An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu - PowerPoint PPT Presentation

An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 12 Haijun Li An Introduction to Stochastic Calculus Week 12 1 / 18

Outline More on Change of Measure 1 Risk-Neutral Measure Construction of Risk-Neutral and Distorted Measures Continuous-Time Interest Rate Models The Forward Risk Adjusted Measure and Bond Option Pricing The World is Incomplete Haijun Li An Introduction to Stochastic Calculus Week 12 2 / 18

Risk-Neutral Measure A risk-neutral measure is a probability measure under which the underlying risky asset has the same expected return as the riskless bond (or money market account). Haijun Li An Introduction to Stochastic Calculus Week 12 3 / 18

Risk-Neutral Measure A risk-neutral measure is a probability measure under which the underlying risky asset has the same expected return as the riskless bond (or money market account). We often demand more for bearing uncertainty. To price assets, the calculated values need to be adjusted for the risk involved. Haijun Li An Introduction to Stochastic Calculus Week 12 3 / 18

Risk-Neutral Measure A risk-neutral measure is a probability measure under which the underlying risky asset has the same expected return as the riskless bond (or money market account). We often demand more for bearing uncertainty. To price assets, the calculated values need to be adjusted for the risk involved. One way of doing this is to first take the expectation under the physical distribution and then adjust for risk. Haijun Li An Introduction to Stochastic Calculus Week 12 3 / 18

Risk-Neutral Measure A risk-neutral measure is a probability measure under which the underlying risky asset has the same expected return as the riskless bond (or money market account). We often demand more for bearing uncertainty. To price assets, the calculated values need to be adjusted for the risk involved. One way of doing this is to first take the expectation under the physical distribution and then adjust for risk. A better way is to first adjust the probabilities of future outcomes by incorporating the effects of risk, and then take the expectation under those adjusted, ‘virtual’ risk-neutral probabilities. Haijun Li An Introduction to Stochastic Calculus Week 12 3 / 18

Risk-Neutral Measure A risk-neutral measure is a probability measure under which the underlying risky asset has the same expected return as the riskless bond (or money market account). We often demand more for bearing uncertainty. To price assets, the calculated values need to be adjusted for the risk involved. One way of doing this is to first take the expectation under the physical distribution and then adjust for risk. A better way is to first adjust the probabilities of future outcomes by incorporating the effects of risk, and then take the expectation under those adjusted, ‘virtual’ risk-neutral probabilities. Definition A risk-neutral measure is a probability measure under which the current value of all financial assets at time t is equal to the expected future payoff of the asset discounted at the risk-free rate, given the information structure available at time t . Haijun Li An Introduction to Stochastic Calculus Week 12 3 / 18

Complete Market The existence of a risk-neutral measure involves absence of arbitrage in a complete market. Haijun Li An Introduction to Stochastic Calculus Week 12 4 / 18

Complete Market The existence of a risk-neutral measure involves absence of arbitrage in a complete market. A market is complete with respect to a trading strategy if all cash flows for the trading strategy can be replicated by a similar synthetic trading strategy. Haijun Li An Introduction to Stochastic Calculus Week 12 4 / 18

Complete Market The existence of a risk-neutral measure involves absence of arbitrage in a complete market. A market is complete with respect to a trading strategy if all cash flows for the trading strategy can be replicated by a similar synthetic trading strategy. For example, consider the put-call parity: A put is synthesized by buying the call, investing the strike at the risk-free rate, and shorting the stock. Haijun Li An Introduction to Stochastic Calculus Week 12 4 / 18

Complete Market The existence of a risk-neutral measure involves absence of arbitrage in a complete market. A market is complete with respect to a trading strategy if all cash flows for the trading strategy can be replicated by a similar synthetic trading strategy. For example, consider the put-call parity: A put is synthesized by buying the call, investing the strike at the risk-free rate, and shorting the stock. If at some time before maturity, they differ, then someone else could purchase the cheaper portfolio and immediately sell the more expensive one to make risk-less profit (since they have the same value at maturity). Haijun Li An Introduction to Stochastic Calculus Week 12 4 / 18

Complete Market The existence of a risk-neutral measure involves absence of arbitrage in a complete market. A market is complete with respect to a trading strategy if all cash flows for the trading strategy can be replicated by a similar synthetic trading strategy. For example, consider the put-call parity: A put is synthesized by buying the call, investing the strike at the risk-free rate, and shorting the stock. If at some time before maturity, they differ, then someone else could purchase the cheaper portfolio and immediately sell the more expensive one to make risk-less profit (since they have the same value at maturity). In insurance markets, a complete market models the situation that agents can buy insurance contracts to protect themselves against any future time and state-of-the-world. Haijun Li An Introduction to Stochastic Calculus Week 12 4 / 18

Fundamental Theorem of Arbitrage-Free Pricing Consider a finite state market. There is no arbitrage if and only if there exists a risk-neutral 1 measure that is equivalent to the physical probability measure. Haijun Li An Introduction to Stochastic Calculus Week 12 5 / 18

Fundamental Theorem of Arbitrage-Free Pricing Consider a finite state market. There is no arbitrage if and only if there exists a risk-neutral 1 measure that is equivalent to the physical probability measure. In absence of arbitrage, a market is complete if and only if there is 2 a unique risk-neutral measure that is equivalent to the physical probability measure. Haijun Li An Introduction to Stochastic Calculus Week 12 5 / 18

Fundamental Theorem of Arbitrage-Free Pricing Consider a finite state market. There is no arbitrage if and only if there exists a risk-neutral 1 measure that is equivalent to the physical probability measure. In absence of arbitrage, a market is complete if and only if there is 2 a unique risk-neutral measure that is equivalent to the physical probability measure. Let B = ( B t , t ≥ 0 ) denote standard Brownian motion and F t the natural filtration generated by B . When risky asset price is driven by a single Brownian motion, there is a unique risk-neutral measure Q . Harrison-Pliska Theorem If ( r t , t ≥ 0 ) is the short rate process driven by Brownian motion, and V t is any F t -adapted contingent claim payable at time t , then its value � T r u du V T |F t � e − � at time t ≤ T is given by V t = E Q . t Haijun Li An Introduction to Stochastic Calculus Week 12 5 / 18

Fundamental Theorem of Arbitrage-Free Pricing Consider a finite state market. There is no arbitrage if and only if there exists a risk-neutral 1 measure that is equivalent to the physical probability measure. In absence of arbitrage, a market is complete if and only if there is 2 a unique risk-neutral measure that is equivalent to the physical probability measure. Let B = ( B t , t ≥ 0 ) denote standard Brownian motion and F t the natural filtration generated by B . When risky asset price is driven by a single Brownian motion, there is a unique risk-neutral measure Q . Harrison-Pliska Theorem If ( r t , t ≥ 0 ) is the short rate process driven by Brownian motion, and V t is any F t -adapted contingent claim payable at time t , then its value � T r u du V T |F t � e − � at time t ≤ T is given by V t = E Q . t The result can be extended to the case when the asset price is driven by a semi-martingale (see Delbaen and Schachermayer 1994). Haijun Li An Introduction to Stochastic Calculus Week 12 5 / 18

A PDE Connection Consider a parabolic partial differential equation 2 σ 2 ( t , x ) ∂ 2 u ∂ u ∂ t + µ ( t , x ) ∂ u ∂ x + 1 ∂ x 2 = r ( x ) u ( t , x ) , x ≥ 0 , t ∈ [ 0 , T ] subject to the terminal condition u ( T , x ) = h ( x ) . Haijun Li An Introduction to Stochastic Calculus Week 12 6 / 18

A PDE Connection Consider a parabolic partial differential equation 2 σ 2 ( t , x ) ∂ 2 u ∂ u ∂ t + µ ( t , x ) ∂ u ∂ x + 1 ∂ x 2 = r ( x ) u ( t , x ) , x ≥ 0 , t ∈ [ 0 , T ] subject to the terminal condition u ( T , x ) = h ( x ) . The functions µ , σ , h and r are known functions, and T is a parameter. Haijun Li An Introduction to Stochastic Calculus Week 12 6 / 18