Question 4. Recall the Black-Scholes dynamics under the physical measure P dBt=rBtdt, Bo = 1, dSt = μStdt+oStdWt, So > 0, where r, µ, o are positive constants. Here, St and Bt respectively denote the time- prices of a share and of a bank account. In this question, you are asked to derive the Black-Scholes PDE using a probabilistic approach. a. Show a change of probability measure from P to the riskneutral probability measure Q using the market price of risk process, as in Girsanov's Theorem. Present the Black-Scholes dynamics under the risk-neutral probability measure Q. b. Let Ct = C(St, t) be the time- t no-arbitrage price of a European call option whose payoff is (St – K)†. Bt Ste[0,T] is a martingale under the - Fact: by the First Fundamental Theorem of Asset Pricing, the process risk-neutral measure Q. Use the above fact to derive the Black-Scholes PDE.
Question 4. Recall the Black-Scholes dynamics under the physical measure P dBt=rBtdt, Bo = 1, dSt = μStdt+oStdWt, So > 0, where r, µ, o are positive constants. Here, St and Bt respectively denote the time- prices of a share and of a bank account. In this question, you are asked to derive the Black-Scholes PDE using a probabilistic approach. a. Show a change of probability measure from P to the riskneutral probability measure Q using the market price of risk process, as in Girsanov's Theorem. Present the Black-Scholes dynamics under the risk-neutral probability measure Q. b. Let Ct = C(St, t) be the time- t no-arbitrage price of a European call option whose payoff is (St – K)†. Bt Ste[0,T] is a martingale under the - Fact: by the First Fundamental Theorem of Asset Pricing, the process risk-neutral measure Q. Use the above fact to derive the Black-Scholes PDE.
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![Question 4. Recall the Black-Scholes dynamics under the physical measure P
dBt=rBtdt, Bo = 1,
dSt = μStdt+oStdWt, So > 0,
where r, μ, o are positive constants. Here, St and Bt respectively denote the time- prices of a share and of a
bank account. In this question, you are asked to derive the Black-Scholes PDE using a probabilistic approach.
a. Show a change of probability measure from P to the riskneutral probability measure Q using the market
price of risk process, as in Girsanov's Theorem. Present the Black-Scholes dynamics under the risk-neutral
probability measure Q.
-
b. Let Ct = C(St, t) be the time- t no-arbitrage price of a European call option whose payoff is (St — K)†.
Fact: by the First Fundamental Theorem of Asset Pricing, the process Bt Ste [0,T]
is a martingale under the
Ct
risk-neutral measure Q.
Use the above fact to derive
Black-Scholes PDE.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F882ce437-6984-48a2-b1c1-0c70a26630a5%2Ffd68dcf1-61e4-40a8-8911-f2b9466d43b8%2Fr9dn5nw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 4. Recall the Black-Scholes dynamics under the physical measure P
dBt=rBtdt, Bo = 1,
dSt = μStdt+oStdWt, So > 0,
where r, μ, o are positive constants. Here, St and Bt respectively denote the time- prices of a share and of a
bank account. In this question, you are asked to derive the Black-Scholes PDE using a probabilistic approach.
a. Show a change of probability measure from P to the riskneutral probability measure Q using the market
price of risk process, as in Girsanov's Theorem. Present the Black-Scholes dynamics under the risk-neutral
probability measure Q.
-
b. Let Ct = C(St, t) be the time- t no-arbitrage price of a European call option whose payoff is (St — K)†.
Fact: by the First Fundamental Theorem of Asset Pricing, the process Bt Ste [0,T]
is a martingale under the
Ct
risk-neutral measure Q.
Use the above fact to derive
Black-Scholes PDE.
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