# Additional questions for chapter 4

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1 Additional questions for chapter 4 1. A stock price is currently \$ 1. Over the next two six-month periods it is expected to go up by 1% or go down by 1%. The risk-free interest rate is 8% per annum with continuous compounding. i) What is the value of a one-year European call option with a strike price of \$ 1. ii) What is the value of a one-year European put option with a strike price of \$ 1. iii) Verify that the European call and the European put satisfy put-call parity. Parameters are u.1, d.1, 1 + r e.5.8. So the risk-neutral probability is p.7. After evaluation of the options at the terminal nodes we use the risk-neutral valuation to get i) π C ) e.5.8) [ ) + 1.7) ] 9.61 and ii) π P ) e.5.8) [ ) ) 19 ] 1.9 iii) For put-call parity one has to verify S π C + π P Ke r, here : e.8.

2 . Assume a standard 3-period CRR binomial model. The price of the stock is currently \$1. The risk-free interest rate with continuous compounding is 6% per annum. Over the next three 4 month periods, the stock is expected to go up by 8% or go down by 7% in each period. a) What is the value of a one-year European call with strike price \$13? b) What is the value of a one-year European put with strike price \$13? c) Verify the Put-Call parity for the European call and the European put. We first calculate the Martingale probability in the tree. We get p r d u d e.6/ a) The tree for the call option looks as follows: S1 C time t t 1/3 b) The tree for the put option is: t /3 t 1 S1 C time t c) The Put-Call parity holds: t 1/3 t /3 t 1 C P e S Ke rt.

3 3. Consider a 3-period Cox-Ross-Rubinstein model. The annual interest rate is r.5 discrete), u.1 and d.1. The initial price of the stock is S) 1. The time horizon is T 3 years. a) Calculate the risk-neutral probability and the stock prices at each node in the binomial tree correct up to decimal places after the decimal point). b) Calculate the value of the European option with payoff sup S t S T S t < 11 t P T ) t T otherwise c) Find a replicating portfolio for the above option for the first trading period. a) For the risk-neutral probability we get p r d prices and the value of the option is S1 P u d 3 4. The tree with the stock time t t 1/3 t /3 t 1 b) The replicating portfolio can be found by solving the equations 1.5 ϕ ϕ 1.5 ϕ ϕ 5.4 As solution we get ϕ and ϕ.6.

4 4. Construct a three period binomial tree using the parameters r.1 discrete, per period), u.15, d.5 and S 1. a) Find the price of a European Put P with strike 15 and maturity date T 3. b) Find the price of the knock in Call option C with knock in level H 11, strike K 9 and maturity date T 3, i.e. { ST ) 9) + t : S t > H 11 C S t H 11 t. The risk neutral probability is S1 P.345 C31.46 p r d u d We first set up a tree with the stock price movements, then compute the values of the two options: time t t 1/3 t /3 t 1

5 5. Assume a 3-period Cox-Ross-Rubinstein model. The annual interest rate with continuous compounding is r.6. The volatility of the stock is σ. with a price of S) 1. Furthermore, there exists an American Put with maturity date T 1 und strike K 9. a) Calculate the risk-neutral probability and the stock prices at each node in the binomial tree correct up to decimal places after the decimal point). b) Calculate the value of the American Put for all nodes in the tree. c) What is the optimal stopping time? Justify your answer. a) The parameter-values are 1 3, 1+r d e r 1., 1+u e σ 1.14, 1+d e σ.899. For the risk-neutral probability we get The tree with the stock prices is 11.4 S time t p r d d u d t 1/3 b) The prices for the american Put are.17 P time t t 1/ t /3 max{1.6, 8.84} t / c) Let Ω {u, d} 3. The optimal exercise date is { n ω {ddu, ddd} τω) n 3 otherwise t 1 t 1 For ω {ddu), ddd)}, we have 1 1+r d E[p f p )f 33 ] < K S 1 + d) ) +. Here f ij denotes the price of the claim in period i with j-down movements.

6 6. Assume that we have a three period CRR model with initial stock price S \$15, interest rate r.5 and volatility σ.. a) What is the value of an American Put with strike \$15, which matures in 6 months? b) What is the value of an American Call with strike \$15, which matures in 6 months? c) Verify that the following inequalities hold: S K C A P A S Ke rt The martingale probability is p.538 with u.85, d.784 and r.84. a), b) For the American Put and Call we get: S15 P A 7.57 C A time t t 1/6 t /6 t.5 c) We have e.5

7 7. Show that a security market is arbitrage-free with respect to Φ iff it is arbitrage-free with respect to Φ a. Here Φ is the set of all self-financing trading strategies and Φ a is the set of all admissible strategies, that means all ϕ Φ with V ϕ t) t,..., T. First note, if ϕ Φ a is an arbitrage strategy, then it is by definition of Φ a also a strategy in Φ. We now have to show that if we have an arbitrage strategy ϕ Φ, then there exists an arbitrage strategy ψ Φ a. Assume that ϕ Φ is an arbitrage strategy. Then we have V ϕ ), P V ϕ T ) ) 1 and P V ϕ T ) > ) >. We have to distinguish between two cases: Case 1: V ϕ t) t,..., T. Then ϕ Φ a and we found the admissible arbitrage strategy. Case : t, A F t with V ϕ t, ω) < ω A and V ϕ t) t > t. Then define a new strategy ψ. Set ψu, ω) ω A c u. Furthermore ψu, ω) ω A and u t. For the remaining possibilities set ψ u, ω) ϕ u, ω) V ϕt, ω) S t, ω) ω A u > t and ψ i u, ω) ϕ i u, ω) ω A i 1,..., d u > t We have to show that this strategy is self-financing and admissible. For ω A c we clearly have no problem. There is nothing to show for ω A, u t. ψ is also clearly self-financing for u > t + 1 as it just replicates the other strategy there. We have to show that ψt )St ) ψt + 1)St ). For ω A we have ψ t + 1)S t ) ϕ t + 1)S t ) V ϕ t ) and ψ i t + 1) ϕ i t + 1) Thus we get ψt +1)St ) 1 A ϕt +1)St ) V ϕ t )) 1 A ϕt )St ) V ϕ t )) ψt )St ) It remains to show that ψ is admissible and an arbitrage opportunity. We get V ψ t) t t and V ψ t) 1 A ϕt)st) V ϕ t ) S ) t) 1 S t A V ϕ t) V ϕ t ) S ) t) ) S t ) and > on A for t T because V ϕ t ) <. We also have that V ψ t) t t. Therefore ψ is admissible and an arbitrage opportunity.

8 8. a) State the Black-Scholes formula for an European Call and Put. Hint: The Put-Call parity C P S Ke rt t) might be useful) b) Replicate the European straddle with payoff DT ) ST ) K using standard European options. c) What is the Black-Scholes price of the straddle? d) What is the of the straddle? How much does the value of the straddle approximately change if the stock price changes from S t to S t + ε? Hint: The of the Call is Nd 1 )) a) The Black-Scholes formula for an European Call and Put is Ct) St)Nd 1 ) Ke rt t) Nd ) P t) Ke rt t) N d ) St)N d 1 ) where d 1 logs/k) + r + σ σ T t d d 1 σ T t. ) T t) b) We can replicate the straddle DT ) ST ) K by buying one call and one put, both with strike K. c) The Black-Scholes price of the straddle is Dt) Ct) + P t) St)Nd 1 ) Ke rt t) Nd ) + Ke rt t) N d ) St)N d 1 ) St)Nd 1 ) 1) Ke rt t) Nd ) 1). d) The Delta of the straddle is D C + P Nd 1 ) N d 1 ) Nd 1 ) 1. When the stock price changes from S t to S t + ε, then the price of the straddle changes about εnd 1 ) 1).

9 9. Consider a financial market in which the Black-Scholes formula for a European call option holds. The risk-free interest rate cont. compounding) is r. The underlying stock has value S with volatility σ. For a European call with strike K and maturity T, show that the following relations hold: C S Nd 1) Γ C S N d 1 ) Sσ T t Θ C t SN d 1 )σ T t rke rt t) Nd ) ρ C r KT t)e rt t) Nd ) ν C σ SN d 1 ) T t Show that the call satisfies the partial differential equation C t C + rs S + 1 σ S C rc. S We first show that SN d 1 ) Ke rt t) N d ): SN d 1 ) Ke rt t) N d ) 1 π Se d 1 / Ke rt t) e d / ) 1 Se d 1 / Ke rt t) e d 1 /+d 1σ ) T t σ T t)/ π 1 π Se d 1 / Se d 1 / K S e rt t) e d 1σ T t σ T t)/ ) 1 Se d 1 / Se d 1 / K ) π S e rt t) e logs/k)+r+σ /)T t) σ T t)/ 1 π Se d 1 / Se d 1 / ) Now we calculate the Greeks: a) b) C S Nd 1) + SN d 1 ) d 1 S Ke rt t) N d ) d S Nd 1 ) + SN d1 d 1 ) S d ) S Nd 1 ) Γ C S S N d 1 ) d 1 S N d 1 ) Sσ T t

10 c) Θ C t SN d 1 ) d 1 t Ke rt t) N d ) d t Kre rt t) Nd ) SN d1 d 1 ) t d ) Kre rt t) Nd ) t SN σ d 1 ) T t rke rt t) Nd ) d) ρ C r SN d 1 ) d 1 r Ke rt t) N d ) d r + KT t)e rt t) Nd ) SN d1 d 1 ) r d ) + KT t)e rt t) Nd ) r KT t)e rt t) Nd ) e) ν C σ SN d 1 ) d 1 σ Ke rt t) N d ) d σ SN d1 d 1 ) σ d ) σ SN d 1 ) T t. The partial differential equation holds because: C t C + rs S + 1 σ S C S rc σ T t rke rt t) Nd )+ SN d 1 ) + rsnd 1 )+ + 1 σ S N d 1 ) Sσ T t + + rc rsnd 1 ) Ke rt t) Nd ) C).

11 1. Prove the following limit relations used in the proof of Proposition 3.5.1, assuming that k n n ): lim ˆp n 1 n, lim k n1 ˆp n ) r n T n σ + σ ) We have the following definitions for the variables: Then, we get for the first limit relation: n T k n u n e σ n 1 d n e σ n 1 r n e r n 1 p n r n d n u n d n ˆp n p 1 + u n n 1 + r n lim ˆp n lim p 1 + u n n lim n n n 1 + r } {{ n} 1n ) lim n In order to show the last equality, it suffices to show that as n + n ). By L Hospital we get e rx e σx lim x + e σ x e 1 σx. e r n e σ n e σ n e σ 1 n. e rx e σx xre rx + σe σx lim lim 1 x + e σ x e σx x + σe σx + σe σx. For the second limit relation we get: lim k n1 ˆp n ) n lim T kn n n lim n 1 er n e σ e σ n e σ n e r n e σ n 1 e σ n + e σ n r n T n 1 kn 1 e σ ) n σ + r )) σ T T For the second to last equation it suffices to show that: T σ + r σ ). e σx 1 + e σx rx lim x x σ T + T 1 e σx ) + r ) σ )

12 as n + n ). We are using L Hospital twice and get: e σx 1 + e σx rx lim x x + T 1 e σx ) lim x + T σe σx σ + rx)e σx rx 1 e σx ) + xσe σx 4σ e σx + σ + rx) e σx rx 4re σx rx lim T x + σe σx + σe σx 4xσ e σx T 4σ + σ 4r σ T 4σ + r ). σ

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