Warm-up & Motivation Probability Theory Stochastic Processes Stochastic Integration & Calculus Arbitrage Pricing Theory Supplements on Stochastic Calculus Departing from Black & Scholes Modelling jumps
Pricing an option by replication
Binomial tree model Pricing an option by replication Cost of replication as expectation What do we have to learn?
-4- 0.4 64
4 e t = 0 the strategy requires an initial investment of o, / C 6 LV0(0) + OS= 0,12066, which is the value of the position we hold at time f from time t = 0 to t = 1 the market goesup\', the strategy generates a profit & loss of — / f (u) ± 002[s? (u) 502] = 0.046 which yields a gross value o time t = 1 if the market went up. To continue from time t = 1 to t = wl our strategy (01(u),19?(u)) we need a capital of Vi(0)(u) = 01(u)51(121+ 0?(u)S?(u) 446462/which fits exactly to what we have generatedfar by our s—itegy! So rearranging the portfolio according to our strategy 0 is costless. We can perform the corresponding calculations for the case of an "down" move from time t = 0 to t = 1. Finally, we continue from time t = 1 to t = 2 with our strategy. By adding up the profit & loss
Stochastic Calculus
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Warm-up & Motivation Probability Theory Stochastic Processes Stochastic Integration & Calculus Arbitrage Pricing Theory Supplements on Stochastic Calculus Departing from Black & Scholes Modelling jumps
Pricing an option by replicati•
n
Binomial tree model Pricing an option by replication Cost of replication as expectation What do we have to learn?
generated, we end up with the following portfolio values at time t = 2
V2(0)(uu) = 0.23
But this is identical to the payoff of our call option. The payoff XT of the option and the result of the above strategy 0 are indistinguishable for all states w of the world. The strategy 0 perfectly replicates the option payoff, it is caller a replicating strategy for XT. i As a consequence, the price of the option and the cost of replication must coincide. Any difference between the two gives raise to risk free profit that is generated by trading the option against its replicating strategy.
Since the replication only required the initial investment V0(6), the fair price of the option is V0(6) = 0.12066.
Stochastic Calculus 10 / 213