European Contingent Claims

European Contingent Claims Seminar: Financial Modelling in Life Insurance organized by Dr. Nikolic and Dr. Meyhöfer Zhiwen Ning 13.05.2016 Zhiwen Ning European Contingent Claims 13.05.2016 1 / 23

outline Recall... What is European contingent claim Examples of European contingent claim Arbitrage-free Price of a European contingent claim Zhiwen Ning European Contingent Claims 13.05.2016 2 / 23

Recall what we have did last time 1. set up the multi-period market model: - d + 1 assets are priced at times t = 0, 1,..., T ; -S i t: the price of the i th asset at time t; - vector S t = (S 0 t, S 1 t,..., S d t ) : the prices of all assets at time t; -trading strategy: a predictable process ξ = (ξ 0 t, ξ 1 t,..., ξ d t ) R d+1, - ξ is self-financing if ξ t S t = ξ t+1 S t, 0 t T 1. -discounted price process: Xt i = Si t, t = 0,..., T ; i = 0,..., d. St 0 -(discouned) value process: V = (V t ) t=0,...,t with V 0 = ξ 1 X 0 and V t = ξ t X t Zhiwen Ning European Contingent Claims 13.05.2016 3 / 23

Recall what we have did last time For a trading strategy ξ the following conditions are equivalent: (a) ξ is self-financing; (b) ξ t S t = ξ t+1 S t, 0 t T 1; (c) V t = V 0 + t k=1 ξ k (X k X k 1 ) for all t. Zhiwen Ning European Contingent Claims 13.05.2016 4 / 23

Recall what we have did last time 2. arbitrage & martingale measures: -arbitrage opportunity: V 0 0, V T 0 P a.s. and P[V T > 0] > 0 -Q is a martingale measure, if the discounted price process X is a Q-martingale, i.e., for 0 s t T and i = 1,..., d, E Q [X i t ] < and X i s = E Q [X i T F s]. -the relationship between arbitrage-free assumption and martingale measure: two theorems Zhiwen Ning European Contingent Claims 13.05.2016 5 / 23

Recall what we have did last time Doob s fundamental systems theorem for martingales For a probability measure Q, the following conditions are equivalent. (a) Q is a maringale measure. (b) If ξ = (ξ 0, ξ) is self-financing and ξ is bounded, then the value process V of ξ is a Q-martingale. (c) If ξ = (ξ 0, ξ) is self-financing and its value process V satisfies E Q [V T ] <, then V is a Q-martingale. (d) If ξ = (ξ 0, ξ) is self-financing and its value process V satisfies V T 0 Q-a.s., then E Q [V T ] = V 0. Zhiwen Ning European Contingent Claims 13.05.2016 6 / 23

Recall what we have did last time Fundamental theorem of asset pricing The market model is arbitrage-free if and only if P. In this case, there exists a P P with bounded density dp /dp This theorem can also be formulated as: the market model is arbitrage-free there exists a P P such that the discounted price process X is a P -martingale. Zhiwen Ning European Contingent Claims 13.05.2016 7 / 23

What will we do this time? the multi-period market model+the nice porperties under the assumption arbitrage-free =? Zhiwen Ning European Contingent Claims 13.05.2016 8 / 23

The definition of European contingent claim Definition 1.1. A non-negative random rariable C on (Ω, F T, P) is called a European contingent claim. A European contingent claim C is said to be a derivative of the underlying assets S 0,S 1,...,S d if C is measurable with respect to the σ-algebra generated by the price process ( S t ) t=0,...,t. -can consider contingent claim as a function of S 0,S 1,...,S d -yields at time T the amount C(ω), T is called the expiration date or the maturity of C. Zhiwen Ning European Contingent Claims 13.05.2016 9 / 23

Examples of European contingent claim European call option has the right, but not the obligation, to buy an asset at time T for a fixed price K, called the strike price. { C call = (ST i K)+ = max{st i K, 0} = ST i K if S T i > K, 0 otherwise. European put option gives the right, but not the obligation, to sell some asset at time T for a strike price K, C put = (K S i T )+. Zhiwen Ning European Contingent Claims 13.05.2016 10 / 23

Examples of European contingent claim Asian option The payoff depends on the average price S i av := 1 T St, i T {0,..., T }. t T Considering the average price S i av instead of S i T average price call: average price put: C call av := (S i av K) +. C put av := (K S i av) +. Zhiwen Ning European Contingent Claims 13.05.2016 11 / 23

Examples of European contingent claim Asian option The payoff depends on the average price S i av := 1 T St, i T {0,..., T }. t T If the strike price is set to be the average price S i av average strike call: (S i T S i av) +, average strike put: (S i av S i T )+. Zhiwen Ning European Contingent Claims 13.05.2016 12 / 23

Pre-work of pricing a contingent claim -discounted European claim or the discounted claim: H := C. ST 0 -assume that our market model is arbitrage-free or, equivalently, that P, where P denotes the set of all equivalent martingale measures. - A contingent claim C is called attainable (replicable, redundant) if there exists a self-financing trading strategy ξ whose teminal portfolio value coincides with C, i.e., C = ξ T S T P-a.s. such a trading strategy ξ is called a replicating strategy for C. Zhiwen Ning European Contingent Claims 13.05.2016 13 / 23

Pre-work of pricing a contingent claim - C is attainable H = C S 0 T = ξ T S T S 0 T = ξ T X T = V T = V 0 + T ξ t (X t X t 1 ), t=1 then we say that the discounted claim H is attainable, and ξ is a replicating strategy for H. - an attainable discounted claim H is automatically integrable with respect to every equivalent martingale measure. Zhiwen Ning European Contingent Claims 13.05.2016 14 / 23

Pre-work of pricing a contingent claim Theorem 1 Any attainable discounted claim H is integrable with respect to each equivalent martingale measure, i.e., E [H] < for all P P. Moreover, for each P P the value process of any replicating strategy satisfies V t = E [H F t ] P-a.s. for t = 0,..., T. In particular, V is a non-negative P -martingale. Proof: This follows from V T = H 0 and the systems theorem. Zhiwen Ning European Contingent Claims 13.05.2016 15 / 23

Pricing a contingent claim A real number π H 0 is called an arbitrage-free price of a discounted claim H, if there exists an adapted stochastic process X d+1 such that X d+1 0 = π H, X d+1 t 0 for t = 1,..., T 1, and X d+1 T = H, and such that the enlarged market model with price process (X 0, X 1,..., X d, X d+1 ) is arbitrage-free. Let Π(H) be the set of all arbitrage-free prices of H. The lower and upper bounds of Π(H) are denoted by π inf (H) := inf Π(H) and π sup (H) := sup Π(H). Zhiwen Ning European Contingent Claims 13.05.2016 16 / 23

Pricing a contingent claim We aim to characterize the set of all arbitrage-free prices of a discounted claim H. Theorem 2 The set of arbitrage-free prices of a discounted claim H is non-empty and given by Π(H) = {E [H] P P and E [H] < }. (1) Moreover, the lower and upper bounds of Π(H) are given by π inf (H) = inf E [H] and π sup (H) = sup E [H]. P P P P Fundamental theorem of asset pricing: the market model is arbitrage-free there exists a P P such that the discounted process X is a P -martingale. Zhiwen Ning European Contingent Claims 13.05.2016 17 / 23

Example of pricing a European call option In an arbitrage-free market model, consider a European call option C calll = (ST 1 K)+ with strike K > 0 and with maturity T. We assume that the numéraire S 0 is the predictable price process of a locally riskless bound, i.e., St 0 = Π t k=1 (1 + r k). Then St 0 is increasing in t and satisfies S0 0 1. For any P P, due to Theorem 2 and Jensen s inequality (x x + is a convex function), an arbitrage-free price π call of C call satisfies the following inequation ] ) + ] ) ] + π call =E [ C call S 0 T = E [ (S 1 T S 0 T (E [ X 1 T K S 0 T ]) + = [ ( K ST 0 = E XT 1 K ST 0 ( [ ]) K + S0 1 E ST 0 (S0 1 K) +. Zhiwen Ning European Contingent Claims 13.05.2016 18 / 23

Example of pricing a European call option (S 1 0 K)+ is called intrinsic value of options. The difference of the price π call and (S 1 0 K)+ is often called time-value of the European call option, see Figure 1. An option s premium is comprised of two components: its intrinsic value and its time value. For a call option, its time-value is nonnegative. Zhiwen Ning European Contingent Claims 13.05.2016 19 / 23

Example of pricing a European call option Figure: The typical price of a call option is always above the opion s intrinsic value (S 1 0 K)+. Zhiwen Ning European Contingent Claims 13.05.2016 20 / 23

Example of pricing a European put option For a Europrean put option C put = (K S 1 T )+. If we consider the same situation, then the analogue of above equation fails unless the neméraire S 0 is a constant. In fact, as a consequence of the put-call parity π call = π put + S 1 0 K Π T k=1 (1 + r k), when the intrinsic value (K S 1 0 )+ of a put option is large, its time value usually becomes negative. Zhiwen Ning European Contingent Claims 13.05.2016 21 / 23

Example of pricing a Eoropean put option Figure: The typical price of a European put option compared to the option s intrinsic value (K S 1 0 )+. Zhiwen Ning European Contingent Claims 13.05.2016 22 / 23

the end Thank you for your attention! Zhiwen Ning European Contingent Claims 13.05.2016 23 / 23