stock options - Stock options are option contract with share of a co. as undertising and --> Value of call opkon - spot foritexnildi) - py of these x ndu) Valm of put option - Prof share (1-(d) - Spotprice (1 ) poria d, & de represent, & lines / area lines however d, &dz are calculated curing the following formula d. - En L x + [Pst Time o Time, where Time = Excrise periods Spa Spot price in years х = gело і тіс R & Risk free return (indecimals) - standud duciation (indeurnal) 8 - Variance,

Rf = 0.06 r = 0.35 x =90 , , 2=0.1225 d. CG) +0.06+ 0.1225 )x% d = 0.35 Se In (0.9778) + 0.0101 0.1010 - 0002248+0.0101 0.1010 = – 0012 d de – d, - & stime = – 0.12 – 0.35 T2 (de a-0.22 (d) = 0.50 - 0.0438 = 0.4562 (de) = 0.50 - 0.0832 a 0.9168 90 or - prof strick price & = 89.551 0.005 0.00) e = (200501257

P-7) call valere - SP Y N (d) - pr of shich & N (dz) = 88 x 0.9562 – 89.5511 X 0.4168 40.1956–37.3269 2.8207 value of call - value of put option = prof stick price (1-Nidz)) - SP (1-(di)) - 89.5511 (1 -0.4168) – 88 (1-0-4562) = 89.5511X0.5832 - 88x0.5438 = 52.2262 - 47.8599 put option = 6.3718

deler spaer , Rp30.06 . X-95, sa doar die es (at) + (0.06 0.125)/ 0.35 Shr In (0.9263) + 0.0101 0.1010 -0.07657 +0.010 = -0.06647 0-1010 001010 7 di = -0.6581] (or) - 0.66 de = – 0.6581 - 001010 de = – 0.75917 (or – 0.76 n (di) = 0.50 - 0.2954 = 0.2546 n (dz) – 0.50 - 0.2766 = 0.2236 = - – pr of strick price . 94.5262 0.005

(PD call value & Spx (d) – profx 1 (d) - 88x0.2546 – 99.5262 (0.2236) = 27.4048 - 21.134 value of cal = 1.2687 put value = prof x (1 – Nite) - St X(1 — 6)) = 94.5262 (1-0.2936) - et (1 -0.2546) 13.3901 - 65.5952 put value = 7.7999

( P-10 working Note: 0.005 e = ? e = 2.71828 0.005 2.71828 = 2 is time Step o 2.11825 orr. ---- 2,11828 step & Step ③ Step 4 E (x00005 HIE E – – – – 15 times step ③ 0.005 get & we can = 1.00501257